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Journal of Functional Analysis 258 (2010) 1427–1451 www.elsevier.com/locate/jfa Square integrability of representations...

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Journal of Functional Analysis 258 (2010) 1427–1451 www.elsevier.com/locate/jfa

Square integrability of representations on p-adic symmetric spaces ✩ Shin-ichi Kato a , Keiji Takano b,∗ a Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan b Department of Arts and Science, Akashi National College of Technology, 679-3 Nishioka, Uozumi-cho,

Akashi 674-8501, Japan Received 12 March 2009; accepted 27 October 2009 Available online 4 November 2009 Communicated by P. Delorme

Abstract A symmetric space analogue of Casselman’s criterion for square integrability of representations of a p-adic group is established. It is described in terms of exponents of Jacquet modules along parabolic subgroups associated to the symmetric space. © 2009 Elsevier Inc. All rights reserved. Keywords: Square integrable representations; Discrete series; Distinguished representations; Reductive p-adic groups; Symmetric spaces

0. Introduction Let G be a connected reductive group defined over a non-archimedean local field F equipped with an involutive F -automorphism σ : G → G, and H the subgroup of all σ -fixed points of G. The quotient space G/H where G = G(F ) and H = H(F ) is called a p-adic symmetric space. We are interested in representations of G which can be realized in the space of functions on G/H . Such representations are often said to be H -distinguished. In this paper, we are concerned ✩

Supported in part by Grant-in-Aid for Scientific Research (No. 18540026), Japan Society for the Promotion of Science. * Corresponding author. E-mail addresses: [email protected] (S. Kato), [email protected] (K. Takano). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.026

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S. Kato, K. Takano / Journal of Functional Analysis 258 (2010) 1427–1451

especially with discrete series for G/H ; roughly speaking, the representations which have realizations in the space of square integrable functions on G/H . Let (π, V ) be a finitely generated admissible representation of G which carries a non-zero H -invariant linear form λ ∈ (V ∗ )H . We consider the functions ϕλ,v on G/H for v ∈ V given by ϕλ,v (g) = λ, π g −1 v

(g ∈ G).

Such functions are called H -matrix coefficients of π defined by λ. Any non-trivial realization of π in the space of functions on G/H is formed by these H -matrix coefficients for some non-zero λ ∈ (V ∗ )H . Note that H -matrix coefficients are not the matrix coefficients in the usual sense, but are generalized matrix coefficients, since H -invariant linear forms are not smooth in general. In a previous work [9], we have studied representations whose H -matrix coefficients have compact support modulo ZG H . Here ZG denotes the center of G. We have called such representations (H, λ)-relatively cuspidal, and established a criterion for (H, λ)-relative cuspidality of π by using Jacquet modules (see [9, 6.2]). In the present work, we shall deal with a different class of representations. For simplicity, suppose that π has a unitary central character. Then |ϕλ,v (·)| is regarded as a function on G/ZG H . We say that π is H -square integrable with respect to λ if |ϕλ,v (·)| is square integrable on G/ZG H for all v ∈ V , namely, if

ϕλ,v (g)2 dg < ∞

G/ZG H

for all v ∈ V . We shall establish a criterion for H -square integrability of π in this paper. Before stating our main theorem, let us recall Casselman’s criterion for the usual square integrability. We say that π is square integrable if all the usual matrix coefficients (defined by smooth linear forms) are square integrable on G/ZG . For each parabolic subgroup P of G with the F -split component AP , let (πP , VP ) be the normalized Jacquet module of π along P and ExpAP (πP ) the set of all quasi-characters χ of AP having non-zero generalized eigenvectors in 1 VP . Let A− P and AP denote the dominant part of AP and the OF -points of AP , respectively. Casselman’s criterion. (See [3, 4.4.6].) The representation π is square integrable if and only if for every parabolic subgroup P of G, the condition |χ(a)| < 1 holds for all χ ∈ ExpAP (πP ) and 1 a ∈ A− P \ ZG AP . Also our criterion for H -square integrability is stated in terms of exponents of Jacquet modules. However we use only those along σ -split parabolic subgroups (see 1.5). In our previous work [9] (and also in Lagier [10]), a canonical mapping rP : (V ∗ )H → (VP∗ )M∩H was introduced for each σ -split parabolic subgroup P = MU (see 3.2). Now, for a given λ ∈ (V ∗ )H and a σ -split parabolic subgroup P with the (σ, F )-split component SP (see 1.5), we put r (λ) = 0 on the generalized P . ExpSP πP , rP (λ) = χ ∈ ExpSP (πP ) χ-eigenspace in VP The main theorem of this paper is the following (Theorem 4.7):


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Main Theorem. The representation π is H -square integrable with respect to λ if and only if for every σ -split parabolic subgroup P of G, the condition |χ(s)| < 1 holds for all χ ∈ ExpSP (πP , rP (λ)) and s ∈ SP− \ ZG SP1 . This is an analogue, and even a generalization, of Casselman’s criterion. Actually, if one applies the above theorem to the symmetric space G/H = (G1 × G1 )/G1 where the involution is the permutation of two factors (referred to as the group case), then one recovers Casselman’s criterion for the group G1 . Let us summarize the contents of this paper. In Section 1 we prepare notation and several definitions used throughout. In Section 2, we recall the analogue of Cartan decomposition for p-adic symmetric spaces given by Benoist and Oh [1] and Delorme and Sécherre [5]. After that, we give two ingredients for the proof of the main theorem; a disjointness assertion (Proposition 2.3) and some volume estimate (Proposition 2.6). Section 3 is essentially a recollection of [9, §5] and [10, §2] on asymptotic behavior of H -matrix coefficients described by the mapping rP . Section 4 is devoted to the proof of the main theorem. We give simple examples of H -square integrable representations in Section 5. We are grateful to the referee for his/her careful reading, useful comments and suggestions. 1. Notation and definitions 1.1. Basic notation Let F be a non-archimedean local field with the absolute value | · |F . The valuation ring of F is denoted by OF and the order of the residue field of F by qF . Throughout this paper, we assume that the residual characteristic of F is not equal to 2. Let νF : F × → Z denote the additive valuation defined by −ν (x) |x|F = qF F x ∈ F× . Let G be a connected reductive group defined over F and σ an F -involution on G. The F subgroup {h ∈ G | σ (h) = h} consisting of all σ -fixed points of G is denoted by H. Let Z be the F -split component of G, that is, the largest F -split torus lying in the center of G. Note that Z is σ -stable. We put 0

Z0 = z ∈ Z σ (z) = z−1 , which we shall call the (σ, F )-split component of G. Here and henceforth, ( )0 stands for the identity component in the Zariski topology. The group G(F ) consisting of all the F -rational points of G is denoted by G. Similarly, for any F -subgroup R of G, we shall write R = R(F ). For a connected F -group M, let X ∗ (M) (resp. X ∗ (M)F ) denote the free Z-module of rational (resp. F -rational) characters of M. If A is an F -split torus, one has X ∗ (A) = X ∗ (A)F . Let M be a connected reductive F -group with its F -split component A. We put aM = Hom X ∗ (M)F , R . The natural homomorphism X ∗ (M)F → X ∗ (A) defined by restriction induces an isomorphism Hom X ∗ (A), R aM .

(1.1.1)

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We define a homomorphism νM : M = M(F ) → aM by

α, νM (m) = νF mα

for all m ∈ M and α ∈ X ∗ (M)F . We can define α, νM (m) also for α ∈ X ∗ (A) through the identification (1.1.1). The kernel of νM is denoted by M 1 . Note that A1 = A(OF ) for an F -split torus A. 1.2. H -matrix coefficients of representations Let C ∞ (G/H ) denote the space of all smooth C-valued functions on G/H on which G acts by left translation. A smooth representation (π, V ) of G is said to be H -distinguished if (V ∗ )H = {0}. Take a non-zero λ ∈ (V ∗ )H and consider the functions on G given by ϕλ,v (g) = λ, π g −1 v

(g ∈ G)

for v ∈ V . We call these functions H -matrix coefficients of π defined by λ. Let us identify right H -invariant functions on G with functions on G/H . Then, H -matrix coefficients belong to C ∞ (G/H ) and the mapping Tλ : V → C ∞ (G/H ),

Tλ (v) = ϕλ,v

gives a G-morphism. Any realization of V in C ∞ (G/H ) is determined by an H -invariant linear form in this way. Let ω0 be a quasi-character of Z0 = Z0 (F ). A smooth representation (π, V ) is called an ω0 -representation if Z0 acts on V by the character ω0 . If (π, V ) is an H -distinguished ω0 -representation, then for any λ ∈ (V ∗ )H , the image Tλ (V ) of V is contained in the space Cω∞0 (G/H ) consisting of functions ϕ ∈ C ∞ (G/H ) which satisfy ϕ(zgH ) = ω0 (z)−1 ϕ(gH )

(z ∈ Z0 , gH ∈ G/H ).

1.3. H -square integrability Both G and H are unimodular groups. Thus the quotient space G/Z0 H carries a left Ginvariant measure, which is denoted by G/Z0 H dg. Let ω0 be a unitary character of Z0 . We define L2ω0 (G/H ) to be the space of (L2 -classes of) all functions ϕ ∈ Cω∞0 (G/H ) satisfying

ϕ(g)2 dg < ∞.

G/Z0 H

Take a non-zero λ ∈ (V ∗ )H . We say that an H -distinguished ω0 -representation (π, V ) is H square integrable with respect to λ if the H -matrix coefficients ϕλ,v defined by λ are square


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integrable on G/Z0 H for all v ∈ V , or equivalently, if Tλ (V ) is contained in L2ω0 (G/H ). Note that this definition agrees with the one given in the Introduction, since ZG H /Z0 H is compact. Irreducible representations which appear as subrepresentations of L2ω0 (G/H ) are said to be in the discrete series for G/H . So, in our terminology, discrete series for G/H are irreducible H -distinguished representations which are H -square integrable with respect to some non-zero λ ∈ (V ∗ )H . In [9], we have put the following definition: An H -distinguished admissible ω0 -representation (π, V ) of G is said to be (H, λ)-relatively cuspidal if all the H -matrix coefficients of π defined by λ are compactly supported modulo Z0 H . We gave examples of such representations in [9, §8]. It is clear from the definition that (H, λ)-relatively cuspidal ω0 -representations are H -square integrable with respect to λ provided that ω0 is unitary. At present, we do not know whether H -square integrability with respect to some λ ∈ (V ∗ )H implies H -square integrability with respect to every λ ∈ (V ∗ )H . The same kind of question concerning relative cuspidality also remains open. 1.4. Tori and roots associated to symmetric spaces We shall fix notation on tori, roots, and parabolic subgroups associated to the involution σ . For the reference, see [7] and also [9, §2]. A torus S is said to be (σ, F )-split if it is F -split and σ (s) = s −1 for all s ∈ S. We fix a maximal (σ, F )-split torus S0 of G and a maximal F -split torus A∅ containing S0 . Then A∅ is necessarily σ -stable, so σ acts naturally on X ∗ (A∅ ). Let Φ ⊂ X∗ (A∅ ) be the root system of (G, A∅ ). It is σ -stable. We choose a σ -basis of Φ that has the property α > 0,

σ (α) = α

⇒

σ (α) < 0

in the corresponding order. The subset of all σ -fixed roots in Φ (resp. ) is denoted by Φσ (resp. σ ). Let p : X∗ (A∅ ) → X∗ (S0 ) be the homomorphism defined by restriction to S0 . It is surjective and its kernel coincides with the submodule of all σ -fixed elements of X∗ (A∅ ). Let us put Φ = p(Φ) \ {0} = p(Φ \ Φσ ) . It is well known that Φ is a root system in X∗ (S0 ) with a basis = p() \ {0} = p( \ σ ) . For each subset I of , we consider the subset [I ] := p −1 (I ) ∩ ∪ σ of . In this paper we say that a subset of is σ -split if it is of the form [I ] for some I ⊂ . This terminology agrees with that in [9, 2.3]. The correspondence I → [I ] is an inclusion-preserving bijection between subsets of and σ -split subsets of . The inverse of this correspondence is given by I → p(I \ σ ) for a σ -split subset I of . ¯ for some Note that maximal proper σ -split subsets of are written in the form [ \ {α}] α¯ ∈ .

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1.5. Parabolic subgroups associated to symmetric spaces A parabolic F -subgroup P of G is said to be σ -split if P and σ (P) are opposite. In such a case, we always take M = P ∩ σ (P) for a (σ -stable) Levi subgroup of P. Let P∅ be the minimal parabolic F -subgroup of G corresponding to the choice of as in 1.4. The centralizer ZG (A∅ ) of A∅ in G is denoted by M∅ , which is a Levi subgroup of P∅ . Put M0 = ZG (S0 ) and P0 = P∅ M0 . Then P0 is a minimal σ -split parabolic F -subgroup of G with the σ -stable Levi subgroup M0 . Let U0 be the unipotent radical of P0 . For each subset I of , let PI denote the standard parabolic F -subgroup (that contains P∅ ) of G corresponding to I . If I is a σ -split subset of , then PI is σ -split. Standard σ -split parabolic F -subgroups of G correspond to σ -split subsets of in this way. Furthermore, it is remarked in [9, 2.5] that any σ -split parabolic F -subgroup of G is of the form γ −1 PI γ for some σ -split I ⊂ and γ ∈ (M0 H)(F ). For each σ -split I ⊂ , we put MI = PI ∩ σ (PI ). Note that Pσ and Mσ coincide with P0 and M0 , respectively. Let UI be the unipotent radical of PI , so that we have a Levi decomposition PI = MI UI . The F -split component (resp. (σ, F )-split component) of MI is denoted by AI (resp. SI ). These are also called the F -split and (σ, F )-split component of PI , respectively. We can describe σ -split parabolic subgroups and their (σ, F )-split components also by using subsets of the restricted basis through the bijective correspondence I ↔ I = [I ] in 1.4. We shall occasionally use notation based on restricted roots if it is convenient. For each subset I of , we put SI =

0 ker(α¯ : S0 → Gm ) .

α∈I ¯

It is easy to see the following equalities: S0 = SI · S\I ,

(SI ∩ S\I )0 = S = Z0 .

(1.5.1)

Observe that if I ⊂ corresponds to a σ -split subset I ⊂ , then SI coincides with SI . For a positive real number ε 1 and a σ -split subset I of , we put

SI− (ε) = s ∈ SI s α F ε (α ∈ \ I ) ,

− (ε) = s ∈ S0 s α F ε (α ∈ \ I ), s α F 1 (α ∈ I ) , S0,I and

− I S0 (ε) = s

∈ S0 s α F ε (α ∈ \ I ), ε < s α F 1 (α ∈ I ) .

We note that if I corresponds to I ⊂ as in 1.4, then SI− (ε) and I S0− (ε) can be written also as

SI− (ε) = s ∈ SI = SI s α¯ F ε (α¯ ∈ \ I )

(1.5.2)

and

− I S0 (ε) = s

∈ S0 s α¯ F ε (α¯ ∈ \ I ), ε < s α¯ F 1 (α¯ ∈ I ) .

(1.5.3)


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We abbreviate SI− (1) as SI− and write

− (1) = s ∈ S0 s α F 1 (α ∈ ) . S0− = S σ It is obvious from the definition that − (ε) ⊂ S0− , SI− (ε) ⊂ I S0− (ε) ⊂ S0,I

and that ε ε

⇒

− − ε . S0,I (ε) ⊂ S0,I

1.6. Lemma. Let I be a σ -split subset of . (1) For any positive real number ε 1, there exists a positive real number ε ε such that − ε ⊂ SI− (ε) · S0− . S0,I (2) There exists a positive real number cI 1 having the following property: For any positive real number ε cI , there exist a positive real number ε 1 and finitely many elements t1 , . . . , tk of S0− such that − I S0 (ε) ⊂

SI− ε ti Z0 S01 .

i

(3) For any positive real number ε 1, one has a decomposition

S0− =

− I S0 (ε)

(disjoint)

I ⊂: σ -split

where I ranges over all σ -split subsets of . Proof. The proof of (1) is exactly the same as that of [3, 4.3.1]. Regard the union in (3) as − I = [I ] I S0 (ε) I ⊂

where I ranges over all subsets of including the empty set. Then the assertion of (3) can be seen by the same way as in [3, remark preceding 4.3.4] according to (1.5.3). For (2), let I be the subset of corresponding to I . First observe that SI · S\I is of finite index in S0 by (1.5.1). We can take a finite set ΓI of representatives of S0 /(SI · S\I ) from S0− . We put c = cI = and take t1 , . . . , tk ∈ S0− so that

1 i ti Z0 S0

min

γ ∈ΓI ,α∈\I ¯

α¯ γ F

contains the subset

s ∈ S0− ε < s α¯ F 1 (α¯ ∈ I ), c s α¯ F 1 (α¯ ∈ \ I ) .

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Now, let us write s ∈ I S0− (ε) as s = s1 s2 γ with s1 ∈ SI = SI , s2 ∈ S\I , and γ ∈ ΓI . We show that s1 ∈ SI− (ε ) for some ε and that s2 γ ∈ i ti Z0 S01 . For any α¯ ∈ \ I , we have α¯ s · c (s1 γ )α¯ = s α¯ ε. 1 F F F Therefore s1 belongs to SI− (ε ) for ε = εc−1 (see (1.5.2)). Note that ε 1 if ε c = cI . On the other hand, we have ε < s α¯ F = (s2 γ )α¯ F 1 for each α¯ ∈ I , while c γ α¯ F = (s2 γ )α¯ F 1 for each α¯ ∈ \ I . This completes the proof.

2

2. Relative Cartan decomposition In this section we recall the analogue of Cartan decomposition for p-adic symmetric spaces given by Benoist and Oh [1] and Delorme and Sécherre [5]. Concerning this decomposition, we shall give a disjointness result (Proposition 2.3) and some volume estimate (Proposition 2.6). These will be key ingredients for the proof of the main theorem in Section 4. 2.1. The analogue of Cartan decomposition + Let us fix the data (S0 , A∅ , ) as in 1.4. We shall write SI+ (ε), S0,I (ε), and I S0+ (ε) for the set − of all elements s such that s −1 belong to SI− (ε), S0,I (ε), and I S0− (ε), respectively. We recall the analogue of Cartan decomposition given in [1] and [5]. Let us state it in the following form as in [9, 3.4]: There exist a compact subset Ω of G and a finite subset Γ of (M0 H)(F ) such that

G = ΩS0+ Γ H. 2.2. Groups with Iwahori factorization Let K be a σ -stable open compact subgroup of G and P = MU a σ -split parabolic subgroup of G standard with respect to . We put P− = σ (P) and U− = σ (U), so that P ∩ P− = M and P− = MU− . We say that K has the Iwahori factorization with respect to P if the product map induces a bijection UK− × MK × UK K, where UK− = U − ∩ K, MK = M ∩ K, and UK = U ∩ K. Let Kmax be an A∅ -good maximal compact subgroup of G. We take a σ -stable open compact subgroup K0 , having the Iwahori factorization with respect to P0 , such that S01 ⊂ K0 ⊂ Kmax .


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For example, it suffices to take K0 = B ∩ σ (B) for an Iwahori subgroup B contained in Kmax . Choose a finite set Ξ ⊂ G such that Ω ⊂ ξ ∈Ξ ξ K0 . We may decompose G as G=

ξ K0 s˙ γ H,

ξ ∈Ξ,γ ∈Γ s˙ ∈S + /S 1 0 0

hence G/Z0 H =

ξ K0 s˙ γ Z0 H /Z0 H.

(2.2.1)

ξ,γ s˙ ∈S + /Z0 S 1 0 0

It is not known whether these are disjoint. However we can assert the following. 2.3. Proposition. For each γ ∈ (M0 H)(F ), the union

K0 s˙ γ Z0 H

s˙ ∈S0+ /Z0 S01

is disjoint. Proof. We use the mapping τ : G → G given by τ (g) = gσ (g)−1 and the action (g, x) → g ∗ x = gxσ (g)−1

g ∈ G, x ∈ τ (G)

of G on τ (G) to study cosets modulo H . Suppose that K0 s1 γ z1 H = K0 s2 γ z2 H for s1 , s2 ∈ S0+ and z1 , z2 ∈ Z0 . Applying τ , we have K0 ∗ s12 z12 mγ = K0 ∗ s22 z22 mγ where mγ = τ (γ ) ∈ M0 . There is an element k ∈ K0 such that k ∗ (s12 z12 mγ ) = s22 z22 mγ . Thus, putting z = z1 z2−1 , we have ks12 z2 mγ = s22 mγ σ (k). Using the Iwahori factorization, write k and σ (k) ∈ K0 as k = u− 1 m1 u1 ,

σ (k) = u− 2 m2 u2

− where u− i ∈ (U0 )K0 , mi ∈ (M0 )K0 , and ui ∈ (U0 )K0 . Then we have − 2 2 2 u− 1 m1 u1 s1 z mγ = s2 mγ u2 m2 u2 ,

hence −1 2 −1 2 2 −1 −1 2 2 u− u1 s1 mγ = mγ s22 u− mγ · mγ s22 m2 · u2 . 1 · m1 s1 z mγ · mγ s1 2 s2

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By the uniqueness of expressions in U0− M0 U0 , we must have m1 s12 z2 mγ = mγ s22 m2 .

(2.3.1)

Now we use the usual Cartan decomposition (see e.g. [12, 1.1(4)]) for M0 to write mγ ∈ M0 as mγ = k1 ak2

σ ,+ k1 , k2 ∈ M0 ∩ Kmax , a ∈ M . ∅

σ ,+ Here M denotes the set of elements m of M∅ satisfying ∅

α, νM∅ (m) 0

σ ,+ . Since z2 , s12 and for all α ∈ σ . Note that m1 , m2 ∈ M0 ∩ K0 ⊂ M0 ∩ Kmax , and S0+ ⊂ M ∅ 2 s2 are central in M0 , we have

m1 k1 · s12 z2 a · k2 = k1 · s22 a · k2 m2 by (2.3.1). From the uniqueness of Cartan decomposition, we may conclude that s12 z2 a ≡ s22 a

mod M∅1 .

Since M∅1 ∩ S0 = S01 , we have (s1 z)2 ≡ s22 mod S01 , and in turn, s1 ≡ s2 mod Z0 S01 .

2

2.4. Some volume computation We have to evaluate the volume of K0 sγ Z0 H /Z0 H with respect to the G-invariant measure on G/Z0 H . For this purpose we first need to take a σ -stable open compact subgroup K from an adapted family given in [9, 4.3]: Besides the Iwahori factorization, we need a good filtration (see [9, 4.3 (2)]) inside K (and its subgroups) to use the result [9, 4.6]. Then we can compute the volume of KsZ0 H /Z0 H for such a K and s ∈ S0+ . Let δP denote the modulus character of P = P(F ) for a parabolic F -subgroup P of G. 2.5. Lemma. If K is a σ -stable open compact subgroup as above, then vol(KsZ0 H /Z0 H ) = δP0 (s) · vol(KeZ0 H /Z0 H ) for all s ∈ S0+ . Proof. First, note that τ (Z0 ) = {z2 | z ∈ Z0 } is a central subgroup of G. We use τ to identify G/Z0 H with τ (G)/τ (Z0 ). There is a natural action of G on τ (G)/τ (Z0 ) induced by the ∗-action. We put e¯ = τ (Z0 ). The left G-invariant measure on G/Z0 H can be transported to the measure on τ (G)/τ (Z0 ) = G ∗ e¯ which is invariant under the ∗-action of G. The subset KsZ0 H /Z0 H is identified with (Ks) ∗ e¯ and vol (Ks) ∗ e¯ = vol s ∗ s −1 Ks ∗ e¯ = vol s −1 Ks ∗ e¯


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by the invariance under ∗. Put Ks = s −1 Ks. The Iwahori factorization − M0,K U0,K K = U0,K

of K with respect to P0 implies − s · M0,K · s −1 U0,K s Ks = s −1 U0,K for s ∈ S0+ with − − s −1 U0,K s ⊃ U0,K ,

s −1 U0,K s ⊂ U0,K .

Put Ks = Ks ∩ σ (Ks ) (= s −1 Ks ∩ sKs −1 ). It is σ -stable and has a factorization − −1 Ks = sU0,K s M0,K s −1 U0,K s . We have Ks ⊂ K. So we may apply [9, 4.6] to the group Ks , which asserts that − −1 s −1 U0,K s ⊂ sU0,K s M0,K H. As a result, we have − −1 − − s M0,K s −1 U0,K s ∗ e¯ ⊂ s −1 U0,K s M0,K sU0,K s M0,K ∗ e¯ Ks ∗ e¯ = s −1 U0,K −1 − ¯ = s U0,K s M0,K ∗ e. − We also note that (s −1 U0,K s)M0,K = P0− ∩ Ks . This shows that the orbit (P0− ∩ Ks ) ∗ e¯ of the ¯ Now, fix Haar measures du− subgroup P0− ∩ Ks of Ks coincides with the whole Ks -orbit Ks ∗ e. − − −1 and dm on s U0,K s and M0,K , respectively. The (P0 ∩ Ks )-invariant measure

f →

f u− ∗ (m ∗ e) ¯ dm du−

− s −1 U0,K s M0,K

on the orbit Ks ∗ e¯ has to be Ks -invariant. Consequently, the volume of Ks ∗ e¯ is proportional to − − · δP0 (s) · vol(M0,K ), vol s −1 U0,K s · vol(M0,K ) = vol U0,K which is a constant multiple of δP0 (s). The constant turns out to be the volume of KeZ0 H /Z0 H , if we consider s = e. 2 2.6. Proposition. Let K be an arbitrary open compact subgroup of G. For each γ ∈ Γ , there exist positive real constants c1 and c2 such that c1 · δP0 (s) vol(Ksγ Z0 H /Z0 H ) c2 · δP0 (s) for all s ∈ S0+ .

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Proof. We put K = γ −1 Kγ , S0 = γ −1 S0 γ , and P0 = γ −1 P0 γ . Then P0 is a σ -split parabolic subgroup with the (σ, F )-split component S0 . We have vol(Ksγ Z0 H /Z0 H ) = vol γ K γ −1 sγ Z0 H /Z0 H = vol K s Z0 H /Z0 H where s = γ −1 sγ ∈ (S0 )+ = γ −1 S0+ γ . Take a member K from the adapted family (corresponding to the γ -conjugated data) such that K ⊂ K . Then we have vol K s Z0 H /Z0 H vol K s Z0 H /Z0 H K : K · vol K s Z0 H /Z0 H . Thus the claim follows from Lemma 2.5.

2

3. Asymptotic behavior of H -matrix coefficients In this section we shall describe asymptotic behavior of H -matrix coefficients through the mapping rP (see 3.2). This section is essentially a recollection of [9, §5] and [10, §2]. From now on, we shall briefly say that P is a σ -split parabolic subgroup of G if P is the group of F -points of a σ -split parabolic F -subgroup P = PI of G etc, by abuse of terminology. 3.1. Normalized Jacquet modules For an admissible representation (π, V ) of G and a parabolic subgroup P = MU of G, the normalized Jacquet module of π along P is denoted by (πP , VP ): The space VP is defined as the quotient V /V (U ) where V (U ) is the subspace of V spanned by all the elements of the form π(u)v − v (u ∈ U , v ∈ V ). The action πP of M on VP is normalized so that −1/2

πP (m)jP (v) = δP

(m)jP π(m)v

for m ∈ M and v ∈ V where jP : V → VP denotes the canonical projection. 3.2. The mapping rP When P = MU is a σ -split parabolic subgroup, we have defined in [9] a linear mapping H M∩H rP : V ∗ → VP∗ between the spaces of invariant linear forms. If v ∈ V is a canonical lifting (in the sense of [3, §4]) of v¯ ∈ VP with respect to a suitable σ -stable open compact subgroup (a member in an adapted family in [9, 4.3]), then rP (λ) for λ ∈ (V ∗ )H is defined by rP (λ), v¯ = λ, v

(3.2.1)

(see [9, 5.3(2) and 5.4]). The same mapping was constructed independently by N. Lagier [10] in a different manner. P. Delorme extended the construction of such mappings to any smooth representations in [4]. In [9, §5], we gave asymptotic behavior of H -matrix coefficients through the mapping rP . The result [9, Proposition 5.5] can be extended to the next proposition. This is a generalization


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of Casselman’s result [3, 4.3.3] to symmetric spaces, and was already proved essentially in [10, Théorème 2]. The proof provided by [10] invokes Casselman’s result itself. We shall give a proof which does not rely on [3, 4.3.3] (but we follow the lines similar to [3] as we did in [9, §5]). It would yield [3, 4.3.3] when we apply this to the group case. 3.3. Proposition. Let I be a σ -split subset of and P = PI the corresponding σ -split parabolic subgroup with the (σ, F )-split component S = SI . Let (π, V ) be an admissible representation of G and V1 ⊂ V a finite dimensional subspace. Then there exists a positive real number ε = εI 1 such that 1/2 λ, π(s)v = δP (s) rP (λ), πP (s)jP (v) − (ε), v ∈ V1 , and λ ∈ (V ∗ )H . for all s ∈ S0,I K Proof. For a compact subgroup K of G, let pK : V → V denote the projection defined by pK (v) = K π(k)v dk. For an open compact subgroup K of G (from the adapted family), there is a positive real number ε 1 such that the space pK (π(s)V K ) does not depend on s ∈ SI− (ε) and is isomorphic to (VP )MK by the restriction of jP : V → VP . The vectors in pK (π(s)V K ) are called canonical liftings over (VP )MK with respect to K. First, using (1) of Lemma 1.6, we can − (ε) (replacing ε even construct the space pK (π(s)V K ) of canonical liftings by taking s from S0,I suitably). This step is entirely the same as the derivation of [3, 4.3.2] from [3, 4.3.1]. Next, choose − − (ε). By K small enough so that V1 ⊂ V K . Then, π(s)v ∈ V M0,K U0,K for each v ∈ V1 and s ∈ S0,I [9, 5.3(1)], we have λ, π(s)v = λ, pU0,K π(s)v = λ, pK π(s)v .

Since pK (π(s)v) is now a vector in the space of canonical liftings, this is further equal to rP (λ), jP pK π(s)v = rP (λ), jP pU0,K π(s)v

(3.3.1)

by the definition (3.2.1) of rP (λ). We use the decomposition U0 = U · U where U = M ∩ U0 . This implies that U0,K = UK ·UK , hence pU0,K = pUK ◦pUK . Since jP ◦pUK = jP and UK ⊂ M, the right-hand side of (3.3.1) is equal to 1/2 rP (λ), jP pUK π(s)v = rP (λ), pUK jP π(s)v = rP (λ), pUK δP (s)πP (s)jP (v) .

Finally, as a vector in the representation πP of M, jP (v) is MK -fixed and thus πP (s)jP (v) ∈ − (VP )M0,K (U0,K ∩M) . Applying [9, 5.3(1)] for the (M ∩ H )-invariant linear form rP (λ), we have rP (λ), pUK πP (s)jP (v) = rP (λ), πP (s)jP (v) . This completes the proof.

2

3.4. Remark. When we take a σ -split parabolic subgroup P without specifying the initial data (S0 , A∅ , ), we may think of P as a standard one PI for a suitable choice of (S0 , A∅ , ) and I ⊂ . However, sometimes we have to fix the data (S0 , A∅ , ) and deal with an arbitrary (possibly non-standard) σ -split parabolic subgroup written in the form P = γ −1 PI γ with

1440


γ ∈ (M0 H)(F ). The relation as in 3.3 for such a P can be derived in the same way, and is presented as follows: There exists a positive real number ε = εγ ,I 1 such that 1/2 λ, π γ −1 sγ v = δP γ −1 sγ rP (λ), πP γ −1 sγ jP (v)

(3.4.1)

− (ε), v ∈ V1 , and λ ∈ (V ∗ )H . for all s ∈ S0,I

4. The main theorem In this section we prepare notation on exponents, give a preliminary result (Proposition 4.3) on the relation between H -square integrability and exponents, and establish a criterion for H -square integrability (Theorem 4.7). We also give a non-trivial relation between H -square integrability and the usual square integrability (Proposition 4.10). 4.1. Exponents Let Z1 be a closed subgroup of the center of G and X (Z1 ) be the set of all quasi-characters of Z1 . For a smooth representation (π, V ) of G and a quasi-character ω ∈ X (Z1 ), we put There exists a d ∈ N such that Vω,∞ = v ∈ V . (π(z) − ω(z))d v = 0 for all z ∈ Z1 This is a G-submodule of V . We put

ExpZ1 (V ) = ExpZ1 (π) = ω ∈ X (Z1 ) Vω,∞ = 0 . If (π, V ) is finitely generated and admissible, then the set ExpZ1 (π) is finite and V has a direct sum decomposition Vω,∞ V= ω∈E xpZ1 (π)

(see [3, 2.1.9]). Let Z1 and Z2 be closed subgroups of the center of G such that Z1 ⊃ Z2 . As is easily seen, the mapping ExpZ1 (π) → ExpZ2 (π) defined by restriction is surjective. 4.2. Exponents along parabolic subgroups Let (π, V ) be a finitely generated admissible representation of G. For each parabolic subgroup P of G with the F -split component A, we consider the finite set ExpA (πP ). The elements of ExpA (πP ) are called exponents of π along P . If P is a σ -split parabolic subgroup with the (σ, F )-split component S, we also consider the finite set ExpS (πP ). The mapping ExpA (πP ) → ExpS (πP ) defined by restriction is surjective.

(4.2.1)


1441

Let P1 = M1 U1 and P2 = M2 U2 be σ -split parabolic subgroups of G with the (σ, F )-split components S1 and S2 , respectively. If P1 ⊂ P2 , then M1 ⊂ M2 and S1 ⊃ S2 . The intersection M2 ∩ P1 is a σ -split parabolic subgroup of M2 having M1 as a σ -stable Levi subgroup. As is well known, (πP2 )M2 ∩P1 is naturally isomorphic to πP1 as an M1 -module. Through this isomorphism, it is easy to see that χ ∈ ExpS1 (πP1 )

⇒

χ|S2 ∈ ExpS2 (πP2 ).

(4.2.2)

In the course of the proofs in this section, we need to fix the initial data (S0 , A∅ , ) as in 1.4. In dealing with an arbitrary σ -split parabolic subgroup P , we have to write it as P = γ −1 PI γ by a σ -split subset I ⊂ and an element γ ∈ (M0 H)(F ). The (σ, F )-split component of P is then written as S = γ −1 SI γ . We also write S − = γ −1 SI− (1)γ in such a case. Now, for a given finitely generated admissible representation (π, V ) of G, let us consider the following condition on a σ -split parabolic subgroup P : (P )

χ(s) < 1 for all χ ∈ ExpS (πP ) and all s ∈ S − \ Z0 S 1 .

4.3. Proposition. Let ω0 be a unitary character of Z0 and (π, V ) a finitely generated H distinguished admissible ω0 -representation of G. If the condition (P ) is satisfied for every σ -split parabolic subgroup P of G, then (π, V ) is H -square integrable with respect to any λ ∈ (V ∗ )H . Proof. Let Γ , K0 , Ξ be as in 2.2 for the data (S0 , A∅ , ). Take a non-zero vector v0 ∈ V and let V0 be the finite dimensional subspace of V generated by π(k −1 ξ −1 )v0 (k ∈ K0 , ξ ∈ Ξ ). Further, let V1 be the finite dimensional subspace of V generated by π(γ −1 )v (γ ∈ Γ , v ∈ V0 ). We take a positive real number ε 1 satisfying: • ε εγ ,I for all γ ∈ Γ and all σ -split I ⊂ where εγ ,I is such that (3.4.1) is valid for all v ∈ V1 , and • ε cI for all σ -split I ⊂ where cI is the constant given in (2) of Lemma 1.6. We use a disjoint decomposition

S0+ =

+ I S0 (ε)

I ⊂: σ -split

obtained from Lemma 1.6(3) for the number ε as above. Let us put

GI,γ =

K0 s˙ γ Z0 H

s˙ ∈I S0+ (ε)/Z0 S01

for each σ -split subset I ⊂ and γ ∈ Γ . Then, by (2.2.1) we have G/Z0 H =

ξ,γ ,I

ξ GI,γ /Z0 H.

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Now we start to evaluate the L2 -norm of the H -matrix coefficient ϕλ,v0 . It is clear that ϕλ,v (g)2 dg 0

ξ,γ ,I

G/Z0 H

ϕλ,v (g)2 dg . 0

ξ GI,γ /Z0 H

It is enough to study the convergence of

ϕλ,v (g)2 dg = 0

ξ GI,γ /Z0 H

2 dg

ϕ

λ,π(ξ −1 )v0 (g)

GI,γ /Z0 H

for each ξ , γ and I . So the proof of the proposition is completed once the following claim is shown. Claim. If (P ) is satisfied for P = γ −1 PI γ , then

ϕλ,v (g)2 dg < ∞

GI,γ /Z0 H

for all v ∈ V0 . Let us prove this. It is obvious that

ϕλ,v (g)2 dg

s˙ ∈I S0+ (ε)/Z0 S01

GI,γ /Z0 H

ϕλ,v (g)2 dg .

(4.3.1)

K0 s˙ γ Z0 H /Z0 H

There is an element k0 ∈ K0 such that ϕλ,v (k s˙ γ )2

(k ∈ K0 )

attains the maximum at k = k0 . We have K0 s˙ γ Z0 H /Z0 H

ϕλ,v (g)2 dg vol(K0 s˙ γ Z0 H /Z0 H ) · ϕλ,v (k0 s˙ γ )2 2 C · δP0 (˙s ) · ϕλ,π(k −1 )v (˙s γ ) 0

(4.3.2)

for some positive real constant C which does not depend on s˙ by 2.6. We may replace π(k0−1 )v ∈ V0 by v. Applying Proposition 3.3 (or 3.4) along P = γ −1 PI γ , we have ϕλ,v (˙s γ ) = λ, π γ −1 s˙ −1 v = λ, π γ −1 s˙ −1 γ π γ −1 v 1/2 = δP γ −1 s˙ −1 γ rP (λ), πP γ −1 s˙ −1 γ jP π γ −1 v


1443

− since s˙ −1 ∈ I S0− (ε) ⊂ S0,I (ε) and π(γ −1 )v ∈ V1 . Next, we use Lemma 1.6(2) to write

s˙ −1 = s˙I · ti ,

s˙I ∈ SI− ε /Z0 S01 = SI− ε /Z0 SI1 ,

where ε 1 by our choice of ε. Putting πP (γ −1 ti γ )jP (π(γ −1 )v) = v¯i for simplicity, we have ϕλ,v (˙s γ ) = δPI (˙sI ti )1/2 rP (λ), πP γ −1 s˙I γ v¯i . The function rP (λ), πP (·)v¯i on S = γ −1 SI γ is S-finite. Thus it can be written as

rP (λ), πP s v¯i =

χ s Pχ,i νS s

(4.3.3)

χ∈E xpS (πP ) −1 for all s ∈ S using suitable polynomials Pχ,i on aS (see [12, I.2]). Let2 us write sI = γ s˙I γ . Then, returning to (4.3.2), we have a bound for K0 s˙γ Z0 H /Z0 H |ϕλ,v (g)| dg by

−1 −1 2 C · δP0 ti s˙I δPI (˙sI ti ) χ sI Pχ,i νS sI χ

2 −1 = C · δP0 (ti ) δPI (ti ) χ sI Pχ,i νS sI χ

χχ s Pχ,i νS s Pχ ,i νS s . C · δP−1 δ (t ) P i I I I I 0 χ,χ

Here we used the fact that δP0 ≡ δPI on SI . Returning further to (4.3.1), it turns out that 2 GI,γ /Z0 H |ϕλ,v (g)| dg is bounded by

s˙I ∈SI− (ε )/Z0 SI1

i

δP−1 δPI (ti ) 0

χχ s Pχ,i νS s Pχ ,i νS s . I

I

I

χ,χ

Now, we may regard SI /Z0 SI1 as a lattice and SI− (ε )/Z0 SI1 as a subset of a suitable positive cone in the lattice. Then the infinite sum

χχ s Pχ,i νS s Pχ ,i νS s I

I

I

s˙I ∈SI− (ε )/Z0 SI1

is essentially a power series whose coefficients are polynomials of indices. This series converges if |χ(sI )| < 1 for all χ and sI = γ −1 s˙I γ with s˙I ∈ SI− (ε )/Z0 SI1 , except for those s˙I ∈ Z0 SI1 −1 which represent the origin in the lattice. Thus the condition (P ) for P = γ PI γ is sufficient for the convergence of GI,γ /Z0 H |ϕλ,v (g)|2 dg. 2

1444


4.4. Exponents with respect to λ Let (π, V ) be a smooth representation of G and take an H -invariant linear form λ on V . For a closed subgroup Z1 of the center of G, we define

ExpZ1 (π, λ) := ω ∈ ExpZ1 (π) λ|Vω,∞ = 0 . Assume that (π, V ) is finitely generated and admissible. For each σ -split parabolic subgroup P of G with the (σ, F )-split component S, we consider the subset ExpS (πP , rP (λ)) of ExpS (πP ). As a relation similar to (4.2.2), we have the following. 4.5. Lemma. Let P1 and P2 be σ -split parabolic subgroups of G such that P1 ⊂ P2 , with the (σ, F )-split components S1 and S2 , respectively. Then, χ ∈ ExpS1 πP1 , rP1 (λ)

⇒

χ|S2 ∈ ExpS2 πP2 , rP2 (λ) .

Proof. Suppose that χ ∈ ExpS1 (πP1 , rP1 (λ)). For each v¯1 ∈ (VP1 )χ,∞ , we can take a v¯2 ∈ (VP2 )χ|S2 ,∞ such that jM2 ∩P1 (v¯2 ) = v¯1 (in the notation of 4.2) regarding (4.2.2). Applying Proposition 3.3 to the M2 ∩ H -matrix coefficients, we have rP2 (λ), πP2 (s)v¯2 = δM2 ∩P1 (s)1/2 rM2 ∩P1 rP2 (λ) , πP1 (s)v¯1

at least for some s ∈ S1 . The right-hand side is written as δM2 ∩P1 (s)1/2 rP1 (λ), πP1 (s)v¯1 by the transitivity result rM2 ∩P1 ◦ rP2 = rP1 given in [9, Proposition 5.9] (and also [10, Théorème 3]). Hence rP2 (λ) cannot be identically zero on (VP2 )χ|S2 ,∞ provided that rP1 (λ)|(VP1 )χ,∞ = 0. 2 Fix a non-zero λ ∈ (V ∗ )H . Let us take up the following condition on a σ -split parabolic subgroup P : (P ,λ )

χ(s) < 1 for all χ ∈ ExpS πP , rP (λ) and all s ∈ S − \ Z0 S 1 .

4.6. Lemma. If the condition (P ,λ ) holds for every maximal σ -split parabolic subgroup P of G, then it holds for every σ -split parabolic subgroup P of G. Proof. It is enough to derive (P ,λ ) for a standard P = PI assuming (P ,λ ) for all maximal standard P . For a given σ -split subset I of , let us write \ I = {α¯ 1 , α¯ 2 , . . . , α¯ r }. For each i, let Pi be the maximal standard σ -split parabolic subgroup corresponding to [ \ {α¯ i }] (see the remark at the end of 1.4). These are maximal σ -split parabolic subgroups containing PI .


1445

The (σ, F )-split component Si of Pi is given by Si = S\{α¯ i } =

0 ker(α) ¯ .

(4.6.1)

α¯ =α¯ i

The (σ, F )-split component S = SI of P = PI can be decomposed as S = S1 S2 . . . Sr , so that S1 S2 . . . Sr is of finite index in S. Thus for any given s ∈ S − , there exists a positive integer m such that s m ∈ S1 S2 . . . Sr . We may write s m = s1 s2 . . . sr (si ∈ Si ). From (4.6.1) it is easy to see that si ∈ Si− for each i. Now, for any χ ∈ ExpS (πP , rP (λ)), we have χ|Si ∈ ExpSi (πPi , rPi (λ)) by Lemma 4.5. Therefore, assumptions (P ,λ ) for all P = Pi imply that χ(s)m = χ s m = χ(s1 )χ(s2 ) · · · χ(sr ) < 1. This completes the proof.

2

Now we shall state the main theorem of this paper. 4.7. Theorem. Let ω0 be a unitary character of Z0 and (π, V ) a finitely generated H distinguished admissible ω0 -representation of G. Then, for a non-zero H -invariant linear form λ on V , the representation (π, V ) is H -square integrable with respect to λ if and only if the condition (P ,λ ) is satisfied for every σ -split parabolic subgroup P of G. Proof. Note that only the exponents in ExpS (πP , rP (λ)) contribute to the evaluation of the integral G/Z0 H |ϕλ,v (g)|2 dg in the proof of Proposition 4.3, specifically at the expression (4.3.3). So the if part is already proved in 4.3. To prove the only if part, suppose that (P ,λ ) fails for some σ -split parabolic subgroup P . By Lemma 4.6, we may suppose that P is a maximal one, say, P = γ −1 PI γ with I = [ \ {α}] ¯ for some α¯ ∈ . Then there exist an exponent χ ∈ ExpS (πP , rP (λ)) and an element s = γ −1 sI γ ∈ S − \ Z0 S 1 , sI ∈ SI− \ Z0 SI1 , such that |χ(s)| 1. Take a vector v¯ ∈ (VP )χ,∞ with rP (λ), v

¯ = 0 and let v ∈ V be such that jP (π(γ −1 )v) = v. ¯ From Proposition 2.3, the union n0 K0 sI−n γ Z0 H is disjoint. So it is enough to see that

ϕλ,v (g)2 dg

n0 K s −n γ Z H /Z H 0 I 0 0

is divergent. Take an open compact subgroup K ⊂ K0 which fixes v. For each n, the function |ϕλ,v (·)|2 is constant on KsI−n γ Z0 H , hence K0 sI−n γ Z0 H /Z0 H

ϕλ,v (g)2 dg

ϕλ,v (g)2 dg

KsI−n γ Z0 H /Z0 H

2 = vol KsI−n γ Z0 H /Z0 H λ, π γ −1 sIn v 2 C · δP0 sI−n λ, π γ −1 sIn γ π γ −1 v

(4.7.1)

1446


for some constant C by Proposition 2.6. Now, let us take ε 1 such that the relation (3.4.1) holds ¯ here, the set SI− (ε) is described as for all s ∈ SI− (ε). Since I = [ \ {α}]

SI− (ε) = s ∈ SI s α¯ F ε . The element sI ∈ SI− \ Z0 SI1 satisfies |(sI )α¯ |F < 1. Thus we can take an integer N such that sIn ∈ SI− (ε) for all n N . As a result, (4.7.1) continues as 2 C · δP0 sI−n δPI sIn rP (λ), πP γ −1 sIn γ jP π γ −1 v 2 = C · rP (λ), πP s n v¯ whenever n N . Since v¯ ∈ (VP )χ,∞ , the function rP (λ), πP (·)v

¯ on S is written just as χ(·)Pχ (νS (·)) by a single non-zero polynomial Pχ on aS . Finally, the series 2 χ s n Pχ νS s n = χ(s)2n Pχ n · νS (s) 2 nN

nN

is obviously divergent, hence the proof is completed.

2

4.8. Remark. We have actually shown that the following three conditions are equivalent: (1) (π, V ) is H -square integrable with respect to λ. (2) (P ,λ ) is satisfied for every σ -split parabolic subgroup P of G. (3) (P ,λ ) is satisfied for every maximal σ -split parabolic subgroup P of G. See [12, III.1.1] for a similar statement on the usual square integrability. 4.9. H -square integrability and the usual square integrability ) be the contragredient of (π, V ). For v ∈ V and , the usual matrix coefficient Let ( π, V v∈V c v ,v is defined by v , π g −1 v . c v ,v (g) = Let ω be a unitary character of the F -split component Z of G. A smooth ω-representation (π, V ) of G is said to be square integrable if the functions |c v ,v (·)| are square integrable on G/Z for all v and v . For the F -split component A = AI of a parabolic subgroup P = PI of G, the dominant part A− of A is defined by

A− = a ∈ A a α 1 (α ∈ \ I ) . There is a well-known criterion for square integrability due to Casselman (see [3, 4.4.6]). In terms of normalized Jacquet modules, it is stated as follows. Casselman’s criterion. A finitely generated admissible ω-representation (π, V ) of G is square integrable if and only if for every parabolic subgroup P , |χ(a)| < 1 holds for all χ ∈ ExpA (πP ) and a ∈ A− \ ZA1 .


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Although the usual square integrability and H -square integrability are different notions, we have the following relation. 4.10. Proposition. Let ω be a unitary character of Z and (π, V ) a finitely generated admissible ω-representation of G. If (π, V ) is square integrable and is H -distinguished, then it is H -square integrable for all λ ∈ (V ∗ )H . Proof. This is immediate from Casselman’s criterion and Proposition 4.3, in view of the surjectivity of (4.2.1). 2 An example of square integrable H -distinguished representations can be found in [6, §7] for the symmetric space GL2 (E)/GL2 (F ) where E/F is a quadratic extension. 5. Examples of H -square integrable representations We shall give two simple examples of H -square integrable representations which are not square integrable. In both cases, there are constructions (see [11] and [8]) of an H -distinguished representation π(ρ) attached to a representation ρ of GL2 (F ). We have observed in [9, §8] that π(ρ) is H -relatively cuspidal if ρ is cuspidal. In what follows, we shall observe that π(ρ) is H -square integrable if ρ is square integrable. 5.1. The symmetric space GL3 (F )/(GL2 (F ) × GL1 (F )) Let G be the group GL3 (F ) and σ the inner involution σ = Int() defined by the anti 1 . Then the σ -fixed point subgroup H is isomorphic to diagonal permutation matrix = 1 1

GL2 (F ) × GL1 (F ). For this symmetric space, all the irreducible H -distinguished representations were determined by D. Prasad [11]. We shall use notation in [9, 8.2] for the case n = 3. Let Q = LU Q be the standard parabolic subgroup of G of type (1, 2). Thus L is isomorphic to GL1 (F ) × GL2 (F ). Let ρ be an infinite dimensional irreducible admissible representation of GL2 (F ) with trivial central character and form the normalized induction π(ρ) = IndG Q (1GL1 (F ) ⊗ ρ) where 1 = 1GL1 (F ) denotes the trivial character of GL1 (F ) = F × . Then π(ρ) is irreducible and H -distinguished (see [11, Theorem 2(2)]). We take a maximal (σ, F )-split torus S0 as the one consisting of diagonal matrices of the form diag(s, 1, s −1 ) with s ∈ F × , and a maximal F -split torus A∅ of all diagonal matrices. The (σ, F )-split component Z0 of G is trivial. As a minimal σ -split parabolic subgroup P0 , we may take the Borel subgroup consisting of upper triangular matrices. In this case, P0 is the only proper σ -split parabolic of G up to H -conjugacy. Let us determine the exponents of π(ρ) along P0 . By the Geometric Lemma of [2, 2.12], we have s.s s.s π(ρ) P = IndG Q (1 ⊗ ρ) P 0

0

w∈[WL \W ]

Fw (1 ⊗ ρ)s.s

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where W and WL denote the Weyl group of A∅ in G and L respectively, [WL \W ] as in [3, 1.1.3], (· · ·)s.s denotes the semisimplified form, and w

M

Fw (1 ⊗ ρ) = IndM0 ∩wQw−1 0

(1GL1 (F ) ⊗ ρ)L∩w−1 P0 w .

The set [WL \W ] consists of three elements. In forms of permutation matrices, those are

1

e=

,

1

1

,

1

1

1

and

1 .

1

1

It is easy to compute the M0 = A∅ -module Fw (1 ⊗ ρ) for each w: Fw (1 ⊗ ρ) = w (1 ⊗ ρB2 ) where B2 denotes the Borel subgroup consisting of upper triangular matrices of GL2 (F ). Now take ρ to be the Steinberg representation St2 of GL2 (F ). We shall observe that the irreducible H -distinguished representation π(St2 ) is H -square integrable. As is well known, 1/2 (St2 )B2 is given by the one-dimensional character δB2 . So the set ExpA∅ (π(St2 )P0 ) of exponents of π(St2 ) along P0 consists of three characters χ1 , χ2 , χ3 respectively given by χ1

a1

1/2 −1/2 = |a2 |F |a3 |F ,

a2 a3

χ3

a1

χ2

1/2

−1/2

= |a1 |F |a2 |F

a2

−1/2

,

a3

a1

1/2

= |a1 |F |a3 |F

a2

.

a3 Finally, since S0− (resp. S01 ) consists of diag(s, 1, s −1 ) with |s|F 1 (resp. |s|F = 1), we may conclude that the restriction to S0 of each exponent χi satisfies |χi (s)| < 1 for all s ∈ S0− \ S01 . Hence the claim follows from Proposition 4.3. 5.2. The symmetric space GL4 (F )/Sp2 (F ) Let G be the group GL4 (F ) and σ the involution on G defined by ⎛

0 1 ⎜ −1 0 σ (g) = ⎝

⎞ 0 1 −1 0

⎛

0 1 ⎟ t −1 ⎜ −1 0 ⎠ g ⎝

⎞−1 0 1 −1 0

⎟ ⎠

(g ∈ G).

Then the σ -fixed point subgroup H is the symplectic group Sp2 (F ). For this symmetric space, H -distinguished representations were studied by Heumos and Rallis [8]. See also [9, 8.3]. We take

S0 = diag(s1 , s1 , s2 , s2 ) s1 , s2 ∈ F ×


1449

as a maximal (σ, F )-split torus of G. The (σ, F )-split component Z0 of G consists of all the scalar matrices of G in this case. Let P0 = M0 U0 be the standard parabolic subgroup of G of type (2, 2). This is the only proper σ -split parabolic subgroup of G up to H -conjugacy. Note that M0 GL2 (F ) × GL2 (F ),

M0 ∩ H SL2 (F ) × SL2 (F ).

(5.2.1)

Let ρ be an irreducible admissible representation of G2 := GL2 (F ) and let us form the normalized induction 1/2 −1/2 I(ρ) = IndG . P0 ρ · det(·) F ⊗ ρ · det(·) F Then I(ρ) is H -distinguished (see [8, 11.1(a)]). If further ρ is square integrable, then I(ρ) has the unique irreducible quotient π(ρ) which also is H -distinguished (see [8, 11.1(b)]). We shall investigate the M0 -module (I(ρ))P0 and M0 ∩ H -distinguished components therein. Note that irreducible M0 ∩ H -distinguished M0 -modules have to be one-dimensional according to (5.2.1). Let W (resp. WM0 ) be the Weyl group of (G, A∅ ) (resp. of (M0 , A∅ )). Following the definition of [3, 1.1.3], we can give the coset representatives [WM0 \W/WM0 ] as ⎛ ⎞ ⎛ ⎞ 1 0 1 0 1⎟ 0 1 ⎜ ⎜ ⎟ e, ⎝ ⎠ and ⎝ ⎠. 1 0 1 0 0 1 1 −1/2

1/2

We use the abbreviation [ρ] = ρ · |det(·)|F ⊗ ρ · |det(·)|F . The Geometric Lemma [2, 2.12] asserts that s.s s.s I(ρ) P Fw [ρ] 0

w∈[WM0 \W/WM0 ]

where M Fw [ρ] = IndM0 ∩wP 0

0w

−1

w [ρ]M0 ∩w−1 P0 w .

It is easy to determine the three pieces Fw ([ρ]) for each w. (i) w = e: Nothing but Fw [ρ] = [ρ], which is notM0 ∩ H-distinguished unless ρ is one-dimensional. (ii) w =

10 01

10 01

: In this case, we have M0 ∩ wP0 w −1 = M0 ∩ w −1 P0 w = M0 ,

so that

−1/2 1/2 Fw [ρ] = w [ρ] = ρ · det(·) ⊗ ρ · det(·) .

Again this cannot be M0 ∩ H -distinguished unless ρ is one-dimensional.

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1 (iii) w =

01 10

: We have 1

⎧⎛ ∗ ⎪ ⎨ ⎜0 −1 −1 M0 ∩ wP0 w = M0 ∩ w P0 w = ⎝ ⎪ ⎩ 0 0

∗ ∗ 0 0

0 0 ∗ 0

⎞⎫ 0 ⎪ ⎬ 0⎟ ⎠ B2 × B2 ∗ ⎪ ⎭ ∗

where B2 is as in 5.1. So we may write 2 ×G2 w Fw [ρ] = IndG [ρ]B2 ×B2 B2 ×B2 in this case. Now, take ρ to be the Steinberg representation St2 of G2 . We claim that the irreducible H distinguished representation π(St2 ) is H -square integrable. It is enough to look at the exponents coming from M0 ∩ H -distinguished components in I(ρ)P0 . So we may look at only the case (iii) above. Put T2 = { ∗0 ∗0 ∈ B2 }. Under this notation, the T2 × T2 -module [ρ]B2 ×B2 is given by 1/2 −1/2 1/2 −1/2 1/2 1/2 ρB2 det(·)F ⊗ ρB2 det(·)F = δB2 det(·)F ⊗ δB2 det(·)F . This is a character of T2 × T2 written as (t1 , t2 ), (t3 , t4 ) → |t1 |F · |t4 |−1 F . Applying w and inducing up to G2 × G2 , we have G2 −1 2 Fw [St2 ] = IndG B2 | · |F ⊗ 1 ⊗ IndB2 1 ⊗ | · |F .

(5.2.2)

This is a reducible principal series having one-dimensional quotient. So the only possible element of ExpS0 (π(St2 )P0 , rP0 (λ)) (for any λ ∈ (π(ρ)∗ )H ) is the restriction of the central character of (5.2.2), which is given by χ diag(s1 , s1 , s2 , s2 ) = |s1 |F · |s2 |−1 F . We may conclude that |χ(s)| < 1 for all s ∈ S0− \ Z0 S01 , hence the claim follows from Theorem 4.7. References [1] Y. Benoist, H. Oh, Polar decomposition for p-adic symmetric spaces, Int. Math. Res. Not. (24) (2007), Art. ID rnm121, 20 pp. [2] I.N. Bernstein, A.V. Zelevinsky, Induced representations of p-adic groups, I, Ann. Sci. École Norm. Sup. 10 (1977) 441–472. [3] W. Casselman, Introduction to the theory of admissible representations of p-adic reductive groups, unpublished manuscript, available at http://www.math.ubc.ca/people/faculty/cass/research.html, 1974. [4] P. Delorme, Constant term of smooth Hψ -spherical functions on a reductive p-adic group, Trans. Amer. Math. Soc. 362 (2) (2010) 933–955. [5] P. Delorme, V. Sécherre, An analogue of the Cartan decomposition for p-adic reductive symmetric spaces, preprint, arXiv:math.RT/0612545, 2006.


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[6] J. Hakim, Distinguished p-adic representations, Duke Math. J. 62 (1) (1991) 1–22. [7] A.G. Helminck, G.F. Helminck, A class of parabolic k-subgroups associated with symmetric k-varieties, Trans. Amer. Math. Soc. 350 (1998) 4669–4691. [8] M.J. Heumos, S. Rallis, Symplectic-Whittaker models for Gln , Pacific J. Math. 146 (1990) 247–279. [9] S. Kato, K. Takano, Subrepresentation theorem for p-adic symmetric spaces, Int. Math. Res. Not. (11) (2008), Art. ID rnn028, 40 pp. [10] N. Lagier, Terme constant de fonctions sur un espace symétrique réductif p-adique, J. Funct. Anal. 254 (4) (2008) 1088–1145. [11] D. Prasad, On the decomposition of a representation of GL(3) restricted to GL(2) over a p-adic field, Duke Math. J. 69 (1) (1993) 167–177. [12] J.-L. Waldspurger, La formule de Plancherel pour les groupes p-adiques, d’après Harish-Chandra, J. Inst. Math. Jussieu 2 (2) (2003) 235–333.


On the isotropy constant of projections of polytopes ✩ David Alonso-Gutiérrez a,∗ , Jesús Bastero a , Julio Bernués a , Paweł Wolff b a Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain b Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106-7058, USA

Received 30 April 2009; accepted 22 October 2009 Available online 30 October 2009 Communicated by K. Ball

Abstract

√ The isotropy constant of any d-dimensional polytope with n vertices is bounded by C n/d where C > 0 is a numerical constant. © 2009 Elsevier Inc. All rights reserved. Keywords: Polytopes; Projections; Isotropy constant

1. Introduction The boundedness of the isotropy constant (see definition below) is a major conjecture in Asymptotic Geometric Analysis. The answer is known to be positive for many families of convex bodies, see for instance [15] or [12] and the references therein. In this paper we focus our attention on the isotropy constant of polytopes or, equivalently, of projections of the unit ball of n1 space (in the symmetric case) and of the regular n-dimensional simplex Sn (in the non-symmetric case). M. Junge [9] proved that the isotropy constant of all orthogonal projections of Bpn , the unit ball of the np space, 1 < p ∞, is bounded by Cp an estimate improved to C p in [13] (p is the ✩

Partially supported by Marie Curie RTN CT-2004-511953. The three first named authors were partially supported by MCYT Grant (Spain) MTM2007-61446 and DGA E-64. The fourth named author was partially supported by National Science Foundation FRG grant DMS-0652722 (USA). * Corresponding author. E-mail addresses: [email protected] (D. Alonso-Gutiérrez), [email protected] (J. Bastero), [email protected] (J. Bernués), [email protected] (P. Wolff). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.019

D. Alonso-Gutiérrez et al. / Journal of Functional Analysis 258 (2010) 1452–1465

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conjugate exponent of p and C a numerical constant). Later [10, Theorem 4] M. Junge showed that the isotropy constant of any symmetric polytope with 2n vertices is bounded by C log n, see also [14]. In a recent paper [12], B. Klartag and G. Kozma show the boundedness of the isotropy constant of random Gaussian polytopes. The integral over a polytope, which defines its isotropy constant, is computed by passing to an integral over its surface (faces). A consequence of their results is that “most” (see precise meaning below) d-dimensional projections of B1n as well as of Sn have bounded isotropy constant. When reading this statement one should have in mind the well-known fact that every symmetric convex body in Rd is “almost” a projection of a B1n ball, with possibly large n. In the same spirit, positive answers for other random d-dimensional polytopes with n ( Cd) vertices were given in [1,6]. Our main theorem (Corollary 3.5) states that for any d-dimensional polytope K with n vertices its isotropy constant LK verifies LK C

n , d

where C > 0 is a numerical constant. We now pass to describe the contents of the paper. The second section introduces the geometric tool (Proposition 2.1) necessary to deal with integration on d-dimensional projections of polytopes (Corollary 2.8). Some time ago, one of the authors learned about this tool from Prof. Franck Barthe. The ideas originate from a paper by U. Betke [3], where a general result was presented, namely a related formula for mixed volumes of two polytopes. However, for the sake of completeness, we provide the proof of the particular result we need. It also seems that the content of the proof is more geometric. In the third section we use these tools to prove our aforementioned main result (Corollary 3.5) by easily reducing it to the cases K = PE B1n or K = PE Sn (Theorem 3.4) where E ⊂ Rn is any d-dimensional subspace and PE is the orthogonal projection onto E. Also in this section we give a proof of the observation that for “most” subspaces, that is, for a subset A of the Grassmann space Gn,d of Haar probability measure 1 − c1 e−c2 max{log n,d} , one has LPE B1n < C and LPE Sn < C for every E ∈ A with numerical constants C, c1 , c2 (Proposition 3.3). The next section studies the isotropy constant of projections of random polytopes with vertices on the sphere S n−1 . Using the techniques from Section 2 and [1] we show that, with high probability,the isotropy constants of all d-dimensional projections of random polytopes are bounded by C dn (Proposition 4.1). In the last section we show that for every isotropic convex body, the isotropy constants of its hyperplane projections are comparable to the isotropy constant of the body itself (Corollary 5.1). Recall that the analogous result for hyperplane sections was already proved in [15]. The proof uses Steiner symmetrization in a similar way as it appears in [5], with better numerical constants. In particular, we have LPH Bpn C for any hyperplane H and 1 < p < ∞ improving Junge’s estimate [9] for the case of hyperplanes. In [2] a different proof of this fact is given with the hope it might be extended to lower-dimensional projections. We recall that a convex body K ⊂ Rn is isotropic if it has volume Voln (K) = 1, the barycenter of K is at the origin and its inertia matrix is a multiple of the identity. Equivalently, there exists a constant LK > 0 called isotropy constant of K such that L2K = K x, θ 2 dx, ∀θ ∈ S n−1 . It is well known [15], that every convex body K ⊂ Rn has an affine transformation K1 isotropic, so we can write LK := LK1 . This is well defined and moreover,

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nL2K = inf

|x|2 dx; a ∈ Rn , T ∈ SL(n) .

1 2

Voln (K) n +1

(1.1)

a+T K

For a convex body K ⊂ Rn , r(K) = min{|x|: x ∈ K} is the inradius of K. We will think of Sn as an n-dimensional regular simplex in Rn with center of mass at the origin. We will write n = conv{e1 , . . . , en+1 } for the natural position of an n-dimensional regular simplex in Rn+1 . The Lebesgue measure on an affine subspace E will be denoted by λE . For a measurable set A ⊆ E, if d is a dimension of E, Vold (A) will stand for λE (A). The notation a ∼ b means a · c1 b a · c2 for some numerical constants c1 , c2 > 0. 2. Projections of polytopes Throughout this section, K ⊆ Rn is a polytope (non-empty but possibly of empty interior), E ⊆ Rn is a linear subspace of dimension d (1 d n − 1) and PE is the orthogonal projection onto E. Let us fix some notation and recall necessary definitions (we follow the book by Schneider [18, Chapters 1, 2]). For a subset A ⊆ Rn , aff A denotes the minimal affine subspace which contains A. The dimension of a convex set A is dim aff A. When writing relint A we mean the relative interior of A w.r.t. the topology of aff A. If G ⊆ Rn is an affine subspace then G0 denotes the linear subspace parallel to G. A convex subset F ⊆ K of a polytope K is called a face if for any x, y ∈ K, (x + y)/2 ∈ F implies x, y ∈ F (see also [18, Section 1.4, p. 18]). The set of j -dimensional faces (j -faces, in short) of K will be denoted as Fj (K) (j = 0, 1, . . . , n), and F (K) = nj=0 Fj (K) ∪ {∅} is the set of all faces of K (∅ is also a face). K can be decomposed into a disjoint union of {relint F ; F ∈ F (K)} (see [18, Theorem 2.1.2]). For that reason for any x ∈ K the unique face F ∈ F (K) such that x ∈ relint F will be denoted by F (K, x). For x ∈ K, a normal cone of K at x is

N (K, x) = u ∈ Rn ; ∀z∈K z − x, u 0 . N(K, x) is a closed convex cone. We shall also consider another closed convex cone, namely S(K, x) =

λ(K − x).

λ>0

(In general, i.e. when K is a convex body, S(K, x) does not have to be closed.) By [18, (2.2.1)], N (K, x)∗ = S(K, x),

(2.2)

where the polarity used here is the polarity of convex cones, namely, if C ⊆ Rn is a convex cone,

C ∗ = y ∈ Rn ; ∀x∈C x, y 0 (see also [18, Section 1.6, p. 34]). We shall also need to consider normal cones taken w.r.t. an affine subspace. If G is an affine subspace of Rn and L ⊆ G is a convex body, then for x ∈ L we define a normal cone for L at x taken w.r.t. G:


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NG (L, x) = u ∈ G0 ; ∀z∈L z − x, u 0 . Note that NG (L, x) ⊆ G0 . The similar duality relation to (2.2) holds: NG (L, x)∗G0 = S(L, x),

(2.3)

where the polarity is taken w.r.t. G0 . For any face ∅ = F ∈ F (K), define N (K, F ) := N (K, x),

where x ∈ relint F.

This definition does not depend on the choice of x (see [18, Section 2.2, p. 72]). NG (L, F ) is analogously defined. For a given polytope K ⊆ Rn and a linear subspace E ⊆ Rn of dimension d, let us fix any u ∈ E ⊥ \ {0} which satisfy u∈ /

PE ⊥ N (K, F ); F ∈ F (K) \ {∅}, dim PE ⊥ N (K, F ) n − d − 1 .

(2.4)

Clearly, such u exists, since (2.4) excludes only a finite union of sets of dimension < n − d from E ⊥ which is of dimension n − d. Consider the following subsets of F (K):

(K, E, u) := F ∈ F (K); u ∈ PE ⊥ N (K, F ) , F

d (K, E, u) := F(K,

F E, u) ∩ Fd (K). Proposition 2.1. Let K ⊂ Rn be a polytope, E a d-dimensional subspace, u ∈ E ⊥ verifying (2.4)

(K, E, u) as described above. Then and F

(K, E, u)} is a family of pair-wise disjoint sets, (a) {PE (relint F ); F ∈ F

(b) {PE F ; F ∈ F(K, E, u)} = PE K.

Moreover, F(K, E, u) ⊆ an affine isomorphism.

0j d

d (K, E, u), PE |F : F → PE F is Fj (K) and for each F ∈ F

In the proof of the proposition we shall use several lemmas. Lemma 2.2. Let L be a polytope in Rn and G ⊆ Rn be an affine subspace. If x ∈ L ∩ G then PG0 N (L, x) = NG (L ∩ G, x). Proof. By taking polars w.r.t. G0 we see that the assertion is equivalent to N (L, x)∗ ∩ G0 = NG (L ∩ G, x)∗G0

(2.5)

(for the l.h.s. we used the fact that for a convex cone C, (PG0 C)∗G0 = C ∗ ∩ G0 ). Since G0 = G − x by (2.2) we get

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N (L, x)∗ ∩ G0 = S(L, x) ∩ (G − x) =

λ(L − x) ∩ (G − x)

λ>0

= S(L ∩ G, x). Applying (2.3) we see that the r.h.s. of (2.5) is also equal to S(L ∩ G, x).

2

Lemma 2.3. (See [18, Section 2.2].) Let L be a polytope contained in an affine subspace G ⊆ Rn . Then

NG (L, F ) = G0 .

F ∈F0 (L)

Lemma 2.4. With the hypothesis as in the previous lemma, for x, y ∈ L, NG (L, x) ∩ NG (L, y) = NG L, (x + y)/2 . Proof. The inclusion ⊆ is immediate from the definition of a normal cone. For the converse x+y inclusion take u ∈ NG (L, (x +y)/2). Then x − x+y 2 , u 0, y − 2 , u 0, so x −y, u = 0. Now, for all z ∈ L, y−x x +y ,u + , u 0, z − x, u = z − 2 2 so u ∈ NG (L, x). Similarly u ∈ NG (L, y).

2

Lemma 2.5. (See [18, Section 2.4].) With the hypothesis as in Lemma 2.3, for ∅ = F ∈ F (L), dim NG (L, F ) = dim G − dim F. Remark 2.6. Actually we shall use only the inequality dim NG (L, F ) dim G − dim F which simply follows from the fact NG (L, F ) ⊆ ((aff F )0 )⊥G0 . Lemma 2.7. Let K ⊆ Rn be a polytope, E ⊆ Rn be a linear subspace of dimension d and u ∈ E ⊥ satisfies (2.4). Let y ∈ K, x = PE y ∈ E, Kx = K ∩ (x + E ⊥ ) (Kx is a polytope in x + E ⊥ ). If one of the equivalent condition holds: (i) u ∈ PE ⊥ N (K, y), (ii) u ∈ Nx+E ⊥ (Kx , y), then {y} ∈ F0 (Kx ) and dim F (K, y) d. Proof. The conditions (i) and (ii) are equivalent by Lemma 2.2. Consider F = F (K, y) and F = F (Kx , y). By the condition (2.4) on u, dim PE ⊥ N (K, F ) n − d,


1457

so dim Nx+E ⊥ (Kx , F ) n − d and also dim N (K, F ) n − d. Therefore Lemma 2.5 applied to Kx and F implies dim F = 0, so {y} = F ∈ F0 (Kx ). Eventually, applying the same lemma to K and F yields dim F d. 2

E, u) such that for some x ∈ E, Proof of Proposition 2.1. (a) Take F1 , F2 ∈ F(K, x ∈ PE (relint F1 ) ∩ PE (relint F2 ) which means that for i = 1, 2 one can find yi ∈ x + E ⊥ that yi ∈ relint Fi and then u ∈ PE ⊥ N (K, Fi ) = PE ⊥ N (K, yi ). Consider a convex polytope Kx = K ∩ (x + E ⊥ ). Lemma 2.7 implies that {y1 }, {y2 } ∈ F0 (Kx ) and also u ∈ Nx+E ⊥ (Kx , y1 ) ∩ Nx+E ⊥ (Kx , y2 ) = Nx+E ⊥ Kx , (y1 + y2 )/2 , where the last equality is due to Lemma 2.4. But again, Lemma 2.7 implies that also {(y1 + y2 )/2} ∈ F0 (Kx ), hence y1 = y2 (see definition of a face) and consequently, F1 = F (K, y1 ) = F2 . (b) The inclusion “⊆” is obvious. For the inclusion “⊇” take arbitrary x ∈ PE K. Put Kx = K ∩ (x + E ⊥ ). Kx is a non-empty polytope in x + E ⊥ . By Lemma 2.3 one can find y ∈ x + E ⊥ such that {y} ∈ F0 (Kx ) and u ∈ Nx+E ⊥ (Kx , y). Lemma 2.7 (or just Lemma 2.2) implies u ∈

(K, E, u). PE ⊥ N(K, F ) where F = F (K, y). Consequently, F ∈ F

(K, E, u) has dimenBy the definition of F (K, E, u) and Lemma 2.7, any face F ∈ F sion d.

d (K, E, u), PE |F : F → PE F is an isomorphism. The condiFinally we show that for F ∈ F tion (2.4) and Lemma 2.5 implies n − d dim PE ⊥ N (K, F ) dim N (K, F ) n − dim F = n − d, so dim PE ⊥ N (K, F ) = dim N (K, F ) = n − d. This means (span N (K, F )) ∩ E = {0} and N(K, F )⊥ ∩ E ⊥ = {0}. The definition of a normal cone yields (aff F )0 ⊆ N (K, F )⊥ , which finally gives (aff F )0 ∩ E ⊥ = {0}. 2 The following corollary is an immediate consequence of Proposition 2.1. Corollary 2.8. Let K ⊆ Rn be a convex polytope, E ⊆ Rn a d-dimensional linear subspace

of Fd (K) such that for any integrable function (1 d n − 1). Then there exists a subset F f : E → R,

Vold (PE F ) f (x) λE (dx) = f (PE y) λaff F (dy). Vold (F )

PE K

In particular (for f ≡ 1),

F ∈F

F

(2.6)

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Vold (PE K) =

Vold (PE F ).

F ∈F

= F

d (K, E, u). By ProposiProof. Choose any u ∈ E ⊥ satisfying (2.4) for K and E and put F tion 2.1, f (x) λE (dx) = f (x) λE (dx)

P F F ∈F E

PE K

=

Vold (PE F ) f (PE y) λaff F (dy). Vold (F )

F ∈F

2

F

For our purposes we shall use above corollary with f (x) = |x|2 . In such case, the obvious Vold (PE F ) inequality |PE y| |y| and the identity 1 = F ∈F Vol lead to the following estimate: if d (PE K) PE K is a body of dimension d (i.e. is non-degenerated) then 1 Vold (PE K)

1 |x| λE (dx) max

Vol d (F ) F ∈F

|y|2 λaff F (dy).

2

PE K

(2.7)

F

3. Projections of the n1 -ball and the regular simplex First of all, we are going to see that “most” projections of B1n on d-dimensional subspaces (d n) have the isotropy constant bounded. It is well known that any symmetric convex polytope in Rd with 2n vertices is linearly equivalent to PE B1n for some E ∈ Gn,d . Indeed, if T : Rn → Rd is a linear transformation of full rank, then taking the d-dimensional subspace E = (ker T )⊥ ⊆ Rn , T can be represented as T |E PE where T |E : E → Rd is a linear isomorphism being a restriction of T to the subspace E. As an immediate consequence we obtain the following Lemma 3.1. Let K = conv{±v1 , . . . , ±vn } ⊆ Rd be a symmetric convex polytope with non-empty interior and let T : Rn → Rd be the linear map such that T ei = vi . Then for E = (ker T )⊥ ∈ Gn,d , PE B1n and K are linearly equivalent. One may also prove a similar lemma in the non-symmetric case. Recall that n = conv{e1 , . . . , en+1 } ⊆ H ⊆ Rn+1 where H , as in the whole of this section, denotes the hyperplane orthogonal to the vector (1, . . . , 1) ∈ Rn+1 . Lemma 3.2. Let K = conv{v1 , . . . , vn+1 } ⊆ Rd (n d) be a convex polytope with non-empty 1 n+1 interior. Let T : Rn+1 → Rd be the linear map that T ei = vi − v0 where v0 = n+1 i=1 vi and E = (ker T )⊥ ⊆ Rn+1 . Then E ⊆ H is a subspace of dimension d and K − v0 is linearly equivalent to PE n . Consequently, K is affinely equivalent to some orthogonal projection of the n-dimensional regular simplex Sn onto a d-dimensional subspace. Proof. Clearly (1, . . . , 1) ∈ ker T , so E ⊆ H . Since K has non-empty interior, vectors vi − v0 span the whole of Rd , so T is of full rank. Therefore dim E = d and the argument given above applies. 2


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Now we can prove the following result concerning the isotropy constant of random projections of B1n and Sn . Proposition 3.3. There exist absolute constants C, c1 , c2 > 0 such that the Haar probability measure of the set of subspaces E ∈ Gn,d verifying LPE B1n < C

and

LPE Sn < C

is greater than 1 − c1 e−c2 max{log n,d} . Proof. For small values of d, namely d c log n, the isotropy constant of a random projection is bounded by an absolute constant with probability greater than 1 − ncc12 as a consequence of Dvoretzky’s theorem. Let G = (gij ) be a d × n Gaussian random matrix, i.e. the gij ’s are i.i.d. N (0, 1) Gaussian random variables. Since (ker G)⊥ = Im(Gt ) ⊆ Rn , Gt being the transpose matrix of G, and the columns of Gt are independent and rotationally invariant random vectors in Rn , then a random subspace E = (ker G)⊥ has dimension d a.s. and is distributed according to the Haar probability measure μ on Gn,d . Therefore for any constant C > 0, μ{E ∈ Gn,d ; LPE B1n < C} = P{LPE B1n < C}. Lemma 3.1 and the affine invariance of the isotropy constant imply LPE B1n = Lconv(±Ge1 ,...,±Gen ) a.s. Klartag and Kozma proved in [12] that if C is a sufficiently large absolute constant, P{Lconv(±Ge1 ,...,±Gen ) < C} > 1 − c1 e−c2 d which completes the proof in the symmetric case. For the non-symmetric case, we proceed n+1analogously. For a d × (n + 1) Gaussian random ¯ = (gij − 1 matrix G = (gij ), take G k=1 gik )id,j n+1 . Since the sum of the columns of n+1 ⊥ t n+1 ¯ ¯ ¯ ¯ (equivalently, columns G is zero, (ker G) = Im(G ) ⊆ H ⊆ R . Moreover, since rows of G t ¯ of G ) are independent canonical Gaussian random vectors in H , the random subspace E = ¯ ⊥ ⊆ H is distributed according to the Haar probability measure on GH,d (Grassmann (ker G) manifold of d-dimensional subspaces of H ). Lemma 3.2 and the affine invariance of the isotropy constant imply LPE n = Lconv(Ge1 ,...,Gen+1 ) a.s. Since PE n = PE (PH n ) and PH n is an n-dimensional regular simplex (in H ), a nonsymmetric counterpart of the result of Klartag and Kozma [12], P{Lconv(Ge1 ,...,Gen+1 ) < C} > 1 − c1 e−c2 d , finishes the proof.

2

In the final part of the section we will use the tools from Section 2 to prove the main result. In particular, whenever d cn the boundedness of the isotropy constant holds not only for “most” projections of B1n and Sn but deterministically for all of them.

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Theorem 3.4. Let E ⊆ Rn be a subspace of dimension 1 d n−1 and K = PE B1n , T = PE Sn . Then LK , LT C n/d, where C > 0 is a universal constant. Proof. As an immediate consequence of (1.1), L2K

1 1 1 d Vold (K)2/d Vold (K)

|x|2 λE (dx).

(3.8)

K

Applying (2.7), we obtain the bound 1 Vold (K)

|x|2 λE (dx) K

1 Vold (d )

|x|2 λaff d (dy) = d

2 d +2

(3.9)

(for the last equality see e.g. [12, Lemma 2.3]). To estimate Vold (K) note that n−1/2 B2n ⊆ B1n , so n−1/2 (B2n ∩ E) ⊆ PE B1n . Therefore c Vold (K)1/d √ . nd

(3.10)

Combining these two, we get L2K

n 1 nd 2 C . 2 d c d +2 d

In the case of the simplex it is convenient to embed E and Sn into H . More precisely, we take Sn = conv{PH ei : i = 1, . . . , n + 1} ⊆ H ⊆ Rn+1 and assume E ⊆ H . Now observe that T = PE Sn = PE n so (2.7) again yields 1 Vold (T )

|x|2 λE (dx) T

2 . d +2

To bound the volume radius of T from below, we use the Rogers–Shephard inequality [17]:

2d d

−1

Vold (T − T ) Vold (T ).

Note that T − T ⊇ conv(T ∪ −T ) = conv(PE n ∪ −PE n ) = PE conv(n ∪ −n ) = PE B1n+1 . Combining with the estimate (3.10),


1/d

Vold (T )

−1/d 1/d 2d Vold PE B1n+1 d −1/d c c 2d √ . √ d (n + 1)d nd

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2

Due to Lemma 3.2, we immediately get the following: Corollary 3.5. Let K ⊆ Rd be a non-degenerated (dim K = d) convex polytope with n vertices. Then n . LK C d 4. Isotropy constant of projections of random polytopes In this section we consider polytopes generated by the convex hull of vertices randomly chosen on the S n−1 . The main result is Proposition 4.1. There exist absolute constants C, c1 and c2 , such that if m n, {Pi }m i=0 are n−1 independent random vectors on S and K = conv{±P1 , . . . , ±Pm } or K = conv{P0 , . . . , Pm }, then n , ∀E ∈ Gn,d , ∀1 d n − 1 1 − c1 e−c2 n . P LPE K C d The proof follows [1]. We shall only sketch the main ideas as the technical computations can be found in that reference. Sketch of the proof. Let E ⊆ Rn denotes a d-dimensional subspace. The ideas in what follows will give us the proof for m cn with an absolute constant c. If m < cn, Corollary 3.5 gives m deterministically LPE K C d C dn . Apply once again (1.1). Writing r(K) the inradius of K and using the inequality (2.7), the main consequence of Proposition 2.1, we obtain that for any polytope K ⊂ Rn and any ddimensional subspace E, L2PE K

C r(K)2

1 F ∈Fd (K) Vold (F )

|x|2 λaff F (dx).

max

F

When K is the symmetric convex hull of m independent random points in S n−1 , it was proved n in [1, Lemma 3.1], that for some constant c such that cn m ne 2 , 1 P r(K) < √ 2 2

log m n n

e−n .

The same proof gives the statement in the non-symmetric case.

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On the other hand, with probability 1, each d-dimensional face of K is a simplex F = conv{Q1 , . . . , Qd+1 } with Qi = εi Pji (or just Qi = Pji in the non-symmetric case) where 1 j1 < · · · < jd+1 m and εi ∈ {−1, 1}. The same proof as in [1,12] shows that with probability 1 we have 1 Vold (F )

|x|2 λaff F (dx) =

d+1 1 2 + Qi1 , Qi2 . d + 2 (d + 1)(d + 2)

(4.11)

i1 =i2

F

In order to give a bound for this quantity for a fixed F ∈ Fd (K) we proceed in the same way as in [1, Theorem 3.1], by using a version of Bernstein’s inequality as stated in [4]. We thus obtain

d+1

P

Qi1 , Qi2 > (d + 1) 2e−cn

(4.12)

i1 =i2

for every > 0 , where 0 is an absolute constant. Now, for each F ∈ Fd (K) let QF1 , . . . , QFd+1 be vertices of F . Applying (4.12) and the union 2m ), we obtain for log m bound over Fd (K) (whose cardinality is clearly bounded by d+1 n > 0 , P

max

F ∈Fd (K)

d+1 i1 =i2

QFi1 , QFi2

m > (d + 1) log n

m 2em m 2m 2e−cn log n 2e−cn log n +(d+1) log d+1 d +1 m

2e−cn log n +n log

2em n

since the function x log Cx is increasing when

, C x

P ∃1 d n − 1 s.t. 2e−cn log

d+1

max

F ∈Fd (K)

m 2em n +n log n +log n

> e. Consequently, by the union bound over d,

i1 =i2

QFi1 , QFi2

m > (d + 1) log n

.

Since m cn, considering the complement set and using (4.11), we can fix > 0 a large enough numerical constant to obtain P ∀1 d n − 1,

1 F ∈Fd (K) Vold (F )

|x|2 λaff F (dx)

max

m m C log 1 − 2e−cn log n . d n

F n

2 Thus, there exist constants c, C > 0 such that if cn m ne then the set of points n (P1 , . . . , Pm ) for which the inequality LPE K C d holds for every d-dimensional subspace m

E and for every 1 d n − 1 has probability greater than 1 − 2e−cn log n − e−n > 1 − c1 e−c2 n .

D. Alonso-Gutiérrez et al. / Journal of Functional Analysis 258 (2010) 1452–1465 n

In case m > ne 2 , for n large enough, r(K) this probability dL2PE K

1 2 Vol (P d E K) Vold (PE K) d 1

and the proof is complete.

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with probability greater than 1 − e−n so with

1 4

|x|2 dx PE K

1 2

Vold ( 14 B2d ) d

cd

2

5. A general result In this section we prove a general relation between the isotropy constant of the hyperplane projections of an isotropic convex body and of the body itself. Corollary 5.1. Let K be an isotropic convex body and let H be a hyperplane. Then LPH K ∼ LK . Its proof relies on the next proposition which improves the numerical constants appearing in a more general statement in [5] for the case of projections onto hyperplanes. Proposition 5.2. Let K ⊂ Rn be an isotropic convex body and let H = ν ⊥ be a hyperplane. If S(K) is the Steiner symmetrization of K with respect to H then, log n LK LS(K) LK 1−c n for some numerical constant c > 0. Proof. Without loss of generality, we may assume that ν = en = (0, . . . , 0, 1). Write E = en for the 1-dimensional subspace generated by en . The Steiner symmetrization of K is defined by 1 S(K) := (y, t) ∈ Rn−1 × R; y ∈ PH K, |t| Vol1 K ∩ (x + E) . 2 Clearly PH K = PH S(K) = S(K) ∩ H . Now we study the inertia matrix of S(K). First notice that for x ∈ PH K, Vol1 (K ∩ (x + E)) = Vol1 (S(K) ∩ (x + E)). For every θ ∈ S n−1 ∩ H , Fubini’s theorem yields

x, θ dx =

y + ten , θ 2 dt dy

2

S(K)

PH K S(K)∩(x+E)

=

y, θ 2 Vol1 K ∩ (x + E) dy = L2K .

PH K

Using the fact that S(K)∩(x+E) t dt = 0 for x ∈ PH K, in the similar fashion we show that for every θ ∈ S n−1 ∩ H ,

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x, θ x, en dx = 0. S(K)

Also

S(K)∩(x+E) t

2 dt

K∩(x+E) t

2 dt

for x ∈ PH K, thus

x, en dx

x, en 2 dx = L2K .

2

K

S(K)

Taking σ > 0 such that σ 2 :=

2 2 S(K) x, en dx/LK ,

⎛ ⎜ M = L2K ⎜ ⎝

1

we obtain that the inertia matrix of S(K) is ⎞

..

⎟ ⎟. ⎠

. 1 σ2

The volume of S(K) is 1, so LS(K) = (det M)1/2n (see [15]), which means LS(K) = σ

1/n

LK =

( S(K) x, en 2 dx)1/2 1/n LK

(5.13)

LK .

Since σ 1, we obtain LS(K) LK . A well-known fact due to Hensley [8] states that Voln−1 (K1 ∩ H ) ∼ ( K1 x, en 2 dx)−1/2 for any convex body K1 with volume 1 and center of mass at the origin. Using this fact for S(K) in (5.13) we obtain that for some absolute constant c > 0, LS(K)

c Voln−1 (S(K) ∩ H )LK

1/n

LK =

c Voln−1 (PH K)LK

1/n LK .

Now we use the following inequality: 1 Voln−1 (PH K)Vol1 (K ∩ E) Voln (K) = 1 n (for the proof, see for instance [16, Lemma 8.8] which works in the non-symmetric case). Hence Voln−1 (PH K)

n n , Vol1 (K ∩ E) 2r(K)

where r(K) is the inradius of K. Since every isotropic convex body verifies r(K) LK (see [7] or [11], for instance) we obtain LS(K)

2c n

1/n LK .

2


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Proof of Corollary 5.1. Since PH K = S(K) ∩ H we have LPH K = LS(K)∩H ∼ LS(K) ∼ LK , where the first equivalence is the corresponding one for sections of convex bodies as proved in [15]. 2 Acknowledgments The fourth named author would like to thank Prof. Stanisław Szarek for some useful discussions. Part of this work has been done while the fourth named author enjoyed the hospitality of University of Zaragoza staying there as Experienced Researcher of the European Network PHD. References [1] D. Alonso-Gutiérrez, On the isotropy constant of random convex sets, Proc. Amer. Math. Soc. 136 (9) (2008) 3293– 3300. [2] D. Alonso-Gutiérrez, J. Bastero, J. Bernués, P. Wolff, The slicing problem for hyperplane sections of Bpn , preprint, available in http://www.unizar.es/matematicas/personales/bernues/ABBWJulioAbad.pdf. [3] U. Betke, Mixed volumes of polytopes, Arch. Math. 58 (1992) 388–391. [4] J. Bourgain, J. Lindenstrauss, V.D. Milman, Minkowski sums and symmetrizations, in: Geometric Aspects of Functional Analysis, 1986/1987, in: Lecture Notes in Math., vol. 1317, Springer, Berlin, 1988, pp. 44–66. [5] J. Bourgain, B. Klartag, V.D. Milman, Symmetrization and isotropic constants of convex bodies, in: Geometric Aspects of Functional Analysis, in: Lecture Notes in Math., vol. 1850, Springer, Berlin, 2004, pp. 101–115. [6] N. Dafnis, A. Giannopoulos, O. Guédon, On the isotropic constant of random polytopes, Adv. Geom., in press. [7] A. Giannopoulos, Notes on Isotropic Convex Bodies, Warsaw, October 2003. [8] D. Hensley, Slicing convex bodies – bounds for slice area in terms of the body’s covariance, Proc. Amer. Math. Soc. 79 (4) (1980) 619–625. [9] M. Junge, Hyperplane conjecture for quotient spaces of Lp , Forum Math. 6 (1994) 617–635. [10] M. Junge, Proportional subspaces of spaces with unconditional basis have good volume properties, in: Geometric Aspects of Functional Analysis, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 121–129. [11] R. Kannan, L. Lovász, M. Simonovits, Isoperimetric problems for convex bodies and a localization lemma, Discrete Comput. Geom. 13 (1995) 541–549. [12] B. Klartag, G. Kozma, On the hyperplane conjecture for random convex sets, Israel J. Math. 170 (1) (2009) 235–268. [13] B. Klartag, E. Milman, On volume distribution in 2-convex bodies, Israel J. Math. 164 (2008) 221–249. [14] E. Milman, Dual mixed volumes and the slicing problem, Adv. Math. 207 (2006) 566–598. [15] V. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: Geometric Aspects of Functional Analysis, 1987/1988, in: Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. [16] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge, 1989. [17] C.A. Rogers, G.C. Shephard, The difference body of a convex body, Arch. Math. (Basel) 8 (1957) 220–233. [18] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.


Equivariant Poincaré duality for quantum group actions Ryszard Nest a , Christian Voigt b,∗ a Institut for Matematiske Fag, Universitet København, Universitetsparken 5, 2100 København, Denmark b Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3-5, 37073 Göttingen, Germany

Received 21 June 2009; accepted 20 October 2009 Available online 6 November 2009 Communicated by S. Vaes

Abstract We extend the notion of Poincaré duality in KK-theory to the setting of quantum group actions. An important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products. Along the way we discuss general properties of equivariant KK-theory for locally compact quantum groups, including the construction of exterior products. As an example, we prove that the standard Podle´s sphere is equivariantly Poincaré dual to itself. © 2009 Elsevier Inc. All rights reserved. Keywords: Quantum groups; Kasparov theory; Poincaré duality

1. Introduction The notion of Poincaré duality in K-theory plays an important rôle in noncommutative geometry. In particular, it is a fundamental ingredient in the theory of noncommutative manifolds due to Connes [11]. A noncommutative manifold is given by a spectral triple (A, H, D) where A is a ∗-algebra represented on a Hilbert space H and D is an unbounded self-adjoint operator on H . The basic requirements on this data are that D has compact resolvent and that the commutators [D, a] are bounded for all a ∈ A. There are further ingredients in the definition of a noncommutative manifold, in particular a grading and the concept of a real structure [12,13]. An important recent result due to Connes is the reconstruction theorem [14], which asserts that in the commutative * Corresponding author.

E-mail addresses: [email protected] (R. Nest), [email protected] (C. Voigt). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.015

R. Nest, C. Voigt / Journal of Functional Analysis 258 (2010) 1466–1503

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case, under some natural conditions, the algebra A is isomorphic to C ∞ (M) for a unique smooth manifold M. The real structure produces a version of KO-Poincaré duality, which is a necessary ingredient for the existence of a smooth structure. Quantum groups and their homogeneous spaces give natural and interesting examples of noncommutative spaces, and several cases of associated spectral triples have been constructed [10,17–19,36]. An important guiding principle in all these constructions is equivariance with respect to the action of a quantum group. In [16,42] a general framework for equivariant spectral triples is formulated, including an equivariance condition for real structures. However, in some examples the original axioms in [12] are only satisfied up to infinitesimals in this setup [17,18]. The K-theoretic interpretation of a real structure up to infinitesimals is unclear. In this paper we introduce a notion of K-theoretic Poincaré duality which is particularly adapted to the symmetry of quantum group actions. More precisely, we generalize the definition of Poincaré duality in KK-theory given by Connes [11] to C ∗ -algebras with a coaction of a quantum group using braided tensor products. Braided tensor products are well known in the algebraic approach to quantum groups [27], in our context they are constructed using coactions of the Drinfeld double of a locally compact quantum group. The example we study in detail is the standard Podle´s sphere, and we prove that it is equivariantly Poincaré dual to itself with respect to the natural action of SU q (2). The Drinfeld double of SU q (2), appearing as the symmetry group in this case, is the quantum Lorentz group [38], a noncompact quantum group built up out of a compact and a discrete part. We remark that the additional symmetry of the Podle´s sphere which is encoded in the discrete part of the quantum Lorentz group is not visible classically. The spectral triple corresponding to the Dirac operator on the standard Podle´s sphere [19] can be equipped with a real structure, and, due to [47], it satisfies Poincaré duality in the sense of [12]. From this point of view the standard Podle´s sphere is very well behaved. However, already in this example the formulation of equivariant Poincaré duality requires the setup proposed in this paper. Usually, the symmetry of an equivariant spectral triple is implemented by the action of a quantized universal enveloping algebra. In our approach we have to work with coactions of the quantized algebra of functions instead. Both descriptions are essentially equivalent, but an advantage of coactions is that the correct definition of equivariant K-theory and K-homology in this setting is already contained in [1]. In particular, we do not need to consider constructions of equivariant K-theory as in [35,47] which do not extend to general quantum groups. Let us now describe how the paper is organized. In the first part of the paper we discuss some results related to locally compact quantum groups and KK-theory. Section 2 contains an introduction to locally compact quantum groups, their coactions and associated crossed products. In particular, we review parts of the foundational work of Vaes on induced coactions [44] which are relevant to this paper. In Section 3 we introduce Yetter–Drinfeld-C ∗ -algebras and braided tensor products and discuss their basic properties, including compatibility with induction and restriction. Then, in Section 4, we review the definition of equivariant KK-theory for quantum groups following Baaj and Skandalis [1]. In particular, we show that KK G for a regular locally compact quantum group G satisfies a universal property as in the group case. A new feature in the quantum setting is the construction of exterior products for KK G . The nontriviality of it is related to the fact that a tensor product of two algebras with a coaction of a quantum group does not inherit a natural coaction in general, in distinction to the case of a group action. We deal with this problem using braided tensor products. Basic facts concerning SU q (2) and the standard Podle´s sphere SU q (2)/T are reviewed in Section 5. The main definition and results are contained in Section 6, where we introduce the

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concept of equivariant Poincaré duality with respect to quantum group actions and show that SU q (2)/T is equivariantly Poincaré dual to itself. As an immediate consequence we determine the equivariant K-homology of the Podle´s sphere. Let us make some remarks on notation. We write L(E, F ) for the space of adjointable operators between Hilbert A-modules E and F . Moreover K(E, F ) denotes the space of compact operators. If E = F we write simply L(E) and K(E), respectively. The closed linear span of a subset X of a Banach space is denoted by [X]. Depending on the context, the symbol ⊗ denotes either the tensor product of Hilbert spaces, the minimal tensor product of C ∗ -algebras, or the tensor product of von Neumann algebras. For operators on multiple tensor products we use the leg numbering notation. It is a pleasure to thank Uli Krähmer for interesting discussions on the subject of this paper. The second author is indebted to Stefaan Vaes for helpful explanations concerning induced coactions and braided tensor products. A part of this work was done during stays of the authors in Warsaw supported by EU-grant MKTD-CT-2004-509794. We are grateful to Piotr Hajac for his kind hospitality. 2. Locally compact quantum groups and their coactions In this section we recall basic definitions and results from the theory of locally compact quantum groups and fix our notation. For more detailed information we refer to the literature [25,26,44]. Let φ be a normal, semifinite and faithful weight on a von Neumann algebra M. We use the standard notation M+ φ = x ∈ M+ φ(x) < ∞ ,

Nφ = x ∈ M φ x ∗ x < ∞

and write M∗+ for the space of positive normal linear functionals on M. Assume that : M → M ⊗ M is a normal unital ∗-homomorphism. The weight φ is called left invariant with respect to if φ (ω ⊗ id)(x) = φ(x)ω(1) + for all x ∈ M+ φ and ω ∈ M∗ . Similarly one defines the notion of a right invariant weight.

Definition 2.1. A locally compact quantum group G is given by a von Neumann algebra L∞ (G) together with a normal unital ∗-homomorphism : L∞ (G) → L∞ (G) ⊗ L∞ (G) satisfying the coassociativity relation ( ⊗ id) = (id ⊗ ) and normal semifinite faithful weights φ and ψ on L∞ (G) which are left and right invariant, respectively. Our notation for locally compact quantum groups is intended to make clear how ordinary locally compact groups can be viewed as quantum groups. Indeed, if G is a locally compact


1469

group, then the algebra L∞ (G) of essentially bounded measurable functions on G together with the comultiplication : L∞ (G) → L∞ (G) ⊗ L∞ (G) given by (f )(s, t) = f (st) defines a locally compact quantum group. The weights φ and ψ are given in this case by left and right Haar measures, respectively. Of course, for a general locally compact quantum group G the notation L∞ (G) is purely formal. Similar remarks apply to the C ∗ -algebras Cf∗ (G), Cr∗ (G) and C0f (G), C0r (G) associated to G that we discuss below. It is convenient to view all of them as different appearances of the quantum group G. Let G be a locally compact quantum group and let Λ : Nφ → HG be a GNS-construction for the weight φ. Throughout the paper we will only consider quantum groups for which HG is a separable Hilbert space. One obtains a unitary WG = W on HG ⊗ HG by W ∗ Λ(x) ⊗ Λ(y) = (Λ ⊗ Λ) (y)(x ⊗ 1) for all x, y ∈ Nφ . This unitary is multiplicative, which means that W satisfies the pentagonal equation W12 W13 W23 = W23 W12 . From W one can recover the von Neumann algebra L∞ (G) as the strong closure of the algebra (id ⊗ L(HG )∗ )(W ) where L(HG )∗ denotes the space of normal linear functionals on L(HG ). Moreover one has (x) = W ∗ (1 ⊗ x)W for all x ∈ M. The algebra L∞ (G) has an antipode which is an unbounded, σ -strong* closed linear map S given by S(id ⊗ ω)(W ) = (id ⊗ ω)(W ∗ ) for ω ∈ L(HG )∗ . Moreover there is a polar decomposition S = Rτ−i/2 where R is an antiautomorphism of L∞ (G) called the unitary antipode and (τt ) is a strongly continuous one-parameter group of automorphisms of L∞ (G) called the scaling group. The unitary antipode satisfies σ (R ⊗ R) = R. The group-von Neumann algebra L(G) of the quantum group G is the strong closure of the ˆ : L(G) → L(G) ⊗ L(G) given by algebra (L(HG )∗ ⊗ id)(W ) with the comultiplication ˆ (y) = Wˆ ∗ (1 ⊗ y)Wˆ where Wˆ = ΣW ∗ Σ and Σ ∈ L(HG ⊗ HG ) is the flip map. It defines a locally compact quantum ˆ which is called the dual of G. The left invariant weight φˆ for the dual quantum group has group G ˆ a GNS-construction Λˆ : Nφˆ → HG , and according to our conventions we have L(G) = L∞ (G). The modular conjugations of the weights φ and φˆ are denoted by J and Jˆ, respectively. These operators implement the unitary antipodes in the sense that R(x) = Jˆx ∗ Jˆ,

ˆ R(y) = Jy ∗ J

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for x ∈ L∞ (G) and y ∈ L(G). Note that L∞ (G) = J L∞ (G)J and L(G) = JˆL(G)Jˆ for the commutants of L∞ (G) and L(G). Using J and Jˆ one obtains multiplicative unitaries V = (Jˆ ⊗ Jˆ)Wˆ (Jˆ ⊗ Jˆ),

Vˆ = (J ⊗ J )W (J ⊗ J )

which satisfy V ∈ L(G) ⊗ L∞ (G) and Vˆ ∈ L∞ (G) ⊗ L(G), respectively. We will mainly work with the C ∗ -algebras associated to the locally compact quantum group G. The algebra [(id ⊗ L(HG )∗ )(W )] is a strongly dense C ∗ -subalgebra of L∞ (G) which we denote by C0r (G). Dually, the algebra [(L(HG )∗ ⊗ id)(W )] is a strongly dense C ∗ -subalgebra of L(G) which we denote by Cr∗ (G). These algebras are the reduced algebra of continuous functions vanishing at infinity on G and the reduced group C ∗ -algebra of G, respectively. One has W ∈ M(C0r (G) ⊗ Cr∗ (G)). Restriction of the comultiplications on L∞ (G) and L(G) turns C0r (G) and Cr∗ (G) into Hopf∗ C -algebras in the following sense. Definition 2.2. A Hopf C ∗ -algebra is a C ∗ -algebra S together with an injective nondegenerate ∗-homomorphism : S → M(S ⊗ S) such that the diagram S

M(S ⊗ S) id⊗

M(S ⊗ S)

⊗id

M(S ⊗ S ⊗ S)

is commutative and [(S)(1 ⊗ S)] = S ⊗ S = [(S ⊗ 1)(S)]. A morphism between Hopf-C ∗ -algebras (S, S ) and (T , T ) is a nondegenerate ∗-homomorphism π : S → M(T ) such that T π = (π ⊗ π)S . If S is a Hopf-C ∗ -algebra we write S cop for the Hopf-C ∗ -algebra obtained by equipping S with the opposite comultiplication cop = σ . A unitary corepresentation of a Hopf-C ∗ -algebra S on a Hilbert B-module E is a unitary X ∈ L(S ⊗ E) satisfying ( ⊗ id)(X) = X13 X23 . A universal dual of S is a Hopf-C ∗ -algebra Sˆ together with a unitary corepresentation X ∈ ˆ satisfying the following universal property. For every Hilbert B-module E and every M(S ⊗ S) unitary corepresentations X ∈ L(S ⊗ E) there exists a unique nondegenerate ∗-homomorphism πX : Sˆ → L(E) such that (id ⊗ πX )(X ) = X. For every locally compact quantum group G there exist a universal dual Cf∗ (G) of C0r (G) and a universal dual C0f (G) of Cr∗ (G), respectively [24]. We call Cf∗ (G) the maximal group C ∗ -algebra of G and C0f (G) the maximal algebra of continuous functions on G vanishing at infinity. Since HG is assumed to be separable the C ∗ -algebras C0f (G), C0r (G) and Cf∗ (G), Cr∗ (G) are separable. The quantum group G is called compact if C0f (G) is unital, and it is called discrete if Cf∗ (G) is unital. In the compact case we also write C f (G) and C r (G) instead of C0f (G) and C0r (G), respectively.


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In general, we have a surjective morphism πˆ : Cf∗ (G) → Cr∗ (G) of Hopf-C ∗ -algebras associated to the left regular corepresentation W ∈ M(C0 (G) ⊗ Cr∗ (G)). Similarly, there is a surjective morphism π : C0f (G) → C0r (G). We will call the quantum group G amenable if πˆ : Cf∗ (G) → Cr∗ (G) is an isomorphism and coamenable if π : C0f (G) → C0r (G) is an isomorphism. If G is amenable or coamenable, respectively, we also write C ∗ (G) and C0 (G) for the corresponding C ∗ -algebras. For more information on amenability for locally compact quantum groups see [6]. Let S be a C ∗ -algebra. The S-relative multiplier algebra MS (S ⊗ A) ⊂ M(S ⊗ A) of a ∗ C -algebra A consists of all x ∈ M(S ⊗ A) such that the relations x(S ⊗ 1) ⊂ S ⊗ A,

(S ⊗ 1)x ⊂ S ⊗ A

hold. In the sequel we tacitly use basic properties of relative multiplier algebras which can be found in [20]. Definition 2.3. A (left) coaction of a Hopf C ∗ -algebra S on a C ∗ -algebra A is an injective nondegenerate ∗-homomorphism α : A → M(S ⊗ A) such that the diagram α

A α

M(S ⊗ A)

M(S ⊗ A) ⊗id

id⊗α

M(S ⊗ S ⊗ A)

is commutative and α(A) ⊂ MS (S ⊗ A). The coaction is called continuous if [α(A)(S ⊗ 1)] = S ⊗ A. If (A, α) and (B, β) are C ∗ -algebras with coactions of S, then a ∗-homomorphism f : A → M(B) is called S-colinear if βf = (id ⊗ f )α. We remark that some authors do not require a coaction to be injective. For a discussion of the continuity condition see [3]. A C ∗ -algebra A equipped with a continuous coaction of the Hopf-C ∗ -algebra S will be called an S-C ∗ -algebra. If S = C0r (G) for a locally compact quantum group G we also say that A is G-C ∗ -algebra. Moreover, in this case S-colinear ∗-homomorphisms will be called G-equivariant or simply equivariant. We write G-Alg for the category of separable G-C ∗ -algebras and equivariant ∗-homomorphisms. A (nondegenerate) covariant representation of a G-C ∗ -algebra A on a Hilbert-B-module E consists of a (nondegenerate) ∗-homomorphism f : A → L(E) and a unitary corepresentation X ∈ L(C0r (G) ⊗ E) such that (id ⊗ f )α(a) = X ∗ 1 ⊗ f (a) X for all a ∈ A. There exists a C ∗ -algebra Cf∗ (G)cop f A, called the full crossed product, together with a nondegenerate covariant representation (jA , XA ) of A on Cf∗ (G)cop f A which satisfies the following universal property. For every nondegenerate covariant representation (f, X) of A

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on a Hilbert-B-module E there exists a unique nondegenerate ∗-homomorphism F : Cf∗ (G)cop f A → L(E), called the integrated form of (f, X), such that X = (id ⊗ F )(XA ),

f = FjA .

Remark that the corepresentation XA corresponds to a unique nondegenerate ∗-homomorphism gA : Cf∗ (G)cop → M(Cf∗ (G)cop f A). On the Hilbert A-module HG ⊗ A we have a covariant representation of A given by the coaction α : A → L(HG ⊗ A) and W ⊗ 1. The reduced crossed product Cr∗ (G)cop r A is the image of Cf∗ (G)cop f A under the corresponding integrated form. Explicitly, we have Cr∗ (G)cop r A =

∗ Cr (G) ⊗ 1 α(A)

inside M(KG ⊗ A) = L(HG ⊗ A) using the notation KG = K(HG ). There is a nondegenerate ∗-homomorphism jA : A → M(Cr∗ (G)cop r A) induced by α. Similarly, we have a canonical nondegenerate ∗-homomorphism gA : Cr∗ (G)cop → M(Cr∗ (G)cop r A). The full and the reduced crossed products admit continuous dual coactions of Cf∗ (G)cop and ∗ Cr (G)cop , respectively. In both cases the dual coaction leaves the copy of A inside the crossed product invariant and acts by the (opposite) comultiplication on the group C ∗ -algebra. If G is amenable then the canonical map Cf∗ (G)cop f A → Cr∗ (G)cop r A is an isomorphism for all G-C ∗ -algebras A, and we will also write C ∗ (G)cop A for the crossed product in this case. The comultiplication : C0r (G) → M(C0r (G) ⊗ C0r (G)) defines a coaction of C0r (G) on itself. On the Hilbert space HG we have a covariant representation of C0r (G) given by the identical representation of C0r (G) and W ∈ M(C0r (G) ⊗ KG ). The quantum group G is called strongly regular if the associated integrated form induces an isomorphism Cf∗ (G)cop f C0r (G) ∼ = KG . Similarly, G is called regular if the corresponding homomorphism on the reduced level gives an isomorphism Cr∗ (G)cop r C0r (G) ∼ = KG . Every strongly regular quantum group is regular, it is not known whether there exist regular quantum groups which are not strongly regular. If G is ˆ is regular as well. regular then the dual G Let EB be a right Hilbert module. The multiplier module M(E) of E is the right Hilbert-M(B)module M(E) = L(B, E). There is a natural embedding E ∼ = K(B, E) → L(B, E) = M(E). If EA and FB are Hilbert modules, then a morphism from E to F is a linear map Φ : E → M(F ) together with a ∗-homomorphism φ : A → M(B) such that

Φ(ξ ), Φ(η) = φ ξ, η

for all ξ, η ∈ E. In this case Φ is automatically norm-decreasing and satisfies Φ(ξ a) = Φ(ξ )φ(a) for all ξ ∈ E and a ∈ A. The morphism Φ is called nondegenerate if φ is nondegenerate and [Φ(E)B] = F . Let S be a C ∗ -algebra and let EA be a Hilbert module. The S-relative multiplier module MS (S ⊗ E) is the submodule of M(S ⊗ E) consisting of all multipliers x satisfying x(S ⊗ 1) ⊂ S ⊗ E and (S ⊗ 1)x ⊂ S ⊗ E. For further information we refer again to [20].


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Definition 2.4. Let S be a Hopf-C ∗ -algebra and let β : B → M(S ⊗ B) be a coaction of S on the C ∗ -algebra B. A coaction of S on a Hilbert module EB is a nondegenerate morphism λ : E → M(S ⊗ E) such that the diagram λ

E

M(S ⊗ E) ⊗id

λ

M(S ⊗ E)

id⊗λ

M(S ⊗ S ⊗ E)

is commutative and λ(E) ⊂ MS (S ⊗ E). The coaction λ is called continuous if [(S ⊗ 1)λ(E)] = S ⊗ E = [λ(E)(S ⊗ 1)]. A morphism Φ : E → M(F ) of Hilbert B-modules with coactions λE and λF , respectively, is called S-colinear if λF Φ = (id ⊗ Φ)λE . If λ : E → M(S ⊗ E) is a coaction on the Hilbert-B-module E then the map λ is automatically isometric and hence injective. Let G be a locally compact quantum group and let B be a G-C ∗ -algebra. A G-Hilbert B-module is a Hilbert module EB with a continuous coaction λ : E → M(S ⊗ E) for S = C0r (G). If G is regular then continuity of the coaction λ is in fact automatic. Instead of S-colinear morphisms we also speak of equivariant morphisms between G-Hilbert B-modules. Let B be a C ∗ -algebra equipped with a coaction of the Hopf-C ∗ -algebra S. Given a Hilbert module EB with coaction λ : E → M(S ⊗ E) one obtains a unitary operator Vλ : E ⊗B (S ⊗ B) → S ⊗ E by Vλ (ξ ⊗ x) = λ(ξ )x for ξ ∈ E and x ∈ S ⊗ B. Here the tensor product over B is formed with respect to the coaction β : B → M(S ⊗ B). This unitary satisfies the relation (id ⊗C Vλ )(Vλ ⊗(id⊗β) id) = Vλ ⊗(⊗id) id in L(E ⊗(⊗id)β (S ⊗ S ⊗ B), S ⊗ S ⊗ E), compare [1]. Moreover, the equation adλ (T ) = Vλ (T ⊗ id)Vλ∗

determines a coaction adλ : K(E) → M(S ⊗ K(E)) = L(S ⊗ E). If the coaction λ is continuous then adλ is continuous as well. In particular, if E is a G-Hilbert B-module with coaction λ, then the associated coaction adλ turns K(E) into a G-C ∗ -algebra. Let B be a C ∗ -algebra equipped with the trivial coaction of the Hopf-C ∗ -algebra S and let λ : E → M(S ⊗ E) be a coaction on the Hilbert module EB . Then using the natural identification E ⊗B (S ⊗ B) ∼ =E⊗S∼ = S ⊗ E the associated unitary Vλ determines a unitary ∗ corepresentation Vλ in L(S ⊗ E). Conversely, if V ∈ L(S ⊗ E) is a unitary corepresentation then λV : E → M(S ⊗ E) given by λV (ξ ) = V ∗ (1 ⊗ ξ ) is a nondegenerate morphism of Hilbert modules satisfying the coaction identity. If S = C0r (G) for a regular quantum group G, then λV defines a continuous coaction on E. As a consequence, for a regular quantum group G and a trivial G-C ∗ -algebra B, continuous coactions on a Hilbert B-module E correspond uniquely to unitary corepresentations of C0r (G) on E .

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Let G be a regular quantum group and let EB be a G-Hilbert module with coaction λE : E → M(C0r (G)⊗E). Then HG ⊗E becomes a G-Hilbert B-module with the coaction λHG ⊗E (x ⊗ξ ) = ∗ Σ (id ⊗ λ )(x ⊗ ξ ) where X = ΣV Σ ∈ L(C r (G) ⊗ K ). In particular, for E = B the X12 12 G E 0 algebra KG ⊗ B = K(HG ⊗ B) can be viewed as a G-C ∗ -algebra. We now state the following version of the Takesaki–Takai duality theorem [2]. Theorem 2.5. Let G be a regular locally compact quantum group and let A be a G-C ∗ -algebra. Then there is a natural isomorphism C0r (G) r Cr∗ (G)cop r A ∼ = KG ⊗ A of G-C ∗ -algebras. An equivariant Morita equivalence between G-C ∗ -algebras A and B is given by an equivariant A-B-imprimitivity bimodule, that is, a full G-Hilbert B-module E together with an isomorphism A∼ = K(E) of G-C ∗ -algebras. Theorem 2.5 shows that the double crossed product C0r (G) r ∗ Cr (G)cop r A is equivariantly Morita equivalent to A for every G-C ∗ -algebra A provided G is regular. A morphism H → G of locally compact quantum groups is a nondegenerate ∗-homomorphism π : C0f (G) → M(C0f (H )) which is compatible with the comultiplications in the sense that (π ⊗ π)G = H π . Every such morphism induces canonically a dual morphism πˆ : Cf∗ (H ) → M(Cf∗ (G)). A closed quantum subgroup H ⊂ G is a morphism H → G for which the latter map is accompanied by a faithful normal ∗-homomorphism L(H ) → L(G) of the group-von Neumann algebras, see [44,45]. In the classical case this notion recovers precisely the closed subgroups of a locally compact group G. Observe that there is in general no associated homomorphism L∞ (G) → L∞ (H ) for a quantum subgroup, this fails already in the group case. Let H → G be a morphism of quantum groups and let B be a G-C ∗ -algebra with coaction β : B → M(C0r (G) ⊗ B). Identifying β with a normal coaction [21] of the full C ∗ -algebra C0f (G), the map π : C0f (G) → M(C0f (H )) induces on B a continuous coaction res(β) : B → ∗ M(C0r (H ) ⊗ B). We write resG H (B) for the resulting H -C -algebra. In this way we obtain a functor resG H : G-Alg → H -Alg. Conversely, let G be a strongly regular quantum group and let H ⊂ G be a closed quantum subgroup. Given an H -C ∗ -algebra B, there exists an induced G-C ∗ -algebra indG H (B) such that the following version of Green’s imprimitivity theorem holds [44]. Theorem 2.6. Let G be a strongly regular quantum group and let H ⊂ G be a closed quantum subgroup. Then there is a natural Cr∗ (G)cop -colinear Morita equivalence ∗ cop Cr∗ (G)cop r indG r B H (B) ∼M Cr (H )

for all H -C ∗ -algebras B. In fact, the induced C ∗ -algebra indG H (B) is defined by Vaes in [44] using a generalized Landstad theorem after construction of its reduced crossed product. A description of Cr∗ (G)cop r indG H (B) can be given as follows. From the quantum subgroup H ⊂ G one first obtains a right coaction L∞ (G) → L∞ (G) ⊗ L∞ (H ) on the level of von Neumann algebras. The von Neumann algebraic homogeneous space L∞ (G/H ) ⊂ L∞ (G) is defined as the subalgebra of invariants


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under this coaction. If πˆ : L(H ) → L(G) is the homomorphism πˆ (x) = JˆG πˆ (JˆH x JˆH )JˆG induced by πˆ : L(H ) → L(G), then I = v ∈ L(HH , HG ) vx = πˆ (x)v for all x ∈ L(H ) defines a von Neumann algebraic imprimitivity bimodule between the von Neumann algebraic crossed product L(G)cop L∞ (G/H ) and L(H )cop . There is a C ∗ -algebraic homogeneous space C0r (G/H ) ⊂ L∞ (G/H ) and a C ∗ -algebraic imprimitivity bimodule I ⊂ I which implements a Morita equivalence between Cr∗ (G)cop r C0r (G/H ) and Cr∗ (H )cop . Explicitly, we have ˆ VˆH∗ Cr∗ (H ) ⊗ HG . I ⊗ HG = VˆG (I ⊗ 1)(id ⊗ π) The crossed product of the induced C ∗ -algebra indG H (B) is then given by ∗ Cr∗ (G)cop r indG H (B) = (I ⊗ 1)β(B) I ⊗ 1 where β : B → M(C0r (H ) ⊗ B) is the coaction on B. At several points of the paper we will rely on techniques developed in [44]. Firstly, as indicated in [44], let us note that we have induction in stages. Proposition 2.7. Let H ⊂ K ⊂ G be strongly regular quantum groups. Then there is a natural G-equivariant isomorphism K ∼ G indG H (B) = indK indH (B)

for every H -C ∗ -algebra B. Proof. Let πˆ HG : L(H ) → L(G) be the normal ∗-homomorphism corresponding to the inclusion G ⊂ I G ⊂ L(H , H ) the associated imprimitivity bimodules. For the H ⊂ G, and denote by IH H G H inclusions H ⊂ K and K ⊂ G we use analogous notation. By assumption we have πˆ KG πˆ HK = πˆ HG , and we observe that IKG IHK ⊂ IHG is strongly dense. Since the ∗-homomorphism πˆ KG is normal and injective we obtain K IH ⊗ HG =

id ⊗ πˆ KG (VˆK ) IHK ⊗ 1 id ⊗ πˆ HG VˆH∗ Cr∗ (H ) ⊗ HG

which yields G K IK IH ⊗ HG = VˆG IKG IHK ⊗ 1 id ⊗ πˆ HG VˆH∗ Cr∗ (H ) ⊗ HG . Using the normality of πˆ HG we see that if (vi )i∈I is a bounded net in IHG converging strongly to zero then VˆG (vi ⊗ 1)(id ⊗ πˆ HG )(VˆH∗ )(x ⊗ ξ ) converges to zero in norm for all x ∈ Cr∗ (H ) G = [I G I K ] for the C ∗ -algebraic imprimitivity and ξ ∈ HG . As a consequence we obtain IH K H bimodules.

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Now let B be an H -C ∗ -algebra with coaction β. Then we have K Cr∗ (G)cop r indG K indH (B) G ∗ G ⊗ id ⊗ id (K ⊗ id) indK IK ⊗ id ⊗ id = IK H (B) G ∗ G ∼ ⊗ id ⊗ id (VK )∗12 (K ⊗ id) indK ⊗ id ⊗ id = IK H (B) (VK )12 IK G K K ∗ G ∗ ∼ IK ⊗ id IH ⊗ id β(B) IH = IK G G ∗ = IH ⊗ id β(B) IH ⊗ id = Cr∗ (G)cop r indG H (B)

using conjugation with the unitary ((πˆ KG ) ⊗ id)(VK∗ )12 in the second step. The resulting isomorK ∗ cop phism between the crossed products Cr∗ (G)cop r indG r indG K indH (B) and Cr (G) H (B) is r ∗ cop Cr (G) -colinear and identifies the natural corepresentations of C0 (G) on both sides. Hence Theorem 6.7 in [44] yields the assertion. 2 Let H ⊂ G be a quantum subgroup of a strongly regular quantum group G and let B be an H -C ∗ -algebra with coaction β. If E denotes the trivial group, then due to Proposition 2.7 we have r G H H r ∼ G H indG H C0 (H ) ⊗ B = indH indE resE (B) = indE resE (B) = C0 (G) ⊗ B where C0r (H ) ⊗ B is viewed as an H -C ∗ -algebra via comultiplication on the first tensor factor. The ∗-homomorphism β : B → M(C0r (H ) ⊗ B) induces an injective G-equivariant G G r ∗-homomorphism ind(β) : indG H (B) → M(indH (C0 (H ) ⊗ B)), and it follows that indH (B) is G r contained in M(C0 (G) ⊗ B). Using that the coaction β is continuous we see that indH (B) is in fact contained in the C0r (G)-relative multiplier algebra of C0r (G) ⊗ B. Now let A and B be H -C ∗ -algebras. According to the previous observations every H -equivariant ∗-homomorphism f : A → B induces a G-equivariant ∗-homomorphism G G indG H (f ) : indH (A) → indH (B) in a natural way. We conclude that induction defines a funcG tor indH : H -Alg → G-Alg. 3. Yetter–Drinfeld algebras and braided tensor products In this section we study Yetter–Drinfeld-C ∗ -algebras and braided tensor products. We remark that these concepts are well known in the algebraic approach to quantum groups [27]. Yetter– Drinfeld modules for compact quantum groups are discussed in [38]. Let us begin with the definition of a Yetter–Drinfeld C ∗ -algebra. Definition 3.1. Let G be a locally compact quantum group and let S = C0r (G) and Sˆ = Cr∗ (G) be the associated reduced Hopf-C ∗ -algebras. A G-Yetter–Drinfeld C ∗ -algebra is a C ∗ -algebra A equipped with continuous coactions α of S and λ of Sˆ such that the diagram λ

A

id⊗ α

M(Sˆ ⊗ A)

M(Sˆ ⊗ S ⊗ A) σ ⊗ id

α

M(S ⊗ A)

id⊗ λ

ad(W )⊗ id

M(S ⊗ Sˆ ⊗ A)

M(S ⊗ Sˆ ⊗ A)


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is commutative. Here ad(W )(x) = W xW ∗ denotes the adjoint action of the fundamental unitary ˆ W ∈ M(S ⊗ S). In order to compare Definition 3.1 with the notion of a Yetter–Drinfeld module in the algebraic setting, one should keep in mind that we work with the opposite comultiplication on the dual. In the sequel we will also refer to G-Yetter–Drinfeld C ∗ -algebras as G-YD-algebras. A homomorphism of G-YD-algebras f : A → B is a ∗-homomorphism which is both G-equivariant ˆ and G-equivariant. We remark moreover that the concept of a Yetter–Drinfeld-C ∗ -algebra is ˆ YD-algebras. self-dual, that is, G-YD-algebras are the same thing as GLet us discuss some basic examples of Yetter–Drinfeld-C ∗ -algebras. Consider first the case that G is an ordinary locally compact group. Since C0 (G) is commutative, every G-C ∗ -algebra ˆ ∗becomes a G-YD-algebra with the trivial coaction of Cr∗ (G). Dually, we may start with a G-C algebra, that is, a reduced coaction of the group G. If G is discrete then such coactions correspond to Fell bundles over G. In this case a Yetter–Drinfeld structure is determined by an action of G on the bundle which is compatible with the adjoint action on the base G. Let G be a locally compact quantum group and consider the G-C ∗ -algebra C0r (G) with coaction . If G is regular the map λ : C0r (G) → M(Cr∗ (G) ⊗ C0r (G)) given by λ(f ) = Wˆ ∗ (1 ⊗ f )Wˆ defines a continuous coaction. Moreover

∗ ∗ ∗ ˆ 23 ad(W ) ⊗ id (id ⊗ λ)(f ) = W12 W W13 (1 ⊗ 1 ⊗ f )W13 Wˆ 23 W12 ∗ ∗ ∗ = Σ23 W13 W23 W12 Σ23 (1 ⊗ 1 ⊗ f )Σ23 W12 W23 W13 Σ23 ∗ ∗ = Σ23 W12 W23 Σ23 (1 ⊗ 1 ⊗ f )Σ23 W23 W12 Σ23 ∗ ˆ∗ = W13 W23 (1 ⊗ 1 ⊗ f )Wˆ 23 W13 = (σ ⊗ id)(id ⊗ )λ(f )

shows that C0r (G) together with and λ is a G-YD-algebra. More generally, we can consider a crossed product C0r (G) r A for a regular quantum group G. The dual coaction together with conjugation by Wˆ ∗ as above yield a G-YD-algebra structure on C0r (G) r A. There is another way to obtain a Yetter–Drinfeld-C ∗ -algebra structure on a crossed product. Let again G be a regular quantum group and let A be a G-YD-algebra. We obtain a continuous coaction λˆ : Cr∗ (G)cop r A → M(Cr∗ (G) ⊗ (Cr∗ (G)cop r A)) by ∗ ˆ λ(x) = Wˆ 12 (id ⊗ λ)(x)213 Wˆ 12

for x ∈ Cr∗ (G)cop r A ⊂ L(HG ⊗ A). On the copy of A in the multiplier algebra of the crossed product this coaction implements λ, and on the copy of Cr∗ (G)cop = Cr∗ (G) it is given by the ˆ of Cr∗ (G). In addition we have a continuous coaction αˆ : Cr∗ (G)cop r A → comultiplication r ∗ M(C0 (G) ⊗ (Cr (G)cop r A)) given by ∗ α(x) ˆ = W12 (1 ⊗ x)W12 .

Remark that on the copy of A in the multiplier algebra this coaction implements α, and on the copy of Cr∗ (G)cop = Cr∗ (G) it implements the adjoint coaction. It is straightforward to check that the crossed product Cr∗ (G)cop r A becomes again a G-YD-algebra in this way.

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The notion of a Yetter–Drinfeld-C ∗ -algebra is closely related to coactions of the Drinfeld double. Let us briefly recall the definition of the Drinfeld double in the context of locally compact quantum groups. It is described as a special case of the double crossed product construction in [4]. If G is a locally compact quantum group, then the reduced C ∗ -algebra of functions on the Drinfeld double D(G) is C0r (D(G)) = C0r (G) ⊗ Cr∗ (G) with the comultiplication ˆ D(G) = (id ⊗ σ ⊗ id) id ⊗ ad(W ) ⊗ id ( ⊗ ). ˆ as closed This yields a locally compact quantum group D(G) which contains both G and G quantum subgroups. If G is regular then D(G) is again regular. Proposition 3.2. Let G be a locally compact quantum group and let D(G) be its Drinfeld double. Then a G-Yetter–Drinfeld C ∗ -algebra is the same thing as a D(G)-C ∗ -algebra. Proof. Let us first assume that A is a D(G)-C ∗ -algebra with coaction γ : A → M(C0r (D(G)) ⊗ ˆ are quantum subgroups of D(G) we obtain associated continuous coactions A). Since G and G α : A → M(C0r (G) ⊗ A) and λ : A → M(Cr∗ (G) ⊗ A) by restriction. These coactions are determined by the conditions (δ ⊗ id)γ = (id ⊗ α)γ ,

(δˆ ⊗ id)γ = (id ⊗ λ)γ

where the maps δ : C0r (D(G)) → M(C0r (D(G)) ⊗ C0r (G)) and δˆ : C0r (D(G)) → M(C0r (D(G)) ⊗ Cr∗ (G)) are given by δ = (id ⊗ σ )ad(W23 )( ⊗ id),

ˆ δˆ = id ⊗ .

We have ad(W23 )(id ⊗ id ⊗ λ)(id ⊗ α)γ

= ad(W23 )(id ⊗ id ⊗ λ)(δ ⊗ id)γ = ad(W23 )(δ ⊗ id ⊗ id)(δˆ ⊗ id)γ ˆ ⊗ id)γ = ad(W34 )(id ⊗ σ ⊗ id ⊗ id)ad(W23 )( ⊗ ˆ ⊗ id)γ = (id ⊗ σ ⊗ id ⊗ id)ad(W24 W23 )( ⊗ ˆ ˆ ⊗ id)γ = (id ⊗ σ ⊗ id ⊗ id)ad((id ⊗ )(W )234 )( ⊗ ˆ ⊗ id ⊗ id)(id ⊗ σ ⊗ id)ad(W23 )( ⊗ id ⊗ id)γ = (idD(G) ⊗ σ ⊗ id)(id ⊗ = (id ⊗ σ ⊗ id)(δˆ ⊗ id ⊗ id)(δ ⊗ id)γ = (id ⊗ σ ⊗ id)(id ⊗ id ⊗ α)(id ⊗ λ)γ , and since the coaction γ is continuous this implies ad(W12 )(id ⊗ λ)α = (σ ⊗ id)(id ⊗ α)λ.

It follows that we have obtained a G-YD-algebra structure on A.


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Conversely, assume that A is equipped with a G-YD-algebra structure. We define a nondegenerate ∗-homomorphism γ : A → M(C0r (D(G)) ⊗ A) by γ = (id ⊗ λ)α and compute (id ⊗ γ )γ = (id ⊗ id ⊗ id ⊗ λ)(id ⊗ id ⊗ α)(id ⊗ λ)α = (id ⊗ id ⊗ id ⊗ λ)(id ⊗ σ ⊗ id)ad(W23 )(id ⊗ id ⊗ λ)(id ⊗ α)α = (id ⊗ σ ⊗ id ⊗ id)ad(W23 )(id ⊗ id ⊗ id ⊗ λ)(id ⊗ id ⊗ λ)( ⊗ id)α ˆ ⊗ id)(id ⊗ λ)α = (id ⊗ σ ⊗ id ⊗ id)ad(W23 )( ⊗ ˆ ⊗ id)γ = (id ⊗ σ ⊗ id ⊗ id)ad(W23 )( ⊗ = (D(G) ⊗ id)γ . It follows that γ is a continuous coaction which turns A into a D(G)-C ∗ -algebra. One checks easily that the two operations above are inverse to each other. 2 We shall now define the braided tensor product A B of a G-YD-algebra A with a G-C ∗ algebra B. Observe first that the C ∗ -algebra B acts on the Hilbert module H ⊗ B by (π ⊗ id)β where π : C0r (G) → L(H) denotes the defining representation on H = HG . Similarly, the C ∗ -algebra A acts on H ⊗ A by (πˆ ⊗ id)λ where πˆ : Cr∗ (G) → L(H) is the defining representation. From this we obtain two ∗-homomorphisms ιA = λ12 : A → L(H ⊗ A ⊗ B) and ιB = β13 : B → L(H ⊗ A ⊗ B) by acting with the identity on the factor B and A, respectively. Definition 3.3. Let G be a locally compact quantum group, let A be a G-YD-algebra and B a G-C ∗ -algebra. With the notation as above, the braided tensor product A G B is the C ∗ -subalgebra of L(H ⊗ A ⊗ B) generated by all elements ιA (a)ιB (b) for a ∈ A and b ∈ B. We will also write A B instead of A G B if the quantum group G is clear from the context. The braided tensor product A B is in fact equal to the closed linear span [ιA (A)ιB (B)]. This follows from Proposition 8.3 in [44], we reproduce the argument for the convenience of the reader. Clearly it suffices to prove [ιA (A)ιB (B)] = [ιB (B)ιA (A)]. Using continuity of the coaction λ and Vˆ = (J Jˆ ⊗ 1)W ∗ (JˆJ ⊗ 1) we get ˆ ⊗ id)λ(A) L(HG )∗ ⊗ id ⊗ id ( ∗ = L(HG )∗ ⊗ id ⊗ id Vˆ12 λ(A)13 Vˆ12 ∗ = L(HG )∗ ⊗ id ⊗ id W12 μ(A)13 W12

λ(A) =

where μ(x) = (JˆJ ⊗ 1)λ(x)(J Jˆ ⊗ 1) for x ∈ A. Since β : B → M(C0r (G) ⊗ B) is a continuous coaction we obtain

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∗ λ(A)12 β(B)13 = L(HG )∗ ⊗ id ⊗ id ⊗ id W12 μ(A)13 W12 β(B)24 ∗ μ(A)13 W12 ( ⊗ id)β(B)124 = L(HG )∗ ⊗ id ⊗ id ⊗ id W12 ∗ μ(A)13 β(B)24 W12 = L(HG )∗ ⊗ id ⊗ id ⊗ id W12 ∗ μ(A)13 W12 = L(HG )∗ ⊗ id ⊗ id ⊗ id ( ⊗ id)β(B)124 W12 ∗ μ(A)13 W12 = L(HG )∗ ⊗ id ⊗ id ⊗ id β(B)24 W12 = β(B)13 λ(A)12 which yields the claim. It follows in particular that we have natural nondegenerate ∗-homomorphisms ιA : A → M(A B) and ιB : B → M(A B). The braided tensor product A B becomes a G-C ∗ -algebra in a canonical way. In fact, we have a nondegenerate ∗-homomorphism α β : A B → M(C0r (G) ⊗ (A B)) given by ∗ (α β) λ(a)12 β(b)13 = W12 (σ ⊗ id) (id ⊗ α)λ(a) 123 β(b)24 W12 = (id ⊗ λ)α(a)123 (id ⊗ β)β(b)124 , and it is straightforward to check that α β defines a continuous coaction of C0r (G) such that the ∗-homomorphisms ιA and ιB are G-equivariant. If B is a G-YD-algebra with coaction γ : B → M(Cr∗ (G)⊗B) then we obtain a nondegenerate ∗-homomorphism λ γ : A B → M(Cr∗ (G) ⊗ (A B)) by the formula ∗ (λ γ ) λ(a)12 β(b)13 = Wˆ 12 λ(a)23 (σ ⊗ id) (id ⊗ γ )β(b) 124 Wˆ 12 = (id ⊗ λ)λ(a)123 (id ⊗ β)γ (b)124 . In the same way as above one finds that λ γ yields a continuous coaction of Cr∗ (G) such that ˆ From the equivariance of ιA and ιB it follows that A B together ιA and ιB are G-equivariant. with the coactions α β and λ γ becomes a G-YD-algebra. If A is a G-YD-algebra and f : B → C a possibly degenerate equivariant ∗-homomorphism of G-C ∗ -algebras, then we obtain an induced ∗-homomorphism MK (K ⊗ A ⊗ B) → MK (K ⊗ A ⊗ C) between the relative multiplier algebras. Since A B ⊂ M(K ⊗ A ⊗ B) is in fact contained in MK (K ⊗ A ⊗ B), this map restricts to an equivariant ∗-homomorphism id f : A B → A C. It follows that the braided tensor product defines a functor A − from G-Alg to G-Alg. Similarly, if f : A → B is a homomorphism of G-YD-algebras we obtain for every G-algebra C an equivariant ∗-homomorphism f id : A C → B C and a functor − C from D(G)-Alg to G-Alg. There are analogous functors A − and − C from D(G)-Alg to D(G)-Alg if we consider G-YD-algebras in the second variable. Assume now that A and B are G-YD-algebras and that C is a G-C ∗ -algebra. According to our previous observations we can form the braided tensor products (A B) C and A (B C), respectively. We have id ⊗ λA λA (A)123 (id ⊗ β)λB (B)124 γ (C)15 ∗ A ∗ ∗ λ (A)23 Wˆ 12 (id ⊗ β)λB (B)124 Wˆ 12 Σ12 W12 Σ12 γ (C)15 = Wˆ 12 ∗ A ∗ λ (A)23 Σ12 id ⊗ λB β(B)124 W12 Σ12 γ (C)15 Σ12 W12 Σ12 Wˆ 12 = Wˆ 12

(A B) C =


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∗ = Wˆ 12 Σ12 λA (A)13 id ⊗ λB β(B)124 (id ⊗ γ )γ (C)125 Σ12 Wˆ 12 ∼ = A (B C), = λA (A)13 (β γ )(B C)1245 ∼ and the resulting isomorphism (A B) C ∼ = A (B C) is G-equivariant. If C is a G-YDˆ algebra then this isomorphism is in addition G-equivariant. We conclude that the braided tensor product is associative in a natural way. If B is a trivial G-algebra then the braided tensor product A B is isomorphic to A ⊗ B with the coaction induced from A. Similarly, if the coaction of Cr∗ (G) on the G-YD-algebra A is trivial then A B is isomorphic to A ⊗ B. Recall that if G is a locally compact group we may view all G-algebras as G-YD-algebras with the trivial coaction of the group C ∗ -algebra. In this case the braided tensor product reduces to the ordinary tensor product of G-C ∗ -algebras with the diagonal G-action. For general quantum groups the braided tensor product should be viewed as a substitute for the latter construction. Following an idea of Vaes, we shall now discuss the compatibility of the braided tensor product with induction and restriction. Let G be a strongly regular quantum group and let H ⊂ G be a closed quantum subgroup determined by the faithful normal ∗-homomorphism πˆ : L(H ) → L(G). Keeping our notation from Section 2, we denote by I the corresponding von Neumann algebraic imprimitivity bimodule for L(G)cop L∞ (G/H ) and L(H )cop , and by I ⊂ I the C ∗ -algebraic imprimitivity bimodule for Cr∗ (G)cop r C0r (G/H ) and Cr∗ (H )cop . Proposition 3.4. Let G be a strongly regular quantum group and let H ⊂ G be a closed quantum subgroup. If A is an H -YD-algebra then the induced C ∗ -algebra indG H (A) is a G-YD-algebra in a natural way. Proof. Let α : A → M(C0r (H ) ⊗ A) be the coaction of C0r (H ) on A. From the construction G r of indG H (A) in [44] we have the induced coaction ind(α) : indH (A) → M(C0 (G) ⊗ A) given G ∗ by ind(α)(x) = (WG )12 (1 ⊗ x)(WG )12 for x ∈ indH (A) ⊂ L(HG ⊗ A). Our task is to define a continuous coaction of Cr∗ (G) on indG H (A) satisfying the YD-condition. Denote by λ : A → M(Cr∗ (H ) ⊗ A) the coaction which determines the H -YD-algebra structure on A. This coaction induces a coaction res(λ) : A → M(Cr∗ (G) ⊗ A) because H ⊂ G is a closed quantum subgroup. Since A is an H -YD-algebra we have in addition the coaction λˆ of Cr∗ (H ) on the crossed product Cr∗ (H )cop r A, and a corresponding coaction res(λˆ ) of Cr∗ (G). We abbreviate B = Cr∗ (H )cop r A and consider the Hilbert B-module E = B with the corepresentation X = WH ⊗ id ∈ M(C0r (H ) ⊗ K(E)) = M(C0r (H ) ⊗ B). The corresponding induced Hilbert B-module indG H (E) is constructed in [44] such that ∼ HG ⊗ indG H (E) = I ⊗πl F where F = HG ⊗ E and the strict ∗-homomorphism πl : L(H ) → L(F ) is determined by (id ⊗ πl )(WH ) = (id ⊗ πˆ )(WH )12 X13 . Let us define a coaction on I ⊗πl F as follows. On I we have the adjoint action η : I → L(G) ⊗ I given by η(v) = Wˆ G∗ (1 ⊗ v)(πˆ ⊗ id)(Wˆ H )

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ˆ H : L(H ) → L(G)⊗L(H ). In addition consider which is compatible with the coaction (πˆ ⊗id) ˆ By construction, βF is the coaction βF of Cr∗ (G) on F given by βF = (σ ⊗ id)(id ⊗ res(λ)). ˆ compatible with the coaction res(λ) on B. Moreover, the ∗-homomorphism πl : L(H ) → L(F ) is covariant in the sense that ˆ H (x) = adβF πl (x) (id ⊗ πl )(πˆ ⊗ id) in L(HG ⊗ F ) for all x ∈ L(H ). According to Proposition 12.13 in [44] we obtain a product coaction of Cr∗ (G) on I ⊗πl F . Under the above isomorphism, this product coaction leaves invariant the natural representations of L∞ (G) and L(G) on the first tensor factor of the left hand side. Hence there is an G G ∗ induced coaction γ : indG H (E) → M(Cr (G) ⊗ indH (E)) on indH (E). Using the identification G ∼ indH (E) = [(I ⊗ 1)α(A)] we see that γ is given by γ (v ⊗ 1)α(a) = η(v) ⊗ 1 (id ⊗ α) res(λ)(a) ∗ cop r indG for v ∈ I and a ∈ A. Since K(indG H (E)) = Cr (G) H (A) we obtain a coaction adγ on G ∗ cop Cr (G) r indH (A). By construction, the coaction adγ commutes with the dual coaction and ˆ on the copy of Cr∗ (G)cop . It follows that adγ induces a continuous coaction δ : is given by G indH (A) → M(Cr∗ (G) ⊗ indG H (A)). Explicitly, this coaction is given by

δ(x) = Wˆ G∗ 12 (σ ⊗ id) id ⊗ res(λ) (x)(Wˆ G )12 ˆ ˆ for x ∈ indG H (A) ⊂ L(HG ⊗ A). Writing WG = W and WG = W we calculate

∗ ∗ ∗ ˆ 23 ad(W12 )(id ⊗ δ) ind(α)(x) = W12 W Σ23 id ⊗ id ⊗ res(λ) W12 (1 ⊗ x)W12 Σ23 Wˆ 23 W12

∗ ∗ ∗ id ⊗ id ⊗ res(λ) (1 ⊗ x)W12 W23 = Σ23 W13 W23 W12 W13 Σ23 ∗ ∗ = Σ23 W12 W23 id ⊗ id ⊗ res(λ) (1 ⊗ x)W23 W12 Σ23 ∗ ˆ∗ = W13 W23 Σ23 id ⊗ id ⊗ res(λ) (1 ⊗ x)Σ23 Wˆ 23 W13 = (σ ⊗ id) id ⊗ ind(α) δ(x)

which shows that ind(α) and δ combine to turn indG H (A) into a G-YD-algebra.

2

Let G be a regular locally compact quantum group and let A be a G-YD-algebra with coactions α and λ. As explained above, the crossed product Cr∗ (G)cop r A is again a G-YD-algebra in a natural way. Moreover let B be a G-algebra with coaction β and observe ∗ Cr (G) ⊗ id ⊗ id ⊗ id (id ⊗ λ)α(A)123 (id ⊗ β)β(B)124 ∗ ∗ ∼ ˆ Cr (G) W12 (id ⊗ λ)α(A)123 W12 β(B)24 = 21 ∗ ˆ Cr (G) (id ⊗ α)λ(A)213 β(B)24 = 21 ∼ = λˆ Cr∗ (G)cop r A 12 β(B)13 = Cr∗ (G)cop r A G B.

Cr∗ (G)cop r (A G B) =


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Under this isomorphism the dual coaction on the left hand side corresponds to the coaction determined by the dual coaction on Cr∗ (G)cop r A and the trivial coaction on B on the right hand side. As a consequence we obtain the following lemma. Lemma 3.5. Let G be a regular locally compact quantum group, let A be a G-YD-algebra and let B be a G-algebra. Then there is a natural Cr∗ (G)cop -colinear isomorphism Cr∗ (G)cop r (A G B) ∼ = Cr∗ (G)cop r A G B. After these preparations we shall now describe the compatibility of restriction, induction and braided tensor products. Theorem 3.6. Let G be a strongly regular quantum group and let H ⊂ G be a closed quantum subgroup. Moreover let A be an H -YD-algebra and let B be a G-algebra. Then there is a natural G-equivariant isomorphism G ∼ G indG H A H resH (B) = indH (A) G B. Proof. Note that the case A = C with the trivial action is treated in [44]. We denote by res(β) the restriction to C0r (H ) of the coaction β : B → M(C0r (G) ⊗ B). Moreover let res(λ) be the push-forward of the coaction λ : A → M(Cr∗ (H ) ⊗ A) to Cr∗ (G). Then (id ⊗ id ⊗ β) λ(A)12 res(β)(B)13 = λ(A)12 (id ⊗ id ⊗ β) res(β)(B)134 ˆ WH∗ 13 β(B)34 (id ⊗ πˆ )(WH )13 = λ(A)12 (id ⊗ π) ˆ H ⊗ id)λ(A) = (id ⊗ π) ˆ WH∗ 13 (πˆ ⊗ id ⊗ id)( β(B)34 (id ⊗ π)(W ˆ H )13 312 ∼ = (πˆ ⊗ id ⊗ id)(id ⊗ λ)λ(A) 312 β(B)34 = (λ ⊗ id ⊗ id) res(λ)(A)21 β(B)23 , and hence ∼ A H resG H (B) = res(λ)(A)12 β(B)13 . Writing WG = W we conclude G Cr∗ (G)cop r indG H A H resH (B) ∼ = (I ⊗ id ⊗ id ⊗ id) id ⊗ res(λ) α(A)123 (id ⊗ β) res(β)(B)124 I ∗ ⊗ id ⊗ id ⊗ id = id ⊗ res(λ) (I ⊗ id)α(A) I ∗ ⊗ id 123 (G ⊗ id)β(B)124 ∗ (W ⊗ id) id ⊗ res(λ) (I ⊗ id)α(A) I ∗ ⊗ id W ∗ ⊗ 1 123 β(B)24 W12 = W12 using that [(I ⊗ id) res(β)(B)] = [β(B)(I ⊗ id)] for the restricted coaction res(β). Moreover

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∗ W12 (W ⊗ id) id ⊗ res(λ) (I ⊗ id)α(A) I ∗ ⊗ id W ∗ ⊗ 1 123 β(B)24 W12 ∼ = (W ⊗ id) id ⊗ res(λ) (I ⊗ id)α(A) I ∗ ⊗ id W ∗ ⊗ 1 123 β(B)24 = δˆ Cr∗ (G)cop r indG H (A) 213 β(B)24 ∼ = Cr∗ (G)cop r indG H (A) G B ∗ ∗ cop where δˆ : Cr∗ (G)cop r indG r indG H (A) → M(Cr (G) ⊗ (Cr (G) H (A))) is the natural coaction G on the crossed product of the G-YD-algebra indH (A). G Under these identifications, the dual coaction on Cr∗ (G)cop r indG H (A H resH (B)) correG sponds on (Cr∗ (G)cop r indH (A)) G B to the dual coaction on the crossed product and the trivial coaction on B. As a consequence, using Lemma 3.5 we obtain a Cr∗ (G)cop -colinear isomorphism

G cop ∼ ∗ Cr∗ (G)cop r indG r indG H A H resH (B) = Cr (G) H (A) G B . G Moreover, the element W ⊗ id ∈ M(C0r (G) ⊗ Cr∗ (G)cop r indG H (A H resH (B))) is mapped to G r ∗ cop W ⊗ id ∈ M(C0 (G) ⊗ Cr (G) r (indH (A) G B)) under this isomorphism. Due to Theorem 6.7 in [44] this shows that there is a G-equivariant isomorphism

G ∼ G indG H A H resH (B) = indH (A) G B as desired.

2

We also need braided tensor products of Hilbert modules. Since the constructions and arguments are similar to the algebra case treated above our discussion will be rather brief. Assume that A is a G-YD-algebra and that B is a G-algebra. Moreover let EA be a D(G)-Hilbert module and let FB be a G-Hilbert module. As in the algebra case, a D(G)-Hilbert module E is the same thing as a Hilbert module equipped with continuous coactions αE of C0r (G) and λE of Cr∗ (G) satisfying the Yetter–Drinfeld compatibility condition in the sense that (σ ⊗ id)(id ⊗ αE )λE = ad(W ) ⊗ id (id ⊗ λE )αE where ad(W ) is the adjoint action. The braided tensor product of E and F is defined as E G F = λE (E)12 βF (F )13 ⊂ MK (K ⊗ E ⊗ F ) where λE denotes the coaction of Cr∗ (G) on E and βF is the coaction of C0r (G) on F . One has [λE (E)12 βF (F )13 ] = [βF (F )13 λE (E)12 ], and E G F is closed under right multiplication by elements from A G B ⊂ MK (K ⊗ A ⊗ B). Moreover the restriction to E G F of the scalar product of MK (K ⊗ E ⊗ F ) takes values in A G B. It follows that E G F is a Hilbert-A G Bmodule. As in the algebra case there is a continuous coaction of C0r (G) on E G F given by

∗ (σ ⊗ id)(id ⊗ αE ⊗ id). ad W12


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Similarly, if B is a G-YD-algebra and F is a D(G)-Hilbert module we have a continuous Cr∗ (G)coaction. The braided tensor product becomes a D(G)-Hilbert module in this case. There are canonical nondegenerate ∗-homomorphisms K(E) → L(E G F ) and K(F ) → L(E G F ). Combining these homomorphisms yields an identification K(E) G K(F ) ∼ = K(E G F ). We conclude this section with a discussion of stability properties. Proposition 3.7. Let G be a regular locally compact quantum group and let A be a G-YDalgebra. a) For every G-C ∗ -algebra B there is a natural G-equivariant Morita equivalence (KD(G) ⊗ A) G B ∼M A G B. If B is a G-YD-algebra this Morita equivalence is D(G)-equivariant. b) For every G-C ∗ -algebra B there is a natural G-equivariant Morita equivalence A G (KG ⊗ B) ∼M A G B. If B is a G-YD-algebra there is a natural D(G)-equivariant Morita equivalence A G (KD(G) ⊗ B) ∼M A G B. Proof. We consider the coaction of D(G) on HD(G) coming from the regular representation. From [4] we know that the corresponding corepresentation of C0r (G) on HD(G) = HG ⊗ HG is W12 ∈ M(C0r (G) ⊗ KD(G) ). The corresponding corepresentation of Cr∗ (G) on HD(G) is given by ∗ W ˆ 13 Z23 where Z = W (J ⊗ Jˆ)W (J ⊗ Jˆ). Z23 ∗ W ˆ 13 Z23 implements a G-equivariant isomorphism To prove a) we observe that Z23 ∗ ∗ (HD(G) ⊗ A) G B = Z23 Wˆ 13 Z23 σ12 (id ⊗ λ)(HD(G) ⊗ A)123 β(B)14 ∼ = HD(G) ⊗ λ(A)12 β(B)13 = HD(G) ⊗ (A G B) of Hilbert modules. This yields (KD(G) ⊗ A) G B ∼ = K (HD(G) ⊗ A) G B ∼ = K HD(G) ⊗ (A G B) ∼M K(A G B) = A G B in a way compatible with the coaction of C0r (G). If B is a G-YD-algebra the above isomorphisms and the Morita equivalence are D(G)-equivariant. The assertions in b) are proved in a similar fashion. 2 4. The equivariant Kasparov category In this section we first review the definition of equivariant Kasparov theory given by Baaj and Skandalis [1]. Then we explain how to extend several standard results from the case of locally compact groups to the setting of regular locally compact quantum groups. In particular, we adapt

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the Cuntz picture of KK-theory [15] to show that equivariant KK-classes can be described by homotopy classes of equivariant homomorphisms. As a consequence, we obtain the universal property of equivariant Kasparov theory. We describe its structure as a triangulated category and discuss the restriction and induction functors. Finally, based on the construction of the braided tensor product in the previous section we construct exterior products in equivariant KK-theory. Let us recall the definition of equivariant Kasparov theory [1]. For simplicity we will assume that all C ∗ -algebras are separable. Let S be a Hopf-C ∗ -algebra and let A and B be graded S-C ∗ algebras. An S-equivariant Kasparov A-B-module is a countably generated graded S-equivariant Hilbert B-module E together with an S-colinear graded ∗-homomorphism φ : A → L(E) and an odd operator F ∈ L(E) such that F, φ(a) ,

2 F − 1 φ(a),

F − F ∗ φ(a)

are contained in K(E) for all a ∈ A and F is almost invariant in the sense that (id ⊗ φ)(x) 1 ⊗ F − adλ (F ) ⊂ S ⊗ K(E) for all x ∈ S ⊗ A. Here S ⊗ K(E) = K(S ⊗ E) is viewed as a subset of L(S ⊗ E) and adλ is the adjoint coaction associated to the given coaction λ : E → M(S ⊗ E) on E. Two S-equivariant Kasparov A-B-modules (E0 , φ0 , F0 ) and (E1 , φ1 , F1 ) are called unitarly equivalent if there is an S-colinear unitary U ∈ L(E0 , E1 ) of degree zero such that U φ0 (a) = φ1 (a)U for all a ∈ A and F1 U = U F0 . We write (E0 , φ0 , F0 ) ∼ = (E1 , φ1 , F1 ) in this case. Let ES (A, B) be the set of unitary equivalence classes of S-equivariant Kasparov A-B-modules. This set is functorial for graded S-colinear ∗-homomorphisms in both variables. If f : B1 → B2 is a graded S-colinear ∗-homomorphism and (E, φ, F ) is an S-equivariant Kasparov A-B1 -module, then ˆ f B2 , φ ⊗id, ˆ F ⊗1) ˆ f∗ (E, φ, F ) = (E ⊗ is the corresponding Kasparov A-B2 -module. A homotopy between S-equivariant Kasparov A-B-modules (E0 , φ0 , F0 ) and (E1 , φ1 , F1 ) is an S-equivariant Kasparov A-B[0, 1]-module (E, φ, F ) such that (evt )∗ (E, φ, F ) ∼ = (Et , φt , Ft ) for t = 0, 1. Here B[0, 1] = B ⊗ C[0, 1] where C[0, 1] is equipped with the trivial action and grading and evt : B[0, 1] → B is evaluation at t. Definition 4.1. Let S be a Hopf-C ∗ -algebra and let A and B be graded S-C ∗ -algebras. The S-equivariant Kasparov group KK S (A, B) is the set of homotopy classes of S-equivariant Kasparov A-B-modules. In the definition of KK S (A, B) one can restrict to Kasparov triples (E, φ, F ) which are essential in the sense that [φ(A)E] = E, compare [29]. We note that KK S (A, B) becomes an abelian group with addition given by the direct sum of Kasparov modules. Many properties of ordinary KK-theory carry over to the S-equivariant situation, in particular the construction of the Kasparov composition product and Bott periodicity [1]. As usual we write KK S0 (A, B) = KK S (A, B) and let KK S1 (A, B) be the odd KK-group obtained by suspension in either variable. In the case S = C0 (G) for a locally compact group G one reobtains the definition of G-equivariant KKtheory [22]. Our first aim is to establish the Cuntz picture of equivariant KK-theory in the setting of regular locally compact quantum groups. This can be done parallel to the account in the group case given


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by Meyer [29]. For convenience we restrict ourselves to trivially graded C ∗ -algebras and present a short argument using Baaj–Skandalis duality. Let S be a Hopf-C ∗ -algebra and let A1 and A2 be S-C ∗ -algebras. Consider the free product A1 ∗ A2 together with the canonical ∗-homomorphisms ιj : Aj → A1 ∗ A2 for j = 1, 2. We compose the coaction αj : Aj → MS (S ⊗ Aj ) with the ∗-homomorphism MS (S ⊗ Aj ) → MS (S ⊗ (A1 ∗ A2 )) induced by ιj and combine these maps to obtain a ∗-homomorphism α : A1 ∗ A2 → MS (S ⊗ (A1 ∗ A2 )). This map satisfies all properties of a continuous coaction in the sense of Definition 2.3 except that it is not obvious whether α is always injective. If necessary, this technicality can be overcome by passing to a quotient of A1 ∗ A2 . More precisely, on A1 ∗S A2 = (A1 ∗ A2 )/ ker(α) the map α induces the structure of an S-C ∗ -algebra, and we have canonical S-colinear ∗-homomorphisms Aj → A1 ∗S A2 for j = 1, 2 again denoted by ιj . The resulting S-C ∗ -algebra is universal for pairs of S-colinear ∗-homomorphisms f1 : A1 → C and f2 : A2 → C into S-C ∗ -algebras C. That is, for any such pair of ∗-homomorphisms there exists a unique S-colinear ∗-homomorphism f : A1 ∗S A2 → C such that f ιj = fj for j = 1, 2. By abuse of notation, we will still write A1 ∗ A2 instead of A1 ∗S A2 in the sequel. We point out that in the arguments below we could equally well work with the ordinary free product together with its possibly noninjective coaction. Let A be an S-C ∗ -algebra and consider QA = A ∗ A. The algebra K ⊗ QA is S-colinearly homotopy equivalent to K ⊗ (A ⊕ A) where K denotes the algebra of compact operators on a separable Hilbert space H. Moreover there is an extension

0

qA

π

QA

A

0

of S-C ∗ -algebras with S-colinear splitting, here π is the homomorphism associated to the pair f1 = f2 = idA and qA its kernel. We shall now restrict attention from general Hopf-C ∗ -algebras to regular locally compact quantum groups and state the Baaj–Skandalis duality theorem [1,2]. Theorem 4.2. Let G be a regular locally compact quantum group and let S = C0r (G) and Sˆ = Cr∗ (G)cop . For all S-C ∗ -algebras A and B there is a canonical isomorphism ˆ

JS : KK S (A, B) → KK S (Sˆ r A, Sˆ r B) which is multiplicative with respect to the composition product. For our purposes it is important that under this isomorphism the class of an S-equivariant ˆ Kasparov A-B-module (E, φ, F ) is mapped to the class of an S-equivariant Kasparov module ˆ (JS (E), JS (φ), JS (F )) with an operator JS (F ) which is exactly invariant under the coaction of S. Let G be a regular locally compact quantum group and let E and F be G-Hilbert B-modules which are isomorphic as Hilbert B-modules. Then we have a G-equivariant isomorphism HG ⊗ E ∼ = HG ⊗ F

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of G-Hilbert B-modules where HG is viewed as a G-Hilbert space using the left regular corepresentation, see [46]. Using the Kasparov stabilization theorem we deduce that there is a G-equivariant Hilbert B-module isomorphism (HG ⊗ E) ⊕ (HG ⊗ H ⊗ B) ∼ = HG ⊗ H ⊗ B for every countably generated G-Hilbert B-module E. This result will be referred to as the equivariant stabilization theorem. In the sequel we will frequently write KK G instead of KK S for S = C0r (G) and call the defining cycles of this group G-equivariant Kasparov modules. It follows from Baaj–Skandalis duality that KK G (A, B) can be represented by homotopy classes of G-equivariant Kasparov (KG ⊗ A)(KG ⊗ B)-modules (E, φ, F ) with G-invariant operator F . Taking Kasparov product with the KG ⊗ B-B imprimitivity bimodule (HG ⊗ B, id, 0) we see that KK G (A, B) can be represented by homotopy classes of equivariant Kasparov (KG ⊗ A)-B modules of the form (HG ⊗ E, φ, F ) with invariant F . Using the equivariant stabilization theorem we can furthermore assume that (HG ⊗ E)± = HG ⊗ H ⊗ B is the standard G-Hilbert B-module. From this point on we follow the arguments in [29]. Writing [A, B]G for the set of equivariant homotopy classes of G-equivariant ∗-homomorphisms between G-C ∗ -algebras A and B, we arrive at the following description of the equivariant KK-groups. Theorem 4.3. Let G be a regular locally compact quantum group. Then there is a natural isomorphism KK G (A, B) ∼ = q(KG ⊗ A), KG ⊗ K ⊗ B G for all separable G-C ∗ -algebras A and B. We also have a natural isomophism KK G (A, B) ∼ = KG ⊗ K ⊗ q(KG ⊗ K ⊗ A), KG ⊗ K ⊗ q(KG ⊗ K ⊗ B) G under which the Kasparov product corresponds to the composition of homomorphisms. Consider the category G-Alg of separable G-C ∗ -algebras for a regular quantum group G. A functor F from G-Alg to an additive category C is called a homotopy functor if F (f0 ) = F (f1 ) whenever f0 and f1 are G-equivariantly homotopic ∗-homomorphisms. It is called stable if for all pairs of separable G-Hilbert spaces H1 , H2 the maps F (K(Hj ) ⊗ A) → F (K(H1 ⊕ H2 ) ⊗ A) induced by the canonical inclusions Hj → H1 ⊕ H2 for j = 1, 2 are isomorphisms. As in the group case, a homotopy functor F is stable iff there exists a natural isomorphism F (A) ∼ = F (KG ⊗ K ⊗ A) for all A. Finally, F is called split exact if for every extension 0

K

E

Q

0

of G-C ∗ -algebras that splits by an equivariant ∗-homomorphism σ : Q → E the induced sequence 0 → F (K) → F (E) → F (Q) → 0 in C is split exact. Equivariant KK-theory can be viewed as an additive category KK G with separable G-C ∗ algebras as objects and KK G (A, B) as the set of morphisms between two objects A and B. Composition of morphisms is given by the Kasparov product. There is a canonical functor ι : GAlg → KK G which is the identity on objects and sends equivariant ∗-homomorphisms to the corresponding KK-elements. This functor is a split exact stable homotopy functor.


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As a consequence of Theorem 4.3 we obtain the following universal property of KK G , see again [29]. We remark that a related assertion is stated in [39], however, some of the arguments in [39] are incorrect. Theorem 4.4. Let G be a regular locally compact quantum group. The functor ι : G-Alg → KK G is the universal split exact stable homotopy functor on the category G-Alg. More precisely, if F : G-Alg → C is any split exact stable homotopy functor with values in an additive category C then there exists a unique functor f : KK G → C such that F = f ι. Let us explain how KK G becomes a triangulated category. We follow the discussion in [30], for the definition of a triangulated category see [34]. Let ΣA denote the suspension C0 (R) ⊗ A of a G-C ∗ -algebra A. Here C0 (R) is equipped with the trivial coaction. The corresponding functor Σ : KK G → KK G determines the translation automorphism. If f : A → B is a G-equivariant ∗-homomorphism then the mapping cone Cf = (a, b) ∈ A × C0 (0, 1], B b(1) = f (a) is a G-C ∗ -algebra in a natural way, and there is a canonical diagram ΣB

Cf

f

A

B

of G-equivariant ∗-homomorphisms. Diagrams of this form are called mapping cone triangles. By definition, an exact triangle is a diagram ΣQ → K → E → Q in KK G which is isomorphic to a mapping cone triangle. The proof of the following proposition is carried out in the same way as for locally compact groups [30]. Proposition 4.5. Let G be a regular locally compact quantum group. Then the category KK G together with the translation functor and the exact triangles described above is triangulated. Several results about the equivariant KK-groups for ordinary groups extend in a straightforward way to the setting of quantum groups. As an example, let us state the Green–Julg theorem for compact quantum groups and its dual version for discrete quantum groups. If G is a locally ∗ compact quantum group and A is a C ∗ -algebra we write resE G (A) for the G-C -algebra A with the trivial coaction. A detailed proof of the following result is contained in [46]. Theorem 4.6. Let G be a compact quantum group. Then there is a natural isomorphism ∗ cop ∼ KK G resE B G (A), B = KK A, C (G) for all C ∗ -algebras A and all G-C ∗ -algebras B. Dually, let G be a discrete quantum group. Then there is a natural isomorphism ∗ cop ∼ KK G A, resE f A, B G (B) = KK Cf (G) for all G-C ∗ -algebras A and all C ∗ -algebras B.

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Let G be a strongly regular quantum group and let H ⊂ G be a regular closed quantum subgroup. It is easy to check that restriction from G to H induces a triangulated functor resG H : G G H ∗ ∗ KK → KK . This functor associates to a G-C -algebra A the H -C -algebra resH (A) = A obtained by restricting the action. Similarly, using the universal property of Theorem 4.4 we obH G ∗ tain a triangulated functor indG H : KK → KK which maps an H -C -algebra A to the induced G ∗ G-C -algebra indH (A). Note that the compatibility of induction with stabilizations follows from Vaes’ imprimitivity theorem stated above as Theorem 2.6. A closed quantum subgroup H ⊂ G is called cocompact if the C ∗ -algebraic quantum homogeneous space C0r (G/H ) is a unital C ∗ -algebra. In this case we write C r (G/H ) instead of C0r (G/H ). Recall that a locally compact quantum group G is coamenable if the natural map C0f (G) → C0r (G) is an isomorphism. Strong regularity is equivalent to regularity in this case. Proposition 4.7. Let H ⊂ G be a cocompact regular quantum subgroup of a strongly regular quantum group G. If G is coamenable there is a natural isomorphism G G ∼ KK H resG H (A), B = KK A, indH (B) for all G-C ∗ -algebras A and all H -C ∗ -algebras B. Proof. We describe the unit η and the counit κ of this adjunction. For a G-C ∗ -algebra A let ηA : G ∼ r A → indG H resH (A) = C (G/H ) G A be the G-equivariant ∗-homomorphism obtained from the embedding of A in the braided tensor product. Here we use Theorem 3.6 and the assumption that H ⊂ G is cocompact. In order to define the counit κ recall that the induced C ∗ -algebra indG H (B) of an H -C ∗ -algebra B is contained in the C0r (G)-relative multiplier algebra of C0r (G) ⊗ B. We G obtain an H -equivariant ∗-homomorphism κB : resG H indH (B) → B as the restriction of ⊗ id : r r M(C0 (G) ⊗ B) → M(B) where : C0 (G) → C is the counit. Here we use coamenability of G. Let A be a G-C ∗ -algebra with coaction α and let res(α) : A → M(C0r (H ) ⊗ A) be the restriction of α to H . Using the relation [(I ⊗ 1) res(α)(A)] = [α(A)(I ⊗ 1)] established in [44] we see that κres(A) is given by id : C r (G/H ) G A → C G A ∼ = A. Note that, although not being G-equivariant, the map is Cr∗ (G)-colinear and hence induces a ∗-homomorphism between the braided tensor products as desired. It follows that the composition resG H (A)

res(ηA )

G G resG H indH resH (A)

κres(A)

resG H (A)

G ∗ is the identity in KK H (resG H (A), resH (A)) for every G-C -algebra A. G G G ∼ Identifying the isomorphism indH resH indH (B) = C r (G/H ) G indG H (B) and using the counit identity (id ⊗ ) = id for C0r (G) we see that

indG H (B)

ηind(B)

G G indG H resH indH (B)

ind(κB )

indG H (B)

G ∗ is the identity in KK G (indG H (B), indH (B)) for every G-C -algebra B. This yields the assertion. 2

Based on the braided tensor product we introduce exterior products in equivariant Kasparov theory.


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Proposition 4.8. Let G be a regular locally compact quantum group, let A and B be G-C ∗ algebras and let D be a G-YD-algebra. Then there exists a natural homomorphism λD : KK G (A, B) → KK G (D G A, D G B) defining a triangulated functor λD : KK G → KK G . If A and B are G-YD-algebras then there is an analogous homomorphism λD : KK D(G) (A, B) → KK D(G) (D G A, D G B) defining a triangulated functor λD : KK D(G) → KK D(G) . Proof. We shall only discuss the first assertion, the case of G-YD-algebras is treated analogously. Taking the braided tensor product with D defines a split exact homotopy functor from G-Alg to KK G . According to Proposition 3.7 this functor is stable. Hence the existence of λD is a consequence of the universal property of KK G established in Theorem 4.4, and the resulting functor is easily seen to be triangulated. 2 The same arguments yield the following right-handed version of Proposition 4.8. Proposition 4.9. Let G be a regular locally compact quantum group, let C and D be G-YDalgebras and let B be a G-C ∗ -algebra. Then there exists a natural homomorphism ρB : KK D(G) (C, D) → KK G (C G B, D G B) defining a triangulated functor ρB : KK D(G) → KK G . If B is a G-YD-algebra we obtain a natural homomorphism ρB : KK D(G) (C, D) → KK D(G) (C G B, D G B) defining a triangulated functor ρB : KK D(G) → KK D(G) . By construction, the class of a G-equivariant ∗-homomorphism f : A → B is mapped to the class of f id under λD : KK G → KK G , and similar remarks apply to the other functors obtained above. Of course one can also give direct definitions on the level of Kasparov modules for the constructions in Propositions 4.8 and 4.9. For instance, let (E, φ, F ) be a G-equivariant Kasparov A-B-module. Then D G E is a D G B-Hilbert module, and the map φ : A → L(E) = M(K(E)) induces a G-equivariant ∗-homomorphism id G φ : D G A → M(D G K(E)) ∼ = L(D G E). Moreover, we obtain id G F ∈ L(D G E) by applying the canonical map L(E) → L(D G E). It is readily checked that this yields a G-equivariant Kasparov module. The construction is compatible with homotopies and induces λD : KK G (A, B) → KK G (D G A, D G B). Let A1 , B1 and D be G-YD algebras and let A2 , B2 be G-C ∗ -algebras. We define the exterior Kasparov product KK D(G) (A1 , B1 G D) × KK G (D G A2 , B2 ) → KK G (A1 G A2 , B1 G B2 )

1492


as the map which sends (x, y) to ρA2 (x) ◦ λB1 (y). Here ◦ denotes the Kasparov composition product, and we use (B1 G D) G A2 ∼ = B1 G (D G A2 ). If A2 , B2 are G-YD-algebras we obtain an exterior product KK D(G) (A1 , B1 G D) × KK D(G) (D G A2 , B2 ) → KK D(G) (A1 G A2 , B1 G B2 ) in the same way. We summarize the main properties of the above exterior Kasparov products in analogy with the ordinary exterior Kasparov product, see [7]. Theorem 4.10. Let G be a regular locally compact quantum group. Moreover let A1 , B1 and D be G-YD algebras and let A2 , B2 be G-C ∗ -algebras. The exterior Kasparov product KK D(G) (A1 , B1 G D) × KK G (D G A2 , B2 ) → KK G (A1 G A2 , B1 G B2 ) is associative and functorial in all possible senses. An analogous statement holds for the product KK D(G) (A1 , B1 G D) × KK D(G) (D G A2 , B2 ) → KK D(G) (A1 G A2 , B1 G B2 ) provided A2 , B2 are G-YD-algebras. Recall that every G-C ∗ -algebra for a locally compact group G can be viewed as a G-YDalgebra with the trivial coaction of Cr∗ (G). In this case our constructions reduce to the classical exterior product in equivariant KK-theory. Still, even for classical groups the products defined above are more general since we may consider G-YD-algebras that are equipped with a nontrivial coaction of the group C ∗ -algebra. 5. The quantum group SUq (2) In this section we recall some definitions and constructions related to the compact quantum group SU q (2) introduced by Woronowicz [48]. For more information on the algebraic aspects of compact quantum groups we refer to [23]. Let us fix a number q ∈ (0, 1] and describe the C ∗ -algebra of continuous functions on SU q (2). Since SU q (2) is coamenable [5,32] there is no need to distinguish between the full and reduced C ∗ -algebras. By definition, C(SU q (2)) is the universal C ∗ -algebra generated by two elements α and γ satisfying the relations αγ = qγ α,

αγ ∗ = qγ ∗ α,

γ γ ∗ = γ ∗γ ,

α ∗ α + γ ∗ γ = 1,

αα ∗ + q 2 γ γ ∗ = 1.

The comultiplication : C(SU q (2)) → C(SU q (2)) ⊗ C(SU q (2)) is given on the generators by (α) = α ⊗ α − qγ ∗ ⊗ γ ,

(γ ) = γ ⊗ α + α ∗ ⊗ γ .

From a conceptual point of view, it is useful to interpret these formulas in terms of the fundamental matrix α −qγ ∗ u= . γ α∗


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In fact, the defining relations for C(SU q (2)) are equivalent to saying that the fundamental matrix is unitary, and the comultiplication of C(SU q (2)) can be written in a concise way as

α γ

−qγ ∗ α∗

=

α γ

−qγ ∗ α∗

⊗

−qγ ∗ α∗

α γ

.

We will also work with the dense ∗-subalgebra C[SU q (2)] ⊂ C(SU q (2)) generated by α and γ . Together with the counit : C[SU q (2)] → C and the antipode S : C[SU q (2)] → C[SU q (2)] determined by ∗ γ∗ α −qγ ∗ α α −qγ ∗ 1 0 = , S = 0 1 γ α∗ γ α∗ −qγ α the algebra C[SU q (2)] becomes a Hopf-∗-algebra. We use the Sweedler notation (x) = x(1) ⊗ x(2) for the comultiplication and write x f = f (x(1) )x(2)

f x = x(1) f (x(2) ),

for elements x ∈ C[SU q (2)] and linear functionals f : C[SU q (2)] → C. The antipode is an algebra antihomomorphism satisfying S(S(x ∗ )∗ ) = x for all x ∈ C[SU q (2)], in particular the map S is invertible. The inverse of S can be written as S −1 (x) = δ S(x) δ −1 where δ : C[SU q (2)] → C is the modular character determined by

α δ γ

−qγ ∗ α∗

=

q −1 0

0 . q

Apart from its role in connection with the antipode, the character δ describes the modular properties of the Haar state φ of C(SU q (2)) in the sense that φ(xy) = φ y(δ x δ) for all x, y ∈ C[SU q (2)]. The Hilbert space HSU q (2) associated to SU q (2) is the GNSconstruction of φ and will be denoted by L2 (SU q (2)) in the sequel. The irreducible corepresentations Vl of C(SU q (2)) are parametrized by l ∈ 12 N, and the dimension of Vl is 2l + 1 as for the classical group SU(2). According to the Peter–Weyl the(l) orem, the Hilbert space L2 (SU q (2)) has an orthonormal basis eij with l ∈ 12 N and i, j ∈ {−l, −l + 1, . . . , l} corresponding to the decomposition of the regular corepresentation. In this picture, the GNS-representation of C(SU q (2)) is given by (l)

αeij = a+ (l, i, j )e (l)

γ eij = c+ (l, i, j )e

(l+ 12 ) i− 12 ,j − 12 (l+ 12 )

i+ 12 ,j − 12

+ a− (l, i, j )e + c− (l, i, j )e

where the explicit form of a± and c± for q ∈ (0, 1) is

(l− 12 ) i− 12 ,j − 12

(l− 12 ) i+ 12 ,j − 12

,

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a+ (l, i, j ) = q 2l+i+j +1 a− (l, i, j ) =

(1 − q 2l−2j +2 )1/2 (1 − q 2l−2i+2 )1/2 , (1 − q 4l+2 )1/2 (1 − q 4l+4 )1/2

(1 − q 2l+2j )1/2 (1 − q 2l+2i )1/2 (1 − q 4l )1/2 (1 − q 4l+2 )1/2

and c+ (l, i, j ) = −q l+j c− (l, i, j ) = q l+i

(1 − q 2l−2j +2 )1/2 (1 − q 2l+2i+2 )1/2 , (1 − q 4l+2 )1/2 (1 − q 4l+4 )1/2

(1 − q 2l+2j )1/2 (1 − q 2l−2i )1/2 . (1 − q 4l )1/2 (1 − q 4l+2 )1/2 (l)

In the above formulas the vectors eij are declared to be zero if one of the indices i, j is not contained in {−l, −l + 1, . . . , l}. We will frequently use the fact that the classical torus T = S 1 is a closed quantum subgroup of SU q (2). The inclusion T ⊂ SU q (2) is determined by the ∗-homomorphism π : C[SU q (2)] → C[T ] = C[z, z−1 ] given by π

α γ

−qγ ∗ α∗

=

z 0 0 z−1

.

By definition, the standard Podle´s sphere C(SU q (2)/T ) is the corresponding homogeneous space [37]. In the algebraic setting, the Podle´s sphere is described by the dense ∗-subalgebra C[SU q (2)/T ] ⊂ C(SU q (2)/T ) of coinvariants in C[SU q (2)] with respect to the right coaction (id ⊗ π) of C[T ]. If V is a finite dimensional left C[T ]-comodule, or equivalently a finite dimensional representation of T , then the cotensor product Γ SU q (2) ×T V = C SU q (2) 2C[T ] V ⊂ C SU q (2) ⊗ V is a noncommutative analogue of the space of sections of the homogeneous vector bundle SU(2) ×T V over SU(2)/T . Clearly Γ (SU q (2) ×T V ) is a C[SU q (2)/T ]-bimodule in a natural way. In accordance with the Serre–Swan theorem, the space of sections Γ (SU q (2) ×T V ) is finitely generated and projective both as a left and right C[SU q (2)/T ]-module. This follows from the fact that C[SU q (2)/T ] ⊂ C[SU q (2)] is a faithfully flat Hopf–Galois extension, see [31,41]. If V = Ck is the irreducible representation of T of weight k ∈ Z we write L2 (SU q (2)×T Ck ) for the SU q (2)-Hilbert space obtained by taking the closure of Γ (SU q (2) ×T Ck ) inside L2 (SU q (2)). We also note the Frobenius reciprocity isomorphism SU (2) HomT resT q (V ), Ck ∼ = HomSU q (2) V , L2 SU q (2) ×T Ck for all finite dimensional corepresentations V of C(SU q (2)).


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6. Equivariant Poincaré duality for the Podle´s sphere Poincaré duality in Kasparov theory plays an important rôle in noncommutative geometry, for instance in connection with the Dirac-dual Dirac method for proving the Novikov conjecture [22]. In this section we extend this concept to the setting of quantum group actions and show that the standard Podle´s sphere is equivariantly Poincaré dual to itself. Let us begin with the following terminology, generalizing the definition given by Connes in [11]. Recall that we write D(G) for the Drinfeld double of a locally compact quantum group G. Definition 6.1. Let G be a regular locally compact quantum group. Two G-YD-algebras P and Q are called G-equivariantly Poincaré dual to each other if there exists a natural isomorphism KK ∗D(G) (P G A, B) ∼ = KK ∗D(G) (A, Q G B) for all G-YD-algebras A and B. Using the notation introduced in Proposition 4.8 we may rephrase this by saying that the G-YD-algebras P and Q are G-equivariantly Poincaré dual to each other iff λP and λQ are adjoint functors. In particular, the unit and counit of the adjunction determine elements α ∈ KK ∗D(G) (P G Q, C),

β ∈ KK ∗D(G) (C, Q G P )

if P and Q are Poincaré dual. In this case one also has a duality on the level of G-equivariant Kasparov theory in the sense that there is a natural isomorphism G ∼ KK G ∗ (P G A, B) = KK ∗ (A, Q G B)

for all G-C ∗ -algebras A and B. In the sequel we restrict attention to Gq = SU q (2). Our aim is to show that the standard Podle´s sphere is SU q (2)-equivariantly Poincaré dual to itself in the sense of Definition 6.1. As a first ingredient we need the K-homology class of the Dirac operator on Gq /T for q ∈ (0, 1). We review briefly the construction in [19], however, instead of working with the action of the quantized universal enveloping algebra we consider the corresponding coaction of C(Gq ). Using the notation from Section 5, the underlying graded Gq -Hilbert space H = H+ ⊕ H− of the spectral triple is given by H± = L2 (Gq ×T C±1 ) with its natural coaction of C(Gq ). The covariant representation φ = φ+ ⊕ φ− of the C(Gq /T ) is given by left multiplication. Finally, the Dirac operator D on H is the odd operator D=

0 D+

D− 0

where D ± |l, m± = [l + 1/2]q |l, m∓

1496


and |l, m± are the standard basis vectors in Vl ⊂ H± and [a]q =

q a − q −a q − q −1

for a nonzero number a ∈ C. Note that H+ and H− are isomorphic corepresentations of C(Gq ) according to Frobenius reciprocity. It follows that the phase F of D can be written as F=

0 1

1 , 0

and the triple (H, φ, F ) is a Gq -equivariant Fredholm module. In this way D determines an G

element in KK 0 q (C(Gq /T ), C).

G

According to Proposition 3.4 the C ∗ -algebra C(Gq /T ) = indT q (C) is a Gq -YD-algebra. For our purposes the following fact is important.

Proposition 6.2. The Dirac operator on the standard Podle´s sphere defines an element in KK D(Gq ) (C(Gq /T ) Gq C(Gq /T ), C) in a natural way. Proof. With the notation as above, we consider the operator F on the Hilbert space H = G G H+ ⊕ H− . Using indT q resT q (C(Gq /T )) ∼ = C(Gq /T ) Gq C(Gq /T ) we obtain a graded Gq equivariant ∗-homomorphism ψ : C(Gq /T ) Gq C(Gq /T ) → L(H) by applying the induction functor to the counit : C(Gq /T ) → C and composing the resulting map with the natural representation of C(Gq /T ) on H. On both copies of C(Gq /T ) the map ψ is given by the homomorphism φ from above. In particular, the commutators of F with elements from C(Gq /T ) Gq C(Gq /T ) are compact. The coaction λ : H → M(C ∗ (Gq ) ⊗ H) which turns H into a D(Gq )-Hilbert space is obtained from the action of C[Gq ] on Γ (Gq ×T C± ) given by f · h = f(1) hδ S(f(2) ) where δ is the modular character. The homomorphism ψ is C ∗ (Gq )-colinear with respect to this coaction, and in order to show ∗ C (Gq ) ⊗ 1 1 ⊗ F − adλ (F ) ⊂ C ∗ (Gq ) ⊗ K(H) it suffices to check that F commutes with the above action of C[Gq ] up to compact operators. This in turn is a lengthy but straightforward calculation based on the explicit formulas for the GNS-representation of C(Gq ) in Section 5. It follows that (H, ψ, F ) is a D(Gq )-equivariant Kasparov module as desired. 2 Note that in the construction of the Dirac cycle in Proposition 6.2 we use two identical representations of C(Gq /T ) as in the case of a classical spin manifold. The difference to the classical situation lies in the replacement of the ordinary tensor product with the braided tensor product. Let us formally write Ek = Gq ×T Ck for the induced vector bundle associated to the representation of weight k, and denote by C(Ek ) the closure of Γ (Ek ) inside C(Gq ). The space


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C(Ek ) is a Gq -equivariant Hilbert C(Gq /T )-module with the coaction induced by comultiplication, and the coaction λ : C(Ek ) → M(C ∗ (Gq ) ⊗ C(Ek )) given by λ(f ) = Wˆ ∗ (1 ⊗ f )Wˆ turns it into a D(Gq )-equivariant Hilbert module. Left multiplication yields a D(Gq )-equivariant ∗-homomorphism μ : C(Gq /T ) → K(C(Ek )). Hence (C(Ek ), μ, 0) defines a class JEk K in D(G )

KK 0 q (C(Gq /T ), C(Gq /T )). Next observe that the unit homomorphism u : C → C(Gq /T ) induces an element [u] ∈ D(G )

D(Gq )

KK 0 q (C, C(Gq /T )). We obtain a class [Ek ] in KK 0 along u, or equivalently, by taking the product

(C, C(Gq /T )) by restricting JEk K

[Ek ] = [u] ◦ JEk K. Under the forgetful map from KK D(Gq ) to KK Gq , this class is mapped to the K-theory class in KK Gq (C, C(Gq /T )) corresponding to Ck in R(T ) under Frobenius reciprocity. G

In addition we define elements [D ⊗ Ek ] ∈ KK 0 q (C(Gq /T ), C) by [D ⊗ Ek ] = JEk K ◦ [D] G

where [D] ∈ KK 0 q (C(Gq /T ), C) is the class of the Dirac operator. We remark that these elements correspond to twisted Dirac operators on Gq /T as studied by Sitarz in [43]. Let us determine the equivariant indices of these twisted Dirac operators. Proposition 6.3. Consider the classes [Ek ] ∈ KK Gq (C, C(Gq /T )) and [D ⊗ El ] ∈ G

KK Gq (C(Gq /T ), C) introduced above. The Kasparov product [Ek ] ◦ [D ⊗ El ] in KK 0 q (C, C) = R(Gq ) is given by ⎧ ⎨ −[V(k+l−1)/2 ] for k + l > 0, for k + l = 0, [Ek ] ◦ [D ⊗ El ] = 0 ⎩ [V for k + l < 0 −(k+l+1)/2 ] for k, l ∈ Z. Proof. This is analogous to calculating the index of a homogeneous differential operator [9]. Since we have JEm K ◦ JEn K = JEm+n K for all m, n ∈ Z it suffices to consider the case k = 0. The product [E0 ] ◦ [D ⊗ El ] is given by the equivariant index of the Gq -equivariant Fredholm operator representing [D ⊗ El ]. This operator can be viewed as an odd operator on L2 (El+1 ) ⊕ L2 (El−1 ). By equivariance, the claim follows from Frobenius reciprocity; we only have to subtract the classes of L2 (El+1 ) and L2 (El−1 ) in the formal representation ring of Gq . 2 We note that for the above computation there is no need to pass to cyclic cohomology or twisted cyclic cohomology. In order to proceed we need a generalization of the Drinfeld double. The relative Drinfeld ˆ q ) is defined as the double crossed product [4] of C(T ) and C ∗ (Gq ) using the double D(T , G matching m(x) = ZxZ ∗ where Z = (π ⊗ id)(WGq ) and π : C(Gq ) → C(T ) is the quotient ˆ q )) = C(T ) ⊗ C ∗ (Gq ) with the comultiplication map. That is, we have C0r (D(T , G ˆ D(T ,Gˆ q ) = (id ⊗ σ ⊗ id)(id ⊗ m ⊗ id)( ⊗ ).

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ˆ q ) is a cocompact closed quantum subgroup of D(Gq ), and The relative Drinfeld double D(T , G ˆ q )) is isomorphic to C(Gq /T ). Under this the quantum homogeneous space C r (D(Gq )/D(T , G identification, the natural D(Gq )-algebra structure on the homogeneous space corresponds to G

the YD-algebra structure on the induced algebra C(Gq /T ) = indT q (C) obtained from Proposition 3.4. Every continuous coaction of C(T ) on a C ∗ -algebra B restricts to a continuous coacˆ q )) in a natural way, and we write resT (B) for the resulting tion of C0r (Dq ) = C0r (D(T , G Dq ˆ q )-C ∗ -algebra. Indeed, since C(T ) is commutative, the canonical ∗-homomorphism D(T , G ˆ q ))) is compatible with the comultiplications. The following result is C(T ) → M(C0r (D(T , G a variant of the dual Green–Julg theorem, see Theorem 4.6. ˆ q ) be the relative Drinfeld double of Gq . Then there is a natural Lemma 6.4. Let Dq = D(T , G isomorphism KK Dq A, resTDq (B) ∼ = KK T C(Gq )cop A, B for all Dq -C ∗ -algebras A and all T -C ∗ -algebras B. Proof. If A is a Dq -C ∗ -algebra then the crossed product C(Gq )cop A becomes a T -C ∗ -algebra using the adjoint action on C(Gq ) and the restriction of the given coaction on A. The natural map ιA : A → C(Gq )cop A is T -equivariant and C ∗ (Gq )-colinear with respect to the coaction on the crossed product induced by the corepresentation Wˆ G . Assume that (E, φ, F ) is a Dq -equivariant Kasparov A-resTDq (B)-module which is essential in the sense that the ∗-homomorphism φ : A → L(E) is nondegenerate. The coaction of C0r (Dq ) on E is determined by a coaction of C(T ) and a unitary corepresentation of C ∗ (Gq ). Together with φ, this corepresentation corresponds to a nondegenerate ∗-homomorphism ψ : C(Gq )cop A → L(E) which yields a T -equivariant Kasparov C(Gq )cop A-B-module (E, ψ, F ). Conversely, assume that (E, ψ, F ) is an essential T -equivariant Kasparov C(Gq )cop A-B-module. Then ψ is determined by a covariant pair consisting of a nondegenerate ∗-homomorphism φ : A → L(E) and a unitary corepresentation of C ∗ (Gq ) on E. In combination with the given C(T )-coaction, this corepresentation determines a coaction of C0r (Dq ) on E such that (E, φ, F ) is a Dq -equivariant Kasparov module. The assertion follows easily from these observations. 2 Before we proceed we need some further facts about the structure of q-deformations. Note that C(Gq ) can be viewed as a T × T -C ∗ -algebra with the action given by left and right translations. The C ∗ -algebras C(Gq ) assemble into a T × T -equivariant continuous field G = (C(Gq ))q∈(0,1] of C ∗ -algebras, compare [8,33]. In particular, the algebra C0 (G) of C0 -sections of the field is a T × T -C ∗ -algebra in a natural way. We can also associate equivariant continuous fields to certain braided tensor products. For instance, the braided tensor products C(Gq ) Gq C(Gq ) yield a continuous field of C ∗ -algebras over (0, 1] whose section algebra we denote by C0 (G) G C0 (G). This is easily seen using that C(Gq ) Gq C(Gq ) ∼ = C(Gq ) ⊗ C(Gq ) as C ∗ -algebras and the fact that C(Gq ) is nuclear for all q ∈ (0, 1]. A similar argument works for the quantum flag manifolds C(Gq /T ) instead of C(Gq ).


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As a consequence of Lemma 6.4 we obtain in particular that the Dirac operator on Gq /T determines an element in KK T (C(Gq /T ) Gq C(Gq /T ) Gq C(Gq ), C) since we have C(Gq )cop A ∼ = A Gq C(Gq ) for every Gq -YD-algebra A by definition of the braided tensor product. In fact, these elements depend in a continuous way on the deformation parameter. More precisely, if we fix q ∈ (0, 1] then the proof of Proposition 6.2 shows that the Dirac operators on Gt /T for different values of t ∈ [q, 1] yield an element [D] ∈ KK T C(G/T ) G C(G/T ) G C(G), C[q, 1] where C(G/T ) G C(G/T ) G C(G) denotes the algebra of sections of the continuous field over [q, 1] with fibers C(Gt /T ) Gt C(Gt /T ) Gt C(Gt ). Similarly, writing C(G/T ) for the algebra of sections of the continuous field over [q, 1] given by the Podle´s spheres, the induced vector bundle Ek determines a class in KK T (C(G/T ), C(G/T )). Composition of this class with the canonical homomorphism C[q, 1] → C(G/T ) yields an element in KK T (C[q, 1], C(G/T )). After these preparations we prove the following main result. Theorem 6.5. The Podle´s sphere C(Gq /T ) is Gq -equivariantly Poincaré dual to itself. That is, there is a natural isomorphism D(Gq )

KK ∗

D(Gq ) C(Gq /T ) Gq A, B ∼ A, C(Gq /T ) Gq B = KK ∗

for all Gq -YD-algebras A and B. Proof. According to Proposition 6.2 the Dirac operator on Gq /T yields an element [Dq ] ∈ D(G )

D(G )

KK 0 q (C(Gq /T ) Gq C(Gq /T ), C). Let us define a dual element ηq in KK 0 q (C, C(Gq /T ) Gq C(Gq /T )) by ηq = [E−1 ] [E0 ] − [E0 ] [E1 ] where we write for the exterior product obtained in Theorem 4.10. In order to show that ηq and [Dq ] are the unit and counit of the desired adjunction we have to study the endomorphisms (id ηq ) ◦ ([Dq ] id) and (ηq id) ◦ (id [Dq ]) of C(Gq /T ) in KK D(Gq ) . First we consider the classical case q = 1. Since all Cr∗ (G1 )-coactions in the construction of [D1 ] and η1 are trivial it suffices to work with the above morphisms at the level of KK G1 . Due to Proposition 4.7 the counit : C(Gq /T ) → C induces an isomorphism 1 T ∼ KK G ∗ (C(G1 /T ), C(G1 /T )) = KK ∗ (C(G1 /T ), C). Hence, according to the universal coefficient theorem for T -equivariant KK-theory [40], in order to identify (id η1 ) ◦ ([D1 ] id) we only have to compute the action of (id η1 ) ◦ ([D1 ] id) ◦ on K∗T (C(G1 /T )). Using Proposition 6.3 we obtain [E0 ] ◦ (id η1 ) ◦ [D1 ] id ◦ = [E0 ] ◦ (id η1 ) ◦ (id id ) ◦ [D1 ] = [E0 ] [E−1 ] ◦ [D1 ] [C0 ] − [E0 ] [E0 ] ◦ [D1 ] [C1 ] = [C0 ]

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in K0T (C) = R(T ). Similarly one checks [E1 ] ◦ (id η1 ) ◦ ([D1 ] id) ◦ = [C1 ]. This implies (id η1 ) ◦ ([D1 ] id) = id since [E0 ] and [E1 ] generate K∗T (G1 /T ) ∼ = R(T ) ⊗R(G1 ) R(T ) due to McLeod’s theorem [28]. In a similar way one shows (η1 id) ◦ (id [D1 ]) = id. As already indicated above, we conclude that these identities hold at the level of KK D(G1 ) as well. For general q ∈ (0, 1] we observe that the Drinfeld double D(Gq ) is coamenable and recall ˆ q ) ⊂ D(Gq ) is a cocompact quantum subgroup. According to Proposition 4.7 this that D(T , G implies D(Gq )

KK ∗

ˆ q ) D(T ,G C(Gq /T ), C(Gq /T ) ∼ C(Gq /T ), C = KK ∗

D(Gq ) (C). Moreover, due to Lemma 6.4 we have since C(Gq /T ) ∼ = ind ˆ D(T ,Gq )

ˆ q ) D(T ,G

KK ∗

C(Gq /T ), C ∼ = KK T∗ C(Gq /T ) Gq C(Gq ), C

using C(Gq )cop C(Gq /T ) ∼ = C(Gq /T ) Gq C(Gq ). Recall that T acts by conjugation on the copy of C(Gq ). The element in KK T∗ (C(Gq /T ) Gq C(Gq ), C) corresponding to (id ηq ) ◦ ([Dq ] id) is given by δq = (id ηq id) ◦ [Dq ] id id ◦ ( id) ◦ . We observe that the individual elements in this composition assemble into KK T -classes for the corresponding continuous fields over [q, 1]. Let us denote by cq ∈ E0T (C(G1 /T ) G1 C(G1 ), C(Gq /T ) Gq C(Gq )) the E-theoretic comparison element for the field C(G/T ) G C(G) over [q, 1]. Using again the universal coefficient theorem for T -equivariant KK-theory we obtain a commutative diagram C(G1 /T ) G1 C(G1 )

cq

C(Gq /T ) Gq C(Gq ) δq

δ1

C

id

C

in KK T where cq is an isomorphism. Moreover, due to our previous considerations in the case q = 1 we have δ1 = ( id) ◦ . This implies δq = ( id) ◦ and hence (id ηq ) ◦ ([Dq ] id) = id in KK D(Gq ) . In a similar way one obtains (ηq id) ◦ (id [Dq ]) = id in KK D(Gq ) . According to the characterization of adjoint functors in terms of unit and counit this yields the assertion. 2 As a corollary we determine the equivariant K-homology of the Podle´s sphere. Corollary 6.6. For the standard Podle´s sphere C(Gq /T ) we have G KK 0 q C(Gq /T ), C ∼ = R(Gq ) ⊕ R(Gq ),

G KK 1 q C(Gq /T ), C = 0.

Let us also discuss the following result which is closely related to Theorem 6.5.


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Theorem 6.7. The standard Podle´s sphere C(Gq /T ) is a direct summand of C ⊕ C in KK D(Gq ) . D(Gq )

Proof. Let us consider the elements αq ∈ KK 0 C, C(Gq /T )) given by αq = [D] ⊕ [D ⊗ E−1 ],

D(G)

(C(Gq /T ), C ⊕ C) and βq ∈ KK 0

(C ⊕

βq = −[E1 ] ⊕ [E0 ],

respectively. Following the proof of Theorem 6.5 we shall show αq ◦ βq = id. Consider first the case q = 1. All C ∗ (G1 )-coactions in the construction of α1 and β1 1 are trivial, and it suffices to check α1 ◦ β1 = id in KK G 0 (C(G1 /T ), C(G1 /T )). Using G1 KK ∗ (C(G1 /T ), C(G1 /T )) ∼ = KK T∗ (C(G1 /T ), C) and the universal coefficient theorem for KK T we only have to compare the corresponding actions on K∗T (C(G1 /T )). One obtains [E0 ] ◦ α1 ◦ β1 ◦ = [E0 ] ◦ [D ⊗ E−1 ] ◦ [E0 ] ◦ − [E0 ] ◦ [D] ◦ [E1 ] ◦ = [E0 ] ◦ [D ⊗ E−1 ] = [C0 ] in R(T ) due to Proposition 6.3, and similarly [E1 ] ◦ α1 ◦ β1 ◦ = [C1 ]. Taking into account McLeod’s theorem [28] this yields the assertion for q = 1. For general q ∈ (0, 1] we recall D(Gq )

KK ∗

C(Gq /T ), C(Gq /T ) ∼ = KK T∗ C(Gq /T ) Gq C(Gq ), C

and notice that the elements corresponding to αt ◦ βt for t ∈ [q, 1] assemble into a class in KK T (C(G/T ) G C(G), C[q, 1]). The comparison argument in the proof of Theorem 6.5 carries over and yields αq ◦ βq = id in KK D(Gq ) . 2 On the level of Gq -equivariant Kasparov theory one can strengthen the assertion of Theorem 6.7 as follows. Proposition 6.8. The standard Podle´s sphere C(Gq /T ) is isomorphic to C ⊕ C in KK Gq . Proof. We have already seen that the elements αq and βq defined in Theorem 6.7 satisfy αq ◦ βq = id in KK D(Gq ) , hence this relation holds in KK Gq as well. Using Proposition 6.3 one immediately calculates βq ◦ αq = id in KK Gq . 2 References [1] S. Baaj, G. Skandalis, C ∗ -algèbres de Hopf et théorie de Kasparov équivariante, K-Theory 2 (1989) 683–721. [2] S. Baaj, G. Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés des C ∗ -algèbres, Ann. Sci. École Norm. Sup. 26 (1993) 425–488. [3] S. Baaj, G. Skandalis, S. Vaes, Non-semi-regular quantum groups coming from number theory, Comm. Math. Phys. 235 (2003) 139–167. [4] S. Baaj, S. Vaes, Double crossed products of locally compact quantum groups, J. Inst. Math. Jussieu 4 (2005) 135–173. [5] T. Banica, Fusion rules for representations of compact quantum groups, Expo. Math. 17 (1999) 313–337. [6] E. Bédos, L. Tuset, Amenability and co-amenability for locally compact quantum groups, Internat. J. Math. 14 (2003) 865–884. [7] B. Blackadar, K-Theory for Operator Algebras, second edition, Cambridge University Press, Cambridge, 1998.

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[8] E. Blanchard, Déformations de C ∗ -algèbres de Hopf, Bull. Soc. Math. France 124 (1996) 141–215. [9] R. Bott, The index theorem for homogeneous differential operators, in: Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, 1965, pp. 167–186. [10] P.S. Chakraborty, A. Pal, Equivariant spectral triples on the quantum SU(2) group, K-Theory 28 (2003) 107–126. [11] A. Connes, Noncommutative Geometry, Academic Press, 1994. [12] A. Connes, Noncommutative geometry and reality, J. Math. Phys. 36 (1995) 6194–6231. [13] A. Connes, Gravity coupled with matter and the foundation of non-commutative geometry, Comm. Math. Phys. 182 (1996) 155–176. [14] A. Connes, On the spectral characterization of manifolds, arXiv:math.OA/0810.2088, 2008. [15] J. Cuntz, A new look at KK-theory, K-Theory 1 (1987) 31–51. [16] L. Dabrowski, ˛ Geometry of quantum spheres, J. Geom. Phys. 56 (2006) 86–107. [17] L. Dabrowski, ˛ F. D’Andrea, G. Landi, E. Wagner, Dirac operators on all Podle´s quantum spheres, J. Noncommut. Geom. 1 (2007) 213–239. [18] L. Dabrowski, ˛ G. Landi, A. Sitarz, W. van Suijlekom, J. Várilly, The Dirac operator on SU q (2), Comm. Math. Phys. 259 (2005) 729–759. [19] L. Dabrowski, ˛ A. Sitarz, Dirac operator on the standard Podle´s quantum sphere, in: Noncommutative Geometry and Quantum Groups, Warsaw, 2001, in: Banach Center Publ., vol. 61, Polish Acad. Sci., Warsaw, 2003, pp. 49–58. [20] S. Echterhoff, S. Kaliszewski, J. Quigg, I. Raeburn, A categorical approach to imprimitivity theorems for C ∗ dynamical systems, Mem. Amer. Math. Soc. 180 (2006). [21] R. Fischer, Volle verschränkte Produkte für Quantengruppen und äquivariante KK-Theorie, PhD thesis, Münster, 2003. [22] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201. [23] A. Klimyk, K. Schmuedgen, Quantum Groups and Their Representations, Texts Monogr. Phys., Springer, Berlin, 1997. [24] J. Kustermans, Locally compact quantum groups in the universal setting, Internat. J. Math. 12 (2001) 289–338. [25] J. Kustermans, S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup. 33 (2000) 837–934. [26] J. Kustermans, S. Vaes, Locally compact quantum groups in the von Neumann algebraic setting, Math. Scand. 92 (2003) 68–92. [27] S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995. [28] J. McLeod, The Künneth formula in equivariant K-theory, in: Algebraic Topology, Proc. Conf., Univ. Waterloo, Waterloo, ON, 1978, in: Lecture Notes in Math., vol. 741, 1978, pp. 316–333. [29] R. Meyer, Equivariant Kasparov theory and generalized homomorphisms, K-Theory 21 (2000) 201–228. [30] R. Meyer, R. Nest, The Baum–Connes conjecture via localisation of categories, Topology 45 (2006) 209–259. [31] E.F. Müller, H.-J. Schneider, Quantum homogeneous spaces with faithfully flat module structures, Israel J. Math. 111 (1999) 157–190. [32] G. Nagy, On the Haar measure of the quantum SU(N ) group, Comm. Math. Phys. 153 (1993) 217–228. [33] G. Nagy, Deformation quantization and K-theory, in: Perspectives on Quantization, South Hadley, MA, 1996, in: Contemp. Math., vol. 214, 1998, pp. 111–134. [34] A. Neeman, Triangulated Categories, Ann. of Math. Stud., vol. 148, Princeton Univ. Press, 2001. [35] S. Neshveyev, L. Tuset, Hopf algebra equivariant cyclic cohomology, K-theory and index formulas, K-Theory 31 (2004) 357–378. [36] S. Neshveyev, L. Tuset, The Dirac operator on compact quantum groups, arXiv:math.OA/0703161, 2007. [37] P. Podle´s, Quantum spheres, Lett. Math. Phys. 14 (1987) 193–202. [38] P. Podle´s, S.L. Woronowicz, Quantum deformation of Lorentz group, Comm. Math. Phys. 130 (1990) 381–431. [39] R. Popescu, Coactions of Hopf-C ∗ -algebras and equivariant E-theory, arXiv:math.KT/0410023, 2004. [40] J. Rosenberg, C. Schochet, The Künneth theorem and the universal coefficient theorem for equivariant K-theory and KK-theory, Mem. Amer. Math. Soc. 62 (1986). [41] P. Schauenburg, Hopf–Galois and bi-Galois extensions, in: Galois Theory, Hopf Algebras, and Semiabelian Categories, in: Fields Inst. Commun., vol. 43, 2004, pp. 469–515. [42] A. Sitarz, Equivariant spectral triples, in: Noncommutative Geometry and Quantum Groups, Warsaw, 2001, in: Banach Center Publ., vol. 61, Polish Acad. Sci., Warsaw, 2003, pp. 231–263. [43] A. Sitarz, Twisted Dirac operators over quantum spheres, J. Math. Phys. 49 (2008) 033509. [44] S. Vaes, A new approach to induction and imprimitivity results, J. Funct. Anal. 229 (2005) 317–374. [45] S. Vaes, L. Vainerman, On low-dimensional locally compact quantum groups, in: Locally Compact Quantum Groups and Groupoids, Strasbourg, 2002, in: IRMA Lect. Math. Theor. Phys., vol. 2, de Gruyter, Berlin, 2003, pp. 127–187. [46] R. Vergnioux, KK-théorie équivariante et opérateur de Julg–Valette pour les groupes quantiques, PhD thesis, Paris, 2002.


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Fredholm composition operators on algebras of analytic functions on Banach spaces P. Galindo a,∗,1 , T.W. Gamelin b , M. Lindström c,2 a Departamento de Análisis Matemático, Universidad de Valencia, 46.100, Burjasot, Valencia, Spain b Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA c Department of Mathematical Sciences, PO Box 3000, FIN-90014, University of Oulu, Finland

Received 22 June 2009; accepted 23 October 2009 Available online 5 November 2009 Communicated by N. Kalton

Abstract We prove that Fredholm composition operators acting on the uniform algebra H ∞ (BE ) of bounded analytic functions on the open unit ball of a complex Banach space E with the approximation property are invertible and arise from analytic automorphisms of the ball. © 2009 Elsevier Inc. All rights reserved. Keywords: Composition operator; Fredholm operator; Bounded analytic function

1. Introduction Let E denote a complex Banach space with open unit ball BE , and let ϕ : BE → BE be a nonconstant analytic map. In this paper, we consider composition operators Cϕ defined by Cϕ (f ) = f ◦ ϕ, acting on the uniform algebra H ∞ (BE ) of bounded analytic functions on BE endowed with the norm f = sup{|f (x)|: x ∈ BE }, f ∈ H ∞ (BE ). Evidently Cϕ is bounded, and Cϕ = 1 = Cϕ (1). * Corresponding author.

E-mail addresses: [email protected] (P. Galindo), [email protected] (T.W. Gamelin), [email protected] (M. Lindström). 1 Supported by Project 2007-64521 (MTM-FEDER, Spain). 2 Partially supported by Project 2007-64521 (MTM-FEDER, Spain). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.020

P. Galindo et al. / Journal of Functional Analysis 258 (2010) 1504–1512

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Recall that an operator T on a Banach space is Fredholm if both the dimension of its kernel and the codimension of its image are finite. This occurs if and only if T is invertible modulo compact operators, that is, there is a bounded operator S such that both T S − I and ST − I are compact. Our goal is to prove the following. Theorem 1.1. Let E be a Banach space with the approximation property. If Cϕ : H ∞ (BE ) → H ∞ (BE ) is a Fredholm operator, then in fact Cϕ is invertible, and ϕ is an analytic automorphism of BE . Fredholm operators have their origins at the roots of functional analysis, where they arose in connection with integral equations. Since the paper [7] by Cima, Thomson and Wogen, where they proved that the Fredholm composition operators on H 2 are invertible, many authors have studied Fredholm composition operators acting on Banach spaces of analytic functions of a finite number of variables (see for example [15] and [14] and references therein). A typical result is that a composition operator is Fredholm only if its symbol is an invertible analytic function. In the finite-dimensional case, two main ingredients of the proofs are local compactness and the inverse function theorem. In the infinite-dimensional setting there is no local compactness and the validity of the inverse function theorem is still an open question. The proof of our main result requires some function theory for analytic functions on infinite-dimensional spaces including the use of linear interpolating sequences for the space at hand, together with a Banach space version of a type of inverse function theorem proved for Hilbert space by Cima and Wogen [6]. It is also useful to realize the existence of a predual of H ∞ (BE ) and to connect it with the composition operator. The paper is organized as follows. In Section 2 we give some background on the function theory we use, and we obtain a result on weak-star continuous homomorphisms of H ∞ (BE ); it yields a characterization of those homomorphisms on H ∞ (BE ) that are composition operators. In Section 3 we treat the bicontinuity of the symbol, and in Section 4 we treat the bianalyticity of the symbol. In Section 5 we give an application to weighted composition operators. For background on analytic maps on Banach spaces, we refer the reader to [5,8,13] and for background on Fredholm operators we refer to [17]. Throughout this paper all appearing homomorphisms are unital. For a given uniform algebra, its spectrum is the set of all complex-valued homomorphisms defined on it. 2. Some preliminaries on H ∞ (BE ) The algebra H ∞ (BE ) is a weak-star closed subalgebra of ∞ (BE ), hence is the dual of the quotient Banach space 1 (BE )/H ∞ (BE )⊥ . (See [13, p. 196].) A bounded net {fα } in H ∞ (BE ) converges weak-star if and only if it converges pointwise on BE , and this occurs if and only if it converges uniformly on compact subsets of BE . Evidently the evaluation functionals δx at points x ∈ BE are weak-star continuous. We denote by i : E ∗ → H ∞ (BE ) the natural embedding map, and for a fixed z ∈ BE , we denote by dz : H ∞ (BE ) → E ∗ the projection operator dz f = f (z), where f (z) is the linear term of the Taylor series of f at z. Thus dz ◦ i is the identity operator on E ∗ . Lemma 2.1. The inclusion operatori : E ∗ → H ∞ (BE ) is the adjoint of the map j : 1 (BE )/ ∞ ⊥ H (BE ) → E defined by j (x) ¯ = ak xk , where x¯ = ak δxk + H ∞ (BE )⊥ .

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Proof. If ak δxk ∈ H ∞ (BE )⊥ , then ak xk = 0, so the operator j is well defined, and j is ¯ L = ak L(xk ) = x, ¯ i(L). It follows that j is bounded and linear. If L ∈ E ∗ , then j (x), j ∗ = i. 2 Lemma 2.2. For fixed z ∈ BE , the projection operator dz f = f (z) of H ∞ (BE ) onto E ∗ is the adjoint of an operator ez : E → 1 (BE )/H ∞ (BE )⊥ . Further, j ◦ ez is the identity operator on E. Proof. It suffices to show that dz is continuous from the weak-star topology of H ∞ (BE ) to the weak-star topology of E ∗ . In view of the Krein–Šmulian theorem (see [9]), it suffices to check the weak-star continuity on bounded subsets of H ∞ (BE ). For this, consider the expression d g(z + λx)|λ=0 . If {fα } is a bounded net in H ∞ (BE ) that converges weak-star to f , (dz g)(x) = dλ then {fα } converges uniformly to f on small λ-disks centered at z, so that the λ-derivatives also converge. This establishes weak-star continuity. Since ez∗ ◦j ∗ = dz ◦i = IE ∗ , also j ◦ez = IE . 2 We return to the analytic self-map ϕ of BE . Evidently Cϕ is continuous with respect to the weak-star topology of H ∞ (BE ), so Cϕ is an adjoint operator. We denote the pre-adjoint operator for Cϕ on 1 (BE )/H ∞ (BE )⊥ by C ϕ . One checks that C ϕ δz + H ∞ (BE )⊥ = δϕ(z) + H ∞ (BE )⊥ ,

z ∈ BE .

We will use the following identity. Lemma 2.3. The composition operator Cϕ and its pre-adjoint C ϕ satisfy dz ◦ Cϕ = ϕ (z)∗ dϕ(z) ,

C ϕ ◦ ez = eϕ(z) ◦ ϕ (z)x.

Proof. For fixed f ∈ H ∞ (BE ) and x ∈ E, the pairing between E ∗ and E satisfies (dz ◦ Cϕ )f, x = dz (f ◦ ϕ), x = dϕ(z) f, ϕ (z)x = ϕ (z)∗ dϕ(z) f, x, where we have used the chain rule for the second equality. This yields the first identity, and the second follows by duality. 2 The hypothesis in the main theorem that E has the approximation property enters the picture only in the proof of the following theorem. We do not know whether the hypothesis is necessary. Theorem 2.4. If E has approximation property, then the only weak-star continuous complexvalued homomorphisms of H ∞ (BE ) are the evaluation functionals at points of BE . Proof. Let u be a weak-star continuous homomorphism of H ∞ (BE ). Let y = u|E ∗ = u ◦ i. Since i is weak-star to weak-star continuous, it turns out that y ∈ E. We claim that y ∈ BE and u = δy . Note that u is also continuous on bounded subsets of H ∞ (BE ) for the compact open topology, τ0 . An application of the Banach–Dieudonné theorem for the space P (m E) of continuous m-homogeneous polynomials [16, Theorem 2.1], leads to the (P (m E), τ0 )-continuity of u. Then u(P ) = P (y) for all finite-type polynomials and hence for all m-homogeneous polynomials because of the approximation property. Choose g ∈ E ∗ such that g = 1 and g(y) = y, and consider the τ0 -null sequence {g n } in the unit ball of H ∞ (BE ). From u(g n ) = u(g)n → 0 it follows that |u(g)| < 1, and therefore y < 1. If f ∈ H ∞ (BE ), the sequence of the Cesàro means (σm f ) of its Taylor series forms a bounded set in H ∞ (BE ) that τ0 -converges to f . Thus u(f ) = lim u(σm f ) = lim σm f (y) = f (y). 2


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Clearly every composition operator is a weak-star to weak-star continuous homomorphism. As a byproduct of the above theorem the converse also holds when the underlying space has the approximation property. Corollary 2.5. Let T : H ∞ (BF ) → H ∞ (BE ) be a weak-star to weak-star continuous homomorphism, where F is a complex Banach space. If F has the approximation property, then there is ϕ : BE → BF analytic such that T = Cϕ . Proof. For each x ∈ BE , the homomorphism δx ◦ T is weak-star continuous. Hence by Theorem 2.4, there is a point, say ϕ(x) ∈ BF such that δx ◦ T = δϕ(x) . We claim that the mapping ϕ : BE → BF is analytic. Firstly, we remark that ϕ(x) − ϕ(y) δϕ(x) − δϕ(y) T δx − δy and then we use the Schwarz type inequality proved in [4], δx − δy 2

x − y 1 − x

to conclude that ϕ is a continuous mapping. Now it is clear that T (f ) = Cϕ (f ). This shows that ϕ is weakly analytic, hence, analytic, and that T = Cϕ . 2 3. Bicontinuity of symbols of Fredholm composition operators Lemma 3.1. If Cϕ : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then ϕ −1 (K) is strictly inside BE for any compact set K ⊂ BE . Proof. Otherwise, there is a compact set K ⊂ BE for which ϕ −1 (K) is not strictly inside BE . In other words, there is a sequence {xn } such that xn → 1 and ϕ(xn ) ∈ K. According to Corollary 7 in [10] (see also Theorem 2.5 in [11]), we can upon passing to a subsequence find functions Fk ∈ H ∞ (BE ) and M > 0 such that Fk (xn ) = 0 if n = k, Fk (xk ) = 1, and |Fk | < M on BE . This latter condition implies that {Fk } converges weakly to zero in H ∞ (BE ). Since Cϕ is Fredholm, there are operators S and Q on H ∞ (BE ), Q compact, such that Cϕ S − I = Q. Then Cϕ (S(Fk )) − Fk = Q(Fk ), and |S(Fk )(ϕ(xk )) − 1| = |Q(Fk )(xk )| Q(Fk ). Since Q is compact and Fk converges weakly to 0, we have Q(Fk ) → 0, and S(Fk )(ϕ(xk )) → 1. However, S(Fk ) → 0 weakly in H ∞ (BE ), which clearly implies that S(Fk ) → 0 uniformly on compact subsets of BE . This leads to |S(Fk )(ϕ(xk ))| → 0, because {ϕ(xk )} lies in the compact set K of BE . Thus we have a contradiction. 2 Lemma 3.2. If Cϕ : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then ϕ −1 (K) is compact in BE for any compact set K ⊂ BE , that is, ϕ is a proper map. In particular, ϕ is a closed map. Proof. Let {xn } be a sequence in BE such that ϕ(xn ) ∈ K. We must show that {xn } has a subsequence that converges to a point x ∈ BE . Passing to a subsequence, we may assume that ϕ(xn ) → w0 ∈ BE . Now an operator is Fredholm if and only if its adjoint is Fredholm (see [17]). Consequently the adjoint map Cϕ∗ : H ∞ (BE )∗ → H ∞ (BE )∗ is Fredholm, and there are operators S and Q on H ∞ (BE )∗ , Q compact, such that SCϕ∗ − I = Q. Since {δxn } is bounded in

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H ∞ (BE )∗ , we may pass to a further subsequence and assume that Q(δxn ) converges in the norm of H ∞ (BE )∗ . Since the map x ∈ BE → δx ∈ H ∞ (BE )∗ is continuous, Cϕ∗ (δxn ) = δϕ(xn ) → δw0 in H ∞ (BE )∗ . Therefore δxn = S(Cϕ∗ (δxn )) − Q(δxn ) converges in the norm of H ∞ (BE )∗ . Since xn − xm δxn − δxm , {xn } is a Cauchy sequence in E, and xn → x for some x ∈ E. Since ϕ(xn ) ∈ K, the preceding lemma shows that the xn ’s lie strictly inside BE , so that xn → x ∈ BE . 2 In the proof of Theorem 3.4 we will use the following result, which follows easily from Theorem 9.8 of [12]. Since the result is not stated formally in [12], and there is a misprint in the proof, we condense the proof given in [12] to give a direct proof for the case at hand. Lemma 3.3. Let A be a uniform algebra, and let B be a subalgebra of A of finite codimension in A. Then the spectrum MB of B is obtained from the spectrum MA of A by identifying a finite number of pairs of points or MB = MA . Proof. We can assume that B is a maximal proper subalgebra of A. Let J be the ideal in A consisting of all f ∈ B such that f A ⊆ B. By the proof of Lemma 9.6 of [12], J is a maximal ideal in B, so that J is the kernel of a complex-valued homomorphism φJ ∈ MB . Choose g ∈ A such that g ∈ / B, and let P (λ) be the minimal polynomial of g over J , so that P (g) ∈ J . Adding a constant to g and P , we can assume P (0) = 0. Write P (λ) = λQ(λ), and define h = Q(g). Then h ∈ A and gh ∈ J . By the definition of J , also g 2 h ∈ J , g 3 h ∈ J , etc., and so h2 = Q(g)h ∈ J . Thus the linear span of J , h, and 1 is a subalgebra of A. We claim that h ∈ / B. Indeed, if h ∈ B, and Q0 (λ) = Q(λ) − φJ (h), then Q0 (g) = h − φJ (h) ∈ J , contradicting the fact that P is the minimal polynomial of g. Since B is maximal, the linear span of J , h, and 1 is A, and B has codimension one in A. Thus the minimal polynomial P of g has degree two, say P (λ) = λ(λ − λ0 ). Each φ ∈ MB , φ = φJ , extends to a unique homomorphism φ˜ ∈ MA . Indeed, choose h ∈ J ˜ ) = φ(hf )/φ(h), f ∈ A. One checks that that satisfies φ(h) = 0, and define φ˜ on f ∈ A by φ(f this definition is independent of the h chosen, and that it is multiplicative, so that φ˜ ∈ MA , and φ˜ extends φ. Since any extension φ˜ satisfies this identity, the extension is unique. If P (λ) = λ2 , one checks directly that the functional defined by φ˜ J (a + bg + k) = a for k ∈ J is a unique extension to A of φJ . In this case, MB = MA . If P (λ) = λ(λ − λ0 ) where λ0 = 0, one checks directly that there are two extensions ψ, ψ0 of φJ to A, determined by ψ(g) = 0 and ψ0 (g) = λ0 . In this case, MB is obtained from MA by identifying the two homomorphisms ψ, ψ0 ∈ MA . 2 Theorem 3.4. Suppose the only weak-star continuous complex-valued homomorphisms of H ∞ (BE ) are the evaluation functionals at points of BE . If Cϕ : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then ϕ maps BE bicontinuously onto BE . Proof. It suffices to show that ϕ is onto and one-to-one. Since ϕ is closed, it is then bicontinuous. We begin by proving that ϕ is onto. Suppose that Cϕ (f ) = 0. Then Cϕ (gf ) = Cϕ (g)Cϕ (f ) = 0 for all g ∈ H ∞ (BE ). Since the kernel of Cϕ is finite-dimensional, we conclude that f = 0. Thus the null space of Cϕ is {0}. Since Cϕ is a Fredholm operator, Cϕ maps H ∞ (BE ) bicontinuously onto a closed subspace of finite codimension in H ∞ (BE ). Let A denote the range of Cϕ . Since Cϕ is multiplicative, A is


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a closed subalgebra of H ∞ (BE ) of finite codimension. Since the kernel of Cϕ is {0}, Cϕ is an algebra isomorphism, hence an isometric isomorphism of H ∞ (BE ) onto A. We denote the spectrum of H ∞ (BE ) by M and the spectrum of A by MA . Since A has finite codimension in H ∞ (BE ), it follows from Lemma 3.3 that the spectrum MA of A is obtained from M by identifying a finite number of pairs of points. We denote by x¯ the point of MA corresponding to x ∈ M. Let ϕˆ be the natural extension of ϕ to a self-map of M, defined by fˆ ϕ(x) ˆ = (C ϕ f )(x),

x ∈ M, f ∈ H ∞ (BE ).

Since Cϕ is an isometric isomorphism of H ∞ (BE ) onto A, the adjoint of Cϕ induces a homeomorphism ψ from MA to M, which satisfies ¯ fˆ ψ(x) ¯ = (C ϕ f )(x),

x¯ ∈ MA , f ∈ H ∞ (BE ).

Comparing these identities, we see that ψ(x) ¯ = ϕ(x) ˆ for x ∈ M. Thus ϕˆ identifies the pairs of points of M that are identified by A, and otherwise ϕˆ is one-to-one. Since ψ maps MA onto M, ϕˆ maps M onto M. Since the weak-star topology on bounded subsets of H ∞ (BE ) is the topology of pointwise convergence, and since Cϕ is a weak-star continuous isometry, it is easy to check that the closed unit ball of A is a weak-star closed subset of H ∞ (BE ), thus A is weak-star closed in H ∞ (BE ) by the Krein–Šmulian theorem. Therefore A is also a dual space, with predual 1 (BE )/A⊥ . Since A⊥ is the direct sum of H ∞ (BE )⊥ and a finite-dimensional space of weak-star continuous linear functionals on H ∞ (BE ), the weak-star topology of A is the restriction to A of the weak-star topology of H ∞ (BE ). The operator Cϕ , regarded as an operator from H ∞ (BE ) to A, is the adjoint of the operator C∗ given by δz + A⊥ → δϕ(z) + H ∞ (BE )⊥ . The operator C∗ is also an isometric isomorphism, and hence Cϕ is bicontinuous with respect to the weak-star topologies of H ∞ (BE ) and A. Under the map ψ , weak-star homomorphisms of A then correspond to weakstar continuous homomorphisms of H ∞ (BE ), that is, x¯ ∈ MA is weak-star continuous if and only if ψ(x) ¯ ∈ M is weak-star continuous. Now x¯ is weak-star continuous on A if and only if x is weak-star continuous on H ∞ (BE ). From our hypothesis, we conclude that x ∈ BE if and only if ϕ(x) ˆ ∈ BE . Since ϕˆ maps M onto M, we conclude that ϕ maps BE onto BE , and ϕ is onto. Since MA is obtained from M by identifying a finite number of pairs of points, in particular ϕ identifies only finitely many pairs of (distinct) points. Fix w0 ∈ BE , and suppose that ϕ −1 (w0 ) = {z1 , . . . , zm }. We may choose a small open neighborhood U of w0 such that ϕ −1 (w) consists of exactly one point for each w ∈ U \{w0 }. Choose δ > 0 such that the closed balls B δ (zj ), j = 1, . . . , m, are pairwise disjoint and contained in ϕ −1 (U ). Since ϕ is closed, the sets ϕ(B δ (zj )) are closed, and the sets ϕ(B δ (zj )\{zj }), j = 1, . . . , m, are pairwise disjoint. Next we claim that there is ε > 0 such that Bε (w0 ) ⊂

ϕ Bδ (zj ) . j

Otherwise, since ϕ is onto, there is a sequence {xk } in BE such that xk ∈ / j Bδ (zj ) and ϕ(xk ) → w0 . Since ϕ is proper, we can pass to a subsequence and assume that xk → x ∈ BE .

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Then ϕ(x) = lim ϕ(xk ) = w0 , and x is one of the zj ’s. However, x ∈ / j Bδ (zj ). This contradiction establishes our claim. Now the sets ϕ(B δ (zj )) ∩ Bε (w0 )\{w0 }, j = 1, . . . , m, decompose the set Bε (w0 )\{w0 } into a finite number of disjoint relatively closed subsets. Since Bε (w0 )\{w0 } is connected, there can be only one set in this decomposition. It follows that m = 1, and w0 has only one preimage. Since w0 ∈ BE is arbitrary, ϕ is one-to-one. 2 4. Bianalyticity of symbols of Fredholm composition operators Let E and F be Banach spaces. The operator T : E → F is upper semi-Fredholm if the dimension of its kernel Ker(T ) is finite and its image Im(T ) is closed. The operator T is lower semi-Fredholm if its image has finite codimension. The index ind(T ) of a (either upper or lower) semi-Fredholm operator T is the dimension of its null space minus the codimension of its image, which might be ±∞. An operator is semi-Fredholm if and only if its adjoint is semi-Fredholm. An operator T is upper semi-Fredholm with complemented image if and only if there is an operator R : F → E such that RT − IE is compact. Similarly, an operator T is lower semiFredholm with complemented kernel if and only if there is an operator S : F → E such that T S − IF is compact. (See [17].) In order to deduce the bianalyticity of the symbol from the bicontinuity, we will use the following version of the inverse function theorem, which was proved in [6] for Hilbert spaces. Our proof follows its pattern. Theorem 4.1. Let E and F be complex Banach spaces, let Ω be an open subset of E, and let a ∈ Ω, b ∈ F . Let ϕ : Ω → F be an analytic mapping such that ϕ(a) = b and such that ϕ (a) : E → F is a semi-Fredholm operator with complemented kernel and image. (i) If ind(ϕ (a)) 0 and ϕ (a) is not invertible, then ϕ is not injective in any neighborhood of a. (ii) If ind(ϕ (a)) < 0, then the ϕ-image of any sufficiently small open neighborhood of a in E is not open in F . Proof. Let p : E → Ker(ϕ (a)) be a projection of E onto Ker(ϕ (a)), and let M be the kernel of p. Let N be a complementary subspace to Im(ϕ (a)). Then E = Ker(T ) ⊕ M and F = N ⊕ Im(ϕ (a)). The matrix representation of ϕ (a) with regard to these decompositions is C=

0 0 0 T

,

where T = ϕ (a)|M : M → Im(ϕ (a)) is invertible. Set P = p ◦ ϕ and Q = q ◦ ϕ. Then dPa = p ◦ T = 0, and dQa = q ◦ T . We will apply the implicit function theorem to Q = q ◦ ϕ : Ω ⊂ Ker(T ) ⊕ M → Im(T ) at the point a ≡ (a1 , a2 ), with ϕ(a) ≡ (b1 , b2 ). Note that Q(a1 , a2 ) = b2 , and D2 Q(a1 , a2 ) = T is invertible. Hence there is an open neighborhood U1 of a1 in Ker(T ), an open neighborhood U2 of a2 in M, and an analytic mapping φ from U1 onto U2 such that for each x ∈ U1 , φ(x) is the unique point y ∈ U2 satisfying Q(x, y) = b2 . Further, φ (x) = −[(D2 Q)−1 ◦ D1 Q](x, φ(x)), so that φ (a1 ) = 0. Let q : F → N be the projection of F onto N whose kernel is Im(ϕ (a)), and define Φ : U1 → N by Φ(x) = q(ϕ(x, φ(x))). Then Φ is analytic, and Φ (a1 ) = 0.


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Now suppose that ind(ϕ (a)) = 0. In this case, both Ker(ϕ (a)) and N have the same nonzero finite-dimension. Since Φ (a1 ) = 0, the analytic function Φ cannot be one-to-one in any neighborhood of a1 (use the inverse function theorem for a finite number of complex variables). Hence there are distinct points x1 , x2 in U1 such that Φ(x1 ) = Φ(x2 ). Then ϕ(x1 , φ(x1 )) = ϕ(x2 , φ(x2 )), and ϕ is not injective in the open neighborhood U1 ⊕ U2 of a in E. If ind(ϕ (a)) > 0, we can take a subspace L of Ker(ϕ (a)) of the same finite-dimension as N and apply the preceding case to the restriction of ϕ to L ⊕ M. This establishes (i). Suppose next that ind(ϕ (a)) < 0. Since dim Ker(T ) < dim N , Φ(U1 ) is nowhere dense in N , and Φ(U1 )⊕b2 is nowhere dense in N ⊕b2 . Now note that for x ∈ U1 , y ∈ U2 , we have ϕ(x, y) ∈ N ⊕ b2 if and only if y = φ(x) is the unique point in U2 satisfying q(ϕ(x, y)) = b2 , which occurs if and only if ϕ(x, y) = Φ(x) ⊕ b2 . Thus the ϕ-image of U1 ⊕ U2 meets N ⊕ b2 in a nowhere dense subset Φ(U1 ) ⊕ b2 of N ⊕ b2 , and ϕ(U1 , U2 ) is not an open subset of F , from which (ii) follows. 2 Now we return to our composition operator Cϕ on H ∞ (BE ). Lemma 4.2. If Cϕ : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then for all z ∈ BE ϕ (z) : E → E is upper semi-Fredholm with complemented image. Proof. As noted previously, the pre-adjoint operator C ϕ on 1 (BE )/H ∞ (BE )⊥ is also a Fredholm operator. Thus there is an operator S on 1 (BE )/H ∞ (BE )⊥ such that SC ϕ − I is compact. Recall (Section 2) the notation for the pre-adjoints j for i and ez for dz . Premultiplying by j , postmultiplying by ez , and using j ◦ ez = IE , we see that j ◦ S ◦ C ϕ ◦ ez − IE is compact. We substitute the identity C ϕ ◦ ez = eϕ(z) ◦ ϕ (z) from Lemma 2.3 to conclude that j ◦ S ◦ eϕ(z) ◦ ϕ (z) − IE is compact. Consequently ϕ (z) has a left inverse modulo compact operators on E, and ϕ (z) is upper semi-Fredholm with complemented image. 2 Proof of Theorem 1.1. The preceding lemma guarantees that ϕ (z) is upper semi-Fredholm with complemented image, and we may apply the inverse function theorem, Theorem 4.1. According to Theorem 3.4, ϕ is a bicontinuous mapping, so we conclude from Theorem 4.1 that ϕ (z) is invertible for all z ∈ BE . This is enough to prove that ϕ −1 is holomorphic (see for instance [5, p. 63]). 2 5. Application to weighted composition operators on H ∞ (BE ) We apply our main result to weighted composition operators. For fixed g ∈ H ∞ (BE ), we denote by Mg the multiplication operator f → gf on H ∞ (BE ). The line of argument of Axler in [2] leads to the following result. Theorem 5.1. Let E be a Banach space of dimension at least two, and let g ∈ H ∞ (BE ). If the multiplication operator Mg : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then 1/g ∈ H ∞ (BE ), and Mg is invertible. Proof. If g(a) = 0, then the evaluation functional at a annihilates the image of Mg . Since the image has finite codimension, there can be at most finitely many points at which g vanishes. Since dim E > 1, g has no isolated zeros, and consequently g does not vanish on BE . Thus 1/g is analytic on BE . We must show that 1/g is bounded.

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Suppose 1/g is not bounded. Then there is a sequence {xn } in BE such that xn → 1 and g(xn ) → 0, say |g(xn )| 1/n. Passing to a subsequence, we can assume that {xn } is an interpolating sequence for H ∞ (BE ) (see [1,10]). For each N 1, choose a function gN ∈ H ∞ (BE ) such that gN (xn ) = 0 for 1 n < N , gN (xn ) = g(xn ) for n N , and gN C/n, where C is the interpolation constant for the interpolating sequence {xn }. Since g − gN vanishes on the tail of the interpolating sequence, the image of Mg−gN has infinite codimension in H ∞ (BE ), and Mg−gN is not Fredholm. Since Mg−gN − Mg gN → 0, and since the set of Fredholm operators is open, Mg cannot be Fredholm. This contradiction establishes the theorem. 2 The assumption dim E > 1 cannot be avoided, as it is pointed out by the case E = C and g(z) = z. The multiplication operator Mg is Fredholm since it is one-to-one and its image is the set of functions in H ∞ that vanish at 0. A characterization of Fredholm multiplication operators on H ∞ was obtained in [3]. Theorem 5.2. Let E be a Banach space of dimension at least two, and suppose E has the approximation property. Let g ∈ H ∞ (BE ). If Mg ◦ Cϕ : H ∞ (BE ) → H ∞ (BE ) is Fredholm, then it is invertible, and both Mg and Cϕ are invertible operators. Proof. The image of Mg includes the image of Mg ◦ Cϕ , hence has finite codimension. It follows that g has at most finitely many zeros, hence no zeros. Thus the kernel of Mg is {0}, and Mg is Fredholm. By the preceding theorem, Mg is invertible. It follows that Cϕ is Fredholm, hence by Theorem 1.1 Cϕ is invertible. 2 References [1] R.M. Aron, B. Cole, T.W. Gamelin, Spectra of algebras of analytic functions on a Banach space, J. Reine Angew. Math. 415 (1991) 51–93. [2] S. Axler, Multiplication operators on Bergman spaces, J. Reine Angew. Math. 336 (1982) 26–44. [3] J. Bonet, P. Domanski, M. Lindström, Pointwise multiplication operators on weighted Banach spaces of analytic functions, Studia Math. 137 (2) (1999) 177–194. [4] T.K. Carne, B. Cole, T.W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (2) (1989) 639–659. [5] S.B. Chae, Holomorphy and Calculus in Normed Spaces, Marcel Dekker, 1985. [6] J. Cima, W. Wogen, On biholomorphy in infinite dimensions, J. Geom. Anal. 18 (2008) 740–745. [7] J. Cima, J. Thomson, W. Wogen, On some properties of composition operators, Indiana Univ. Math. J. 24 (1974) 215–220. [8] S. Dineen, Complex Analysis on Infinite Dimensional Spaces, Springer, 1999. [9] N. Dunford, J. Schwartz, Linear Operators, vol. I, Interscience, 1958. [10] P. Galindo, A. Miralles, Interpolating sequences for bounded analytic functions, Proc. Amer. Math. Soc. 135 (10) (2007) 3225–3231. [11] P. Galindo, T. Gamelin, M. Lindström, Composition operators on uniform algebras and the pseudohyperbolic metric, J. Korean Math. Soc. 41 (2004) 1–20. [12] T. Gamelin, Embedding Riemann surfaces in maximal ideal spaces, J. Funct. Anal. 2 (1968) 123–146. [13] T. Gamelin, Analytic functions on Banach spaces, in: Complex Potential Theory, Kluwer Academic Publishers, Netherlands, 1994, pp. 187–233. [14] O. Hatori, Fredholm composition operators on spaces of holomorphic functions, Integral Equations Operator Theory 18 (1994) 202–210. [15] B. MacCluer, Fredholm composition operators, Proc. Amer. Math. Soc. 125 (1997) 163–166. [16] J. Mujica, Complex homomorphisms on the algebras of holomorphic functions on Fréchet spaces, Math. Ann. 241 (1979) 73–82. [17] V. Müller, Spectral Theory of Linear Operators, Birkhäuser, 2003.


Free holomorphic functions on the unit ball of B(H)n, II ✩ Gelu Popescu Department of Mathematics, The University of Texas at San Antonio, San Antonio, TX 78249, USA Received 23 June 2009; accepted 21 October 2009 Available online 31 October 2009 Communicated by D. Voiculescu

Abstract In this paper we continue the study of free holomorphic functions on the noncommutative ball

1/2 0, 1/2 B(H)n γ := (X1 , . . . , Xn ) ∈ B(H)n : X1 Xn∗ + · · · + Xn Xn∗ < γ , as the set of all power series α∈F+n aα Zα with radius of convergence γ , i.e., {aα }α∈F+n are complex numbers with lim supk→∞ ( |α|=k |aα |2 )1/2k γ1 . A free holomorphic function on [B(H)n ]γ is the representation of an element F ∈ Hballγ on the Hilbert space H, that is, the mapping

G. Popescu / Journal of Functional Analysis 258 (2010) 1513–1578

B(H)n

γ

(X1 , . . . , Xn ) → F (X1 , . . . , Xn ) :=

∞

1515

aα Xα ∈ B(H),

k=0 |α|=k

where the convergence is in the operator norm topology. Due to the fact that a free holomorphic function is uniquely determined by its representation on an infinite-dimensional Hilbert space, we identify, throughout this paper, a free holomorphic function with its representation on a separable infinite-dimensional Hilbert space. We recall that a free holomorphic function F on [B(H)n ]1 is bounded if F ∞ := supF (X) < ∞, where the supremum is taken over all X ∈ [B(H)n ]1 and H is an infinite∞ be the set of all bounded free holomorphic functions and let dimensional Hilbert space. Let Hball Aball be the set of all elements F such that the mapping B(H)n 1 (X1 , . . . , Xn ) → F (X1 , . . . , Xn ) ∈ B(H) ∞ has a continuous extension to the closed unit ball [B(H)n ]− 1 . We showed in [37] that Hball and Aball are Banach algebras under pointwise multiplication and the norm · ∞ , which can be identified with the noncommutative analytic Toeplitz algebra Fn∞ and the noncommutative disc algebra An , respectively. In Section 1, we present new results concerning the composition of free holomorphic functions with operator-valued coefficients and the behavior of their model boundary functions, which will play an important role throughout this paper. Fractional maps of the operatorial unit ball [B(E, G)]− 1 are due to Siegel [50] and Phillips [28] (see also [55]). We should mention that the noncommutative ball [B(H)n ]1 can be identified with the open unit ball of B(Hn , H), which is one of the infinite-dimensional Cartan domains studied by L. Harris [18]. He has obtained several results, related to our topic, in the setting of J B ∗ algebras. We also remark that the group of all free holomorphic automorphisms of [B(H)n ]1 (see [43]), can be identified with a subgroup of the group of automorphisms of [B(Hn , H)]1 considered by R.S. Phillips [28] (see also [55]). Following these ideas, fractional transforms of free holomorphic functions were recently considered in [19,41,45]. In Section 1, we continue to investigate these transforms and work out several of their properties. A fractional transform ΨA is associated with each strict contraction A = I ⊗ A0 , A0 ∈ B(E, G). We show that ΨA : Sball (B(E, G)) → Sball (B(E, G)) defined by

−1 ΨA [F ] := A − DA∗ I − FA∗ F DA is a homeomorphism of the noncommutative Schur class Sball (B(E, G)) of all free holomorphic functions F on [B(H)n ]1 with coefficients in B(E, G) such that F ∞ 1. Among other properties, we prove that F is inner if and only if its fractional transform ΨA [F ] is inner, and that the is in An ⊗ ¯ min B(E, G) if and only if Ψ ¯ min B(E, G). model boundary function F A [F ] is in An ⊗ We mention that the noncommutative Schur class Sball (B(E, G)) was introduced in [32] in connection with a noncommutative von Neumann inequality for row contractions. This class was extended to more general settings by Ball, Groenewald and Malakorn (see [4,5]), and by Muhly and Solel (see [24–26]). The Muhly–Solel paper [26] gives an intrinsic characterization for the Schur class Sball (B(E, G)) in terms of completely positive kernels, and presents a description of the automorphism group of their Hardy algebra H ∞ (E), which has some overlap with [43] and Theorem 1.3 of the present paper.

1516


Using fractional transforms and a noncommutative version of Schwarz’s lemma [37], we prove a maximum principle for free holomorphic functions with operator-valued coefficients (see [43] for the scalar case). On the other hand, using fractional transforms, the noncommutative Cayley transforms of [39], and [42], we obtain results concerning the geometric structure of bounded free holomorphic functions. More precisely, we prove that a map F : [B(H)n ]1 → ¯ min B(E) is a bounded free holomorphic function such that F ∞ 1 and F (0) < 1, B(H) ⊗ if and only if there exist a strict contraction A0 ∈ B(E), an n-tuple of isometries (V1 , . . . , Vn ) on a Hilbert space K, with orthogonal ranges, and an isometry W : E → K, such that F = (ΨI ⊗A0 ◦ C)(G), where C is the noncommutative Cayley transform and G is defined by

−1 G(X1 , . . . , Xn ) := I ⊗ W ∗ 2 I − X1 ⊗ V1∗ − · · · − Xn ⊗ Vn∗ − I (I ⊗ W ) for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 . In particular, in the scalar case, we obtain a characterization and parametrization of all bounded free holomorphic functions on the unit ball [B(H)n ]1 . We mention that, for the noncommutative polydisc, a representation theorem of the same flavor was obtained in [22] and [1]. In Section 2, we provide a Vitali type convergence theorem [20] for uniformly bounded sequences of free holomorphic functions with operator-valued coefficients. As a consequence, we show that two free holomorphic functions F , G coincide if and only if there exists a sen (k) quence {A(k) }∞ k=1 ⊂ [B(H) ]1 of bounded-bellow operators such that limk→∞ A = 0 and F (A(k) ) = G(A(k) ) for any k = 1, 2, . . . . In Section 3, we introduce the class of free holomorphic functions with the radial infimum property. A function F is in this class if lim inf inf F (rS1 , . . . , rSn )x = F ∞ , r→1

x=1

where S1 , . . . , Sn are the left creation operators on the full Fock space F 2 (Hn ) with n generators. We obtain several characterizations for this class of functions and consider several examples. We is in the noncommutative disc algebra An show that if F is inner and its boundary function F then F has the radial infimum property. In particular, any free holomorphic automorphism of [B(H)n ]1 has the property. We study the radial infimum property in connection with products, direct sums, and compositions of free holomorphic functions. We also show that the class of functions with the radial infimum property is invariant under the fractional transforms of Section 1. These results are important in the following sections. It is well known that if f ∈ H ∞ (D), a bounded analytic function on the open unit disc D := {z ∈ C: |z| < 1}, is such that f ∞ 1 and f (z) = θ (z)g(z),

z ∈ D,

where θ is an inner function in the disc algebra A(D) and g is analytic in D, then g∞ 1. If, in addition, f ∈ A(D), then g ∈ A(D). Moreover, if f ∈ A(D) is inner, then so is g. These facts are fundamental for the theory of bounded analytic functions (see [8,15]).


1517

In Section 4, we obtain analogues of these results in the context of free holomorphic functions. Let F , Θ, and G be free holomorphic functions on [B(H)n ]1 such that F (X) = Θ(X)G(X),

X ∈ B(H)n 1 .

Assume that F is bounded with F ∞ 1 and Θ has the radial infimum property with Θ∞ = 1. Then we prove that G∞ 1 and F (X)F (X)∗ Θ(X)Θ(X)∗ ,

X ∈ B(H)n 1 .

and Θ are in the noncommutative disc algeMoreover, we show that if the boundary functions F bra, then so is G. When we add the condition that F is inner, then we deduce that G is also inner. In particular, if F is a bounded free holomorphic function with F ∞ 1 and representation F (X) =

∞

Xα ⊗ A(α) ,

k=m |α|=k

X = (X1 , . . . , Xn ) ∈ B(H)n 1 ,

for some m = 1, 2, . . . , and A(α) ∈ B(E, G), then F (X)F (X)∗

Xβ Xβ∗ ⊗ IG ,

|β|=m

X ∈ B(H)n 1 .

Consequently, we recover the corresponding version of Schwarz’s lemma from [37] and, when m = 1, the one from [19]. The classical Schwarz’s lemma (see [9,48]) states that if f : D → C is a bounded analytic function with f (0) = 0 and |f (z)| 1 for z ∈ D, then |f (0)| 1 and |f (z)| |z| for z ∈ D. Moreover, if |f (0)| = 1 or if |f (z)| = |z| for some z = 0, then there is a constant c with |c| = 1 such that f (w) = cw for any w ∈ D. A faithful generalization of this result is obtained (see Theorem 4.5) when f, θ , and g are free holomorphic functions on [B(H)n ]1 with scalar coefficients such that: (i) f (X) = θ (X)g(X), X ∈ [B(H)n ]1 ; (ii) f is bounded with f ∞ 1; (iii) θ has the radial infimum property and θ ∞ = 1. In the particular case when n = 1 and θ (z) = z, we recover the Schwarz’s lemma. We remark that Schwarz’s lemma has been extended to various settings by several authors (e.g. [13,17,21, 25,28,43,44,47,50]). In Section 4, we also obtain noncommutative extensions of Harnack’s double inequality (see Theorem 4.9) for a class of free holomorphic functions F = I + ΘΓ with positive real parts. In the particular case when Θ(X) = X, we deduce that if F is a free holomorphic function on [B(H)n ]1 with coefficients in B(E) such that F (0) = I and F 0, then 1 + X 1 − X F (X) , 1 + X 1 − X

X ∈ B(H)n 1 .

1518


The Borel–Carathéodory theorem [52] establishes an upper bound for the modulus of a function on the circle |z| = r from bounds for its real (or imaginary) parts on larger circles |z| = R. More precisely, if f is an analytic function for |z| R and 0 < r < R, then sup f (z)

|z|=r

2r R + r sup f (z) + f (0) . R − r |z|=R R−r

In Section 5, we obtain an analogue of this result for free holomorphic functions (see Theorem 5.4). We also obtain a Borel–Carathéodory type result for free holomorphic functions which admit factorizations F = ΘΓ , where Θ is an inner function with the radial infimum property and Θ(0) < 1. We show that if F I then F (X) 2Θ(X) , 1 − Θ(X)

X ∈ B(H)n 1 .

Let f : D → D be a nonconstant analytic function and let z0 ∈ D and w0 = f (z0 ). Pick’s theorem [29] (see also [8]) asserts that z0 − z w0 − f (z) = g(z), 1 − w¯ 0 f (z) 1 − z¯ 0 z

z ∈ D,

for some analytic function g : D → D. In Section 6, we provide a generalization of Pick’s theorem, for bounded free holomorphic functions. We show that if F : [B(H)n ]1 → [B(H)m ]− 1 is a free holomorphic function with F (0) < 1 and a ∈ Bn , then there exists a free holomorphic function Γ with Γ ∞ 1 such that

ΦF (a) F (X) = Φa (X)(Γ ◦ Φa )(X),

X ∈ B(H)n 1 ,

where Φa and ΦF (a) are the corresponding free holomorphic automorphisms of the noncommutative balls [B(H)n ]1 and [B(H)m ]1 , respectively. Consequently,

∗ ΦF (a) F (X) ΦF (a) F (X) Φa (X)Φa (X)∗ ,

X ∈ B(H)n 1 .

We mention that the group Aut([B(H)n ]1 ) of all free holomorphic automorphisms of [B(H)n ]1 was determined in [43], using the theory of characteristic functions for row contractions [31]. We also remark that the group of all free holomorphic automorphisms of [B(H)n ]1 , can be identified with a subgroup of the group of automorphisms of [B(Hn , H)]1 considered by R.S. Phillips [28] (see also [55]). We recall that Julia’s lemma [21] (see also [7]) says that if f : D → D is an analytic function (zk )| and there is a sequence {zk } ⊂ D with zk → 1, f (zk ) → 1, and such that 1−|f 1−|zk | is bounded, then f maps each disc in D tangent to ∂D at 1 into a disc of the same kind. Julia’s lemma has been extended to analytic functions of a single operator variable by Fan [14] and to the setting of function algebras by Glicksberg [16]. Using the above-mentioned noncommutative analogue of Pick’s theorem and basic facts concerning the involutive free holomorphic automorphisms of [B(H)n ]1 , we obtain a free analogue of Julia’s lemma (see Theorem 6.3). In particular, we prove the following result.


1519

Let F : [B(H)n ]1 → [B(H)m ]1 be a free holomorphic function. Let zk ∈ Bn be such that limk→∞ zk = (1, 0, . . . , 0) ∈ ∂Bn , limk→∞ F (zk ) = (1, 0, . . . , 0) ∈ ∂Bm , and 1 − F (zk )2 = L < ∞. k→∞ 1 − zk 2 lim

If F := (F1 , . . . , Fm ), then L > 0 and

−1

−1

I − F1 (X) L I − X1∗ I − XX ∗ (I − X1 ) I − F1 (X)∗ I − F (X)F (X)∗ for any X = (X1 , . . . , Xn ) ∈ [B(H)n ]1 . Moreover, if 0 < c < 1, then F (Ec ) ⊂ Eγ ,

where γ :=

Lc 1 + Lc − c

and Ec and Eγ are certain noncommutative ellipsoids. A similar result holds if we replace the ellipsoids with some noncommutative Korany type regions [49] in the unit ball [B(H)n ]1 (see Corollary 6.5). In Section 7, we use fractional transforms and a version of the noncommutative Schwarz’s lemma to obtain Pick–Julia theorems for free holomorphic functions F with operator-valued coefficients such that F ∞ 1 (resp. F 0) (see Theorems 7.1 and 7.2). As a consequence, we obtain a Julia type lemma for free holomorphic functions with positive real parts (see Theorem 6.4). We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space (see Corollary 7.4). When n = 1, we recover (with different proofs) the corresponding results obtained by Potapov [47] and Ando and Fan [2]. In Section 8, we provide a noncommutative extension of a classical result due to Lindelöf (see [15,23]). We prove that if F : [B(H)n ]1 → [B(H)m ]− 1 is a free holomorphic function, then F (X) X + F (0) , 1 + XF (0)

X ∈ B(H)n 1 .

If, in addition, the boundary function of F has its entries in the noncommutative disc algebra An , then the inequality above holds for any X ∈ [B(H)n ]− 1 . We remark that if F (0) < 1, then the inequality above is sharper than the noncommutative von Neumann inequality (see [32,33]). In Section 9, we introduce a pseudohyperbolic metric d on [B(H)n ]1 which is invariant under the action of the free holomorphic automorphism group of [B(H)n ]1 and turns out to be a noncommutative extension of the pseudohyperbolic distance (see [56]) on Bn , the open unit ball of Cn , i.e., dn (z, w) := ψz (w)2 ,

z, w ∈ Bn ,

where ψz is the involutive automorphism of Bn that interchanges 0 and z. We show that d(X, Y ) = tanh δ(X, Y ),

X, Y ∈ B(H)n 1 ,

where δ is the hyperbolic (Poincaré–Bergman [6] type) metric on [B(H)n ]1 introduced and studied in [45]. As a consequence, we obtain a Schwarz–Pick lemma for free holomorphic functions

1520


on [B(H)n ]1 with operator-valued coefficients, with respect to the pseudohyperbolic metric. More precisely, if F = (F1 , . . . , Fm ) and Fj are free holomorphic functions with operator-valued coefficients such that F ∞ 1, then

d F (X), F (Y ) d(X, Y ),

X, Y ∈ B(H)n 1 .

It is well known (see [35,37,42]) that if F is a contractive (F ∞ 1) free holomorphic function with coefficients in B(E), then the evaluation map Bn z → F (z) ∈ B(E) is a contractive operator-valued multiplier of the Drury–Arveson space [3,12]. Moreover, any such a contractive multiplier has this kind of representation. Due to this reason, several results of the present paper have commutative versions for operator-valued multipliers of the Drury–Arveson space. It would be interesting to see if the results of this paper can be extended to more general infinite-dimensional bounded domains such as the J B ∗ -algebras of Harris [18], or the noncommutative domains from [46] and [19]. Since our results are based on the power series representation of free holomorphic functions we are inclined to believe in a positive answer for the domains considered in [46] and [19]. 1. Free holomorphic functions: fractional transforms, maximum principle, and geometric structure In this section, we present results concerning the composition and fractional transforms of free holomorphic functions, and the behavior of their model boundary functions. These results are used to prove a maximum principle and a Naimark type representation theorem for free holomorphic functions with operator-valued coefficients. Let Hn be an n-dimensional complex Hilbert space with orthonormal basis e1 , e2 , . . . , en , where n = 1, 2, . . . , or n = ∞. We consider the full Fock space of Hn defined by F 2 (Hn ) := C1 ⊕

Hn⊗k ,

k1

where Hn⊗k is the (Hilbert) tensor product of k copies of Hn . Define the left (resp. right) creation operators Si (resp. Ri ), i = 1, . . . , n, acting on F 2 (Hn ) by setting Si ϕ := ei ⊗ ϕ,

ϕ ∈ F 2 (Hn )

(resp. Ri ϕ := ϕ ⊗ ei , ϕ ∈ F 2 (Hn )). The noncommutative disc algebra An (resp. Rn ) is the norm closed algebra generated by the left (resp. right) creation operators and the identity. The noncommutative analytic Toeplitz algebra Fn∞ (resp. R∞ n ) is the weakly closed version of An (resp. Rn ). These algebras were introduced in [32] in connection with a noncommutative von Neumann type inequality [54], and have been intensively studied in recent years (see [11,24, 33–36,44], and the references therein). is We denote eα := ei1 ⊗ · · · ⊗ eik if α = gi1 · · · gik ∈ F+ n and eg0 := 1. Note that {eα }α∈F+ n 2 ∗ ∗ an orthonormal basis for F (Hn ). Let C (S1 , . . . , Sn ) be the Cuntz–Toeplitz C -algebra generated by the left creation operators (see [10]). The noncommutative Poisson transform at T := ∗ (T1 , . . . , Tn ) ∈ [B(H)n ]− 1 is the unital completely contractive linear map PT : C (S1 , . . . , Sn ) → B(H) defined by


PT [f ] := lim KT∗ ,r (IH ⊗ f )KT ,r ,

1521

f ∈ C ∗ (S1 , . . . , Sn ),

r→1

where the limit exists in the norm topology of B(H). Here, the noncommutative Poisson kernel KT ,r : H → T ,r H ⊗ F 2 (Hn ),

0 < r 1,

is defined by KT ,r h :=

∞

r |α| T ,r Tα∗ h ⊗ eα ,

h ∈ H,

k=0 |α|=k

where T ,r := (IH − r 2 T1 T1∗ − · · · − r 2 Tn Tn∗ )1/2 and T := T ,1 . We recall that PT Sα Sβ∗ = Tα Tβ∗ ,

α, β ∈ F+ n.

When T := (T1 , . . . , Tn ) is a pure row contraction, i.e., SOT- limk→∞ have PT [f ] = KT∗ (IDT ⊗ f )KT ,

∗ |α|=k Tα Tα

= 0, then we

f ∈ C ∗ (S1 , . . . , Sn ) or f ∈ Fn∞ ,

where DT := T H. We refer to [35,36,44] for more on noncommutative Poisson transforms on C ∗ -algebras generated by isometries. Let E, G be Hilbert spaces and let B(E, G) be the set of all bounded linear operators from E ¯ min B(E, G) is a free holomorphic function on [B(H)n ]γ to G. A map F : [B(H)n ]γ → B(H) ⊗ with coefficients in B(E, G) if there exist A(α) ∈ B(E, G), α ∈ F+ n , such that F (X1 , . . . , Xn ) =

∞

Xα ⊗ A(α) ,

k=0 |α|=k n where the series converges in the operator norm topology for any (X1 , . . . , Xn ) ∈ [B(H) ]γ . According to [37], a power series F := α∈F+n Zα ⊗ A(α) represents a free holomorphic function on the open operatorial n-ball of radius γ , with coefficients in B(E, G), if and only if 1 lim supk→∞ |α|=k A∗(α) A(α) 2k γ1 . This is also equivalent to the fact that the series ∞

r |α| Sα ⊗ A(α)

k=0 |α|=k

is convergent in the operator norm topology for any r ∈ [0, γ ), where S1 , . . . , Sn are the left creation operators on the Fock space F 2 (Hn ). We denote by Hball (B(E, G)) the set of all free holomorphic functions on the noncommutative ball [B(H)n ]1 and coefficients in B(E, G). Let ∞ (B(E, G)) denote the set of all elements F in H Hball ball (B(E, G)) such that F ∞ := supF (X1 , . . . , Xn ) < ∞,

1522


where the supremum is taken over all n-tuples of operators (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where H is an infinite-dimensional Hilbert space. ∞ (B(E, G)) can be identified According to [37] and [42], the noncommutative Hardy space Hball ¯ B(E, G) (the weakly closed operator space generated by the spatial to the operator space Fn∞ ⊗ tensor product), where Fn∞ is the noncommutative analytic Toeplitz algebra. More precisely, a bounded free holomorphic function F is uniquely determined by its (model) boundary function (S1 , . . . , Sn ) ∈ Fn∞ ⊗ ¯ B(E, G) defined by F = F (S1 , . . . , Sn ) := SOT- lim F (rS1 , . . . , rSn ). F r→1

(S1 , . . . , Sn ) at X := (X1 , . . . , Xn ) ∈ Moreover, F is the noncommutative Poisson transform of F n [B(H) ]1 , i.e., (S1 , . . . , Sn ) . F (X1 , . . . , Xn ) = (PX ⊗ I ) F Similar results hold for bounded free holomorphic functions on the noncommutative ball [B(H)n ]γ , γ > 0. We recall from [43] some facts concerning the composition of free holomorphic functions ¯ min B(Y)]m be a free holomorwith operator-valued coefficients. Let Φ : [B(H)n ]γ1 → [B(H) ⊗ ¯ min B(Y), phic function with Φ(X) := (Φ1 (X), . . . , Φm (X)), where Φj : [B(H)n ]γ1 → B(H) ⊗ j = 1, . . . , m, are free holomorphic functions with standard representations Φj (X) =

∞

(j )

Xα ⊗ B(α) ,

k=0 α∈F+ n ,|α|=k

X := (X1 , . . . , Xn ) ∈ B(H)n γ , 2

for some B(α) ∈ B(Y), where α ∈ F+ n , j = 1, . . . , m. Assume that (j )

Φ(X) < γ2

for any X ∈ B(H)n γ . 1

This is equivalent to Φ(rS1 , . . . , rSn ) < γ2

for any r ∈ [0, γ1 ).

¯ min B(E, G) be a free holomorphic function with standard repreLet F : [B(K)m ]γ2 → B(K) ⊗ sentation F (Y1 , . . . , Ym ) :=

∞

k=0 α∈F+ m ,|α|=k

Yα ⊗ A(α) ,

(Y1 , . . . , Ym ) ∈ B(K)m γ , 2

for some operators A(α) ∈ B(E, G), α ∈ F+ m . Then it makes sense to define the map F ◦ ¯ min B(Y) ⊗ ¯ min B(E, G) by setting Φ : [B(H)n ]γ1 → B(H) ⊗


1523

(F ◦ Φ)(X1 , . . . , Xn ) :=

∞

Φα (X1 , . . . , Xn ) ⊗ A(α) ,

k=0 α∈F+ m ,|α|=k

(X1 , . . . , Xn ) ∈ B(H)n γ , 1

where the convergence is in the operator norm topology. We proved in [43] that F ◦ Φ is a free holomorphic function on [B(H)n ]1 with standard representation (F ◦ Φ)(X1 , . . . , Xn ) =

∞

Xσ ⊗ C(σ ) ,

k=0 σ ∈F+ n ,|σ |=k

where C(σ ) x, y =

1 ∗ S ⊗ IY ⊗G (F ◦ Φ)(rS1 , . . . , rSn )(1 ⊗ x), 1 ⊗ y r |σ | σ

for any σ ∈ F+ n , x ∈ Y ⊗E, and y ∈ Y ⊗G. Actually, this is a slight extension of the corresponding result from [43]. However, the proof is basically the same. For simplicity, throughout this paper, [X1 , . . . , Xn ] denotes either the n-tuple (X1 , . . . , Xn ) ∈ B(H)n or the operator row matrix [X1 . . . Xn ] acting from H(n) , the direct sum of n copies of a Hilbert space H, to H. Now, we present new results concerning the composition of bounded free holomorphic functions with operator-valued coefficients. ¯ min B(E, G) and Φ : [B(H)n ]γ1 → [B(H) ⊗ ¯ min Theorem 1.1. Let F : [B(K)m ]γ2 → B(K) ⊗ m B(Y)] be bounded free holomorphic functions such that Φ(X) < γ2

for any X ∈ B(H)n γ . 1

Then the boundary function of the bounded free holomorphic function F ◦Φ satisfies the equation 1 , . . . , r Φ m ). F ◦ Φ = SOT- lim F (r Φ r→1

∈ An ⊗ := [Φ 1 , . . . , Φ m ] is such that Φ j ∈ An ⊗ ¯ min B(E, G) and Φ ¯ min B(Y), Moreover, if F ¯ j = 1, . . . , m, and Φ < γ2 , then F ◦ Φ ∈ An ⊗min B(Y ⊗ E, Y ⊗ G). Proof. Using the fact that a function X → G(X) is free holomorphic on [B(K)m ]γ , γ > 0, if and only if the mapping Y → G(γ Y ) is free holomorphic on [B(K)m ]1 , we can assume, without loss of generality, that γ1 = γ2 = 1. Due to [43] (see the considerations preceding this theorem), F ◦ Φ is a bounded free holomorphic function. Let F have the representation F (Y1 , . . . , Ym ) :=

∞

k=0 α∈F+ m ,|α|=k

Yα ⊗ A(α) ,

(Y1 , . . . , Ym ) ∈ B(K)m 1 .

1524


Since F is bounded on [B(K)n ]1 , we have

1/2 A(β) h

F ∞ h,

2

h ∈ E.

β∈F+ m

Given > 0 and h ∈ E, we choose q ∈ N such that

A(β) h2 < 2 .

(1.1)

β∈F+ m ,|β|q

For any x ∈ F 2 (Hn ), we have ∞

k=q β∈F+ m ,|β|=k

Φβ (rS1 , . . . , rSn ) ⊗ A(β) (x ⊗ h)

∞ I ⊗ A(β) (x ⊗ h) : Φβ (rS1 , . . . , rSn ) ⊗ I : |β| = k |β| = k k=q 1/2 ∞ I ⊗ A(β) ∞ A(β) h2 x (x ⊗ h) x : |β| = k k=q k=q β∈F+ m ,|β|=k

for any r ∈ (0, 1). Here we used the fact that [Φ1 (rS1 , . . . , rSn ), . . . , Φn (rS1 , . . . , rSn )] is a contraction and, therefore, the operator row matrix [Φβ (rS1 , . . . , rSn ) ⊗ I : |β| = k] is also a contraction. Now denote Fr (Y1 , . . . , Ym ) := F (rY1 , . . . , rYm ), 0 < r < 1, and note that Fr is a bounded := [Φ 1 , . . . , Φ m ] is a free holomorphic function on [B(K)n ]1/r . Since the boundary function Φ row contraction, we have 1 , . . . , Φ m ) = Fr (Φ

∞

k=0

α∈F+ m ,|α|=k

α ⊗ A(α) , r |α| Φ

where the convergence is in the operator norm topology. β ⊗ I : |β| = k] is a row contraction, one can show, as Using relation (1.1) and that [r |β| Φ above, that ∞

k=q β∈F+ m ,|β|=k

|β|

β ⊗ A(β) (x ⊗ h) r Φ < x

for any r ∈ (0, 1). On the other hand, we have lim

r→1

β∈F+ m ,|β| 0, there exists p ∈ N such that quently, G is in An ⊗ ∞

k=p α∈F+ m ,|α|=k

α ⊗ A(α) Φ < .

Due to the noncommutative von Neumann inequality (see [32]), we have ∞


∞ Φα (rS1 , . . . , rSn ) ⊗ A(α)


Φα ⊗ A(α)

for any k ∈ N. Consequently, we have (F ◦ Φ)(rS1 , . . . , rSn ) − G p Φ ⊗ A (rS , . . . , rS ) − Φ α 1 n α (α) + 2. k=0 α∈F+ m ,|α|=k

i ∈ An ⊗ ¯ min B(Y), i = 1, . . . , n, we have On the other hand, since Φ α , lim Φα (rS1 , . . . , rSn ) = Φ

r→1

α ∈ F+ m,

in the operator norm topology. Now, using relation (1.3), we deduce that

(1.3)

1526


F ◦ Φ := lim (F ◦ Φ)(rS1 , . . . , rSn ) = G, r→1

¯ min B(Y ⊗ E, Y ⊗ G). where the limit is in the operator norm topology. Therefore F ◦ Φ is in An ⊗ This completes the proof. 2 Using Theorem 1.1, and Theorem 4.1 from [27], we can prove the following result for bounded free holomorphic functions with operator-valued coefficients. We recall that a bounded free holomorphic function is called inner (resp. outer) if its model boundary function is an isometry (resp. has dense range). ¯ min B(E, G) and Φ : [B(H)n ]1 → [B(H)]m be Theorem 1.2. Let F : [B(K)m ]1 → B(K) ⊗ 1 is nonbounded free holomorphic functions. Assume that Φ = (Φ1 , . . . , Φm ) is inner and Φ unitary if m = 1. Then the following statements hold: (a) F ◦ Φ∞ = F ∞ ; (b) if F is inner, then F ◦ Φ is inner; (c) if F is outer, then F ◦ Φ is outer. Proof. Let Φj : [B(H)n ]1 → B(H), j = 1, . . . , m, be free holomorphic functions with scalar := [Φ 1 , . . . , Φ n ] is an isometry. coefficients and assume that Φ = [Φ1 , . . . , Φn ] is inner, i.e., Φ is a pure isometry, i.e., According to Theorem 4.1 from [27], Φ

WOT- lim

k→∞

ω Φ ω∗ = 0. Φ

ω∈F+ m ,|ω|=k

Due to the noncommutative Wold-type decomposition for sequences of isometries with orthog ], where S , . . . , S are the is unitarily equivalent to [IL ⊗ S , . . . , IL ⊗ Sm onal ranges [30], Φ m 1 1 2 left creation operators on the full Fock space F (Hm ), and L is a separable Hilbert space. Consequently, there is a unitary operator U : F 2 (Hn ) → L ⊗ F 2 (Hm ) such that

j = IL ⊗ Sj U, UΦ

j = 1, . . . , m.

Hence, if F has the representation F (Y1 , . . . , Ym ) :=

∞

(Y1 , . . . , Ym ) ∈ B(K)m 1 ,

Yα ⊗ A(α) ,

k=0 α∈F+ m ,|α|=k

we deduce that 1 , . . . , r Φ m ) = (U ⊗ I )F (r Φ

∞

α ⊗ A(α) r |α| (U ⊗ I )Φ

k=0 α∈F+ m ,|α|=k

=

∞

k=0 α∈F+ m ,|α|=k

r

|α|

IL ⊗ Sα ⊗ A(α) (U ⊗ I )

= IL ⊗ F rS1 , . . . , rSm (U ⊗ I ).


1527

Now, using Theorem 1.1, we obtain 1 , . . . , r Φ m ) (U ⊗ I )(F ◦ Φ) = (U ⊗ I ) SOT- lim F (r Φ r→1

(U ⊗ I ). = SOT- lim IL ⊗ F rS1 , . . . , rSm r→1

(1.4)

)=F , and F (rS , . . . , Since the Hilbert space L is separable, SOT- limr→1 F (rS1 , . . . , rSm 1 ) F , we also have rSm ∞

. SOT- lim IL ⊗ F rS1 , . . . , rSm = IL ⊗ F r→1

Combining the result with relation (1.4), we conclude that )(U ⊗ I ). (U ⊗ I )F ◦ Φ = (IL ⊗ F

(1.5)

Hence, we deduce that = F ∞ . F ◦ Φ∞ = F ◦ Φ∞ = F = I , then relation (1.5) implies ∗ F Now, if we assume that F is inner, i.e., F ∗ (F ◦ Φ) F ◦ Φ = I . Therefore, F ◦ Φ is inner. Finally, assume that Φ is inner and F is outer, has dense range. Using again relation (1.5), we deduce that F i.e., F ◦ Φ has dense range and, therefore, F ◦ Φ is outer. The proof is complete. 2 We recall a few well-known facts (see [28,50,55]) about fractional maps on the unit ball − B(X , Y) 1 := W ∈ B(X , Y): W 1 , where X and Y are Hilbert spaces. We denote by [B(X , Y)]1 the open ball of strict contractions. Let A, B ∈ [B(X , Y)]− 1 be such that A < 1 and define ΨA (B) ∈ B(X , Y) by setting

−1 ΨA (B) := A − DA∗ I − BA∗ BDA ,

(1.6)

where DA := (I − A∗ A)1/2 and DA∗ := (I − AA∗ )1/2 . One can show that, for any contractions A, B, C ∈ B(X , Y) with A < 1,

−1

−1 I − BC ∗ I − AC ∗ DA∗ , I − ΨA (B)ΨA (C)∗ = DA∗ I − BA∗

−1

−1 I − B ∗ C I − A∗ C D A . I − ΨA (B)∗ ΨA (C) = DA I − B ∗ A

(1.7)

Hence, we deduce that ΨA (B) 1 and ΨA (B) < 1 when B < 1. Straightforward calculations reveal that ΨA (0) = A,

ΨA (A) = 0,

and ΨA ΨA (B) = B

− for any B ∈ B(X , Y) 1 .

(1.8)

1528


− Consequently, the fractional map ΨA : [B(X , Y)]− 1 → [B(X , Y)]1 is a homeomorphism and, moreover, ΨA ([B(X , Y)]1 ) = [B(X , Y)]1 . Consider the noncommutative Schur class

∞ Sball B(E, G) := G ∈ Hball B(E, G) : G∞ 1 , ¯ B(E, G). We also use the which can be identified to the unit ball of the operator space Fn∞ ⊗ notation

0 Sball B(E, G) := G ∈ Sball B(E, G) : G(0) = 0 . Fractional transforms of free holomorphic functions were considered in [41] (see the proof of Theorem 6.1). In what follows we expand on those ideas and provide new properties. ¯ min B(E, G) be a bounded free holomorphic function Theorem 1.3. Let F : [B(H)n ]1 → B(H) ⊗ with F ∞ 1 and let F be its model boundary function. For each operator A = IH ⊗ A0 with A0 ∈ B(E, G) and A0 < 1, we define the map ¯ min B(E, G) ΨA [F ] : B(H)n 1 → B(H) ⊗ by setting

−1 ΨA [F ] := A − DA∗ I − FA∗ F DA . Then the following statements hold: (i) ΨA [F ] is a bounded free holomorphic function with ΨA [F ]∞ 1 and its boundary func), tion has the following properties: Ψ A [F ] = ΨA (F ∗

A∗ −1 I − F F ∗ I − AF ∗ −1 DA∗ , I − Ψ A [F ]Ψ A [F ] = DA∗ I − F ∗

∗ A −1 I − F ∗ F I − A∗ F −1 DA ; I − Ψ A [F ] Ψ A [F ] = DA I − F

(1.9)

(ii) for any X ∈ [B(H)n ]1 , ) = A − DA∗ I − F (X)A∗ −1 F (X)DA = ΨA F (X) , ΨA [F ](X) = (PX ⊗ I ) ΨA (F

(iii) (iv) (v) (vi)

where PX is the noncommutative Poisson at X; ΨA [0] = A, ΨA [A] = 0, and ΨA [ΨA [F ]] = F ; ΨA : Sball (B(E, G)) → Sball (B(E, G)) is a homeomorphism; is inner if and only if Ψ F A [F ] is inner; is in An ⊗ ¯ min B(E, G) if and only if Ψ ¯ min B(E, G). F A [F ] is in An ⊗


1529

Proof. To prove (i), note that FA∗ is a bounded free holomorphic function on [B(H)n ]1 and FA∗ ∞ A < 1. Since the map Y → (I − Y )−1 is a free holomorphic on [B(K)]1 , Theorem 1.1 implies that (I − FA∗ )−1 and, consequently, ΨA [F ] are bounded free holomorphic functions on [B(H)n ]1 . On the other hand, since F ∞ = sup F (rS1 , . . . , rSn ) 1 r∈[0,1)

and using the properties of the fractional transform ΨA , we deduce that

ΨA [F ](rS1 , . . . , rSn ) = ΨA F (rS1 , . . . , rSn ) 1 for any r ∈ [0, 1). Hence ΨA [F ]∞ 1. Since F is a bounded free holomorphic function, we know (see [37,42]) that the boundary function := SOT- lim F (rS1 , . . . , rSn ) F r→1

1, one can easily see exists. Taking into account that A < 1 and F (rS1 , . . . , rSn ) F that

−1

A∗ −1 = I −F SOT- lim I − F (rS1 , . . . , rSn )A∗ r→1

and, moreover,

). Ψ A [F ] = SOT- lim ΨA F (rS1 , . . . , rSn ) = ΨA (F r→1

Now, notice that relation (1.9) follows from (1.7). This proves part (i). Using the Poisson representation for bounded free holomorphic functions and the continuity of the Poisson transform in the operator norm topology, we obtain ] ΨA [F ](X) = (PX ⊗ I ) Ψ A [F ] = (PX ⊗ I ) ΨA [F −1 = A − DA∗ I − F (X)A∗ F (X)DA = ΨA F (X) for any X ∈ [B(H)n ]1 , which proves part (ii). Hence and using relation (1.8), one can deduce (iii). Now let us prove item (iv). Let F, Fm ∈ Sball (B(E, G)) and assume that Fm − F ∞ → 0 as → 0. Using the fact that m − F m → ∞, which is equivalent to F

I −F m∗ A −1 F m∗ A −1 − I − F m A∗ − F m∗ A −1 ∗ A −1 I − F A∗ I − F

A m − F , F (1 − A)2

m ) → ΨA (F ), as m → ∞. Due to (i), we have we deduce that ΨA (F m ) − ΨA (F ) = 0. lim ΨA [Fm ] − ΨA [F ]∞ = lim ΨA (F

m→∞

m→∞

1530


Moreover, since ΨA [ΨA [F ]] = F , we deduce that ΨA−1 is continuous, as well, in the uniform norm · ∞ . is in An ⊗ ¯ min Note that item (v) follows from relation (1.9). To prove (vi), we assume that F = limr→1 F (rS1 , . . . , rSn ) in the operator norm topology. Since B(E, G). Then, due to [37], F F (rS1 , . . . , rSn )A∗ F ∞ A < 1, we deduce that

−1

A∗ −1 = I −F lim I − F (rS1 , . . . , rSn )A∗

r→1

and, due to (iv),

) Ψ A [F ] = lim ΨA [F ](rS1 , . . . , rSn ) = lim ΨA F (rS1 , . . . , rSn ) = ΨA (F r→1

r→1

where the limits are in the operator norm topology. Using again [37], we conclude that Ψ A [F ] ¯ min B(E, G). The converse follows using item (iv) and the fact that ΨA [ΨA [F ]] = F . is in An ⊗ ¯ min B(E, G), then Indeed, if Ψ A [F ] is in An ⊗ lim ΨA Ψ [F ] = Ψ Ψ [F ](rS , . . . , rS ) A A A 1 n r→1

= lim ΨA ΨA F (rS1 , . . . , rSn ) r→1

, = lim F (rS1 , . . . , rSn ) = F r→1

is in An ⊗ ¯ min B(E, G). The where the limits are in the operator norm topology. Consequently, F proof is complete. 2 Note that under the conditions of Theorem 1.3, we have −1 −1 I − ΨA [F ](X)ΨA [F ](X)∗ = DA∗ I − F (X)A∗ I − F (X)F (X)∗ I − AF (X)∗ DA∗ , I − ΨA [F ](X)∗ ΨA [F ](X) −1 −1 = DA I − F (X)∗ A I − F (X)∗ F (X) I − A∗ F (X) DA

(1.10)

for any X ∈ [B(H)n ]1 . Moreover, we have (i) F (X) < 1 if and only if ΨA [F ](X) < 1; (ii) if X ∈ [B(H)n ]1 , then F (X) is an isometry (resp. co-isometry) if and only if ΨA [F ](X) has the same property. ¯ min B(E, G) is a free holomorphic funcWe recall (see [37,42]) that if F : [B(H)n ]1 → B(H) ⊗ tion with coefficients in B(E, G) and Fr (X) := F (rX) for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1/r , r ∈ (0, 1), then Fr is free holomorphic on [B(H)n ]1/r and Fr ∞ = sup F (X) = sup F (X) = F (rS1 , . . . , rSn ), Xr

X=r


1531

where S1 , . . . , Sn are the left creation operators. Moreover, the map [0, 1) r → Fr ∞ is increasing. This result can be improved for free holomorphic functions with scalar coefficients. We recall that, in [43], we proved a maximum principle for free holomorphic functions on the noncommutative ball [B(H)n ]1 , with scalar coefficients. More precisely, we showed that if f : [B(H)n ]1 → B(H) is a free holomorphic function and there exists X0 ∈ [B(H)n ]1 such that f (X0 ) f (X) for any X ∈ [B(H)n ]1 , the f must be a constant. As a consequence of this principle and the noncommutative von Neumann inequality, one can easily obtain the following. Proposition 1.4. Let f : [B(H)n ]1 → B(H) be a non-constant free holomorphic function with f ∞ 1. Then the following statements hold: (i) f (X1 , . . . , Xn ) < 1 for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 ; (ii) the map [0, 1) r → fr ∞ is strictly increasing. Proof. The first part follows immediately from the maximum principle for free holomorphic functions on the noncommutative ball [B(H)n ]1 . To prove the second part, let 0 r1 < r2 < 1. We recall that, if r ∈ [0, 1), then the boundary function fr is in the noncommutative disc algebra An and fr ∞ = fr = fr (S1 , . . . , rSn ). Applying part (i) to fr2 and (X1 , . . . , Xn ) := ( rr12 S1 , . . . , rr12 Sn ), we obtain r1 r1 < fr2 (S1 , . . . , Sn ) = fr2 ∞ , S1 , . . . , Sn fr1 ∞ = fr1 (S1 , . . . , Sn ) = fr2 r2 r2 which completes the proof.

2

Now, using fractional transforms, and the noncommutative version of Schwarz’s lemma [37], we extend the maximum principle to free holomorphic functions with operator-valued coefficients. ¯ min B(E, G) be a bounded free holomorphic function Theorem 1.5. Let F : [B(H)n ]1 → B(H) ⊗ with F (0) < F ∞ . Then there is no X0 ∈ [B(H)n ]1 such that F (X0 ) = F ∞ . Proof. Without loss of generality, we can assume that F ∞ = 1. Set A := F (0) and let G := ΨA [F ]. Due to Theorem 1.3, G is a bounded free holomorphic function with G∞ 1 and G(0) = ΨA (A) = 0. Applying the noncommutative Schwarz lemma (see [37]), we obtain

G(X) = ΨA F (X) X < 1,

X ∈ B(H)n 1 .

Using again Theorem 1.3, we have (ΨA ◦ ΨA )[F ] = F and, therefore, F (X) < 1 = F ∞ , The proof is complete.

2

X ∈ B(H)n 1 .

1532


We need to make a few remarks, which are familiar in the case n = 1 (see [51]). First, we recall (see [43]) that in the scalar case, E = G = C, if f : [B(H)n ]1 → B(H) is a free holomorphic function and f ∞ = |f (0)|, then f must be a constant. On the other hand, if F is not a scalar free holomorphic function and F ∞ = F (0), then Theorem 1.5 fails. Indeed, take E = G = C2 , and

I F (X) = 0

0 , g(X)

where g is a scalar free holomorphic function with g(X) < 1 for X = (X1 , . . . , Xn ) ∈ [B(H)n ]1 (for example, g(X) = Xα , α ∈ F+ n with |α| 1). Note that F (X) = 1 = F (0) for any X ∈ [B(H)n ]1 . We also mention that, if F ∞ = 1 andF (0)is an isometry, then F must be a constant. Indeed, if F has the representation f (X) = ∞ k=0 |α|=k Xα ⊗ A(α) , then, due to [38], we have

A∗(α) A(α) I − F (0)∗ F (0)

for k = 1, 2, . . . .

|α|=k

Hence, we deduce our assertion. Using Theorem 1.5, one can prove the following result. Since the proof is similar to that of Proposition 1.4, we shall omit it. ¯ min B(E, G) be a bounded free holomorphic function Corollary 1.6. Let F : [B(H)n ]1 → B(H) ⊗ with F ∞ 1 and F (0) < 1. Then F (X) < F ∞

for any X ∈ B(H)n 1 .

If F (0) = 0, then the map [0, 1) r → Fr ∞ is strictly increasing. We remark that, in general, under the conditions of Corollary 1.6, but without the condition F (0) = 0, the map [0, 1) r → Fr ∞ is not necessarily strictly increasing. Indeed, take 1 F (X1 , . . . , Xn ) = and note that F ∞ =

1 2

3I

0

0

1 2 X1

and F (rS1 , . . . , rSn ) =

1

3, r 2,

r ∈ [0, 23 ], r ∈ ( 23 , 1].

+ (B(E)) the set of all free holomorphic functions f on the noncommutative Denote by Hball ball [B(H)n ]1 with coefficients in B(E), where E is a separable Hilbert space, such that f 0, where

(F )(X) :=

F (X)∗ + F (X) , 2

X ∈ B(H)n 1 .


1533

∞ (B(E))]inv denote the set of all bounded free holomorphic functions on [B(H)n ] with Let [Hball 1 representation F (X1 , . . . , Xn ) = α∈F+n A(α) ⊗ Xα such that IE − A(0) is an invertible operator in B(E). According to [39], the noncommutative Cayley transform defined by

C[F ] := [F − 1][1 + F ]−1 + ∞ (B(E))]inv . In this case, we have is a bijection between Hball (B(E)) and the unit ball of [Hball

C −1 [G] = [I + G][I − G]−1 . Consider also the set

+ H+ 1 B(E) := f ∈ Hball B(E) : f (0) = I . Now, we recall that the restriction to H+ 1 (B(E)) of the noncommutative Cayley transform is a 0 (B(E)), where the noncommutative Schur class S 0 (B(E)) was (B(E)) → S bijection C : H+ 1 ball ball introduced before Theorem 1.3. Using fractional transforms, we can prove the following theorem concerning the structure of bounded free holomorphic functions. ¯ min B(E) is a bounded free holomorphic function Theorem 1.7. A map F : [B(H)n ]1 → B(H) ⊗ such that F ∞ 1 and F (0) < 1, if and only if there exist a strict contraction A0 ∈ B(E), an n-tuple of isometries (V1 , . . . , Vn ) on a Hilbert space K, with orthogonal ranges, and an isometry W : E → K, such that F = (ΨI ⊗A0 ◦ C)[G], where C is the noncommutative Cayley transform and G is defined by

−1 − I (I ⊗ W ) G(X1 , . . . , Xn ) = I ⊗ W ∗ 2 I − X1 ⊗ V1∗ − · · · − Xn ⊗ Vn∗ for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . In this case, F (0) = I ⊗ A0 . ¯ min B(E) be a bounded free holomorphic function with Proof. Let F : [B(H)n ]1 → B(H) ⊗ F ∞ 1 and F (0) < 1. Then F ∈ Sball (B(E)) and, due to Theorem 1.3, ΨF (0) [F ] ∈ 0 (B(E)). Since the noncommutative Cayley transform C : H+ (B(E)) → S 0 (B(E)) is a biSball 1 ball (B(E)). jection, we deduce that C −1 (ΨF (0) [F ]) ∈ H+ 1 According to [42], a free holomorphic function G is in H+ 1 (B(E)), i.e., G(0) = I and G 0, if and only if there exists an n-tuple of isometries (V1 , . . . , Vn ) on a Hilbert space K, with orthogonal ranges, and an isometry W : E → K such that

−1 − I (I ⊗ W ). G(X1 , . . . , Xn ) = I ⊗ W ∗ 2 I − X1 ⊗ V1∗ − · · · − Xn ⊗ Vn∗ This completes the proof.

2

We remark that, in the scalar case, i.e., E = C, due to the maximum principle for free holomorphic functions, any nonconstant free holomorphic function f such that f ∞ 1, has the

1534


property that |f (0)| < 1. Therefore, we can apply Theorem 1.7 and obtain a characterization and a parametrization of all bounded free holomorphic functions on [B(H)n ]1 . 2. Vitali convergence and identity theorem for free holomorphic functions In this section, we provide a Vitali type convergence theorem for uniformly bounded sequences of free holomorphic functions with operator-valued coefficients. Theorem 2.1. Let {Fm }∞ m=1 be a uniformly bounded sequence of free holomorphic functions on n n [B(H) ]1 with coefficients in B(E). Let {A(k) }∞ k=1 ⊂ [B(H) ]1 be a sequence of operators with the following properties: (i) A(k) is bounded below, for each k = 1, 2, . . .; (ii) limk→∞ A(k) = 0; (iii) limm→∞ Fm (A(k) ) exists in the operator norm topology, for each k = 1, 2, . . . . Then there exists a free holomorphic function F on [B(H)n ]1 with coefficients in B(E) such that Fm converges to F uniformly on any closed ball [B(H)n ]− r , r ∈ [0, 1). Proof. For each m = 1, 2, . . . , let Fm have the representation Fm (X1 , . . . , Xn ) =

∞

(m)

Xα ⊗ C(α) ,

k=0 |α|=k

where the series converges in the operator norm topology for any X = (X1 , . . . , Xn ) ∈ [B(H)n ]1 . Let M > 0 be such that Fm (X) M for any X ∈ [B(H)n ]1 and m = 1, 2, . . . . Due to the Cauchy type estimation of Theorem 2.1 from [37], we have (m) ∗ (m) 1/2 M C C (α) (α)

for any m = 1, 2, . . . , and j = 0, 1, . . . .

(2.1)

|α|=j

Since Fm (X) − Fm (0) 2M for X ∈ [B(H)n ]1 , the Schwarz type result for free holomorphic functions [37] implies (k)

(m) Fm A − IH ⊗ A 2M A(k) (0)

for any m, k = 1, 2, . . . . Hence, we deduce that (m)

A − A(q) IH ⊗ A(m) − Fm A(k) + Fm A(k) − Fq A(k) (0) (0) (0)

(m) + Fq A(k) − IH ⊗ A(0)

4M A(k) + Fm A(k) − Fq A(k) . Since limk→∞ A(k) = 0 and limm→∞ Fm (A(k) ) exists in the operator norm topology, for each (m) k = 1, 2, . . . , we deduce that C(0) := limm→∞ C(0) exists.


1535

Let Fm,0 := Fm and note that (m)

Fm,0 = I ⊗ C(0) + [X1 ⊗ IE , . . . , Xn ⊗ IE ]Fm,1 (X) where ⎤ (g ) Fm,11 (X) ⎥ ⎢ .. Fm,1 (X) = ⎣ ⎦ . ⎡

(g )

Fm,1n (X) and (g )

(m)

Fm,1i (X) = IH ⊗ C(gi ) +

j =1 |β|=j

(m)

Xβ ⊗ C(gi β) ,

i = 1, . . . , n.

By induction over q = 0, 1, 2, . . . , we can easily prove that

Xα ⊗ C(α) = Xβ ⊗ IE : |β| = q Fm,q (X),

|α|q

where ⎡

⎤ (β) Fm,q (X) ⎦ Fm,q (X) := ⎣ : |β| = q and (m)

(β) Fm,q (X) := IH ⊗ C(β) +

(m)

Xγ ⊗ C(βγ )

for |β| = q.

j =1 |γ |=j

Now, note that, for each i = 1, . . . , n, we have ⎤ (g g ) Fm,2i 1 (X) ⎥ ⎢ (g ) (m) .. Fm,1i (X) = IH ⊗ C(gi ) + [X1 ⊗ IE , . . . , Xn ⊗ IE ] ⎣ ⎦. . (gi gn ) Fm,2 (X)| ⎡

Consequently, we have (m) ⎤ IH ⊗ C(g1 )

⎥ ⎢ .. Fm,1 (X) = ⎣ ⎦ + [X1 ⊗ IE , . . . , Xn ⊗ IE ] ⊗ ICN1 Fm,2 (X), . (m) IH ⊗ C(g n)

⎡

where N1 := n. One can easily prove by induction over q = 0, 1, . . . that

(2.2)

1536


(m)

IH ⊗ C(β) Fm,q (X) = : |β| = q

+ [X1 ⊗ IE , . . . , Xn ⊗ IE ] ⊗ ICNq Fm,q+1 (X)

(2.3)

for any X ∈ [B(H)n ]1 and m = 1, 2, . . . , where Nq := card{β ∈ F+ n : |β| = q}. In what follows we prove by induction over p = 0, 1, . . . the following statements: (a) limm→∞ Fm,p (A(k) ) exists in the operator norm topology, for each k = 1, 2, . . . ; (b) Fm,p (X) (p + 1)M for any X ∈ [B(H)n ]1 and m = 1, 2, . . . ; # IH ⊗C (m) $ (β) (p + 2)MA(k) for any k, m = 1, 2, . . . ; (c) Fm,p (A(k) ) − : # (d)

|β|=p # C (m) $ C(β) $ (β) : := limm→∞ : |β|=p |β|=p

exists in the operator norm topology.

Assume that these relations hold for p = q. Using relation (2.3) when X = A(k) and taking into account (a) and (d) (when p = q), we deduce that the sequence {(A(k) ⊗ IE ⊗CNq+1 )Fm,q+1 (A(k) )}∞ m=1 is convergent in the operator norm topology and, consequently, (k) a Cauchy sequence. On the other %nhand, since A is bounded below, there exists C > 0 such that (k) A y Cy for any y ∈ i=1 H. This implies that (k)

(k)

(k)

(k) A ⊗I x − A x ⊗ I Nq+1 Fm,q+1 A Nq+1 Ft,q+1 A E ⊗C E ⊗C

C Fm,q+1 A(k) x − Ft,q+1 A(k) x for any x ∈ H ⊗ E ⊗ CNq+1 and m, t = 1, 2, . . . . Hence, we deduce that {Fm,q+1 (A(k) )}∞ m=1 is a Cauchy sequence and, therefore, limm→∞ Fm,q+1 (A(k) ) exists. Now, due to relation (2.1) and (b), we have (m) IH ⊗ C(β) : (q + 2)M, Fm,q (X) − |β| = q

X ∈ B(H)n 1 .

Using relation (2.2) and the noncommutative Schwarz lemma, we obtain (X ⊗ I

(m) IH ⊗ C(β) Fm,q (X) − : (q + 2)MX E ⊗CNq )Fm,q+1 (X) = |β| = q

for any X ∈ [B(H)n ]1 , which implies Fm,q+1 (X) (q + 2)M. Hence and using again (2.1), we obtain (m) IH ⊗ C(β) : Fm,q+1 (X) − (q + 3)M |β| = q + 1


1537

for any X ∈ [B(H)n ]1 . Once again, applying the Schwarz’s lemma for free holomorphic functions, we deduce that (m) IH ⊗ C(β) (k)

− : (q + 3)M A(k) Fm,q+1 A |β| = q + 1 for any k, m = 1, 2, . . . , which is condition (c), when p = q + 1. Now, note that (s) (m) (s) C(β) C(β) − C(β) (k)

− Fs,q+1 A + Fs,q+1 A(k) − Fm,q+1 A(k) : : |β| = q + 1 |β| = q + 1 (m) C(β) (k)

+ Fm,q+1 A − : |β| = q + 1

2(q + 3)M A(k) + Fs,q+1 A(k) − Fm,q+1 A(k) (k) for any k, s, m = 1, 2, . . . . Since {Fm,q+1 (A(k) )}∞ m=1 is a Cauchy sequence and limk→∞ A = 0, we deduce condition (d) when p = q + 1, i.e.,

(m) C(β) C(β) := lim : : m→∞ |β| = q + 1 |β| = q + 1

exists in the operator norm topology. This concludes our proof by induction. Now, due to (2.1), we deduce that 1/2 ∗ C(α) C(α) M

for any j = 0, 1, . . . ,

|α|=j

1 ∗ C 2k which implies lim supk→∞ |α|=k C(α) 1. Consequently, the mapping F (X) := (α) ∞ n j =0 |α|=j Xα ⊗ C(α) is a free holomorphic function on [B(H) ]1 with coefficients in B(E). If X r < 1, then we have ∞ p−1

(m) (m) Fm (X) − F (X) Xα ⊗ C(α) − C(α) + Xα ⊗ C(α) − C(α) j =0 |α|=j (m) C (α)

j =p |α|=j

(m) ∞ − C(α) C(α) − C(α) j r : : + |α| = j |α| = j j =p j =0 (m) p−1 C − C(α) rp (α) . : + 2M 1−r |α| = j p−1

j =0

Hence and due to relation (d), we deduce that Fm (X) − F (X) → 0, as m → ∞, uniformly for X ∈ [B(H)n ]− r , r ∈ [0, 1). The proof is complete. 2

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Corollary 2.2. Let F , G be free holomorphic functions on [B(H)n ]1 with coefficients in n B(E). If there exists a sequence {A(k) }∞ k=1 ⊂ [B(H) ]1 of bounded bellow operators such that (k) (k) (k) limk→∞ A = 0 and F (A ) = G(A ) for any k = 1, 2, . . . , then F = G. Proof. When F and G are bounded on [B(H)n ]1 , we can apply Theorem 2.1 to the sequence F, G, F, G, . . . , and deduce that F = G. Otherwise, let r ∈ (0, 1) be such that A(k) < r and consider Fr (X) := F (rX) and Gr (X) := G(rX) for X ∈ [B(H)n ]1 . Since Fr and Gr are bounded free holomorphic functions on [B(H)n ]1 and

Fr r −1 A(k) = F A(k) = G A(k) = Gr r −1 A(k) we can apply the first part of the proof and deduce that Fr = Gr . Consequently, F (rS1 , . . . , rSn ) = G(rS1 , . . . , rSn ), where S1 , . . . , Sn are the left creation operators. Hence F = G. 2 Remark 2.3. Theorem 2.1 fails if the operators A(k) are not bounded bellow. Proof. Let m = 2, 3, . . . , and consider the sequence of strict row contractions 1 1 := PPm−1 S1 |Pm−1 , . . . , PPm−1 Sn |Pm−1 , k k

(k)

A

k = 1, 2, . . . ,

where S1 , . . . , Sn are the left creation operators and Pm−1 is the subspace of F 2 (Hn ) spanned by the vectors eα , with α ∈ F+ n and |α| m − 1. Let F and G be any free holomorphic functions (k) on [B(H)n ]1 such that F (X) − G(X) = Xβ for some β ∈ F+ n with |β| = m. Since Aα = 0 for |α| m, we have

F A(k) = G A(k) , and limk→∞ A(k) = 0. However, F = G.

k 2,

2

We should mention that in the particular case when n = 1, E = C, and {A(k) } is a sequence of invertible strict contractions, we recover the corresponding results obtained by Fan [13]. 3. Free holomorphic functions with the radial infimum property We introduce the class of free holomorphic functions with the radial infimum property, obtain several characterizations, and consider several examples We study the radial infimum property in connection with products, direct sums, and compositions of free holomorphic functions. We also show that the class of functions with the radial infimum property is invariant under the fractional transforms of Section 1. These results are important in the following sections. ¯ min B(E, G) be a bounded free holomorphic function on Let F : [B(H)n ]1 → B(H) ⊗ [B(H)n ]1 . Due to [37] and [42], the model boundary function (S1 , . . . , Sn ) := SOT- lim F (rS1 , . . . , rSn ) F r→1


1539

exists, and F has the radial supremum property, i.e., lim sup F (rS1 , . . . , rSn )x = F ∞ . r→1 x=1

We introduce now the class of free holomorphic functions with the radial infimum property. We say that F has the radial infimum property if lim inf inf F (rS1 , . . . , rSn )x = F ∞ . r→1 x=1

¯ min B(E, G) is a bounded free holomorphic function Proposition 3.1. If F : [B(H)n ]1 → B(H) ⊗ with the radial infimum property such that F ∞ = 1, then F is inner. (S1 , . . . , Sn ) is an isometry. To this end, Proof. We have to show that the boundary function F denote μ(r) := inf F (rS1 , . . . , rSn )x for r ∈ [0, 1). x=1

Due to the noncommutative von Neumann inequality, we have μ(r)

F (rS1 , . . . , rSn )y F (rS1 , . . . , rSn ) F ∞ y

for any y ∈ F 2 (Hn ) ⊗ E , y = 0 and r ∈ [0, 1). Taking into account that lim infr→1 μ(r) = F ∞ , (S1 , . . . , Sn ) := SOT- limr→1 F (rS1 , we deduce that limr→1 F (rS1 , . . . , rSn )y = y. Since F . . . , rSn ), it is clear that F (S1 , . . . , Sn )y = y for any y ∈ F 2 (Hn ) ⊗ E, which Shows that F is inner and completes the proof. 2 Now, we present several characterizations for free holomorphic functions with the radial infimum property. ¯ min B(E, G) be a bounded free holomorphic function Theorem 3.2. Let F : [B(H)n ]1 → B(H) ⊗ with F ∞ = 1. Then the following statements are equivalent: (i) F has the radial infimum property. (ii) limr→1 infx=1 F (rS1 , . . . , rSn )x = 1. (iii) For every ∈ (0, 1) there is δ ∈ (0, 1) such that F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) (1 − )I

for any r ∈ (δ, 1).

(iv) There exist constants c(r) ∈ (0, 1], r ∈ (0, 1), with limr→1 c(r) = 1 such that F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) c(r)I. (v) There is δ ∈ (0, 1) such that F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) is invertible for any r ∈ (δ, 1) and −1 lim F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) = 1.

r→1

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Proof. The equivalence of (i) with (ii) is clear if one takes into account the inequality F (rS1 , . . . , rSn ) F ∞ ,

r ∈ [0, 1).

Since the equivalence of (ii) with (iii) is straightforward, we leave it the reader. To prove the implication (ii) ⇒ (iv), define μ(r) := inf F (rS1 , . . . , rSn )x , x=1

r ∈ [0, 1),

and note that 0 μ(r) F (rS1 , . . . , rSn ) F ∞ = 1 and F (rS1 , . . . , rSn )x μ(r)x

for any x ∈ F 2 (Hn ) ⊗ E.

Since the latter inequality is equivalent to F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) μ(r)2 I, if (ii) holds, then limr→1 μ(r) = 1 and (iv) follows. Conversely, assume that (iv) holds. Then we have F (rS1 , . . . , rSn )x 2 c(r)x2

for any x ∈ F 2 (Hn ) ⊗ E,

which implies c(r)1/2 inf F (rS1 , . . . , rSn )x F (rS1 , . . . , rSn ) F ∞ = 1. x=1

Since limr→1 c(r) = 1, we deduce item (ii). It remains to prove that (iv) ↔ (v). First, assume that condition (iv) holds. Note that the inequality F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) c(r)I

(3.1)

is equivalent to F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )x c(r)x

for any x ∈ F 2 (Hn ) ⊗ E.

Indeed, if (3.1) holds, then F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )x x F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )x, x c(r)x2 , which proves one implication. Conversely, if (3.2) holds, then, by squaring, we deduce that 2 F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) c(r)2 I.

(3.2)


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Hence, we obtain relation (3.1). Now, denote d(r) := inf F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )x , x=1

r ∈ (0, 1),

(3.3)

and note that 0 < c(r) d(r) 1. Hence, using (3.2) and condition (iv), we deduce that the positive operator F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) is invertible and −1 1 1 F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) = 1. c(r) d(r) Since limr→1 c(r) = 1, we obtain item (v). Conversely, assume now that condition (v) holds. 1 ,where d(r) is given by (3.3), we have Since [F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )]−1 = d(r) limr→1 d(r) = 1 On the other hand, due to (3.3), we also have F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )x d(r)x for any x ∈ F 2 (Hn ) ⊗ E, which, as proved above, is equivalent to F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) d(r)I . Since limr→1 d(r) = 1, we deduce (iv) and complete the proof. 2 Now we consider several examples of bounded free holomorphic functions with the radial infimum property. Another notation is necessary. If ω, γ ∈ F+ n , we say that ω γ if there is σ ∈ F+ n such that ω = γ σ . Example 3.3. Let A(α) ∈ B(E, G) and let F, G, ϕ be free holomorphic function on [B(H)n ]1 having the following forms: (i) F (X1 , . . . , Xn ) = |α|=m Xα ⊗ A(α) , where |α|=m A∗α Aα = IE and m ∈ N; n (ii) G(X1 , . . . , Xn ) = k=1 |β|=k,βgk Xβ ⊗ A(β) , where nk=1 |β|=k,βgk A∗(β) A(β) = I +; and g1 , . . . , gn are the generators of the free semigroup n F ∞ ∞ k (iii) ϕ(X1 , . . . , Xn ) = k=0 ak X2 X1 , where ak ∈ C with k=0 |ak |2 = 1. Then, F, G, ϕ have the radial infimum property. Proof. Since S1 , . . . , Sn satisfy the relation Sj∗ Si = δij I for i, j = 1, . . . , n, one can easily see that {Sα }|α|=m is a sequence of isometries with orthogonal ranges. Consequently, we have F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) = r m I. Applying Theorem 3.2, we deduce that F has the radial infimum property. Similarly, one can prove that G(rS1 , . . . , rSn )∗ G(rS1 , . . . , rSn ) =

n

r 2|β| A∗(β) A(β)

k=1 |β|=k,βgk

r

2n

n

k=1 |β|=k,βgk

A∗(β) A(β) = r 2n I.

1542


Consequently, G has the radial infimum property. Finally, note that {S2k S1 }∞ k=0 is a sequence of isometries with orthogonal ranges and, consequently ϕ(rS1 , . . . , rSn )∗ ϕ(rS1 , . . . , rSn ) = r 2

∞

r 2k |ak |2

k=0

∞

|ak |2 = 1.

k=1

Hence ϕ is a bounded free holomorphic function and, taking into account that 2k 2 limr→1 r 2 ∞ k=0 r |ak | = 1, Theorem 3.2 shows that ϕ has the radial infimum property. 2 Proposition 3.4. If F, G are bounded free holomorphic functions with the radial infimum property, then so is their product F G. If, in addition, F ∞ = G∞ , then F0 G0 has the radial infimum property. Proof. Without loss of generality, one can assume that F ∞ = G∞ = 1. According to Theorem 3.2, there exist constants c(r), d(r) ∈ (0, 1], r ∈ (0, 1), with limr→1 c(r) = limr→1 d(r) = 1 such that F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) c(r)I

and

∗

G(rS1 , . . . , rSn ) G(rS1 , . . . , rSn ) c(r)I. Hence, we deduce that G(rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )G(rS1 , . . . , rSn ) c(r)d(r)I. Applying again Theorem 3.2, we conclude that the product F G has the radial infimum property. To prove the second part of this proposition, note that

F (rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) 0 min c(r), d(r) I

0 G(rS1 , . . . , rSn )∗ G(rS1 , . . . , rSn )

and limr→1 min{c(r), d(r)} = 1. Applying again Theorem 3.2, we complete the proof.

2

The next result will provide several classes of free holomorphic functions with the radial infimum property. ¯ min B(E, G) be a bounded free holomorphic function, Theorem 3.5. Let F : [B(H)m ]1 → B(H) ⊗ and let ϕ : [B(H)n ]1 → [B(H)m ]1 be an inner free holomorphic function. Then the following statements hold: ∈ Am ⊗ ¯ min B(E, G), then F has the radial infimum property. (i) If F is inner and F ¯ min B(E, G), and ¯ min B(Cm , C) with ϕ = ( ϕ1 , . . . , ϕm ) is in An ⊗ ϕ1 (ii) If F is inner, F ∈ Am ⊗ non-unitary if m = 1, then the composition F ◦ ϕ has the radial infimum property. (iii) If F has the radial infimum property and ϕ is homogeneous of degree q 1, then F ◦ ϕ has the radial infimum property.


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(iv) If A := I ⊗ A0 , A0 ∈ B(E, G), with A0 < 1, then F has the radial infimum property if and only if the fractional transform ΨA [F ] has the radial infimum property. ) in the is the limit of F (rS , . . . , rSm Proof. According to [37], the model boundary function F 1 operator norm, as r → 1, where S1 , . . . , Sm are the left creation operators on the full Fock space F 2 (Hm ), with m generators. Consequently, for any ∈ (0, 1) there exists δ ∈ (0, 1) such that

F rS , . . . , rS − F < m 1


is an isometry, we deduce that, for any r ∈ (δ, 1) and x ∈ Hence, and due to the fact that F F 2 (Hm ) ⊗ E,

1/2

∗ x, x F rS1 , . . . , rSm F rS1 , . . . , rSm = F rS1 , . . . , rSm

x − F rS1 , . . . , rSm x F −F x − x = (1 − )x. Consequently,

∗ F rS1 , . . . , rSm (1 − )2 I F rS1 , . . . , rSm

for any r ∈ (δ, 1)

and, due to Theorem 3.2, F has the radial infimum property. Therefore, item (i) holds. ∈ Am ⊗ ¯ min B(E, G), To prove (ii), note first that, due to Theorem 1.2, F ◦ ϕ is inner. Since F m ¯ min B(C , C), Theorem 1.1 implies that F ¯ B(E, G). Applying now and ϕ ∈ An ⊗ ◦ ϕ is in An ⊗ item (i) to F ◦ ϕ, we deduce part (ii). Now, we prove (iii). Since ϕ := [ϕ1 , . . . , ϕm ] is homogeneous of degree m 1, we deduce that each ϕj is a homogeneous noncommutative polynomial of degree q. Therefore, ϕj = ϕj (S1 , . . . , Sn ) and ϕα (rS1 , . . . , rSn ) = r q|α| ϕα (S1 , . . . , Sn ),

α ∈ F+ m,

(3.4)

where S1 , . . . , Sn are the left creation operators on the full Fock space F 2 (Hn ). As in the proof of Theorem 1.2, we have )(U ⊗ I ), (U ⊗ I )F ◦ ϕ = (IL ⊗ F where L is a separable Hilbert space and U : F 2 (Hn ) → L ⊗ F 2 (Hm ) is a unitary operator. Hence, we deduce that r )(U ⊗ I ), (U ⊗ I )F r ◦ ϕ = (IL ⊗ F

r ∈ (0, 1),

(3.5)

where Fr (X) := F (rX), X ∈ [B(H)m ]1/r . Since Fr is a bounded free holomorphic function on [B(H)m ]1/r , we have

r = Fr S1 , . . . , Sm F

and F r ◦ ϕ = Fr ϕ1 (S1 , . . . , Sm ), . . . , ϕm (S1 , . . . , Sn ) .

(3.6)

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Since F has the radial infimum property, for any ∈ (0, 1) there is δ ∈ (0, 1) such that

∗ Fr S1 , . . . , Sm (1 − )I Fr S 1 , . . . , S m


Hence, and using relations (3.5) and (3.6), we obtain ∗ (F r ◦ ϕ) F r ◦ ϕ (1 − )I


(3.7)

On the other hand, due to relation (3.4), we have

F r ◦ ϕ = F rϕ1 (S1 , . . . , Sn ), . . . , rϕm (S1 , . . . , Sn )

= F ϕ1 r 1/q S1 , . . . , r 1/q Sn , . . . , ϕm r 1/q S1 , . . . , r 1/q Sn

= (F ◦ ϕ) r S1 , . . . , r Sn , where r := r 1/q . Now inequality (3.7) becomes

∗

(F ◦ ϕ) r S1 , . . . , r Sn (F ◦ ϕ) r S1 , . . . , r Sn (1 − )I

for any r ∈ δ 1/k , 1 .

Applying Theorem 3.2, we conclude that F ◦ ϕ has the radial infimum property. To prove item (iv), assume that F has the radial infimum property. Applying Theorem 1.3 to ΨA [Fr ], r ∈ (0, 1), we obtain ∗

I − ΨA [F ] rS ΨA [F ] rS ∗ −1 ∗ −1 I − F rS F rS I − A∗ F rS DA , = DA I − F rS A ). Since F has the radial infimum property, there exist constants where rS := (rS1 , . . . , rSm c(r) ∈ (0, 1], r ∈ (0, 1), with limr→1 c(r) = 1 such that

∗

F rS F rS c(r)I. Note also that, since A < 1 and F (rS ) 1, we have I − F rS ∗ A −1 1 + F rS ∗ A + F rS ∗ A2 + · · · 1 + A + A2 + · · · 1 . = 1 − A Using all these relations, we deduce that ∗

1 − c(r) ΨA [F ] rS ΨA [F ] rS 1 − I. (1 − A)2 Since limr→1 c(r) = 1, Theorem 3.2 shows that ΨA [F ] has the radial infimum property. Now, using the fact that ΨF [ΨA [F ]] = F , one can prove the converse. The proof is complete. 2


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∈ Am ⊗ ¯ min B(E, G) with We remark that under the hypothesis of Theorem 3.5, if F F ∞ = 1, then F is inner if and only if it has the radial infimum property. This suggests the following open question. Is there any bounded free holomorphic function with the radial infimum property so that its boundary function is not in the noncommutative disc algebra? Corollary 3.6. Any free holomorphic automorphism of the noncommutative ball [B(H)n ]1 has the radial infimum property. Proof. According to [43], if Ψ ∈ Aut([B(H)n ]1 ), the automorphism group of all free holomor = [Ψ 1 , . . . , Ψ n ] is an isometry and phic functions on [B(H)n ]1 , then its boundary function Ψ Ψi ∈ An , the noncommutative disc algebra. Applying Theorem 3.5, part (i), we deduce that Ψ has the radial infimum property. The proof is complete. 2 4. Factorizations and free holomorphic versions of classical inequalities In this section we study the class of free holomorphic functions with the radial infimum property in connection with factorizations and noncommutative generalizations of Schwarz’s lemma and Harnack’s double inequality from complex analysis. If A, B ∈ B(K) are self-adjoint operators, we say that A 0 such that (B − A)h, h γ h2 for any h ∈ K. Note that C ∈ B(K) is a strict contraction (C < 1) if and only if C ∗ C < I . Theorem 4.1. Let F, Θ, and G be free holomorphic functions on [B(H)n ]1 with coefficients in B(E, G), B(Y, G), and B(E, Y), respectively, such that F (X) = Θ(X)G(X),

X ∈ B(H)n 1 .

Assume that F is bounded with F ∞ 1 and Θ has the radial infimum property with Θ∞ = 1. Then G∞ 1, F (X)F (X)∗ Θ(X)Θ(X)∗ ,

X ∈ B(H)n 1 ,

and F (X) Θ(X),

X ∈ B(H)n 1 .

If, in addition, G(0) < 1 and X0 ∈ [B(H)n ]1 , then: (i) F (X0 )F (X0 )∗ < Θ(X0 )Θ ∗ (X0 ) if and only if Θ(X0 )Θ ∗ (X0 ) > 0; (ii) F (X0 ) < Θ(X0 ) if and only if G(X0 ) = 0. Proof. Since Θ has the radial infimum property and Θ∞ = 1, Theorem 3.2 shows that there exist constants c(r) ∈ (0, 1], r ∈ (0, 1), with limr→1 c(r) = 1 and such that Θ(rS1 , . . . , rSn )∗ Θ(rS1 , . . . , rSn ) c(r)I.

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Consequently, taking into account that F (rS1 , . . . , rSn ) = Θ(rS1 , . . . , rSn )G(rS1 , . . . , rSn ) for any r ∈ [0, 1), we deduce that Θ(rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn )y c(r)G(rS1 , . . . , rSn )y for any y ∈ F 2 (Hn ) ⊗ E . Since Θ(rS1 , . . . , rSn ) Θ∞ = 1 and F (rS1 , . . . , rSn ) 1, the inequality above implies c(r)G(rS1 , . . . , rSn ) 1 for any r ∈ [0, 1). Using the fact the map r → G(rS1 , . . . , rSn ) is increasing and that limr→1 c(r) = 1, we deduce that limr→1 G(rS1 , . . . , rSn ) 1. Hence, G is bounded and G∞ = lim G(rS1 , . . . , rSn ) 1. r→1

Consequently, G(X)G(X)∗ I and F (X)F (X)∗ = Θ(X)G(X)G(X)∗ Θ(X)∗ Θ(X)Θ(X)∗ ,

X ∈ B(H)n 1 .

Hence, we have F (X)F (X)∗ Θ(X)Θ(X)∗ for all X ∈ [B(H)n ]1 . To prove the second part of this theorem, assume that G(0) < 1. According to Corollary 1.6, we have G(X) < 1 for any X ∈ [B(H)n ]1 . Since F (X) = Θ(X)G(X), X ∈ [B(H)n ]1 , we deduce that 2

Θ(X)Θ(X)∗ − F (X)F (X)∗ 1 − G(X) Θ(X)Θ(X)∗ .

(4.1)

Since G(X) < 1, we have (1 − G(X)2 )Θ(X)Θ(X)∗ 0. Note also that if X0 ∈ [B(H)n ]1 is such that Θ(X0 )Θ(X0 )∗ > 0 then relation (4.1) implies Θ(X0 )Θ(X0 )∗ − F (X0 )F (X0 )∗ > 0. The converse is obviously true. To prove item (ii), note that when G(X0 ) < 1 and G(X0 ) = 0, we have F (X0 ) = Θ(X0 )G(X0 ) Θ(X0 )G(X0 ) < Θ(X0 ). Consequently, since F (X0 ) = Θ(X0 )G(X0 ), we deduce that F (X0 ) < Θ(X0 ) if and only if G(X0 ) = 0. This completes the proof. 2 Proposition 4.2. Let F, Θ, and G be free holomorphic functions on [B(H)n ]1 with coefficients in B(E, G), B(Y, G), and B(E, Y), respectively, such that F (X) = Θ(X)G(X), Assume that: ∈ An ⊗ ¯ min B(E, G); (i) F ∞ 1 and F ¯ min B(Y, G). (ii) Θ is inner and Θ ∈ An ⊗

X ∈ B(H)n 1 .


1547

∈ An ⊗ ¯ min B(E, Y), and Then G∞ 1, G F (X)F (X)∗ Θ(X)Θ(X)∗ ,

X ∈ B(H)n 1 .

If, in addition, F is inner, then so is G. = I , and Θ ∈ An ⊗ ∗ Θ ¯ min B(Y, G), Theorem 3.5 implies that Proof. Since Θ is inner, i.e., Θ Θ has the radial infimum property. Now, due to Theorem 4.1, G is bounded and G∞ 1. Consequently, inequality F (X)F (X)∗ Θ(X)Θ(X)∗ for all X ∈ [B(H)n ]1 . ∈ An ⊗ ∈ An ⊗ ¯ min B(E, G) and Θ ¯ min B(Y, G), according to On the other hand, since F [37,42], we have = lim Θ(rS1 , . . . , rSn ) Θ r→1

= lim F (rS1 , . . . , rSn ), and F r→1

(4.2)

= SOT- limr→1 G(rS1 , in the operator norm topology. Since G∞ 1, its boundary function G . . . , rSn ) exists. Now, for any r ∈ [0, 1), we have Θ(rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) = Θ(rS1 , . . . , rSn )∗ Θ(rS1 , . . . , rSn )G(rS1 , . . . , rSn ). Taking the SOT-limit in this equality and using the fact that Θ(rS1 , . . . , rSn ) 1 and = G. Now, due to relation (4.2), we have ∗ F F (rS1 , . . . , rSn ) 1, we deduce that Θ = lim Θ(rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) ∗ F Θ r→1

and

lim Θ(rS1 , . . . , rSn )∗ Θ(rS1 , . . . , rSn ) = I

r→1

in the operator norm. Consequently, since G − Θ(rS1 , . . . , rSn )∗ F (rS1 , . . . , rSn ) G − G(rS1 , . . . , rSn ) + Θ(rS1 , . . . , rSn )∗ Θ(rS1 , . . . , rSn ) − I G(rS1 , . . . , rSn ) = limr→1 G(rS1 , . . . , rSn ) in the operator norm and G(rS1 , . . . , rSn ) 1, we deduce that G ∈ An ⊗ ¯ min B(E, Y). The proof is complete. topology. Hence, we deduce that G ˜ implies = Θ G If in addition, F is inner, then relation F =G ∗ G, = G ∗ Θ ∗ F ∗ Θ G I =F which proves that G is inner.

2

We remark that the second part of Theorem 4.1 holds also under the hypothesis of Proposition 4.2. In [37,43,44], we obtained analogues of Schwarz’s lemma for free holomorphic functions. We mention the following. Let F (X) = α∈F+n Xα ⊗ A(α) , A(α) ∈ B(E, G), be a free holomorphic function on [B(H)n ]1 with F ∞ 1 and F (0) = 0. Then F (X) X for any X ∈ B(H)n . 1

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Note that Theorem 4.1 can be seen as a generalization of Schwarz’s lemma. Let us consider a few important particular cases. ¯ min B(E, G) is a bounded free holomorphic function Corollary 4.3. If F : [B(H)n ]1 → B(H) ⊗ with F ∞ 1 and representation F (X1 , . . . , Xn ) =

∞

Xα ⊗ A(α) ,

k=m |α|=k

where m = 1, 2, . . . , then F (X1 , . . . , Xn )F (X1 , . . . , Xn )∗

Xβ Xβ∗ ⊗ IG

|β|=m

for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 . If, in addition, equality above is strict for any X = 0.

∗ |β|=m A(β) Aβ)

< 1, then the in-

Proof. As in the proof of Theorem 3.4 from [37], we have the Gleason type factorization F = ΘG, where Θ and G are free holomorphic functions given by Θ(X1 , . . . , Xn ) := Xβ ⊗ IG : |β| = m

Φ(β) (X1 , . . . , Xn ) and G(X1 , . . . , Xn ) = . : |β| = m

Due to Section 3 (see Example 3.3), Θ is inner and has the radial infimum property. Applying now Theorem 4.1, we deduce that Xβ Xβ∗ ⊗ IG F (X1 , . . . , Xn )F (X1 , . . . , Xn )∗ Θ(X1 , . . . , Xn )Θ(X1 , . . . , Xn )∗ = |β|=m

for any (X1 , . . . , Xn ) ∈ [B(H)n ]1 . On the other hand, since A(β) ∗ G(0) = A(β) Aβ) < 1, : = |β| = m |β|=m we can use the second part of Theorem 4.1, to complete the proof.

2

We remark that Corollary 4.3 implies the version of Schwarz’s lemma obtained in [37] and, when m = 1, the corresponding result from [19]. ¯ min B(E, G) is a bounded free holomorphic function Corollary 4.4. If F : [B(H)n ]1 → B(H) ⊗ with F ∞ 1 and F (0) < 1, then −1 −1 I − F (X)F (X)∗ I − F (0)F (X)∗ DF (0)∗ DF (0)∗ I − F (X)F (0)∗ I− Xα Xα∗ ⊗ IG |α|=1

for any X := (X1 , . . . , Xn ) ∈ [B(H)n ]1 .


1549

Proof. According to Theorem 1.3, the mapping G := ΨF (0) [F ] is a bounded free holomorphic function with G∞ 1 and G(0) = ΨF (0) [F (0)] = 0. Applying Corollary 4.3 to G (when m = 1), we deduce that G(X)G(X)∗ XX ∗ ⊗ IG . On the other hand, using relation (1.10), we have I − G(X)G(X)∗ −1 −1 I − F (X)F (X)∗ I − F (0)F (X)∗ DF (0)∗ . = DF (0)∗ I − F (X)F (0)∗ Now, one can easily complete the proof.

2

We remark that Corollary 4.4 can be seen as an extension on Corollary 4.3 (case m = 1) to the case when F (0) < 1. When dealing with free holomorphic functions with scalar coefficients, Theorem 4.1 can be improved, as follows. Theorem 4.5. Let f, θ , and g be free holomorphic functions on [B(H)n ]1 with scalar coefficients such that: (i) f (X) = θ (X)g(X), X ∈ [B(H)n ]1 ; (ii) f is bounded with f ∞ 1; (iii) θ has the radial infimum property and θ ∞ = 1. Then g∞ 1 and, consequently, f (X)f (X)∗ θ (X)θ (X)∗ , X ∈ B(H)n 1 , f (X) θ (X), X ∈ B(H)n , 1 and g(0) 1. Moreover, (a) f (X0 )f (X0 )∗ < θ (X0 )θ (X0 )∗ for some X0 ∈ [B(H)n ]1 if and only if θ (X0 )θ (X0 )∗ > 0 and g is not a constant c with |c| = 1. (b) f (X0 ) = θ (X0 ) for some X0 ∈ [B(H)n ]1 if and only if either θ (X0 ) = 0 or f = cθ for some constant c with |c| = 1. (c) If |g(0)| = 1, then f = cθ for some constant c with |c| = 1. Proof. Due to Proposition 1.4, if g : [B(H)n ]1 → B(H) is a non-constant free holomorphic function with g∞ 1, then g(X) < 1 for any X ∈ [B(H)n ]1 . Using this result and Theorem 4.1, in the particular case when E = G = Y = C, one can complete the proof. 2 We remark that in the particular case when n = 1 and θ (z) = z, we recover Schwarz’s lemma (see [9]).

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In what follows we obtain generalizations of some classical results from complex analysis for certain classes of free holomorphic functions with positive real parts and of the form F = I + ΘΓ . Theorem 4.6. Let F , Θ, and Γ be free holomorphic functions on [B(H)n ]1 with coefficients in B(E), B(G, E), and B(E, G), respectively. If (i) F 0, (ii) Θ has the radial infimum property, Θ∞ = 1, and Θ(0) < 1, (iii) F = I + ΘΓ , then I − F (X) I − F (X)∗ I + F (X) Θ(X)Θ(X)∗ I + F (X)∗ and F (X) 1 + Θ(X) 1 − Θ(X) for any X ∈ [B(H)n ]1 . Proof. Since F (X) 0, X ∈ [B(H)n ]1 , its noncommutative Cayley transform G := (F − ∞ (B(E)), thus G(X) 1. Due to item (iii), we have I )(I + F )−1 is in the unit ball of Hball G = ΘΓ (I + F )−1 . Now, since Θ has the radial infimum property and Θ∞ = 1, we can apply Theorem 4.1 to G and obtain G(X)G(X)∗ Θ(X)Θ(X)∗ for all X ∈ [B(H)n ]1 . Hence, we deduce that −1 −1 F (X) − I F (X)∗ − I I + F (X)∗ Θ(X)Θ(X)∗ , I + F (X) which is equivalent to I − F (X) I − F (X)∗ I + F (X) Θ(X)Θ(X)∗ I + F (X)∗ . The latter inequality implies

F (X) − 1 I − F (X) Θ(X) 1 + F (X) , which leads to

F (X) 1 − Θ(X) 1 + Θ(X).

(4.3)

Since Θ∞ = 1 and Θ(0) < 1, the maximum principle for free holomorphic functions with operator-valued coefficients (see Theorem 1.5) implies that Θ(X) < 1. Now, inequality (4.3) implies the desired inequality. 2 Taking into account Theorems 4.5 and 4.6, we can make the following observation.


1551

Remark 4.7. In the scalar case, when E = G = C, the first inequality in Theorem 4.6 is strict if and only if Θ(X)Θ(X)∗ > 0 and F is not of the form F = (I + ηΘ)(I − ηΘ)−1 for some constant η with |η| = 1. >0 ∞ (B(E))] Consider the set [Hball 0) for any >0 (B(E)) of the noncommutative Cayley X ∈ [B(H)n ]1 . We remark that the restriction to Hball transform, defined by

C[F ] := [F − 1][1 + F ]−1 , −1 >0 (B(E)) → [H ∞ (B(E))] −1 is a bijection C : Hball 0 (B(E)) if and taking into account Theorem 1.5 from [39], it is enough to show that F ∈ Hball ∞ −1 only if G ∈ [Hball (B(E))] 0 for any r ∈ [0, 1) if and only if G(rS1 , . . . , rSn ) < 1 for any r ∈ [0, 1). Using the noncommutative Poisson transform, we deduce that F (X) > 0 for any X ∈ [B(H)n ]1 if and only if G(X) < 1 for any X ∈ [B(H)n ]1 , which proves our assertion. In what follows we need the following result. ¯ min B(E) be a free holomorphic function with coeffiLemma 4.8. Let F : [B(H)n ]1 → B(H) ⊗ ¯ min B(E) such cients in B(E). Then there is a free holomorphic function G : [B(H)n ]1 → B(H) ⊗ that F (X)G(X) = G(X)F (X) = I,

X ∈ B(H)n 1 ,

if and only if F (rS1 , . . . , rSn ) is an invertible operator for any r ∈ [0, 1). Moreover, in this case, G(rS1 , . . . , rSn ) = F (rS1 , . . . , rSn )−1 ,

r ∈ [0, 1),

where S1 , . . . , Sn are the left creation operators. Proof. One implication is obvious. Assume that F (rS1 , . . . , rSn ) is an invertible operator for any ¯ B(E), the weakly closed algebra r ∈ [0, 1). First we prove that F (rS1 , . . . , rSn )−1 is in Fn∞ ⊗ generated by the spatial tensor product. Since F is a free holomorphic function, F (rS1 , . . . , rSn ) ¯ min B(E) for any r ∈ [0, 1). In particular, we have is in An ⊗ F (rS1 , . . . , rSn )(Ri ⊗ I ) = (Ri ⊗ I )F (rS1 , . . . , rSn ),

i = 1, . . . , n,

where R1 , . . . , Rn are the right creation operators. Hence, we deduce that (Ri ⊗ I )F (rS1 , . . . , rSn )−1 = F (rS1 , . . . , rSn )−1 (Ri ⊗ I ),

i = 1, . . . , n.

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¯ B(E). ConseAccording to [34], the commutant of {Ri ⊗ I, i = 1, . . . , n} is equal to Fn∞ ⊗ ¯ B(E) and has a unique Fourier representation quently, F (rS1 , . . . , rSn )−1 is in Fn∞ ⊗ F (rS1 , . . . , rSn )−1 ∼

Sα ⊗ r |α| B(α) (r)

α∈F+ n

for some operators B(α) (r) ∈ B(E). We prove now that the operators B(α) (r), α ∈ F+ n , don’t depend on r ∈ [0, 1). Assume that F has the representation F (X1 , . . . , Xn ) :=

∞

Xα ⊗ Aα) ,

k=0 |α|=k

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

where the convergence is in the operator norm topology. Since I = F (rS1 , . . . , rSn )F (rS1 , . . . , rSn )−1 ∞ |α| |α| = Sα ⊗ r B(α) (r) r Sα ⊗ Aα) α∈F+ n

k=0 |α|=k

for any β ∈ F+ n , we have

∗ Sβ ⊗ IE F (rS1 , . . . , rSn )F (rS1 , . . . , rSn )−1 (1 ⊗ x), (1 ⊗ y) r |β| A(α) B(ω) (r)x, y = α,ω∈F+ n ,αω=β

for any x, y ∈ E. Therefore, A(0) B(0) (r) = I and

A(α) B(ω) (r) = 0

(4.4)

α,ω∈F+ n ,αω=β

if |β| 1. Now, we proceed by induction. Note that B(0) (r) = A−1 (0) and assume that the operators + B(α) (r) don’t depend on r ∈ [0, 1) for any α ∈ Fn with |β| m. We prove that the property holds if |β| = m + 1. To this end, let β := gi1 gi2 · · · gim gim+1 ∈ F+ n . Due to relation (4.4), we have A(0) B(β) + A(gi1 ) B(gi2 ···gim+1 ) + · · · + A(gi1 ···gim ) B(gim+1 ) + A(β) B(0) = 0. Hence and due to the induction hypothesis, we deduce that B(β) (r) does not depend on r ∈ + −1 [0, 1). Thus we can write B(β) := B(β) (r) for any β ∈ F (rS1 , . .|α|. , rSn ) has the Fourier F∞n and |α| representation α∈F+n Sα ⊗ r B(α) and the series k=0 |α|=k (sr) Sα ⊗ B(α) converges in the operator norm topology for any s, r ∈ [0, 1). Hence, we deduce that the map G : [B(H)n ]1 → ¯ min B(E) defined by B(H) ⊗ G(X1 , . . . , Xn ) :=

∞ k=0 |α|=k

Xα ⊗ B(α) ,

(X1 , . . . , Xn ) ∈ B(H)n 1 ,


1553

is a free holomorphic function. Here the convergence is in the operator norm topology. Due to (4.4) and the similar relation that can be deduced from the equation F (rS1 , . . . , rSn )−1 F (rS1 , . . . , rSn ) = I , one can easily see that F (X)G(X) = G(X)F (X) = I for any X ∈ [B(H)n ]1 . Moreover, we have G(rS1 , . . . , rSn ) = F (rS1 , . . . , rSn )−1 for any r ∈ [0, 1). The proof is complete.

2

The next result is a noncommutative extension of Harnack’s double inequality. Theorem 4.9. Let F , Θ, and Γ be free holomorphic functions on [B(H)n ]1 with coefficients in B(E), B(G, E), and B(E, G), respectively. If (i) F > 0, (ii) Θ has the radial infimum property, Θ∞ 1, and Θ(0) < 1, (iii) F = I + ΘΓ and F −1 = I + ΘL for some free holomorphic function L on [B(H)n ]1 , then 1 + Θ(X) 1 − Θ(X) F (X) 1 + Θ(X) 1 − Θ(X) for any X ∈ [B(H)n ]1 . Proof. The inequality F (X) 1+Θ(X) 1−Θ(X) is due to Theorem 4.6. We prove now the first inequality. Since F (X) > 0, X ∈ [B(H)n ]1 , there exit constants γ (r) ∈ (0, 1) such that F (rS1 , . . . , rSn ) γ (r)I,

r ∈ (0, 1).

Hence, we deduce that F (rS1 , . . . , rSn )∗ x + F (rS1 , . . . , rSn )x 2γ (r)x,

x ∈ F 2 (Hn ) ⊗ E,

which shows that F (rS1 , . . . , rSn ) and F (rS1 , . . . , rSn )∗ are bounded below. Therefore, the operator F (rS1 , . . . , rSn ) is invertible for all r ∈ [0, 1). Due to Lemma 4.8, there is a free holo¯ min B(E) such that morphic function Λ : [B(H)n ]1 → B(H) ⊗ F (X)Λ(X) = Λ(X)F (X) = I,

X ∈ B(H)n 1 ,

and Λ(rS1 , . . . , rSn ) = F (rS1 , . . . , rSn )−1 for all r ∈ [0, 1). Since ∗ Λ(rS1 , . . . , rSn ) = F (rS1 , . . . , rSn )−1 F (rS1 , . . . , rSn ) F (rS1 , . . . , rSn )−1 and F (rS1 , . . . , rSn ) > 0, we deduce that Λ(rS1 , . . . , rSn ) > 0. Therefore Λ > 0. Due to item (iii), we have Λ = I + ΘL for some free holomorphic function L on [B(H)n ]1 . Applying now Theorem 4.6 to Λ, we obtain

1554


Λ(X) 1 + Θ(X) 1 − Θ(X) for any X ∈ [B(H)n ]1 . Since Λ(X) = F (X)−1 , we have F (X) inequalities, we deduce that

1 Λ(X) .

Combining these

1 − Θ(X) F (X) 1 + Θ(X) for any X ∈ [B(H)n ]1 . The proof is complete.

2

We remark that when n = 1 and E = G = C, then the condition F −1 = I +ΘL in Theorem 4.9 is redundant, so we can drop it. Corollary 4.10. Let F be a free holomorphic function on [B(H)n ]1 with coefficients in B(E) and standard representation F (X1 , . . . , Xn ) = I +

∞ k=m |α|=k

Xα ⊗ A(α) ,

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

where m = 1, 2 . . . . If F 0, then

∗ 1/2 |β|=m Xβ Xβ 1 + |β|=m Xβ Xβ∗ 1/2

1−

1 + |β|=m Xβ Xβ∗ 1/2 F (X) 1 − |β|=m Xβ Xβ∗ 1/2

for any X ∈ [B(H)n ]1 . Proof. First, we consider the case when F > 0. As in the proof of Theorem 2.4 from [37], we have a decomposition F = I + ΘΓ , where Θ(X1 , . . . , Xn ) := Xβ ⊗ I : |β| = m

Φ(β) (X1 , . . . , Xn ) and Γ (X1 , . . . , Xn ) := : |β| = m

are free holomorphic functions. Due to Section 3 (see Example 3.3), Θ is inner and has the radial infimum property. On the other hand, due to the proof of Theorem 4.9, X → F (X)−1 exists as a free holomorphic function on [B(H)n ]1 . Since

I = F (X)−1 F (X) = F (X)−1 I + Θ(X)Γ (X) we deduce that F (X)−1 = I − F (X)−1 Θ(X)Γ (X). Taking into account that Θ is a homogeneous polynomial of degree m, it is easy to see that X → F (X)−1 Θ(X)Γ (X) is a free holomorphic function so that each monomial in its standard representation has degree greater than or equal to m. This implies that F −1 has a decomposition of the form I + ΘL for some free holomorphic function L. Since we are under the hypotheses of Theorem 4.9, we can apply this theorem to F and obtain the desired inequalities. In the case when F 0, the map 1 (F − I ), > 0, has the property G > 0, so that we can use the first part of G := I + 1+


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the proof and obtain the corresponding inequalities. Taking the limit as → 0, we complete the proof. 2 From Corollary 4.10, we can deduce the following remarkable particular case, which should be compared to Theorem 1.4 from [45]. Corollary 4.11. If F is a free holomorphic function on [B(H)n ]1 with coefficients in B(E) such that F (0) = I and F 0, then 1 + X 1 − X F (X) , 1 + X 1 − X

X ∈ B(H)n 1 .

5. Noncommutative Borel–Carathéodory theorems In this section, we obtain Borel–Carathéodory type results for free holomorphic functions with operator-valued coefficients. We start with a Carathéodory type result for free holomorphic functions which admit factorizations F = ΘΓ , where Θ is an inner function with the radial infimum property. Theorem 5.1. Let F , Θ, and Γ be free holomorphic functions on [B(H)n ]1 with coefficients in B(E), B(G, E), and B(E, G), respectively. If (i) F (X) I for any X ∈ [B(H)n ]1 , (ii) Θ has the radial infimum property, Θ∞ = 1, and Θ(0) < 1, (iii) F = ΘΓ , then F (X) 2Θ(X) , 1 − Θ(X)

X ∈ B(H)n 1 .

Proof. Since G := I − F = I − ΘΓ has the property that G 0, we can apply Theorem 4.6 to G and obtain I − G(X) I − G(X)∗ I + G(X) Θ(X)Θ(X)∗ I + G(X)∗ ,

X ∈ B(H)n 1 .

Consequently, we deduce that F (X) = I − G(X) (G(X) + I Θ(X) = 2I − F (X)Θ(X) 2 + F (X) Θ(X). Hence, we have

F (X) 1 − Θ(X) 2Θ(X). We recall that, since Θ∞ = 1 and Θ(0) < 1, the maximum principle for free holomorphic functions with operator-valued coefficients (see Theorem 1.5) implies that Θ(X) < 1. Now we can complete the proof. 2

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Corollary 5.2. Let F be a free holomorphic function on [B(H)n ]1 with coefficients in B(E) and standard representation F (X1 , . . . , Xn ) =

∞

Xα ⊗ A(α) ,

k=m |α|=k

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

where m = 1, 2 . . . . If F I , then 2 |β|=m Xβ Xβ∗ 1/2 F (X) , 1 − |β|=m Xβ Xβ∗ 1/2

X ∈ B(H)n 1 .

Proof. As in the proof of Theorem 2.4 from [37], we have a Gleason type decomposition F = ΘΓ , where Θ(X1 , . . . , Xn ) := Xβ ⊗ I : |β| = m

Φ(β) (X1 , . . . , Xn ) and Γ (X1 , . . . , Xn ) := : |β| = m

are free holomorphic functions. Since Θ is inner with the radial infimum property and Θ(0) = 0, we apply Theorem 5.1 and complete the proof. 2 From Corollary 5.2, we can deduce the following particular case. Corollary 5.3. If F is a free holomorphic function on [B(H)n ]1 with coefficients in B(E) such that F (0) = 0 and F I , then F (X) 2X , 1 − X

X ∈ B(H)n 1 .

The next result is a generalization of the Borel–Carathéodory theorem, mentioned in the introduction, for free holomorphic functions with operator-valued coefficients. ¯ Theorem 5.4. Let F : [B(H)n ]− γ → B(H) ⊗min B(E) be a free holomorphic function with coefficients in B(E) and let r ∈ (0, γ ). Then 2r γ +r F (0), A(γ ) + sup F (X) γ −r γ −r X=r where A(γ ) := supy=1 F (γ S1 , . . . , γ Sn )y, y and S1 , . . . , Sn are the left creation operators. Proof. If F is constant, i.e., F = F (0), then the inequality holds due to the fact that F (0) −F (0)IH⊗E . Assume that F is not constant and F (0) = 0. First we show that A(γ ) > 0.

(5.1)


1557

Indeed, if we assume that A(γ ) 0, then F (γ S1 , . . . , γ Sn ) F (0) = 0. Applying the noncommutative Poisson transform at [ γt X1 , . . . , γt Xn ], where 0 t < γ and (X1 , . . . , Xn ) ∈ [B(H)n ]− 1 , we obtain F (tX1 , . . . , tXn ) = P[ t X1 ,..., t Xn ] F (γ S1 , . . . , γ Sn ) F (0) = 0 γ

γ

for any t ∈ [0, γ ) and (X1 , . . . , Xn ) ∈ [B(H)n ]− 1 . According to Theorem 2.9 from [42], we deduce that F = F (0), which contradicts our assumption. Therefore, inequality (5.1) holds. Since F (γ S1 , . . . , γ Sn ) A(γ )I , we can use again the noncommutative Poisson transform to deduce that F (X) A(γ )I for X ∈ [B(H)n ]− γ . Now, let > 0 and define the free holomorphic function on a noncommutative ball [B(H)n ]s with s > γ , by ϕ (X) := 2 A(γ ) + IH⊗E − F (X),

X ∈ B(H)n s .

Note that, for any y ∈ H ⊗ E and X ∈ [B(H)n ]s , we have ϕ (X)y 2 = 4 A(γ ) + 2 y2 − 4 A(γ ) + F (X)y, y + F (X)y 2 2 4 A(γ ) + y2 + F (X)y 4 A(γ ) + y2 .

(5.2)

Similar calculations show that ϕ (X)∗ y 2 4 A(γ ) + y2 . Replacing X by (tS1 , . . . , tSn ), t s, and taking y ∈ F 2 (Hn ) ⊗ E in the inequalities above, we deduce that ϕ (tS1 , . . . , tSn ) and ϕ (tS1 , . . . , tSn )∗ are bounded below and, consequently, invertible for any t ∈ [0, s). Applying Lemma 4.8 to ϕ , we deduce that there is a free holomorphic function ψ on [B(H)n ]s such that ϕ (X)ψ (X) = ψ (X)ϕ (X) = I,

X ∈ B(H)n s .

Using relation (5.2) and replacing y with ψ (X)y, we obtain that 2 2 y2 = ϕ (X)ψ (X)y F (X)ψ(X)y + 4 A(γ ) + ψ (X)y . Hence, we deduce that the map Λ (X) := F (X)ψ (X),

X ∈ B(H)n s ,

(5.3)

is a contractive free holomorphic function on [B(H)n ]s . Since Λ (0) = 0, Theorem 1.5 implies that Λ (X) < 1. Hence, and due to relation (5.3), we deduce that

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−1 F (X) = 2 A(γ ) + Λ I + Λ (X) ,

(5.4)

which implies

F (X) 2 A(γ ) + Λ (X) 1 + Λ (X) + Λ (X)2 + · · · Λ (X) . 2 A(γ ) + 1 − Λ (X) On the other hand, applying the Schwarz type lemma for free holomorphic functions (see [37]) to Λ , we deduce that Λ (X) r γ

(5.5)

for any X ∈ [B(H)n ]γ with X = r, where 0 r < γ . Combining this with the previous inequality, we obtain F (X) 2[A(γ ) + ]r . γ −r Taking → 0, we deduce that 2r sup F (X) A(γ ), γ −r X=r which proves the theorem when F (0) = 0. Now, we consider the case when F (0) = 0. Applying the result above to F − F (0), we obtain

2r sup F (X) − F (0) sup F (γ S1 , . . . , γ Sn ) − F (0) y, y γ − r X=r y=1

2r A(γ ) + F (0) . γ −r

Consequently, we have sup F (X) sup F (X) − F (0) + F (0)

X=r

X=r

= The proof is complete.

2r γ +r F (0). A(γ ) + γ −r γ −r

2

We remark that if A(γ ) 0 in Theorem 5.4, then we can deduce that γ +r sup F (X) A(γ ) + F (0) . γ −r X=r


1559

A closer look at the proof of Theorem 5.4 reveals another Carathéodory type inequality. More precisely, applying Corollary 4.3 to the free holomorphic function Λ , we deduce that Λ (X)Λ (X)∗

XX ∗ γ2

for any X ∈ [B(H)n ]γ with X = r, where 0 r < γ . Using now relations (5.4) and (5.5), we obtain 2 Λ (X)Λ (X)∗ F (X)F (X)∗ 4 A(γ ) + (1 − Λ (X))2

4[A(γ ) + ]2 XX ∗ ⊗ IE . 2 (γ − r) 2

4A(γ ) Taking → 0, we deduce that F (X)F (X)∗ (γ (XX ∗ ⊗ IE ). Now, in the general case when −r)2 F (0) is not necessarily 0, we obtain the following result:

Corollary 5.5. Under the hypotheses of Theorem 5.4, we have

∗ 4A(γ )2 ∗ XX ⊗ I F (X) − F (0) F (X) − F (0) E (γ − r)2 for any X ∈ B(H)n with X = r. 6. Julia’s lemma for holomorphic functions on noncommutative balls In this section, we provide a noncommutative generalization of Pick’s theorem for bounded free holomorphic functions. Using this result and basic facts concerning the involutive free holomorphic automorphisms of [B(H)n ]1 , we obtain a free analogue of Julia’s lemma from complex analysis. A map F : [B(H)n ]1 → [B(H)n ]1 is called free biholomorphic if F is free holomorphic, one-to-one and onto, and has free holomorphic inverse. The automorphism group of [B(H)n ]1 , denoted by Aut([B(H)n ]1 ), consists of all free biholomorphic functions of [B(H)n ]1 . It is clear that Aut([B(H)n ]1 ) is a group with respect to the composition of free holomorphic functions. Inspired by the classical results of Siegel [50] and Phillips [28] (see also [55]), we used, in [43], the theory of noncommutative characteristic functions for row contractions (see [31]) to find all the involutive free holomorphic automorphisms of [B(H)n ]1 , which turn out to be of the form Φλ (X1 , . . . , Xn ) = −Θλ (X1 , . . . , Xn ),

(X1 , . . . , Xn ) ∈ B(H)n 1 ,

for some λ = [λ1 , . . . , λn ] ∈ Bn , where Θλ is the characteristic function of the row contraction λ, acting as an operator from Cn to C. We recall that the characteristic function of the row contraction λ := (λ1 , . . . , λn ) ∈ Bn is the boundary function Θ˜ λ , with respect to R1 , . . . , Rn , of the free holomorphic function Θλ : [B(H)n ]1 → [B(H)n ]1 given by

1560


Θλ (X1 , . . . , Xn ) := −λ + λ IH −

n

−1 λ¯ i Xi

[X1 , . . . , Xn ]λ∗

i=1

for (X1 , . . . , Xn ) ∈ [B(H)n ]1 , where λ = (1 − λ22 )1/2 IC and λ∗ = (IK − λ∗ λ)1/2 . For simplicity, we also used the notation λ := [λ1 IG , . . . , λn IG ] for the row contraction acting from G (n) to G, where G is a Hilbert space. 1 In [43], we proved that if λ := (λ1 , . . . , λn ) ∈ Bn \{0} and γ := λ , then Φλ := −Θλ is a free 2 n holomorphic function on [B(H) ]γ which has the following properties: (i) Φλ (0) = λ and Φλ (λ) = 0; (ii) the identities

−1

−1 I − XY ∗ I − λY ∗ λ , IH − Φλ (X)Φλ (Y )∗ = λ I − Xλ∗

−1

−1 IH⊗Cn − Φλ (X)∗ Φλ (Y ) = λ∗ I − X ∗ λ I − X ∗ Y I − λ∗ Y λ∗ ,

(6.1)

hold for all X and Y in [B(H)n ]γ ; (iii) Φλ is an involution, i.e., Φλ (Φλ (X)) = X for any X ∈ [B(H)n ]γ ; (iv) Φλ is a free holomorphic automorphism of the noncommutative unit ball [B(H)n ]1 ; n − (v) Φλ is a homeomorphism of [B(H)n ]− 1 onto [B(H) ]1 . Moreover, we determined all the free holomorphic automorphisms of the noncommutative ball [B(H)n ]1 by showing that if Φ ∈ Aut([B(H)n ]1 ) and λ := Φ −1 (0), then there is a unitary operator U on Cn such that Φ = ΦU ◦ Φλ , where ΦU (X1 , . . . , Xn ) := [X1 , . . . , Xn ]U,

(X1 , . . . , Xn ) ∈ B(H)n 1 .

The first result of this section is following extension of Pick’s theorem (see [8,29]), for bounded free holomorphic functions. Let Mn×m be the set of all n × m matrices with scalar coefficients. Theorem 6.1. Let F : [B(H)n ]1 → [B(H)m ]− 1 be a free holomorphic function with F (0) < 1 ¯ min Mn×m and let a ∈ Bn . Then there exists a free holomorphic function Γ : [B(H)n ]1 → B(H) ⊗ with Γ ∞ 1 such that

ΦF (a) F (X) = Φa (X)(Γ ◦ Φa )(X),

X ∈ B(H)n 1 ,

where Φa and ΦF (a) are the corresponding free holomorphic automorphisms of [B(H)n ]1 and [B(H)m ]1 , respectively. Consequently,

∗ ΦF (a) F (X) ΦF (a) F (X) Φa (X)Φa (X)∗ ,

X ∈ B(H)n 1 ,


1561

and

ΦF (a) F (X) Φa (X),

X ∈ B(H)n 1 .

Proof. Since F is a free holomorphic function with F ∞ 1 and F (0) < 1, Corollary 1.6 implies that F (X) < 1 for any X ∈ [B(H)n ]1 . We know that Φa ∈ Aut([B(H)n ]1 ) and ΦF (a) ∈ Aut([B(H)m ]1 ). Due to Section 1 and the properties of the free holomorphic automorphisms of [B(H)m ]1 , the composition map G := ΦF (a) ◦ F ◦ Φa : [B(H)n ]1 → [B(H)m ]1 is a free holomorphic function with G(0) = 0. Therefore, it has a representation of the form G(X1 , . . . , Xn ) =

∞

Xα ⊗ A(α) = [X1 , . . . , Xn ]Γ (X1 , . . . , Xn )

(6.2)

k=1 |α|=k

for any [X1 , . . . , Xn ] ∈ [B(H)n ]1 , for some matrices A(α) ∈ M1×m and a free holomorphic function Γ with coefficients in Mn×m . Since G∞ 1 with G(0) < 1, and Θ(X) := [X1 , . . . , Xn ] is inner and has the radial infimum property, Theorem 4.1 implies that Γ ∞ 1, G(X)G(X)∗ XX ∗ ,

X ∈ B(H)n 1 ,

and G(X) X,

X ∈ B(H)n 1 .

Replacing X by Φa (X) in these inequalities and in relation (6.2), and using the fact that Φa ◦ Φa = id, we complete the proof. 2 Corollary 6.2. If F : [B(H)n ]1 → [B(H)m ]− 1 is a free holomorphic function with F (0) < 1 and a ∈ Bn , then, for any X ∈ [B(H)n ]1 , −1 −1 I − F (X)F (X)∗ I − F (a)F (X)∗ F (a) F (a) I − F (X)F (a)∗

−1

−1 I − XX ∗ I − aX ∗ a a I − Xa ∗

(6.3)

and I − F (a)F (X)∗ I − F (X)F (X)∗ −1 I − F (X)F (a)∗

2F (a)

I − a ∗ X I − XX ∗ −1 I − Xa ∗ . 2 a

(6.4)

Proof. The first inequality follows from Theorem 6.1 and relation (6.1). Since F (0) < 1, Corollary 1.6 implies that F (X) < 1 for any X ∈ [B(H)n ]1 . Note that each side of inequality (6.3) is a positive invertible operator. It is well known that if A, B are two positive invertible operator such that A B then B −1 A−1 . Applying this result to inequality (6.3), we deduce that

1562


−1 I − F (a)F (X)∗ I − F (X)F (X)∗ I − F (X)F (a)∗

2F (a)

−1

I − aX ∗ I − XX ∗ I − Xa ∗ . 2 a

Hence, the second inequality follows. The proof is complete.

2

Let F : [B(H)n ]1 → [B(H)m ]− 1 be a free holomorphic function. Let ξ ∈ ∂Bn := {z ∈ Cn : z2 = 1} and assume that L := lim inf z→ξ

1 − F (z)2 < ∞. 1 − z2

Then there is a sequence {zk }∞ k=1 ⊂ Bn such that limk→∞ zk = ξ and limk→∞ F (zk ) = η for some η ∈ ∂Bm , and 1 − F (zk )2 = L. k→∞ 1 − zk 2 lim

Now, we can present our first generalization of Julia’s lemma for free holomorphic functions on noncommutative balls. Theorem 6.3. Let F : [B(H)n ]1 → [B(H)m ]− 1 be a free holomorphic function with F (0) < 1. Let {zk }∞ ⊂ B be a sequence such that lim n k→∞ zk = ξ , limk→∞ F (zk ) = η for some ξ, ∈ ∂Bn , k=1 η ∈ ∂Bm , and 1 − F (zk )2 = L < ∞. k→∞ 1 − zk 2 lim

Then the following statements hold: (i) For any X ∈ [B(H)n ]1 , I − ηF (X)∗ I − F (X)F (X)∗ −1 I − F (X)η∗

−1

L I − ξ ∗ X I − XX ∗ I − Xξ ∗ . (ii) If β > 0 and X ∈ [B(H)n ]1 is such that

I − Xξ ∗ I − ξ X ∗ < β I − XX ∗ , then I − F (X)η∗ I − ηF (X)∗ < βL I − F (X)F (X)∗ .


1563

Proof. Due to inequality (6.4), we have I − F (zk )F (X)∗ I − F (X)F (X)∗ −1 I − F (X)F (zk )∗

1 − F (zk )2 I − z∗ X I − XX ∗ −1 I − Xz∗ . k k 2 1 − zk

Taking the limit as k → ∞, we deduce item (i). To prove (ii), note first that, using the same inequality (6.4), when a = zk and X = 0, we obtain I − F (zk )F (0)∗ I − F (0)F (0)∗ −1 I − F (0)F (zk )∗

1 − F (zk )2 . 1 − zk 2

Taking zk → ξ and due to the fact that F (0) < 1, we deduce that −1 L I − ηF (0)∗ I − F (0)F (0)∗ I − F (0)η∗ > 0. Notice also that, for any X ∈ [B(H)n ]1 , the following inequalities are equivalent: (a) (I − Xξ ∗ )(I − ξ X ∗ ) < β(I − XX ∗ ), (b) (I − ξ X ∗ )(I − XX ∗ )−1 (I − Xξ ∗ ) < β. Indeed, inequality (b) holds if and only if (I −ξ X ∗ )(I −XX ∗ )−1/2 < β 1/2 , which is equivalent to

−1/2

−1/2 I − Xξ ∗ I − ξ X ∗ I − XX ∗ < β. I − XX ∗ The latter inequality is clearly equivalent to (a). Now, to prove (ii), we assume that

I − Xξ ∗ I − ξ X ∗ < β I − XX ∗ . Due to the equivalence of (a) with (b), and using the inequality from (i) and the fact that L > 0, we obtain I − ηF (X)∗ I − F (X)F (X)∗ −1 I − F (X)η∗ < βL. Once again using the equivalence of (a) with (b) when X is replaced by F (X), we obtain that I − F (X)η∗ I − ηF (X)∗ < βL I − F (X)F (X)∗ . This completes the proof.

2

We mention that, using unitary transformations in B(Cn ) and B(Cm ), respectively, we can choose the coordinates such that ξ = (1, 0, . . . , 0) ∈ ∂Bn and η = (1, 0, . . . , 0) ∈ ∂Bm , in Theorem 6.3. For 0 < c < 1, we define the noncommutative ellipsoid

1564


& [X1 − (1 − c)I ][X1∗ − (1 − c)I ] Ec := (X1 , . . . , Xn ) ∈ B(H)n : c2 ' X2 X2∗ Xn Xn∗ + + ··· + 1, where Dα = Dα (1, 0, . . . , 0). Proof. Since F (0) = 0, due to Schwarz’s lemma for free holomorphic functions, we have F (X) X for all X ∈ [B(H)n ]1 . Consequently, 1 − F (zk )2 1 k→∞ 1 − zk 2

L = lim

Lc Lc, therefore Eγ ⊂ EcL . Due to Theorem 6.4, we deduce that which implies γ := 1+Lc−c 1 F (Ec ) ⊆ EcL when 0 < c < L . To prove item (ii), let X = (X1 , . . . , Xn ) ∈ Dα , i.e.,

α2

1 − X2 I − XX ∗ . (I − X1 ) I − X1∗ < 4 Applying Theorem 6.3, part (ii), when ξ = (1, 0, . . . , 0) and β =

α2 2 4 (1 − X )

we deduce that

Lα 2

1 − X2 I − F (X)F (X)∗ . I − F1 (X) I − F1 (X)∗ < 4

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Since F (X) X, we obtain 2 Lα 2 1 − F (X) I − F (X)F (X)∗ , I − F1 (X) I − F1 (X)∗ < 4 which shows that F (X) ∈ Dα √L and completes the proof.

2

7. Pick–Julia theorems for free holomorphic functions with operator-valued coefficients In this section, we use fractional transforms and a version of the noncommutative Schwarz’s lemma to obtain Pick–Julia theorems for free holomorphic functions F with operator-valued coefficients such that F ∞ 1 (resp. F 0). As a consequence, we obtain a Julia type lemma for free holomorphic functions with positive real parts. We also provide commutative versions of these results for operator-valued multipliers of the Drury–Arveson space. ¯ min B(E, G) be a free holomorphic function with Theorem 7.1. Let F : [B(H)n ]1 → B(H) ⊗ F ∞ 1 and F (0) < 1. If z ∈ Bn , then

∗ ΨF (z) F (X) ΨF (z) F (X) Φz (X)Φz (X)∗ ⊗ IG ,

X ∈ B(H)n 1 ,

where ΨF (z) is the fractional transform defined by (1.6) and Φz is the corresponding free holomorphic automorphisms of [B(H)n ]1 . Moreover, we have −1 −1 DF (z)∗ I − F (X)F (z)∗ I − F (X)F (X)∗ I − F (z)F (X)∗ DF (z)∗

−1

−1 I − XX ∗ I − zX ∗ z ⊗ IG z I − Xz∗ for any z ∈ Bn and X ∈ [B(H)n ]1 . Proof. Since F (0) < 1, Corollary 1.6 implies that F (X) < 1 for any X ∈ [B(H)n ]1 . According to Theorem 1.1 and Theorem 1.3, the mapping X → (ΨF (z) ◦ F ◦ Φz )(X) is a bounded free holomorphic function on [B(H)n ]1 with (ΨF (z) ◦ F ◦ Φz )(X) < 1 for any X ∈ [B(H)n ]1 . On the other hand, since we have (ΨF (z) ◦ F ◦ Φz )(0) = ΨF (z) (F (z)) = 0, we can apply Corollary 4.3 and obtain ∗ (ΨF (z) ◦ F ◦ Φz )(Y ) (ΨF (z) ◦ F ◦ Φz )(Y ) Y Y ∗ ⊗ IG for any Y ∈ [B(H)n ]1 . Taking Y = Φz (X), X ∈ [B(H)n ]1 , and due to the identity Φz ◦ Φz = id, we obtain

∗ ΨF (z) F (X) ΨF (z) F (X) Φz (X)Φz (X)∗ ⊗ IG , X ∈ B(H)n 1 . Using now relations (1.7) and (6.1), we complete the proof.

2

We remark that under the conditions of Theorem 7.1, one can show, as in the proof of Theorem 6.1, that there is a free holomorphic function G with operator-valued coefficients and G∞ 1 such that ΨF (z) F (X) = Φz (X)(G ◦ Φz )(X), X ∈ B(H)n 1 .


1567

Our Pick–Julia type result for free holomorphic functions with positive real parts is the following. ¯ min B(E) is a free holomorphic function with G > 0, Theorem 7.2. If G : [B(H)n ]1 → B(H) ⊗ then −1 −1 G(X) G(z) + G(X)∗ I + G(z) Γ (z) Γ (z) I + G(z)∗ G(X) + G(z)∗

−1

−1 1 − z22 I − Xz∗ I − XX ∗ I − zX ∗ ⊗ IE for any z ∈ Bn and X ∈ [B(H)n ]1 , where −1 1/2 −1 G(z) I + G(z)∗ . Γ (z) := 2 I + G(z) Proof. According to the considerations preceding Lemma 4.8, since G > 0, the noncommutative Cayley transform F := C[G] := (G − I )(I + G)−1 is a bounded free holomorphic function with F (X) < 1 for any X ∈ [B(H)n ]1 . Due to Theorem 7.1, we obtain −1 −1 DF (z)∗ I − F (X)F (z)∗ I − F (X)F (X)∗ I − F (z)F (X)∗ DF (z)∗

−1

−1 I − XX ∗ I − zX ∗ z ⊗ IG . z I − Xz∗

(7.1)

Note that −1 −1 I − F (X)F (z)∗ = I − I + G(X) G(X) − I G(z)∗ − I I + G(z)∗ −1 I + G(X) I + G(z)∗ = I + G(X) −1 − G(X) − I G(z)∗ − I I + G(z)∗ −1 −1 G(X) + G(z)∗ I + G(z)∗ = 2 I + G(X) and, similarly, −1 −1 I − F (X)F (X)∗ = 2 I + G(X) G(X) + G(X)∗ I + G(X)∗ for any X ∈ [B(H)n ]1 and z ∈ Bn . Using these identities, we deduce that −1 −1 I − F (X)F (z)∗ I − F (X)F (X)∗ I − F (z)F (X)∗ =

−1 1 I + G(z)∗ G(X) + G(z)∗ I + G(X) 2 −1 −1 × 2 I + G(X) G(X) + G(X)∗ I + G(X)∗

−1 1 I + G(X)∗ G(z) + G(X)∗ I + G(z) 2 −1 −1 1 = I + G(z)∗ G(X) + G(z)∗ G(X) + G(X)∗ G(z) + G(X)∗ I + G(z) . 2 ×

1568


Now, since 1/2 −1 1/2 −1 DF (z)∗ = I − F (z)F (z)∗ G(z) + G(z)∗ I + G(z)∗ = 2 I + G(z) −1 1/2 −1 G(z) I + G(z)∗ = 2 I + G(z) the inequality (7.1) implies the inequality of the theorem. The proof is complete.

2

The next result is a Julia type lemma for free holomorphic functions with scalar coefficients and positive real parts. Theorem 7.3. Let G : [B(H)n ]1 → B(H) be a free holomorphic function with G > 0. Let {zk }∞ k=1 ⊂ Bn be a sequence such that limk→∞ zk = ξ ∈ ∂Bn , limk→∞ |G(zk )| = ∞, and such that G(zk ) = M < ∞. k→∞ (1 − zk 2 )|G(zk )|2 2 lim

Then M > 0 and G(X)

−1

−1 1 I − Xξ ∗ I − XX ∗ I − ξ X ∗ 4M

for any X ∈ [B(H)n ]1 . Proof. According to Theorem 7.2, when E = C, we have Γ (zk ) =

2[G(zk )]1/2 |1 + G(zk )|

and −1 −1 4 G(zk ) G(X) + G(zk )∗ G(X) G(zk ) + G(X)∗

−1

−1 I − XX ∗ I − zk X ∗ . 1 − zk 2 I − Xzk∗ Hence, we obtain 1 4G(zk ) ∗ ∗ G(X) ) ) + G(X) G(X) + G(z A(k) G(z , k k (1 − zk 2 )|G(zk )|2 |G(zk )|2

(7.2)

where A(k) := (I − Xzk∗ )−1 (I − XX ∗ )(I − zk X ∗ )−1 . Taking X = 0 in inequality (7.2), we obtain 4G(zk ) 1 G(0) + G(zk )∗ G(zk ) + G(0)∗ , G(0) 2 2 2 (1 − zk )|G(zk )| |G(zk )|


1569

whence −1 1 G(zk ) G(0) + G(zk )∗ G(0) G(zk ) + G(0)∗ I. |G(zk )|2 (1 − zk 22 )|G(zk )|2 Since limk→∞ |G(zk )| = ∞ and taking the limit in the latter inequality, we obtain −1 lim G(0)

G(zk ) I = MI. k→∞ (1 − zk 2 )|G(zk )|2 2

Consequently, we have M > 0. Now, due to inequality (7.2) and the fact that 1 G(X) + G(zk )∗ A(k) G(zk ) + G(X)∗ |G(zk )|2

−1

−1 = lim A(k) = I − Xξ ∗ I − XX ∗ I − ξ X ∗ ,

lim

k→∞

k→∞

we deduce that

−1

−1 I − XX ∗ I − ξ X ∗ 4MG(X) I − Xξ ∗ for any X ∈ [B(H)n ]1 , which completes the proof.

2

We recall (see [35,37,42]) that if F is a contractive (F ∞ 1) free holomorphic func¯ B(E). Consequently, the tion with coefficients in B(E), then its boundary function is in Fn∞ ⊗ evaluation map Bn z → F (z) ∈ B(E) is a contractive operator-valued multiplier of the Drury– Arveson space [3,12], and any such a contractive multiplier has this type of representation. Corollary 7.4. The following statements hold: ¯ min B(E) is a free holomorphic function with F ∞ 1 and (i) If F : [B(H)n ]1 → B(H) ⊗ F (0) < 1 then −1 I − F (w)F (z)∗ I − F (z)F (w)∗ I − F (w)F (w)∗

|1 − w, z|2 I − F (z)F (z)∗ (1 − z2 )(1 − w2 )

for any z, w ∈ Bn . ¯ min B(E) is a free holomorphic function with G > 0, then (ii) If G : [B(H)n ]1 → B(H) ⊗ −1 G(z) + G(w)∗ G(w) G(w) + G(z)∗ for any z, w ∈ Bn .

4|1 − w, z|2 G(z) (1 − z2 )(1 − w2 )

1570


Proof. Taking X = w ∈ Bn in Theorem 7.1, we deduce that −1 −1 I − F (w)F (z)∗ I − F (w)F (w)∗ I − F (z)F (w)∗

−1 (1 − z22 )(1 − w22 ) I − F (w)F (w)∗ 2 |1 − w, z|

for any z, w ∈ Bn . We recall that if A, B are two positive invertible operator such that A B then B −1 A−1 . Applying this result to the inequality above, we complete the proof of item (i). To prove part (ii), take X = w ∈ Bn in Theorem 7.2. We obtain −1 −1 I + G(z)∗ G(w) + G(z)∗ G(w) G(z) + G(w)∗ I + G(z)

−1 (1 − z22 )(1 − w2 ) I + G(z)∗ G(z) I + G(z) . 2 4|1 − w, z|

Multiplying to the left by [I + G(z)∗ ]−1 and to the right by [I + G(z)]−1 , and passing to inverses, as above, we obtain the desired inequality. The proof is complete. 2 8. Lindelöf inequality and sharpened forms of the noncommutative von Neumann inequality In this section, we provide a noncommutative generalization of a classical inequality due to Lindelöf, which turns out to be sharper then the noncommutative von Neumann inequality. Theorem 8.1. If F : [B(H)n ]1 → [B(H)m ]− 1 is a free holomorphic function, then F (X) X + F (0) , 1 + XF (0)

X ∈ B(H)n 1 .

If, in addition, the boundary function of F has its entries in the noncommutative disc algebra An , then the inequality above can be extended to any X ∈ [B(H)n ]− 1. Proof. First, we consider the case when F (0) < 1. Using the first inequality of Corollary 6.2, in the particular case when a = 0, we obtain −1 −1 I − F (X)F (X)∗ I − F (0)F (X)∗ F (0) I − XX ∗ . F (0) I − F (X)F (0)∗ Hence, we deduce that I − F (X)F (X)∗

1 − X2 I − F (X)F (0)∗ I − F (0)F (X)∗ . 2 1 − F (0)

On the other hand, since F (0) < 1, the operator I − F (X)F (0)∗ is invertible and I − F (X)F (0)∗ −1 1 + F (X)F (0) + F (X)2 F (0)2 + · · · =

1 . 1 − F (X)F (0)

(8.1)


Similarly, we have [I − F (0)F (X)∗ ]−1

1 1−F (X)F (0)

1571

and deduce that

−1 −1 −1 −1 I − F (0)F (X)∗ I − F (X)F (0)∗ I − F (0)F (X)∗ I − F (X)F (0)∗ I

1 I. (1 − F (X)F (0))2

Hence, we obtain 2 I − F (X)F (0)∗ I − F (0)F (X)∗ 1 − F (X)F (0) I, which combined with inequality (8.1), leads to 2 1 − X2 I. 1 − F (X) F (0) F (X)F (X) 1 − 1 − F (0)2 ∗

This inequality implies 2 2 F (X)2 1 − 1 − X 1 − F (X)F (0) , 2 1 − F (0)

which is equivalent to 2 2 2

1 − F (X) 1 − F (0) 1 − X2 1 − F (X)F (0) . Straightforward calculations show that the latter inequality is equivalent to

F (X) − F (0) 2 X2 I − F (X)F (0) 2 . Hence, we obtain F (X) − F (0) X − XF (X)F (0), which is equivalent to F (X) X + F (0) , 1 + XF (0)

X ∈ B(H)n 1 .

Now, we consider the case when F (0) = 1. Applying our result above to F , where ∈ (0, 1), we get X + F (0) . F (X) 1 + XF (0) Taking → 0, the result follows. Now, consider the case when the boundary function of F has its entries in the noncommutative disc algebra An . According to [37], we have F (X) = lim F (rX1 , . . . , rXn ), r→1

− X = (X1 , . . . , Xn ) ∈ B(H)n 1 ,

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in the operator norm topology. Applying inequality of this theorem to the free holomorphic function X → F (rX1 , . . . , rXn ) and taking r → 1, we complete the proof. 2 A few remarks are necessary. First, notice that in the particular case when F (0) = 0, Theorem 8.1 implies the noncommutative Schwarz type result. We also remark that if F (0) < 1, then X + F (0) < 1, 1 + XF (0)

X ∈ B(H)n 1 .

Therefore, the inequality in Theorem 8.1 is sharper than the noncommutative von Neumann inequality, which gives only F (X) 1, when F ∞ 1. We recall that if F := (F1 , . . . , Fm ) is a contractive (F ∞ 1) free holomorphic function, then the evaluation map Bn z → F (z) ∈ Bm is a contractive matrix-valued multiplier of the Drury–Arveson space and, moreover, any such a contractive multiplier has this kind of representation. In particular, Theorem 8.1 implies that F (z) z + F (0) , 1 + zF (0)

z ∈ Bn ,

for any contractive multiplier F : Bn → Bm of the Drury–Arveson space. We consider now the particular case when m = 1. Here is a sharpened form of the noncommutative von Neumann inequality (see [32]). Corollary 8.2. If f : [B(H)n ]1 → B(H) is a nonconstant free holomorphic function with f ∞ 1, then f (X) X + |f (0)| < 1, 1 + X|f (0)|

X ∈ B(H)n 1 .

If, in addition, f is in the noncommutative disc algebra An , then the left inequality holds for any X ∈ [B(H)n ]− 1. Another consequence of Theorem 8.1 is the following. Corollary 8.3. Let F : [B(H)n ]1 → [B(H)m ]− 1 be a free holomorphic function and let z ∈ Bn , then F (X) Φz (X) + F (z) , 1 + Φz (X)F (z)

X ∈ B(H)n 1 ,

where Φz is the free holomorphic automorphism of the noncommutative unit ball [B(H)n ]1 associated z ∈ Bn . Proof. Applying Theorem 8.1 to the free holomorphic function F ◦Φz : [B(H)n ]1 → [B(H)m ]− 1, we obtain (F ◦ Φz )(Y ) Y + (F ◦ Φz )(0) , 1 + Y (F ◦ Φz )(0)

Y ∈ B(H)n 1 .


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Taking into account that Φz (0) = z, Φz ◦ Φz = id, and setting Y = Φz (X), X ∈ [B(H)n ]1 , in the inequality above, we obtain the desired inequality. 2 9. Pseudohyperbolic metric on the unit ball of B(H)n and an invariant Schwarz–Pick lemma The pseudohyperbolic distance on the open unit disc D := {z ∈ C: |z| < 1} of the complex plane is defined by z−w , z, w ∈ D. d1 (z, w) := 1 − z¯ w Some of the basic properties of the pseudohyperbolic distance are the following: (i) the pseudohyperbolic distance is invariant under the conformal automorphisms of D, i.e.,

d1 ϕ(z), ϕ(w) = d1 (z, w),

z, w ∈ D,

for all ϕ ∈ Aut(D); (ii) the d1 -topology induced on the open disc is the usual planar topology; (iii) any analytic function f : D → D is distance-decreasing, i.e., satisfies

d1 f (z), f (w) d1 (z, w),

z, w ∈ D.

The analogue of the pseudohyperbolic distance for the open unit ball of Cn , Bn := z = (z1 , . . . , zn ) ∈ Cn : z2 < 1 , is defined by dn (z, w) = ψz (w)2 ,

z, w ∈ Bn ,

where ψz is the involutive automorphism of Bn that interchanges 0 and z. This distance has properties similar to those of d1 (see [49,56]). In what follows, we introduce a pseudohyperbolic metric on the noncommutative ball [B(H)n ]1 , which satisfies properties similar to those of the pseudohyperbolic metric d1 on the unit disc D and which is a noncommutative extension of dn on the open unit ball of Cn . In particular, we obtain a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with respect to this pseudohyperbolic metric. H

B) if and We recall [45] that A, B ∈ [B(H)n ]− 1 are called Harnack equivalent (and denote A ∼ c only if there exists c 1 such that 1 Re p(B1 , . . . , Bn ) Re p(A1 , . . . , An ) c2 Re p(B1 , . . . , Bn ) c2 for any noncommutative polynomial with matrix-valued coefficients p ∈ C[X1 , . . . , Xn ] ⊗ H

Mm×m , m ∈ N, such that Re p 0. The equivalence classes with respect to ∼ are called Harnarck

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n parts of [B(H)n ]− 1 . We proved that the open unit ball [B(H) ]1 is a distinguished Harnack part − n of [B(H) ]1 , namely, the Harnack part of 0. Now, let be a Harnarck part of [B(H)n ]− 1 and define the map d : × → [0, ∞) by setting

d(A, B) :=

ω(A, B)2 − 1 , ω(A, B)2 + 1

A, B ∈ ,

(9.1)

where H ω(A, B) := inf c 1: A ∼ B , c

A, B ∈ .

The first result of this section is the following. Theorem 9.1. Let be a Harnarck part of [B(H)n ]− 1 . Then the map d defined by relation (9.1) has the following properties: (i) d is a bounded metric on ; (ii) for any free holomorphic automorphism Φ of the noncommutative unit ball [B(H)n ]1 ,

d(X, Y ) = d Φ(X), Φ(X) ,

X, Y ∈ .

Proof. According to Lemma 2.1 from [45], if is a Harnack part of [B(H)n ]− 1 and A, B, C ∈ , then the following properties hold: (a) ω(A, B) 1 and ω(A, B) = 1 if and only if A = B; (b) ω(A, B) = ω(B, A); (c) ω(A, C) ω(A, B)ω(B, C). Part (c) can be used to show that d(A, C) d(A, B) + d(B, C). . Since f (x) = Indeed, define the function f : [1, ∞) → [0, ∞) by f (x) := xx 2 −1 +1 we deduce that f is increasing. Hence, and due to inequality (c), we have 2

f ω(A, C) f ω(A, B)ω(B, C) . Since f (ω(A, C)) = d(A, C), it remains to prove that

f ω(A, B)ω(B, C) f ω(A, B) + f ω(B, C) . Setting x := ω(A, B) and y := ω(B, C), the inequality above is equivalent to x2y2 − 1 x2 − 1 y2 − 1 + . x2y2 + 1 x2 + 1 y2 + 1

2x (x 2 +1)2

0,


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Straightforward calculations reveal that the latter inequality is equivalent to 2 2

x y − 1 x 2 − 1 y 2 − 1 0, which holds for any x, y 1. Using (a) and (b), one can deduce that d is a metric. Now, we prove part (ii). According to Lemma 2.3 from [45], if A and B are in [B(H)n ]− 1, H

H

c

c

c 1, and Ψ ∈ Aut([B(H)n ]1 ), then A ≺ B if and only if Φ(A) ≺ Φ(B). Consequently, if A, B ∈ , then we deduce that ω(A, B) = ω(Φ(A), Φ(B)), which implies (ii). The proof is complete. 2 We introduced in [45] a hyperbolic (Poincaré–Bergman [6] type) metric δ on any Harnack part of [B(H)n ]− 1 by setting δ(A, B) := ln ω(A, B).

(9.2)

We will use the properties of δ to deduce the following result concerning the pseudohyperbolic distance on the open noncommutative ball [B(H)n ]1 . Theorem 9.2. The pseudohyperbolic metric d : [B(H)n ]1 × [B(H)n ]1 → [0, ∞) has the following properties: (i) for any X, Y ∈ [B(H)n ]1 , d(X, Y ) = tanh δ(X, Y ); (ii) d|Bn ×Bn coincides with the pseudohyperbolic distance on Bn , i.e., d(z, w) = ψz (w)2 , z, w ∈ Bn , where ψz is the involutive automorphism of Bn that interchanges 0 and z; (iii) the d-topology coincides with the norm topology on the open unit ball [B(H)n ]1 ; (iv) for any X, Y ∈ [B(H)n ]1 , d(X, Y ) :=

max{Γ , Γ −1 } − 1 , max{Γ , Γ −1 } + 1

where

∗

Γ := CX CY−1 CX CY−1 ,

CX := (X ⊗ I )(I − RX )−1 ,

and RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator. Proof. Part (i) follows from relations (9.1) and (9.2). Part (ii) follows from part (i) and the fact that, according to [45], δ|Bn ×Bn coincides with the Poincaré–Bergman distance on Bn , i.e., δ(z, w) =

1 1 + ψz (w)2 ln , 2 1 − ψz (w)2

z, w ∈ Bn ,

where ψz is the involutive automorphism of Bn that interchanges 0 and z.

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Since the δ-topology coincides with the norm topology on the open unit ball [B(H)n ]1 , part (i) implies (iii). In [45], we proved that δ(A, B) = ln max CA CB−1 , CB CA−1 ,

A, B ∈ B(H)n 1 ,

where CX := (X ⊗ I )(I − RX )−1 and RX := X1∗ ⊗ R1 + · · · + Xn∗ ⊗ Rn is the reconstruction operator. Hence and due to (i), part (iv) follows. 2 H

H

We showed in [45] that if A, B ∈ [B(H)n ]− 1 , then A ∼ B if and only if rA ∼ rB for any r ∈ [0, 1) and supr∈[0,1) ω(rA, rB) < ∞. Moreover, in this case, the function r → ω(rA, rB) is ancreasing on [0,1) and ω(A, B) = supr∈[0,1) ω(rA, rB). As a consequence, one can see H

that if A ∼ B, then the function r → d(rA, rB) are increasing on [0, 1) and d(A, B) = supr∈[0,1) d(rA, rB). This result together with Theorem 9.2 can be used to obtain an explicit formula for the pseudohyperbolic metric on any Harnack part of [B(H)n ]− 1. Now we provide a Schwarz–Pick lemma for free holomorphic functions on [B(H)n ]1 with operator-valued coefficients, with respect to the pseudohyperbolic metric. ¯ min B(E), j = 1, . . . , m, be free holomorphic funcTheorem 9.3. Let Fj : [B(H)n ]1 → B(H) ⊗ tions with coefficients in B(E), and assume that F := (F1 , . . . , Fm ) is a contractive free holoH

morphic function. If X, Y ∈ [B(H)n ]1 , then F (X) ∼ F (Y ) and

d F (X), F (Y ) d(X, Y ), where d is the pseudohyperbolic metric defined on the Harnack parts of [B(H)n ]− 1. H

Proof. According to the proof of Theorem 4.2 from [45], we have F (X) ∼ F (Y ) and

ω F (X), F (Y ) ω(X, Y ). Using the definition (9.1) and the fact that the function f (x) := [1, ∞), the result follows. 2

x 2 −1 x 2 +1

is increasing on the interval

If F := (F1 , . . . , Fm ) is a contractive (F ∞ 1) free holomorphic function with coefficients in B(E), then the evaluation map Bn z → F (z) ∈ B(E)(m) is a contractive operator-valued multiplier of the Drury–Arveson space, and any such a contractive multiplier has this type of representation. Corollary 9.4. Let F := (F1 , . . . , Fm ) be a contractive free holomorphic function with coeffiH

cients in B(E). If z, w ∈ Bn , then F (z) ∼ F (w) and

d F (z), F (w) d(z, w).


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Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor H. Bercovici a,∗,1 , B. Collins b,c,2 , K. Dykema d,1 , W.S. Li e,1 , D. Timotin f,3 a Department of Mathematics, Indiana University, Bloomington, IN 47405, USA b Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada c CNRS, Department of Mathematics, Université Claude Bernard, Lyon 1, Lyon, France d Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA e School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-1060, USA f Simion Stoilow Institute of Mathematics of the Romanian Academy, PO Box 1-764, Bucharest 014700, Romania

Received 24 June 2009; accepted 30 September 2009 Available online 9 October 2009 Communicated by D. Voiculescu

Abstract The intersection ring of a complex Grassmann manifold is generated by Schubert varieties, and its structure is governed by the Littlewood–Richardson rule. Given three Schubert varieties S1 , S2 , S3 with intersection number equal to one, we show how to construct an explicit element in their intersection. This element is obtained generically as the result of a sequence of lattice operations on the spaces of the corresponding flags, and is therefore well defined over an arbitrary field of scalars. Moreover, this result also applies to appropriately defined analogues of Schubert varieties in the Grassmann manifolds associated with a finite von Neumann algebra. The arguments require the combinatorial structure of honeycombs, particularly the structure of the rigid extremal honeycombs. It is known that the eigenvalue distributions of self-adjoint elements a, b, c with a + b + c = 0 in the factor Rω are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. In particular, if x, y, z are self-adjoint elements of such a factor and x + y + z = 0, then there exist self-adjoint a, b, c ∈ Rω such that a + b + c = 0 and a (respectively, b, c) has the same eigenvalue distribution as x (respectively, y, z). A (‘complete’) matricial form of * Corresponding author.

E-mail addresses: [email protected] (H. Bercovici), [email protected] (B. Collins), [email protected] (K. Dykema), [email protected] (W.S. Li), [email protected] (D. Timotin). 1 Supported in part by grants from the National Science Foundation. 2 Supported in part by an NSERC grant. 3 Supported in part by grant PNII of the Romanian Government – Programme Idei (code 1194). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.023

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H. Bercovici et al. / Journal of Functional Analysis 258 (2010) 1579–1627

this result is known to imply an affirmative answer to an embedding question formulated by Connes. The critical point in the proof of this result is the production of elements in the intersection of three Schubert varieties. When the factor under consideration is the algebra of n × n complex matrices, our arguments provide new and elementary proofs of the Horn inequalities, which do not require knowledge of the structure of the cohomology of the Grassmann manifolds. © 2009 Elsevier Inc. All rights reserved. Keywords: Schubert variety; Hive; Honeycomb; Factor

0. Introduction Assume that A, B, C are complex self-adjoint n × n matrices, and A + B + C = 0. A. Horn proposed in [23] the question of characterizing the possible eigenvalues of these matrices, and indeed he conjectured an answer which was eventually proved correct due to efforts of A. Klyachko [24] and A. Knutson and T. Tao [25]. To explain this characterization, list the eigenvalues of A, repeated according to multiplicity, in nonincreasing order λA (1) λA (2) · · · λA (n), choose an orthonormal basis xj ∈ Cn such that Axj = λA (j )xj , and denote by EA (j ) the space generated by {x1 , x2 , . . . , xj }. Horn’s conjecture involves, in addition to the trace identity n λA (j ) + λB (j ) + λC (j ) = 0, j =1

a collection of inequalities of the form i∈I

λA (i) +

λB (j ) +

j ∈J

λC (k) 0,

k∈K

where I, J, K ⊂ {1, 2, . . . , n} are sets with equal cardinalities. One way to prove such inequalities is to observe that Tr(P AP + P BP + P CP ) = 0 for any orthogonal projection P , and to find a projection P such that Tr(P AP )

i∈I

λA (i),

Tr(P BP )

λB (j ),

Tr(P CP )

j ∈J

λC (k).

k∈K

Now, if I = {i1 < i2 < · · · < ir }, the first condition is guaranteed provided that the range M of P has dimension r, and dim M ∩ EA (i ) ,

= 1, 2, . . . , r.


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These conditions describe the Schubert variety S(EA , I ) determined by the flag {EA ()}n=1 and the set I . Thus, such a projection can be found provided that S(EA , I ) ∩ S(EB , J ) ∩ S(EC , K) = ∅. Klyachko [24] proved that the collection of all inequalities obtained this way is sufficient to answer Horn’s question, and observed that Horn’s conjectured answer would also be proved if a certain ‘saturation conjecture’ were true. This conjecture was proved by Knutson and Tao [25]. (See also [27] for a direct proof of Horn’s conjecture, [26] for a brief presentation of the combinatorics of honeycombs, and [18] for an excellent survey of the problem, its history and ramifications. Another exposition is in [17], while [13] presents the state of the art before the work of Klyachko [24].) There are several infinite dimensional analogues of the Horn problem. One can for instance consider compact self-adjoint operators A, B, C on a Hilbert space and their eigenvalues. This analogue was considered by several authors [16,19], and a complete solution can be found in [5] for operators such that A, B, and −C are positive, and [6] for the general case. Without going into detail, let us say that these solutions are based on an understanding of the behavior of the Horn inequalities as the dimension of the space tends to infinity. The analogue we are interested in here replaces the algebra Mn (C) of n × n matrices by a finite factor. This is simply a self-adjoint algebra A of operators on a complex Hilbert space H such that A ∩ A = C1H (where A = {T : AT = T A for all A ∈ A}), A = A, and for which there exists a linear functional τ : A → C such that τ (X ∗ X) = τ (XX ∗ ) > 0 for all X ∈ A \ {0}. The algebras Mn (C) are finite factors. When A is an infinite dimensional finite factor, it is called a factor of type II1 . A complete flag in a II1 factor A is a family of orthogonal projections {E(t): 0 t τ (1H )} such that τ (E(t)) = t, and E(t) E(s) for t s. For any self-adjoint operator A ∈ A there exist a nonincreasing function λA : [0, τ (1H )] → R, and a complete flag {EA (t): 0 t τ (1H )} such that τ(1H )

A=

λA (t) dEA (t). 0

This is basically a restatement of the spectral theorem. The function λA is uniquely determined at its points of continuity, but the space EA (t) is not uniquely determined on the open intervals where λA is constant. Note that τ(1H )

τ (A) =

λA (t) dt, 0

and therefore we have a trace identity τ(1H )

λA (t) + λB (t) + λC (t) dt = 0

0

whenever A + B + C = 0.

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Many factors of type II1 can be approximated in a weak sense by matrix algebras. These are the factors that embed in the ultrapower Rω of the hyperfinite II1 factor R. For elements in such factors one can easily prove analogues of Horn’s inequalities. More precisely, assume that λ : [0, T ] → R is a nonincreasing function. The sequence λ(n) (1) λ(n) (2) · · · λ(n) (n) j T /n is defined by λ(n) (j ) = (j −1)T /n λ(t) dt. The following result was proved in [4]. We use the normalization τ (1H ) = 1 for the factor Rω . Theorem 0.1. Let α, β, γ : [0, 1] → R be nonincreasing functions. The following are equivalent: (1) There exist self-adjoint operators A, B, C ∈ Rω such that λA = α, λB = β, λC = γ , and A + B + C = 0. (2) For every integer n 1, there exist matrices An , Bn , Cn ∈ Mn (C) such that λAn = α (n) , λBn = β (n) , λCn = γ (n) , and An + Bn + Cn = 0. Note that condition (2) above requires, in addition to the trace identity, an infinite (and infinitely redundant) collection of Horn inequalities. We will show that these inequalities are in fact satisfied in any factor of type II1 . Theorem 0.2. Given a factor A of type II1 , self-adjoint elements A, B, C ∈ A such that A + B + C = 0, and an integer n 1, there exist matrices An , Bn , Cn ∈ Mn (C) such that λAn = (n) (n) (n) λA , λBn = λB , λCn = λC , and An + Bn + Cn = 0. The proof of the relevant inequalities relies, as in finite dimensions, on finding projections with prescribed intersection properties. In order to state our main result in this direction we need a more precise description of the Horn inequalities. Assume that the subsets I = {i1 < i2 < · · · < ir }, J = {j1 < j2 < · · · < jr }, and K = {k1 < k2 < · · · < kr } of {1, 2, . . . , n} satisfy the identity r (i − ) + (j − ) + (k − ) = 2r(n − r). =1

One associates to these sets a nonnegative integer cI J K , called the Littlewood–Richardson coefficient. The sets I , J , K yield an eigenvalue inequality in Horn’s conjecture if cI J K = 0. Moreover, as shown by P. Belkale [1], the inequalities corresponding with cI J K > 1 are in fact redundant. Thus, the preceding theorem follows from the next result. Theorem 0.3. Given a factor A of type II1 , self-adjoint elements A, B, C ∈ A such that A + B + C = 0, an integer n 1, and sets I, J, K ⊂ {1, 2, . . . , n} such that cI J K = 1, we have i∈I

(n)

λA (i) +

j ∈J

(n)

λB (j ) +

k∈K

(n)

λC (k) 0.


1583

This result follows from the existence of projections satisfying specific intersection requirements. Before stating our result in this direction, we need to specify a notion of genericity. Fix a finite factor A with trace normalized so that τ (1) = n. We will deal with flags of projections with integer dimensions, i.e., with collections E = {0 = E0 < E1 < · · · < En = 1} of orthogonal projections in A such that τ (Ej ) = j for all j . Given such a flag and a unitary operator U ∈ A, the projections U EU ∗ = {U Ej U ∗ : 0 j n} form another flag. In fact, all flags with integer dimensions are obtained this way. A statement about a collection of three flags {E, F , G} will be said to hold generically (or for generic flags) if it holds for the flags {U EU ∗ , V EV ∗ , W EW ∗ } with (U, V , W ) in a norm-dense open subset of U(A)3 (where U(A) denotes the group of unitaries in A). In finite dimensions, this set of unitaries can usually be taken to be Zariski open. Note that flags with integer dimensions always exist if A is of type II1 . If A = Mm (C), such flags only exist when n divides m. One final piece of notation. Given variables {ej , fj , gj : 1 j n}, we consider the free lattice L = L({ej , fj , gj : 1 j n}). This is simply the smallest collection which contains the given variables, and has the property that, given p, q ∈ L, the expressions (p) ∧ (q) and (p) ∨ (q) also belong to L. We refer to the elements of L as lattice polynomials; see [20] for more details on L. If p is a lattice polynomial and {Ej , Fj , Gj : 1 j n} is a collection of orthogonal projections in a factor A, we can substitute projections for the variables of p to obtain a new projection p({Ej , Fj , Gj : 1 j n}). The lattice operations are interpreted as usual: P ∨ Q is the projection onto the closed linear span of the ranges of P and Q, and P ∧ Q is the projection onto the intersection of the ranges of P and Q. Note that we did not impose any algebraic relations on L. When we work with flags, we can always reduce lattice polynomials using the relations ej ∧ ek = emin{j,k} and ej ∨ ek = emax{j,k} . Further manipulations are possible: for instance, the lattice of projections in a finite factor is modular, i.e. (P ∨ Q) ∧ R = P ∨ (Q ∧ R) provided that P R. As in finite dimensions, the Horn inequalities follow from the intersection result below. Given a flag E = (Ej )nj=0 ⊂ A such that τ (Ej ) = j , and a set I = {i1 < i2 < · · · < ir } ⊂ {1, 2, . . . , n}, we denote by S(E, I ) the collection of projections P ∈ A satisfying τ (P ) = r and τ (P ∧ Ei ) ,

= 1, 2, . . . , r.

Theorem 0.4. Given subsets I, J, K ⊂ {1, 2, . . . , n} with cardinality r, and with the property that cI J K = 1, a finite factor A with τ (1) = n, and arbitrary flags E = (Ej )nj=0 , F = (Fj )nj=0 , G = (Gj )nj=0 such that τ (Ej ) = τ (Fj ) = τ (Gj ) = j , the intersection S(E, I ) ∩ S(F , J ) ∩ S(G, K) is not empty. For generic flags, more is true.

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Theorem 0.5. Given subsets I, J, K ⊂ {1, 2, . . . , n} with cardinality r, and with the property that cI J K = 1, there exists a lattice polynomial p ∈ L({ej , fj , gj : 0 j n}) with the following property: for any finite factor A with τ (1) = n, and for generic flags E = (Ej )nj=0 , F = (Fj )nj=0 , G = (Gj )nj=0 such that τ (Ej ) = τ (Fj ) = τ (Gj ) = j , the projection P = p(E, F , G) has trace τ (P ) = r and, in addition τ (P ∧ Ei ) = τ (P ∧ Fj ) = τ (P ∧ Gk ) = when i i < i+1 , j j < j+1 , k k < k+1 and = 0, 1, . . . , r, where i0 = j0 = k0 = 0 and ir+1 = jr+1 = kr+1 = n + 1. When A = Mn (C), the existence and generic uniqueness of a projection P satisfying the trace conditions in the statement is well known. In fact cI J K serves as an algebraic way to count these projections. Our argument works equally well for linear subspaces of Fn for any field F (except that orthogonal complements 1 − P must be replaced by annihilators in the dual). In the following result, a generic set is simply an open set in the Zariski topology, that is its complement is determined by a finite number of polynomial equations. The result is generally false when cI J K > 1. Theorem 0.6. Fix a field F, and complete flags E = (Ej )nj=0 , F = (Fj )nj=0 , G = (Gj )nj=0 of subspaces in Fn . Given subsets I, J, K ⊂ {1, 2, . . . , n} with cardinality r, and with the property that cI J K = 1, there exists a subspace M ⊂ Fn such that dim M = r, and dim(M ∩ Ei ) ,

dim(M ∩ Fj ) ,

dim(M ∩ Gk )

for = 1, 2, . . . , r. For generic flags E, F , G, such a space M can be obtained by evaluating a lattice polynomial on the spaces of these flags. When F is either C or R, the preceding result follows, of course, from the structure of the cohomology ring of the Grassmannian (calculated modulo 2 in the real case). The new feature in our proof is the explicit construction of M via a lattice polynomial, with appropriate adjustments in nongeneric situations. Naturally, this leads to new proofs of the Horn inequalities for sums of self-adjoint or real symmetric matrices. These proofs proceed by producing explicitly, via lattice operations, a ‘witness’ projection in the intersection of three Schubert cells, and thus rely on methods that were available when this type of problem was first approached by Weyl [34]. An immediate consequence of Theorems 0.1 and 0.2 is the following result, where all traces are normalized so that τ (I ) = 1. Theorem 0.7. Given a factor A of type II1 and self-adjoint elements A, B ∈ A, there exist selfadjoint A , B ∈ Rω such that λA = λA , λB = λB and λA +B = λA+B . As shown in [9], this result is intimately related with the Connes embedding problem [11], which asks whether every separably acting factor of type II1 can be embedded in Rω . It was shown in [9] that this problem can be reformulated as follows: given a factor A of type II1 , self-adjoint elements A, B ∈ A, and complex self-adjoint n × n matrices α, β, do there exist selfadjoint A , B ∈ Rω such that λα⊗A = λα⊗A , λβ⊗B = λβ⊗B and λα⊗A +β⊗B = λα⊗A+β⊗B ?


1585

The preceding result answers this in the affirmative when n = 1. The answer to this question for larger values of n will certainly require new ideas. The search for projections P satisfying the conclusion of Theorem 0.4 is much more difficult when cI J K > 1. One of the simplest cases of this problem is equivalent to the invariant subspace problem relative to a II1 factor A; this case was first discussed in [10] where an approximate solution is found. This relative invariant subspace problem remains open but there was spectacular progress in the work of U. Haagerup and H. Schultz [21,22]. The function λA can be defined more generally for a self-adjoint element of a von Neumann algebra A endowed with a faithful, normal trace τ . The inequalities in Theorem 0.3 are in fact true in this more general context. Rather than prove this fact directly, we can embed any such von Neumann algebra in a II1 factor, in such a way that the trace is preserved. The existence of such an embedding was proved in [15]; we also provide an argument based on free probability theory in Section 8. An alternative proof can be obtained using von Neumann’s reduction theory. The remainder of this paper is organized as follows. In Section 1 we describe an enumeration of the sets I, J, K with cI J K > 0 in terms of a class of measures on the plane. This enumeration is essentially the one indicated in [27]; the measures we use can be viewed as the second derivatives of hives, or the first derivatives of honeycombs. (The fact that honeycombs provide an equivalent form of the Littlewood–Richardson rule is proved in a direct way in the appendix of [8]; see also Tao’s ‘proof without words’ illustrated in [31].) We also describe the duality observed in [27, Remark 2 on p. 42], realized by inflation to a puzzle, and ∗-deflation to a dual measure. In Section 2, we use then the puzzle characterization of rigidity from [27] to formulate the condition cI J K = 1 in terms of the support of the corresponding measure m. This result may be viewed as the N = 0 version of [27, Lemma 8]. This characterization is used in Section 3 to show that a measure m corresponding to sets with cI J K = 1 (also called a rigid measure) can be written uniquely as a sum m1 + m2 + · · · + mp of extremal measures, and to introduce a partial order relation ‘≺’ on the set {mj : 1 j p}. In Section 4 we provide an extension of the concept of clockwise overlay from [27], and show that ‘≺’ provides examples of clockwise overlays. The main results are proved in Section 5. The most important observation is that general Schubert intersection problems can be reduced to problems corresponding to extremal measures. The order relation is essential here as the minimal measures mj (relative to ‘≺’) must be considered first. An intersection problem corresponding to an extremal measure has then a dual form (obtained by taking orthogonal complements) which is no longer extremal, except for essentially one trivial example. Section 6 contains a number of illustrations of this reduction procedure, including explicit expressions for the corresponding lattice polynomials which yield the solution for generic flags. In Section 7 we describe a particular intersection problem which is equivalent to the invariant subspace problem relative to a II1 factor. In Section 8 we embed any algebra with a trace in a factor of type II1 , and we show that projections can be moved to general position by letting them evolve according to free unitary Brownian motion. The latter procedure was used earlier by Voiculescu [33] in the process of liberation. Here the goal is merely to obtain projections in general position, and the argument is correspondingly simpler. There has been quite a bit of recent work on the geometry and intersection of Schubert cells. Belkale [2] shows that the inductive structure of the intersection ring of the Grassmannians can be justified geometrically. R. Vakil [31] provides an approach to the structure of this ring by a process of flag degenerations. He also indicates [32,31] that this can be used in order to solve effectively all Schubert intersection problems, at least for generic flags. More precisely, [32, Remark 2.10] suggests that these solutions can be found, after an appropriate parametrization, by an application of the implicit function theorem which can be made numerically effective.

1586


These methods would apply to arbitrary values of cI J K , and it is not clear that they would yield the formulas of Theorem 0.5 when cI J K = 1. The method of flag degeneration of [31] depends essentially on finite dimensionality. A prospective analogue in a II1 factor would require a checkerboard with a continuum of squares, and would be played with a continuum of pieces. This kind of game is difficult to organize, and some of the difficulties may be illustrated by the cumbersome argument used in [3]. The result proved with so much labor in that paper is deduced very simply from our current methods, as shown in Section 6 below. One might view Theorem 0.5 as the beginning of an intersection theory for the Grassmannian G(r) = {P ∈ A: P = P ∗ = P 2 , τ (P ) = r} associated to a II1 factor A and a number r ∈ (0, τ (1)). This would be closer in spirit to algebraic intersection theory, since varieties of the form S(E, I ) cannot generally be viewed as cycles in a classical homology theory. As pointed out earlier, a thorough study of such an intersection theory will require significant advances in our understanding of factors of type II1 . 1. Horn inequalities and measures Fix integers 1 r n, and subsets I, J, K ⊂ {1, 2, . . . , n} of cardinality r. We will find it useful on occasion to view the set I as an increasing function I : {1, 2, . . . , r} → {1, 2, . . . , n}, i.e., I = {I (1) 0 if and only if there exist self-adjoint matrices X, Y, Z ∈ Mr (C) such that X + Y + Z = 2(n − r)1Cr and λX (r + 1 − ) = I () − , λY (r + 1 − ) = J () − , λZ (r + 1 − ) = K() − for = 1, 2, . . . , r. Thus, as conjectured by Horn, such sets can be described inductively, using Horn inequalities with fewer terms. In other words, we have cI J K > 0 if and only if r I () − + J () − + K() − = 2r(n − r), =1

and s I I () − I () + J J () − J () + K K () − K () 2s(n − r) =1

whenever s ∈ {1, 2, . . . , r − 1} and I , J , K ⊂ {1, 2, . . . , r} are sets of cardinality s such that cI J K > 0. The last inequality can also be written as s I I () − + J J () − + K K () − 2s(n − s). =1

The numbers cI J K can be calculated using the Littlewood–Richardson rule which we discuss next. We use the form of the rule described in [27], so we need first to describe a set of measures on the plane. Begin by choosing three unit vectors u, v, w in the plane such that u + v + w = 0.


1587

Consider the lattice points iu + j v with integer i, j . A segment joining two nearest lattice points will be called a small edge. We are interested in positive measures m which are supported by a union of small edges, whose restriction to each small edge is a multiple of linear measure, and which satisfy the balance condition (called zero tension in [27]) m(AB) − m(AB ) = m(AC) − m(AC ) = m(AD) − m(AD )

(1.1)

whenever A is a lattice point and the lattice points B, C , D, B , C, D are in cyclic order around A.

If e is a small edge, the value m(e) is equal to the density of m relative to linear measure on that edge. Fix now an integer r 1, and denote by r the (closed) triangle with vertices 0, ru, and ru + rv = −rw. We will use the notation Aj = j u, Bj = ru + j v, and Cj = (r − j )w for the lattice points on the boundary of r . We also set Xj = Aj + w,

Yj = Bj + u,

Zj = Cj + v

for j = 0, 1, 2, . . . , r + 1. The following picture represents 5 and the points just defined; the labels are placed on the left. Given a measure m, a branch point is a lattice point incident to at least three edges in the support of m. We will only consider measures with at least one branch point. This excludes measures whose support consists of one or more parallel lines. We will denote by Mr the collection of all measures m satisfying the balance condition (1.1), whose branch points are contained in r , and such that m(Aj Xj +1 ) = m(Bj Yj +1 ) = m(Cj Zj +1 ) = 0,

j = 0, 1, . . . , r.

The numbers m(Aj Xj ),

m(Bj Yj ),

m(Cj Zj ),

j = 0, 1, . . . , r,

will be called the exit densities of the measure m ∈ Mr . Analogously M∗r consists of measures m whose branch points are contained in Mr , and such that m(Aj Xj ) = m(Bj Yj ) = m(Cj Zj ) = 0,

j = 0, 1, . . . , r,

1588


Fig. 1. Boundary of r .

and the exit densities of such a measure are the numbers m(Aj Xj +1 ),

m(Bj Yj +1 ),

j = 0, 1, . . . , r.

m(Cj Zj +1 ),

Clearly, M∗r can be obtained from Mr by reflection relative to one of the angle bisectors of r . Given a measure m ∈ Mr , we define its weight ω(m) ∈ R+ to be

ω(m) =

r

m(Aj Xj ) =

j =0

r

m(Bj Yj ) =

j =0

r

m(Cj Zj ),

j =0

and its boundary sum ∂m = (α, β, γ ) ∈ (Rr )3 , where

α =

−1

m(Aj Xj ),

j =0

β =

−1

m(Bj Yj ),

j =0

γ =

−1

m(Cj Zj ),

= 1, 2, . . . , r.

j =0

The equality of the three sums giving ω(m) is an easy consequence of the balance condition. The results of [25,27] imply that the sets I, J, K ⊂ {1, 2, . . . , n} of cardinality r satisfy cI J K > 0 if and only if there exists a measure m ∈ Mr with weight ω(m) = n − r, and with boundary sum ∂m = (α, β, γ ) such that α = I () − ,

β = J () − ,

γ = K() − ,

= 1, 2, . . . , r.

The number cI J K is equal to the number of measures in Mr satisfying these conditions, and with integer densities on all edges. Moreover, as shown in [27], if cI J K = 1, there is only one measure m satisfying these conditions, and its densities must naturally be integers. In general, we will say that a measure m ∈ Mr is rigid if it is entirely determined by its exit densities or, equivalently, by its weight and boundary sum.


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We will also use the version of these results in terms of M∗r , so we define for m ∈ M∗r the weight ω(m) =

r

m(Aj Xj +1 ) =

j =0

r

m(Bj Yj +1 ) =

j =0

r

m(Cj Zj +1 ),

j =0

and boundary sum ∂m = (α, β, γ ), where α =

r j =r+1−

m(Aj Xj +1 ),

β =

r

m(Bj Yj +1 ),

γ =

j =r+1−

r

m(Cj Zj +1 )

j =r+1−

for = 1, 2, . . . , r. Measures in Mr or M∗r are entirely determined by their restrictions to r and, when the corners of r are not branch points, even by their restrictions to the interior of r . Indeed, the lack of branch points outside r implies that the densities are constant on the half-lines starting with Aj Xj , Bj Yj and Cj Zj . Note that a restriction m|r with m ∈ Mr is not generally of the form m |r for some m ∈ M∗r . The first picture below represents r (dotted lines), and the support (solid lines) of a measure in r . The second one represents the support of a measure in M∗r .

To conclude this section, we establish a connection between measures in Mr and the honeycombs of [25]. A honeycomb is a real-valued function h defined on the set of small edges contained in r satisfying the following two properties: (i) If ABC is a small triangle contained in r , we have h(AB) + h(AC) + h(BC) = 0. (ii) If A, B, C, D are lattice vertices in r such that B = A + u, C = A − v, D = A + w (or B = A + v, C = A − w, D = A + u, or B = A + w, C = A − u, D = A + v), then h(AB) − h(CD) = h(BC) − h(AD) 0. The reason for the term honeycomb is not apparent in our definition. One can associate to each small triangle ABC ⊂ r , whose sides AB, BC, CA are parallel to u, v, w, respectively, the point (h(AB), h(BC), h(AC)) in the plane {(x, y, z) ∈ R3 : x + y + z = 0}. These points form the vertices of a graph, with two adjacent triangles determining an edge. This graph looks like a honeycomb if it is not too degenerate (cf. [25]). The following result will be required for our discussion of the Horn inequalities in Section 4. Lemma 1.1. Let m ∈ Mr be a measure with weight ω and ∂m = (α, β, γ ). There exists a honeycomb h with the following properties.

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(1) h(A−1 A ) = α − 2ω/3, h(B−1 B ) = β − 2ω/3, h(C−1 C ) = γ − 2ω/3 for = 1, 2, . . . , r. (2) If B = A + u, C = A − v, D = A + w (or B = A + v, C = A − w, D = A + u, or B = A + w, C = A − u, D = A + v) then h(AB) − h(CD) = m(AC). Proof. This is routine. Condition (2) allows us to calculate all the values of h starting from the boundary of r . To verify (ii) one must use the balance condition for measures in Mr . 2 2. Inflation, duality, and rigidity Let A be a finite factor with trace normalized so that τ (1) = n, and let E = {E : = 0, 1, . . . , n} be a flag so that τ (E ) = for all . Fix also a set I ⊂ {1, 2, . . . , n} of cardinality r and a projection P ∈ A with τ (P ) = r. We have P ∈ S(E, I ), i.e. τ (P ∧ EI () ) for = 1, 2, . . . , r, if and only if τ (P ∧ E ) ϕI () for = 0, 1, . . . , n, where ϕI () = p

if I (p) < I (p + 1)

for p = 0, 1, . . . , r, and I (0) = 1, I (r + 1) = n + 1. With the notation P ⊥ = 1 − P , these conditions imply τ P ⊥ ∧ E⊥ = n − τ (P ∨ E ) = n − τ (P ) − τ (E ) + τ (P ∧ E ) n − r − + ϕI (). ⊥ : = 0, 1, This implies that P ∈ S(E, I ) if and only if P ⊥ ∈ S(E ⊥ , I ∗ ), where E ⊥ = {En− ∗ / I }. In general, we will have cI ∗ J ∗ K ∗ = cI J K , and this equality . . . , n}, and I = {n + 1 − i: i ∈ is realized by a duality considered in [27]. More precisely, assume that m ∈ Mr . We define the inflation of m as follows. Cut r along the edges in the support of m to obtain a collection of (white) puzzle pieces, and translate these pieces away from each other in the following way: the parallelogram formed by the two translates of a side AB of a white puzzle piece has two sides of length equal to the density of m on AB and 60◦ clockwise from AB. The original puzzle pieces and these parallelograms fit together, and leave a space corresponding to each branch point in the support of m. Here is an illustration of the process; the thinner lines in the support of the measure have density one, and the thicker ones density 2. The original pieces of the triangle r are white, the added parallelogram pieces are dark gray, and the branch points become light gray pieces. Each light gray piece has as many sides as there are branches at the original branch point (counting the branches outside r , which are not represented in this figure, though their number and densities are dictated by the balance condition, and the fact that m belongs to Mr ). The dark gray parallelograms will be referred to simply as dark parallelograms.


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The original triangle r has been inflated to a triangle of size r + ω(m), and the decomposition of this triangle into white, gray, and light gray pieces is known as the puzzle associated to m. The sides of each dark parallelogram will be colored white or light gray, according to the color of the neighboring puzzle piece. A side contained in ∂r+ω(m) is given the color of the opposite side. Thus, each dark parallelogram has two light gray sides, i.e. sides bordering a light gray piece, and two white sides. The length of the light gray side equals the density of the white sides in the original support of m in r . This process can be applied to the entire support of m, but we are only interested in its effect on r . (The white regions in the puzzle are called ‘zero regions’, and the light gray ones ‘one regions’ in [27]. In our drawings we use a different color scheme from the one in [27].) We can now apply a dual deflation, or ∗-deflation, to the puzzle of m as follows: discard all the white pieces, and shrink the dark parallelograms by reducing their white sides to points. The segments obtained this way are assigned densities equal to the lengths of the white sides of the corresponding dark parallelograms. In the picture below, the shrunken dark parallelograms are represented as solid lines.

The result of this deflation is a triangle with sides ω(m), endowed with a measure supported by the solid lines which will be denoted m∗ . The support of m∗ can be obtained directly from the support of m as follows: take every edge of a white puzzle piece, rotate it 60◦ clockwise, and change its length to the density of m on the original edge. The new segments must now be translated so that the edges originating from the sides of a white puzzle piece become incident. Thus, the dual picture depends primarily on the combinatorial structure of the support of m. More precisely, let us say that the measures m ∈ Mr and m ∈ Mr are homologous if there is a bijection between the edges determined by the support of m and the edges determined by the support of m such that corresponding edges are parallel, and incident edges correspond with incident edges (the intersection point being precisely the one dictated by the correspondence of the edges). Then m and m are homologous if and only if m∗ and m ∗ are homologous. For instance, measures in Mr that have the same support are homologous. Assume now that the measure m ∈ Mr has integer densities, and I, J, K are the corresponding sets in {1, 2, . . . , n = r + ω(m)}. Then the triangle obtained by inflating m can be identified with n . Under this identification the small edges Ai−1 Ai are either white (if they border a white piece, or they belong to a white edge of a dark parallelogram) or light gray. It is easy to see that the white small edges are precisely Ai −1 Ai for = 1, 2, . . . , r, and therefore the light gray edges correspond with the complement of I . Furthermore, the light gray triangle obtained by ∗deflation can be identified with n−r , and m∗ ∈ M∗n−r is a measure satisfying ω(m∗ ) = r. This measure determines subsets of {1, 2, . . . , n} which are precisely I ∗ , J ∗ , K ∗ . This observation gives a bijective proof of the equality cI J K = cI ∗ J ∗ K ∗ . The passage from m to m∗ can be reversed by applying ∗-inflation to m∗ , and then applying deflation to the resulting puzzle. Another important application of the inflation process is a characterization of rigidity. Orient the edges of the dark parallelograms in a puzzle so that they point away from the acute angles. Some of the border edges do not have a neighboring dark parallelogram and will not be oriented.

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It was shown in [27] that a measure m is rigid if and only if the associated directed graph contains no gentle loops, i.e., loops which never turn more than 60◦ . Note that the number and relative position of the puzzle pieces depend only on the support of the measure m. The following result follows immediately. Proposition 2.1. Let m1 , m2 ∈ Mr be such that the support of m1 is contained in the support of m2 . If m2 is rigid then m1 is rigid as well. It is also easy to see that the support of a rigid measure does not contain six edges which meet at the same point. Indeed, the inflation reveals immediately a gentle loop.

We will need to characterize rigidity in terms of the support of the original measure. Let A1 A2 · · · Ak A1 be a loop consisting of small edges Aj Aj +1 contained in the support of a measure m ∈ Mr . We will say that this loop is evil if each three consecutive points Aj −1 Aj Aj +1 = ABC form an evil turn, i.e. one of the following situations occurs: (1) C = A, and the small edges BX, BY , BZ which are 120◦ , 180◦ , and 240◦ clockwise from AB are in the support of m. (2) BC is 120◦ clockwise from AB. (3) C = A and A, B, C are collinear. (4) BC is 120◦ counterclockwise from AB and the edge BX which is 120◦ clockwise from AB is in the support of m. (5) BC is 60◦ counterclockwise from AB and the edges BX, BY which are 120◦ and 180◦ clockwise from AB are in the support of m. Proposition 2.2. A measure m ∈ Mr is rigid if and only if its support contains none of the following configurations: (1) Six edges meeting in one lattice point. (2) An evil loop. Proof. Assume first that m is not rigid, and consider a gentle loop of minimal length in its puzzle. Use ai to denote white parallelogram sides in this loop, and bi light gray parallelogram sides. The sides ai , bi may consist of several small edges. The gentle loop is of one of the following three forms:


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(1) a1 a2 a3 a4 a5 a6 , (2) b1 b2 b3 b4 b5 b6 , (3) a1 a2 · · · ai1 b1 b2 · · · bj1 ai1 +1 · · · ai2 bj1 +1 · · · bj2 · · · aip−1 +1 · · · aip bjp−1 +1 · · · bjp , with at most five consecutive a or b symbols. In case (1), the loop runs counterclockwise around a white piece, and it deflates to a translation of itself which is obviously evil. In case (2), the loop deflates to a single point where six edges in the support of m meet. In case (3), the loop deflates to a1 a2 · · · aip , where each aj is a translate of aj . The turns in this loop are obtained by deflating a path of the form a1 b1 · · · bj a2 with 0 j 5. The edges b1 · · · bj run clockwise around a light gray piece which must have at most five edges because the gentle loop was taken to have minimal length. When j = 0, the edges a1 and a2 border the same white puzzle piece, and it is obvious that a1 a2 is an evil turn. The remaining cases will be enumerated according to the number of edges in the light gray piece next to the edges bj . When this piece is a triangle, we can only have j = 1, and the situation is illustrated below. The dashed line in the deflation indicates a portion of the support of m.

When the light gray piece is a parallelogram, we have j = 1 or j = 2. The three possible deflations are as follows.

Next, the light gray piece may be a trapezoid, and 1 j 3. For j = 1 we have these four possibilities:

For j = 2, 3 there are three more possibilities.

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Finally, if the light gray piece is a pentagon, there are five situations when j = 1,

four situations when j = 2,

three situations when j = 3,

and three more when j = 4, 5.


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Of course, the last case does not occur in a minimal gentle loop. Thus, in all situations, the deflated turns are evil, and therefore the support of m contains an evil loop. Conversely, if the support of m contains an evil loop, the above figures show that one can obtain a gentle loop in the puzzle of m. 2 The following figure represents the support of a rigid measure in M8 , along with a loop which may seem evil but is not evil when traversed in either direction.

3. Extremal measures and skeletons For fixed r 1, the collection Mr is a convex polyhedral cone. Recall that a measure m ∈ Mr is extremal (or belongs to an extreme ray) if any measure m m is a positive multiple of m. The support of an extremal measure will be called a skeleton. Clearly, an extremal measure is entirely determined by its value on any small edge contained in its skeleton. Checking extremality is easily done by using the balance condition (1.1) at all the branch points of the support to see how the density propagates from one edge to the rest of the support. In the following figure of a skeleton, the thicker edges must be assigned twice the density of the other edges. (This skeleton contains an evil loop, hence the measures it supports are not rigid.)

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It is not always obvious when a collection of edges supports a nonzero measure in Mr . The reader may find it amusing to verify that the following figure represents sets which do not support any nonzero measure.

We will be mostly interested in the supports of rigid extremal measures, which we will call rigid skeletons. When r = 1, there are only the three possible skeletons, all of them rigid, pictured below.

The following figure shows some rigid skeletons for r = 2, 3, 4, 5.

A greater variety of rigid skeletons is available for r = 6. In addition to larger versions (plus rotations and reflections) of the above skeletons, we have the ones in the next figure.

For larger r, rigid skeletons can be quite involved. We provide just one more example for r = 8. This skeleton has edges with densities 2 and 3 which we did not indicate.


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An important consequence of the characterization of rigid measures in Mr is the fact that such measures can be written uniquely as sums of extremal measures; see Corollary 3.6. Fix a rigid measure m ∈ Mr , and let e = AB and f = BC be two distinct small edges. We will write e →m f , or simply e → f when m is understood, if one of these two situations arises: (1) ABC = 120◦ and the edge opposite e at B has m measure equal to zero; (2) e and f are opposite, and one of the edges making an angle of 60◦ with f has m measure equal to zero. Note that in both cases we may also have f → e. The significance of this relation is that e → f implies that m(e) m(f ), with strict inequality unless f → e as well. More precisely, if e → f and we do not have f → e, there is at least one edge g such that g → f and the angle between e and g is 60◦ . In this case we have m(f ) = m(e) + m(g).

(3.1)

Indeed, if e and f are opposite, the edge opposite g must have m measure equal to zero. A useful observation is that if XY → Y Z but Y Z XY , then the edges Y A, Y B, Y C which are 120◦ , 180◦ , 240◦ clockwise from Y Z must be in the support of m. In other words, ZY Z is an evil turn. The following result is a simple consequence of the fact that the measure m exists. Lemma 3.1. Assume that a sequence of edges e1 , e2 , . . . , en in the support of m is such that e1 → e2 → e3 → · · · → en → e1 . Then we also have en → en−1 → · · · → e1 → en . Proof. Indeed, if one of the arrows cannot be reversed, then m(e1 ) < m(e1 ).

2

We can therefore define a preorder relation on the set of small edges as follows. Given two small edges e, f , we write e ⇒ f if either e = f , or there exist edges e1 , e2 , . . . , en such that e = e1 → e2 → · · · → en = f.

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In this case, we will say that f is a descendant of e and e is an ancestor of f . Two edges are equivalent, e ⇔ f , if e ⇒ f and f ⇒ e. The relation of descendance becomes an order relation on the equivalence classes of small edges. An edge e will be called a root edge (or simply a root) if m(e) = 0 and e belongs to a minimal class relative to descendance. Clearly, every edge in the support of m is a descendant of at least one root. If m(e) = 0 and e ⇒ f , then there exists a path A0 A1 · · · Ak in the support of m such that Aj −1 Aj → Aj Aj +1 for all j , A0 A1 = e, and Ak−1 Ak = f . Paths of this form will be referred to as descendance paths from e to f . All the turns Aj −1 Aj Aj +1 and Aj +1 Aj Aj −1 in a descendance path are evil. Lemma 3.2. Assume that e and f are in the support of a rigid measure m ∈ Mr , and A0 A1 · · · Ak and B0 B1 · · · B are two descendance paths from e to f . Then Ak−1 = B−1 and Ak = B . Proof. Assume to the contrary that Ak−1 = B . There are two cases to consider, according to whether A0 = B1 or A0 = B0 . In the first case, the loop A0 A1 · · · Ak−1 B−1 B−2 · · · B1 is evil, contradicting the rigidity of m. In the second case, there is a first index p such that Ap+1 = Bp+1 . Then the loop Ap Ap+1 · · · Ak−1 B−1 B−2 · · · Bp is evil, yielding again a contradiction.

2

Lemma 3.3. Let e and e be inequivalent root edges in the support of a rigid measure m ∈ Mr , and let f be an edge which is a descendant of both e and e . Consider a descendance path A0 A1 · · · Ak from e to f , and a descendance path B0 B1 · · · B from e to f . Then Ak−1 = B−1 and Ak = B . Proof. Assume to the contrary that Ak = B−1 . The edge f is not equivalent to either e or e . Therefore there exist indices p, q such that Ap Ap+1 Ap−1 Ap and Bq Bq+1 Bq−1 Bq . It follows that Ap Ap+1 · · · Ak−1 B−1 B−2 · · · Bq Bq−1 · · · B−2 B−1 Ak−1 · · · Ap+1 Ap is an evil loop, contradicting rigidity.

2

These lemmas show that all the non-root edges in the support of a rigid measure m can be given an orientation. More precisely, given a relation e ⇒ f with e a root edge, choose a descendance path A0 A1 · · · Ak from e to f , and assign f the orientation Ak−1 Ak . This will be called the orientation of f away from the root edges. Any common edge of two skeletons in the support of m can be oriented away from the root edges; indeed, such an edge is not a root edge. In the proofs of the next two results, we will be concerned with the descendants of a fixed root edge e, and it will be convenient to orient the other root edges equivalent to e away from e. The edge e can be oriented either way, as needed.


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Lemma 3.4. Fix a root edge in the support of a rigid measure m, and suppose that two descendants f = CX and g = DX have orientations pointing toward X. Then the turns CXD and DXC are not evil. In particular, the angle between f and g is 60◦ , and at least one of the edges C X, D X opposite f and g has m measure equal to zero. Proof. Assume to the contrary that either CXD or DXC is an evil turn. The assumed orientations imply that f g and g f . Since one of the edges incident to X must have measure zero, it follows f and g are not collinear. Moreover, the edges f = XC and g = XD opposite to f and g, respectively, must be in the support of m; in the contrary case we would have f → g or g → f if the angle between f and g is 120◦ , or the turn CXD would not be evil if the angle is 60◦ . Let A0 A1 · · · Ak and B0 B1 · · · B be descendance paths from e to f and g, respectively. By assumption, we have Ak−1 = C, B−1 = D, and Ak = B = X. If A0 = B0 , then A0 = B1 , A1 = B0 , and clearly A0 A1 · · · Ak B−1 B−2 · · · B1 or its reverse is an evil loop, contradicting the rigidity of m. Thus we must have A0 = B0 . Let p be the largest integer such that Aj = Bj for j p. If p < min{k, }, the loop Ap Ap+1 · · · Ak B−1 B−2 · · · Bp or its reverse is evil. Indeed, since Ap−1 Ap → Ap Ap+1 and Ap−1 Ap → Bp Bp+1 , the turns Ap+1 Bp Bp+1 and Bp+1 Bp Ap+1 are both evil. We conclude that p = min{k, }. If p = k, it follows that the Bk−1 Bk · · · B is a descendance path from f to g. Since f g, we must have Bk+1 = C , and then the loop Bk Bk+1 · · · B or its reverse is obviously evil, leading to a contradiction. The case p = similarly leads to a contradiction. 2 Theorem 3.5. Let m ∈ Mr be a rigid measure, and e a root edge in the support of m. Then the collection of all descendants of e is a skeleton. Proof. Since m can be written as a sum of extremal measures, there exists an extremal measure m m such that m (e) = 0. Since f →m g implies that f →m g, the support of m is a skeleton containing all the descendants of e. Therefore it will suffice to show that the descendants of e form the support of some measure in Mr . We set μ(e) = 1, μ(f ) = 0 if f is not a descendant of e, and for each descendant f = e of e we define μ(f ) to be the number of descendance paths from e to f . Clearly, no edge occurs twice in such a path; such an occurrence would imply the existence of an evil loop. Thus the number μ(f ) is finite. To conclude the proof, it suffices to show that μ ∈ Mr . The support of μ is contained in the support of m. Therefore all the branch points are in r , and μ(Aj Xj +1 ) = μ(Bj Yj +1 ) = μ(Cj Zj +1 ) = 0

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for all j . It remains to verify the balance conditions. Consider a lattice point X in r . If no descendant of e is incident to X, the six edges meeting at X have μ measure zero, and the balance condition is trivial. Otherwise, the number of descendants of e incident to X can be 1, 2, 3, 4 or 5; the value 6 is excluded by the rigidity of m. We first exclude the case where this number is 1. Assume indeed that there is only one descendant incident to X, and let A0 A1 · · · Ak be a descendance path from e with Ak = X. Since Ak−1 Ak has no descendants of the form XY , it follows that the turn Ak−1 Ak Ak−1 is evil. On the other hand, since rigidity of m insures that one of the edges around X has measure zero, the edge Ak−1 Ak is a strict descendant of some other edge XZ. In particular, Ak−1 Ak is not a root edge, and therefore it is not equivalent to e. It follows that, for some p, we do not have Ap Ap+1 → Ap−1 Ap , and this implies that the turn Ap+1 Ap Ap+1 is evil. Thus Ap Ap+1 · · · Ak−1 Ak Ak−1 · · · Ap+1 Ap is an evil loop, contrary to the rigidity of m. Consider now the case when there are exactly two descendants of e incident to X, call them f and g. They cannot both point away from X since this would require the existence of a third descendant pointing toward X. They cannot both point toward X. Indeed, if this were the case, Lemma 3.4 insures that one of the edges opposite f and g has measure zero, and therefore f or g has another descendant pointing away from X. Thus we can assume that f points toward X, and g away from X, in which case we have f → g. Then f and g must be collinear, and every descendance path for g passes through f . Thus μ(f ) = μ(g), which is the required balance condition. Assume next that there are exactly three descendants incident to X. Two of them must be noncollinear and of the form W X → XY , and the third descendant must be W X → XZ, with the three edges forming 120◦ angles. Every descendance path for either XY or XZ passes through W X, showing that μ(W X) = μ(XY ) = μ(XZ), and therefore satisfying the balance requirement at X. Now, consider the case of exactly four descendants incident to X. If these four edges form two collinear pairs, then two of them must point towards X, and they will form an evil turn, contrary to Lemma 3.4. Therefore we can find among the four descendants two noncollinear edges W X → XY , in which case we also have W X → XZ with these three edges forming 120◦ angles. The fourth descendant is not collinear with W X, so it makes a 60◦ angle with W X. If it points away from X, it must be a descendant of the only incoming edge W X, and this is not possible. Therefore this fourth edge must be V X with V X → XY or V X → XZ. Assume V X → XY for definiteness. In this case, all descendance paths for XZ pass through W X, so that μ(XZ) = μ(W X). On the other hand, descendance paths for XY pass either through W X or through V X, showing that μ(XY ) = μ(W X) + μ(V X). The balance requirement is again verified. Finally, if 5 descendants of e are incident to X, then the sixth edge must have mass equal to zero, and it is impossible to orient the five edges so that every pair of incoming edges form a 60◦ angle, and every outgoing edge is a descendant of an incoming edge. Thus, this situation does not occur. 2 The preceding proof shows that a rigid skeleton does not cross itself transversely. In other words, a rigid skeleton does not contain four edges meeting at the same point, such that they form two collinear pairs of edges. Fig. 2 shows the possible ways (up to rotation) that the edges


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Fig. 2. Local structure of a rigid skeleton.

of a rigid skeleton can meet, along with the possible orientations. In each case, a (or the) dotted edge must have density equal to zero. The measure constructed in the above argument is the only measure supported by the descendants of e such that μ(e) = 1. We will denote this measure by μe . Corollary 3.6. Let m ∈ Mr be a rigid measure, and let e1 , e2 , . . . , ek be a maximal collection of inequivalent root edges. (1) We have m = kj =1 m(ej )μej . (2) More generally, for every measure μ ∈ Mr with support contained in the support of m, we have μ = kj =1 μ(ej )μej . (3) The only extremal measures μ ∈ Mr with support contained in the support of m are positive multiples of the measures μej . (4) The decomposition of m as a sum of extremal measures is unique up to a permutation of the summands. Proof. Fix a maximal set S of inequivalent root edges of m. Let us say that an edge f in the support of m has height p 2 if there exists a descendance path A0 A1 · · · Ap from some root edge e ∈ S to f . If f has height p, we say that f has height equal to p if it does not have height p + 1. The rigidity of m implies that there are no evil loops, and this in turn implies that the height of any edge is finite. The requirement that m ∈ Mr shows that the measure of any edge can be calculated in terms of the measures of edges of smaller height, as can be seen from (3.1). Therefore, m is entirely determined by the values m(ej ), j = 1, 2, . . . , k. On the other hand, the measure m = kj =1 m(ej )μej has support contained in the support of m. Since m (ej ) = m(ej ), we conclude that m = m . This argument proves (1). Assume next that supp(μ) ⊂ supp(m), and observe that any descendance path for m is a descendance path for μ. It follows that the preceding argument proves (2) as well. The remaining assertions follow immediately from (2). 2 We mention one more useful property of rigid skeletons. Lemma 3.7. Let e and f be two edges in a rigid skeleton. There exists a path C0 C1 · · · Cp in this skeleton such that C0 C1 = e, Cp−1 Cp = f , and all the turns Cj −1 Cj Cj +1 and Cj +1 Cj Cj −1 are evil. Proof. The result is obvious if e is a root edge. If it is not, choose descendance paths A0 A1 · · · Ak from a root edge to e, and B0 B1 · · · B from the same root edge to f . If A0 = B1 , the path Ak Ak−1 · · · A1 B1 B2 · · · B satisfies the requirements. If A0 = B0 one chooses instead the path

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Ak Ak−1 · · · Ar+1 Br Br−1 · · · B , where r is the first integer such that Ar+1 = Br+1 . If no such integer exists, one of the paths is contained in the other. For instance, k < and Aj = Bj for j k. In this case the desired path is Bk−1 Bk · · · B . 2 The path provided by this lemma is not generally a descendance path. We conclude this section by introducing an order relation on the set of skeletons contained in the support of a rigid measure m. Given two rigid skeletons S1 and S2 , we will write S1 ≺0 S2 if S1 has collinear edges AX, XB and S2 has collinear edges CX, XD such that XA is 60◦ clockwise from XC. The following figure shows the four possible configurations of S1 and S2 around the point X, up to rotation, and assuming that the two skeletons are contained in the support of a rigid measure. The edges in S1 \ S2 are dashed, the edges in S2 \ S1 are solid without arrows, and the common edges are oriented away from the root edges.

The four turns AXC, AXD, BXC, and BXD are evil. Note that the point X could be on the boundary of r , but not one of the three corner vertices. It is possible that S1 ≺0 S2 and S2 ≺0 S1 , as illustrated in the picture below (with S1 in dashed lines).

We will show that this does not occur when the skeletons are associated with a fixed rigid measure. Theorem 3.8. Fix a rigid measure m ∈ Mr and an integer n 1. There do not exist skeletons S1 , S2 , . . . , Sn contained in the support of m such that S1 ≺0 S2 ≺0 · · · ≺0 Sn ≺0 S1 . Proof. Assume to the contrary that such skeletons exist. Choose for each j collinear edges Aj Xj , Xj Bj in the support of Sj and collinear edges Cj Xj , Xj Dj in the support of Sj +1 (with Sn+1 = S1 ) such that XAj is 60◦ clockwise from XCj . The rigidity of m implies that one of the


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edges XY has measure zero. Therefore we must have either Xj Aj → Xj Bj or Xj Bj → Xj Aj . Label these two edges fj and fj so that fj → fj , and note that fj is not the descendant of any edge XY , except possibly fj . Analogously, denote the edges Xj −1 Cj −1 and Xj −1 Dj −1 by ej and ej so that ej → ej . Since both ej and fj are contained in Sj , Lemma 3.7 provides a path with evil turns joining ej and fj . A moment’s thought shows that this path either begins at Xj −1 , or it begins with one of Cj −1 Xj −1 Dj −1 , Dj −1 Xj −1 Cj −1 . If the second alternative holds, remove the first edge from the path. Performing the analogous operation at the other endpoint, we obtain a path γj with only evil turns which starts at Xj −1 ends at Xj (with Xj −1 = Xn if j = 0), its first edge is one of Xj −1 Cj −1 , Xj −1 Dj −1 , and its last edge is one of Aj Xj , Bj Xj . As noted above, the turn formed by the last edge of γj and the first edge of γj +1 is evil. We conclude that the loop γ1 γ2 · · · γn is evil, contradicting rigidity. 2 The preceding result shows that there is a well-defined order relation on the set of skeletons in the support of a rigid measure m defined as follows: S ≺ S if there exist skeletons S1 , S2 , . . . , Sk such that S = S1 ≺0 S2 ≺0 · · · ≺0 Sk = S . The following figure shows the support of a rigid measure m ∈ M6 . The elements of a maximal collection of mutually inequivalent root edges have been indicated with dots.

For this measure, there is a smallest skeleton (relative to ≺) pictured below. The reader can easily draw all the other skeletons and determine the order relation.

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4. Horn inequalities and clockwise overlays In this section we discuss some technical details which are essential in the proof of our main result. We begin with a discussion of the well-known Horn inequalities, leading to the important fact that any measure, for which one of these inequalities is not strict, exhibits a special structure extending the clockwise overlays of [25,27]. We then construct a special class of clockwise overlays arising from the precedence of rigid skeletons. The results of [25,27] show that the triples (α, β, γ ) = ∂m with m ∈ Mr have the following property: there exist self-adjoint matrices A, B, C ∈ Mr (C) such that λA () = αr+1− ,

λB () = βr+1− ,

λC () = γr+1− ,

= 1, 2, . . . , r,

and A + B + C is a multiple of the identity, namely, 2ω(m)1Cr . (An appropriately formulated converse of this statement is true. Given self-adjoint matrices A, B, C ∈ Mr (C) such that A + B + C is a constant multiple of the identity, there exists a measure m ∈ Mr such that, setting (α, β, γ ) = ∂m, the differences λA () − αr+1− , λB () − βr+1− and λC () − γr+1− do not depend on .) The Horn inequalities for these matrices are

λA (i) +

λB (j ) +

j ∈J

i∈I

λC (k) 2sω(m)

k∈K

when I, J, K ⊂ {1, 2, . . . , r} have cardinality s and cI J K > 0. Applying this inequality to the matrices −A, −B, −C instead and switching signs, we obtain

λA (r + 1 − i) +

λB (r + 1 − j ) +

j ∈J

i∈I

λC (r + 1 − k) 2sω(m).

k∈K

Equivalently,

αi +

i∈I

βj +

j ∈J

γk 2sω(m).

(4.1)

k∈K

Now, each triple of sets {I, J, K} ⊂ {1, 2, . . . , r} such that cI J K > 0 is obtained from some measure ν ∈ Ms with weight r − s, and we will see how this inequality follows from the superposition of the support of m and the puzzle associated with ν. A similar proof is provided in [12]; the argument below yields some useful additional information. Let h be the honeycomb associated to m by Lemma 1.1. Let D ⊂ r be a region bounded by small edges, and let Xj Yj , j = 1, 2, . . . , p be an enumeration of the small edges of ∂D, oriented so that D lies on the left of Xj Yj . For each j , there is εj = εXj Yj = ±1 such that Yj − Xj ∈ {εj u, εj v, εj w}. The definition of honeycombs implies then the identity XY ⊂∂D

εXY h(XY ) =

p

εj h(Xj Yj ) = 0,

j =1

which is easily deduced by induction on the size of D. Inequality (4.1) amounts to verifying that the sum


S=

h(Ai−1 Ai ) +

h(Bj −1 Bj ) +

j ∈J

i∈I

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h(Ck−1 Ck )

k∈K

is nonnegative, where I, J, K are given by a measure ν ∈ Ms with ω(ν) = r − s. Here we use the labels Ai , Bi , Ci for the lattice points on ∂r (see Fig. 1 in Section 1). The inflation of the measure ν yields a partition of r into white pieces (the translated parts of s ), dark parallelograms, and light gray pieces; see the picture below for an illustration. The smaller triangle represents the support of ν, and the larger one represents the inflation of ν, along with the support of m. For this figure, the density of ν was assumed to be 4 on all edges in its support.

Denote by D the union of the dark parallelograms P1 , P2 , . . . , Pσ and white pieces W1 , W2 , . . . , Wτ . The boundary ∂D (oriented as above) consists of the edges Ai−1 Ai , i ∈ I , Bj −1 Bj , j ∈ J , Ck−1 Ck , k ∈ K, represented by dashed lines in the above figure, and of the light gray edges of the parallelograms P1 , P2 , . . . , Pσ . It is easy to see that εAi−1 Ai = εBj −1 Bj = εCk−1 Ck = 1, and thus 0=

εXY h(XY ) = S +

XY ⊂∂D

σ

S ,

=1

where S =

εXY h(XY ),

= 1, 2, . . . , σ.

gray XY ⊂∂P

Here the sum is taken over the light gray edges of P . Condition (2) of Lemma 1.1 implies the equality S = −

e

m(e),

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where the sum is taken over those small edges contained in P which are not parallel to the sides of P . Thus S=−

σ

S 0,

=1

as claimed. (For the preceding figure, the sum S consists of only one term, corresponding to the thicker edge.) This also tells us when the equality S = 0 occurs: this happens if and only if all the edges e ⊂ P for which m(e) > 0 are parallel to the edges of P , = 1, 2, . . . , s. In other words, the support of m must cross each P along lines parallel to the edges of P . The following figure illustrates the support of a measure ν, the inflation of ν, and the support of a measure m which satisfies the Horn equality associated with ν. In this example the support of the measure m never crosses the white puzzle pieces.

The next example involves a similar measure m, and its support never crosses the light gray pieces.

Assume that we are in a case of equality i∈I

h(Ai−1 Ai ) +

j ∈J

h(Bj −1 Bj ) +

h(Ck−1 Ck ) = 0.

k∈K

In this case, we can define a measure μ ∈ Ms by moving the support of m to s in the following way: those parts which are contained in white puzzle pieces are simply translated back to s (along with their densities); the segments in the support of m which cross dark parallelograms


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between white pieces are deleted; the segments which cross dark parallelograms between light gray pieces are replaced by the corresponding parallel sides of white pieces, and the density is preserved. It may be that several segments cross a single dark parallelogram between light gray pieces, in which case the density of the corresponding side of a white piece is the sum of their densities. When the measure μ can be obtained using this procedure, we will say that μ is obtained by contracting m, and that μ is clockwise from ν (or that (μ, ν) form a clockwise overlay). This is easily seen to be an extension of the notion of clockwise overlay introduced in [27] (see also item (1), second case, in the proof of Theorem 4.3). Generally, a clockwise overlay (μ, ν) can be obtained by shrinking more than one measure m. Indeed, the shrinking operation loses all the information about the branch points of m in the light gray puzzle pieces. In the case illustrated above, the support of the measure μ is actually contained in the support of ν; this is what happens when the support of m does not cross the white pieces. In the second case illustrated above we obtain the following figure for the supports of ν and μ.

For future reference, we emphasize one aspect of the preceding discussion in the following statement. Proposition 4.1. Let m ∈ Mr be a measure, and let h be the honeycomb provided by Lemma 1.1. Assume also that ν ∈ Ms is a measure with integer densities such that r = s + ω(ν), and i∈I

h(Ai−1 Ai ) +

j ∈J

h(Bj −1 Bj ) +

h(Ck−1 Ck ) = 0,

k∈K

where the sets I, J, K ⊂ {1, 2, . . . , r} are constructed using the measure ν. Then m can be contracted to a measure μ ∈ Ms so that (μ, ν) is a clockwise overlay. We will need one more important property of clockwise overlays. Proposition 4.2. Let (μ, ν) be a clockwise overlay obtained by contracting a measure m. Then ω(μ) = ω(m), and μ∗ m∗ . Proof. Consider the puzzle obtained by inflating the measure εν for ε > 0. The white pieces of the puzzle are independent of ε. Since the support of m intersects any dark parallelogram in the puzzle of ν only on segments parallel to the edges of the parallelogram, it follows that there exists a measure mε on the puzzle of εν obtained by translating the support of m in each white piece, and applying appropriate translation and/or shrinking in the dark parallelograms and light gray puzzle pieces. Clearly m1 = m, and all the measures mε are homologous to m, as defined in Section 2; in fact, homologous sides have equal densities, and therefore ω(m) = ω(mε ) for all

1608


ε > 0. (Here it may be useful to recall that ω(m) is defined in terms of its densities outside r , and the outside edges are not generally present in our drawings.) Moreover, all the measures m∗ε have the same support, except that some of the densities are decreased for ε < 1. The measure μ is simply the limit of mε as ε → 0, and the statement follows immediately from this observation. The following pictures illustrate the process as applied to the above examples for ε = 2/3 and ε = 1/3.

2 Unfortunately, the definition of clockwise overlays is not quite explicit since they are seen as the result of a process – something akin to defining a car as the end product of car manufacture. We can however use the relation ≺0 between skeletons to produce an important class of clockwise overlays. Theorem 4.3. Let μ1 , μ2 ∈ Mr be such that μ1 + μ2 is rigid, the support Sj of μj is a skeleton, and S2 ⊀0 S1 . Then (μ1 , μ2 ) is a clockwise overlay. Proof. We need to inflate μ2 , and construct a measure m1 such that μ1 is obtained from m1 by the shrinking process described above. It is clear what the measure m1 should be on the interior of every white puzzle piece. The common edges of S1 and S2 cannot be root edges; orient them away from the root edges, and attach them (along with their μ1 masses) to the white puzzle piece on their right side. What remains to be proved is that this partially defined measure can be extended so as to satisfy the balance condition at all points. For this purpose we only need to analyze the situation at lattice points where S1 and S2 meet. For each such lattice point, there will be 2, 3, or 4 edges of each skeleton meeting at that point, and this gives rise to many possibilities. In order to reduce the number of cases we need to study, observe that the inflation construction is invariant relative to rotations of 60◦ , and therefore the position (but perhaps not the orientation) of the edges in one of the skeletons can be fixed. In the following figures, the arrows indicate


1609

the orientation on the edges in S1 ∩ S2 . The other edges of S1 are dashed, and the other edges of S2 are solid without arrows. In each case, the extension required after inflation is indicated by dashed lines crossing (or on the boundary of) dark parallelogram pieces. In the following enumeration, the label (p, q) signifies that S1 has p and S2 has q edges meeting at one point. (See Fig. 2 for the possible configurations of Sj around the given meeting point.) (1) (2, 2) The edges of the skeletons may overlap, and after a rotation the orientation is as in the figure below.

No extensions are required in this case. If the skeletons do not overlap, we have two possibilities:

and finally

which would imply S2 ≺0 S1 , contrary to the hypothesis. (2) (3, 2) In this case there is (up to rotations) only the case illustrated in the figure.

(3) (4, 2) Up to rotations, there are three possibilities. In the first one we have an extension as shown.

The orientation shown above is the only one which is compatible with the rigidity of μ1 +μ2 . In the following figure, the orientation given is also the only possible one.

1610


The third situation

implies S2 ≺0 S1 . (4) (2, 3) The case illustrated is the only one up to rotations.

(5) (3, 3) There are two cases up to rotations.

The second case is not compatible with the rigidity of μ1 + μ2 .

(6) (4, 3) There is only one relative position of S1 and S2 compatible with rigidity, but there are two possible orientations.

The second orientation requires a different extension.


1611

(7) (2, 4) There are three possibilities up to rotation.

In the figure above, there is no ambiguity in the orientation. For the illustration we assigned μ2 masses of 1 and 2 to the edges.

The orientation is also clear in this case. The third case implies S2 ≺0 S1 .

(8) (3, 4) There is only one position compatible with rigidity, and there are only two possible orientations.

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(9) (4, 4) When the two skeletons overlap completely, there are two possible orientations.

When the skeletons do not overlap completely, there is only one relative position of the two skeletons which is compatible both with rigidity and with S2 ⊀ S1 . There is only one possible orientation.

2 The following result follows easily by induction, inflating successively the measures μp , μp−1 , . . . , μt+1 . p Corollary 4.4. Let m ∈ Mr be a rigid measure, and write it as m = =1 μ , where each is supported μ ∈ M r pon some skeleton S . Assume also that Si ≺ Sj implies that i j . Then the pair ( t=1 μ , =t+1 μ ) is a clockwise overlay for 1 t < p. For the clockwise overlays (μ1 , μ2 ) considered in the preceding two results there is a canonical construction for the measure m1 ∈ Mr+ω(μ2 ) . We will call this measure m1 the stretch of μ1 to the puzzle of μ2 . 5. Proof of the main results Fix a triple (I, J, K) of subsets with cardinality r of {0, 1, . . . , n} such that cI J K = 1, and let m ∈ Mr be the corresponding measure. It will be convenient now to use the normalization τ (1) = n in a finite factor. This will not require renormalizations when passing to a subfactor, and has the added benefit of working in finite dimensions as well, with the usual matrix trace. In order to prove the intersection results in the introduction, we will want to prove the following related properties:


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Property A(I , J , K) or A(m). Given a II1 factor A with τ (1) = n, and given flags E, F , G with τ (E ) = τ (F ) = τ (G ) = , = 0, 1, 2, . . . , n, the intersection S(E, I ) ∩ S(F , J ) ∩ S(G, K) is not empty. Property B(I , J , K) or B(m). There exists a lattice polynomial p ∈ L({ej , fj , gj : 1 j n}) with the following property: for any finite factor A with τ (1) = n, and for generic flags E = (Ej )nj=0 , F = (Fj )nj=0 , G = (Gj )nj=0 such that τ (Ej ) = τ (Fj ) = τ (Gj ) = j , the projection P = p(E, F , G) has trace τ (P ) = r and, in addition τ (P ∧ Ei ) = τ (P ∧ Fj ) = τ (P ∧ Gk ) = when i i < i+1 , j j < j+1 , k k < k+1 and = 0, 1, . . . , r, where i0 = j0 = k0 = 0 and ir+1 = jr+1 = kr+1 = n + 1. We will prove these properties by reducing them to simpler measures for which they are trivial. The basic reduction is from an arbitrary measure to a skeleton. p Proposition 5.1. Let m ∈ Mr be a rigid measure with integer densities, and write m = =1 μ , where

1 ∈ Mr , r=r+ p μ is supported by a skeleton S , and Si ≺ S j implies p i j . Let μ ) (resp., ω(μ ) be the stretch of μ to the puzzle of m = μ . If A(

μ ) and A(m 1 1 =2 =2 B(

μ1 ) and B(m )) are true, then A(m) (resp., B(m)) is true as well. Proof. Pick a root edge e for μ1 which is not contained in the support of m . With the usual notation Ai = iu, Xi = Ai + w, the edges Ai Xi are oriented in the direction of w (if they belong (q) to the support of m). Let us set ai = m(Ai Xi ), ai = μq (Ai Xi ), and ai = m (Ai Xi ), so that

ai =

p

(q)

ai

(1)

= ai

+ ai

and n = r +

q=1

r

ai .

i=0

1 can only exit the left side of r at the points We have μ

1 ∈ Mr , and the support of μ An(i) , where n(0) = 0, and n(i) = i + i−1 s=0 as for i > 0. This follows from the way the in flation of m is constructed, and from the outward orientation of the segments Ai Xi . Moreover, (1) μ

1 (An(i) Xn(i) ) = ai for i = 0, 1, . . . , r. Denote by I (1) , J (1) , K (1) ⊂ {1, 2, . . . , n} the sets determined by the measure μ

1 , and by I , J , K ⊂ {1, 2, . . . , n1 } those corresponding to m , where n = r = r + ω(m ) = n − ω(μ1 ). For instance, we have I (1) =

r j =1

where

(1)

Ij ,

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(1) (1) I0 = s + a0 : s = 1, 2, . . . , a0 + 1 , j j −1 (1) (1) a + 1 : s = 1, 2, . . . , aj + 1 , a + Ij = s + Ir(1)

= s+

=0 r

0 < j < r,

=0

(1) a

r−1 a + 1 : s = 1, 2, . . . , ar , +

=0

=0

and I = it −

t−1

(1) a :

t = 1, 2, . . . , r ,

=0 (1)

where I = {i1 , i2 , . . . , ir }. Observe that the th element i of I (1) equals it for = t + t−1 s=0 as = it . Assume first that A(

μ1 ) and A(m ) are true, and let E, F , G be arbitrary flags in a II1 factor μ1 ) implies such that τ (Ei ) = τ (Fi ) = τ (Gi ) = i for i = 0, 1, . . . , n, and τ (1) = n. Property A(

r and the existence of a projection P1 ∈ A such that τ (P1 ) = τ (P1 ∧ Ei (1) ) ,

= 1, 2, . . . , r.

As noted above, we have i = it for = it , and therefore we have (1)

τ (P1 ∧ Ei t ) it ,

t = 1, 2, . . . , r,

with analogous inequalities for F and G. Consider now the factor A1 = P1 AP1 with the trace r. The inequalities above imply the existence of a flag E τ1 = τ |A1 , so that τ1 (1A1 ) = τ (P1 ) = in A1 such that τ1 (Ej ) = j for j = 1, 2, . . . , n1 , and Ei P1 ∧ Eip , p

p = 1, 2, . . . , r.

Analogous considerations lead to the construction of flags F and G . Property A(m ) implies now the existence of a projection P ∈ A1 such that τ1 (P ) = r, τ1 P ∧ Ei p, p

p = 1, 2, . . . , r,

and analogous inequalities are satisfied for F and G . Clearly the projection P satisfies τ (P ∧ Eip ) p,

p = 1, 2, . . . , r,

so that it solves the intersection problem for the sets I, J, K. The case of property B is settled analogously. The difference is that P1 is given as a lattice polynomial P1 = p1 (E, F , G), and the projections Ej can be taken to be of the form P1 ∧ Ei , and hence they too are lattice polynomials in E, F , G. Finally, the solution P is given as P =


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p (E , F , G ), where the existence of p is given by property B(m ). One must however assume that E , F , G are generic flags, and this simply amounts to an additional genericity condition on the original flags. 2 The preceding proposition shows that proving property A(m) or B(m) can be reduced to proving it for simpler measures, at least when m is not extremal. A dual reduction is obtained by recalling that a projection P belongs to S(E, I ) if and only if P ⊥ = 1 − P belongs to S(E ⊥ , I ∗ ). Moreover, if the sets I, J, K are associated to the measure m ∈ Mr , then I ∗ , J ∗ , K ∗ are the sets associated to the measure m∗ . Therefore A(m) is equivalent to A(m∗ ) and B(m) is equivalent to B(m∗ ). To quantify these reductions, we define for each measure m ∈ Mr the positive integer κ(m) as the number of dark parallelograms in the puzzle obtained by inflating m. This is equal to the number of white piece edges which have positive measure. Analogously, for m ∈ M∗r , we define κ ∗ (m) to be the number of dark parallelograms in the puzzle obtained by ∗-inflating m. With this definition it is clear that κ(m) = κ ∗ m∗ ,

m ∈ Mr .

Indeed, the two numbers count pieces of the same puzzle. With the notation of the preceding proposition, we have κ(

μ1 ) = κ(μ1 ) < κ(m),

κ(m ) < κ(m),

μ1 ) = κ(μ1 ) because μ1 and μ

1 are homologous, and the supports of unless m = μ1 . Indeed, κ(

μ1 and m are strictly contained in the support of m. In fact, the support of m does not contain the root edges of μ1 , and the support of μ1 does not contain the root edges of the extremal summands of m . Thus the preceding proposition also allows us to reduce the proof of these properties to measures with smaller values of κ in case either m or m∗ is not extremal. The exceptional situations in which both m and m∗ are extremal are very few in number. To see this we need to use the structure of the convex polyhedral cone Cr = {∂m: m ∈ Mr }, whose facets were determined in [27]. If ∂m = (α, β, γ ), these facets are of two kinds. The first kind are the chamber facets determined by one of the following inequalities: (1) α = α+1 , β = β+1 or γ = γ+1 for 1 < r, (2) αr = ω(m), βr = ω(m) or γr = ω(m). The second kind are the regular facets determined by Horn identities i∈I

αi +

j ∈J

βj +

γk = ω(m),

k∈K

where I, J, K ⊂ {1, 2, . . . , r} have s < r elements and cI J K = 1.

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For a given measure m ∈ Mr , we define the number of attachment points Γ (m) to be the number of chamber facets to which m does not belong. The reason for this terminology is that Γ (m) is precisely the number of points on the sides of r which are endpoints of interior edges in the support of m. The vertices of r should also be counted as attachment points when they are branch points of the measure. Proposition 5.2. Let m ∈ Mr be an extremal rigid measure. If m∗ is extremal as well, then Γ (m) = 1. Proof. Assume that m and m∗ are both extremal, and Γ (m) > 1. Note first that ∂m is extremal in Cr . Indeed, in the contrary case, we would have ∂m = ∂μ1 + ∂μ2 with ∂μ1 not a positive multiple of ∂m. This would however imply m = μ1 + μ2 by rigidity, and hence μ1 is a multiple of m, a contradiction. Next, since Γ (m) = Γ (m ) for homologous m, m , we may assume that m∗ = μe for some root edge e. Indeed, m∗ = m∗ (e)μe is homologous to μe , and therefore m is homologous to μ∗e . The definition of Γ (m) implies that ∂m belongs to 3r − Γ (m) = dim Cr − Γ (m) chamber facets. However, an extremal measure must belong to at least dim Cr − 1 facets, and hence ∂m belongs to at least one regular facet. As seen in Proposition 4.1, there must then exist a clockwise overlay (m1 , m2 ) such that m1 is obtained by contracting m. It follows from Proposition 4.2 that 0 = m∗1 m∗ . Since m∗1 has integer densities, we must have m∗1 = m∗ , and this implies that m1 = m, a contradiction. 2 Thus the repeated application of the reduction procedure to m and m∗ leads eventually to one of the three skeletons pictured below.

For these, the intersection problem is completely trivial. Indeed, consider the first of the three on r , and with ω(m) = s. We have then I = {1, 2, . . . , r} and J = K = {s + 1, s + 2, . . . , s + r}, and the desired element in S(E, I ) ∩ S(F , J ) ∩ S(G, K) is simply Er . Thus A(m) is true for this measure. To show that B(m) is true as well, we must verify that generically we also have τ (Er ∧ F ) = τ (Er ∧ G ) = max{0, r + − n} for = 1, 2, . . . , n. This follows easily from the following result.


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Proposition 5.3. Let E and F be two projections in a finite factor A. There is an open dense set O ⊂ U(A) such that

τ E ∧ U F U ∗ = max 0, τ (E) + τ (F ) − τ (1) for U ∈ O. Proof. Replacing E and F by E ⊥ and F ⊥ if necessary, we may assume that τ (E) + τ (F ) τ (1). Since A is a factor, we can replace F with any other projection with the same trace. In particular, we may assume that F E ⊥ . The condition τ (E ∧ U F U ∗ ) = 0 is satisfied if the operator F U F is invertible on the range of F . The proposition follows because the set O of unitaries satisfying this condition is a dense open set in U(A). To verify this fact, it suffices to consider the case in which the algebra A is of the form A = B ⊗ M2 (C) for some other finite factor B, and F=

1 0

0 . 0

An arbitrary unitary U ∈ A can be written as

T U= V (I − T ∗ T )1/2

(I − T T ∗ )1/2 W , −V T ∗ W

where T , V , W ∈ B, V and W are unitary, and T 1. Since B is a finite von Neumann algebra, T can be approximated arbitrarily well in norm by an invertible operator T , in which case U is approximated in norm by the operator U =

T V (I − T ∗ T )1/2

(I − T T ∗ )1/2 W −V T ∗ W

with F U F invertible. In finite dimensions, the complement of O is defined by the single homogeneous polynomial equation det(F U F + F ⊥ ) = 0. Thus O is open in the Zariski topology. 2 Corollary 5.4. Properties A(m) and B(m) are true for all rigid measures m. This proves finally Theorems 0.4 and 0.5. The fact that Theorem 0.3 follows from Theorem 0.4 was already shown in [4]. 6. Some illustrations We have just seen that proving property A(I, J, K) or B(I, J, K) can be reduced, when cI J K = 1, to the case in which the associated measure m has precisely one attachment point. We will illustrate how this reduction works in a few cases. Given a measure m ∈ Mr with integer densities, a point A , = 1, 2, . . . , r, is an attachment point of m precisely when m(A X ) > 0. The solution to the associated Schubert intersection problem will only depend on the projections Ei() where is an attachment point. These projections, and the analogous Fj () , Gk() , will be called the attachment projections for this problem.

1618


As noted already in Section 2, the Schubert problem associated with m and the flags E, F , G is equivalent to problem associated with m∗ and the flags E ⊥ , F ⊥ , G ⊥ . Moreover, the attachment projections for this dual problem are of the form I − Q, where Q is an attachment projection for the original problem. With the notation of Proposition 5.1, the attachment projections of m corresponding to the

1 . These observations allow us to exit points of μ1 are exactly the attachment projections of μ construct solutions to intersection problems without actually having to construct the measure μ

1 and focus instead on the attachment projections of m. Assume, for instance, that μ1 is a measure with only one attachment point, and X is the corresponding attachment projection of m. (Note

1 also has one attachment point, and that X is one of the projections Er , Fr and Gr .) Then μ the corresponding attachment projection is X, so that the solution of the corresponding Schubert problem is precisely X. In order to continue with the solution of the Schubert problem associated with m − μ1 , we need to construct new flags consisting of the spaces X ∧ Ei , X ∧ Fi and X ∧ Gi , and work in the algebra XAX. We proceed now to solve the intersection problems associated with more complicated skeletons. Consider first an extreme measure m ∈ Mr with two attachment points. Assume, for instance, that these attachment points are Ax and Cr−x , and the density of m is y. The following picture shows the supports of m and m∗ .

For the illustration we took r = 3, x = 2 and y = 3. Note that m∗ is a sum of two extremal measures μ and ν with one attachment point each. More precisely, the horizontal segment has density r − x, and the other segment in the support has density x. If X and Z are the attachment projections of m, the attachment projections of these skeletons are X ⊥ and Z ⊥ . Neither of the two skeletons precedes the other. Thus, following the method of Proposition 5.1, we see that the solution of the intersection problem associated with μ is generically X ⊥ , and the attachment ∗ ⊥ ⊥ point of ν = m − μ is X ∧ Z . Thus the intersection problem associated with m∗ has the generic solution X ⊥ ∧ Z ⊥ . It follows that the intersection problem associated with m has the generic solution X ∨ Z. The problem just discussed can easily be solved without reference to our general method. Indeed, observe that X = Ex and Z = Gr−x , and the only relevant conditions on the solution P to the intersection problem are τ (P ∧ Ex ) x,

τ (P ∧ Gr−x ) r − x.

Since τ (Ex ) = x and τ (Gr−x ) = r − x, these conditions amount to P X ∨ Z. Thus P solves the intersection problem if and only if P X ∨ Z and τ (P ) = r. Since τ (X) + τ (Z) = r, we will have τ (X ∨ Z) = r if X ∧ Y = 0. Thus P = X ∨ Y if the genericity condition X ∧ Z = 0 is satisfied. When this condition is not satisfied, one can choose an arbitrary projection P X ∨ Z with τ (P ) = r. There are two kinds of skeletons with three attachment points. The first one, and its dual, are illustrated below.


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Assume that the attachment projections are X, Y and Z. As in the preceding situation, m∗ is a sum of three extremal measures with one attachment point each, and there are no precedence relations among the skeletons. It follows that the generic solution of the intersection problem is X ∨ Y ∨ Z. The two cases just mentioned correspond to the reductions considered in [30] for finite dimensions, and in [10] for the factor case. Note however that these papers also apply these reductions when cI J K > 1. Consider next the other kind of skeleton with three attachment points, and with attachment projections X, Y, Z.

In this case, m∗ is the sum of three extremal measures with two attachment points each, and with no precedence relations. The intersection problems associated with the three skeletons have then generic solutions X ⊥ ∨ Y ⊥ , X ⊥ ∨ Z ⊥ , and Y ⊥ ∨ Z ⊥ . According to Proposition 5.1, the solution of the intersection problem for m∗ will be (generically) the intersection of these three projections, so that the problem associated with m has the solution (X ∧ Y ) ∨ (X ∧ Z) ∨ (Y ∧ Z). Several Horn inequalities proved in the literature can now be deduced by considering rigid measures which are sums of extremal measures with 1, 2 or 3 attachment points. Consider, for instance, a measure m ∈ Mr defined by

m=ρ+

r (μ + ν ), =1

where ρ has attachment point Cr , μ1 has attachment point Ar , ν1 has attachment point Br , μ has attachment points Ar−+1 and C−1 , and ν has attachment points Br−+1 and C−1 for > 1.

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The only precedence relations are μ ≺0 νk and ν ≺0 μk for < k. Generically, the associated intersection problem is solved as follows. Set P0 = Gr and P+1 = (G ∧ P ) ∨ (Fr− ∧ P ) ∧ (G ∧ P ) ∨ (Er− ∧ P ) for = 1, 2, . . . , r − 1. The space Pr is the generic solution. The sets I, J, K associated with m are easily calculated. Using the notations c = ω(ρ),

a = ω(μ ),

b = ω(ν )

for = 1, 2, . . . , r,

we have r (a + b ), n=r +c+ =1

and I = {n + 1 − (a1 + a2 + · · · + a + ): = 1, 2, . . . , r}, J = {n + 1 − (b1 + b2 + · · · + b + ): = 1, 2, . . . , r}, and K = {a1 + b1 + · · · + a + b + : = 1, 2, . . . , r}. These sets yield the eigenvalue inequalities proved in [29]. Consider next sequences of integers 0 z1 z2 · · · zq ,

0 w1 w2 · · · wq

such that zq + wq r, and consider the measure m ∈ Mr defined by m=

q

μi ,

i=1

where μ has attachment points Az , Bw , and Cr−z −w .


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The illustration uses q = 3, r = 6, z1 = 1, z2 = 2, z3 = 3, w1 = w2 = 1, and w3 = 2. We have μ ≺0 μk only when < k, w < wk and z < zk . If we set P1 = Ez1 ∨ Fw1 ∨ Gr−z1 −w1 and P+1 = (Ez+1 ∧ P ) ∨ (Fw+1 ∧ P ) ∨ (Gr−z+1 −w+1 ∧ P ) for = 1, 2, . . . , d − 1, then Pq is the generic solution of the intersection problem. Assume that ω(μi ) = 1 for all i, and use the notation 1x 0, Ut∗ Ut+ε is free from Us for all s < t. For the purposes of the following result, we will say that P and Q are in general position if τ (P ∧ Q) = max{0, τ (P ) + τ (Q) − 1}. Theorem 8.2. The projections Ut P Ut∗ and Q are in general position for every t > 0. Proof. Fix t > 0, and set Pt = Ut P Ut∗ . As in the proof of Proposition 5.3, we may and shall assume that and τ (P ) + τ (Q) 1. Arguing by contradiction, assume that Pt ∧ Q = 0. Setting R = Pt ∧ Q, observe that the function f (s) = τ (RPs R − R)2 ,

s 0,

is nonnegative, and therefore f has a minimum at s = t. The fact that Ut is a free Brownian motion, and Ito calculus, imply that f is a differentiable function, and f (t) = τ (R) −1 + τ (P ) + τ (R) − τ (R)τ (P ) , where we used the fact that (RPt R)2 = RPt R = R. Now, we have 0 < τ (R) τ (P ) and 1 − τ (P ) τ (R), so that this relation implies f (t) −τ (R)2 τ (P ) < 0. This however is not compatible with f (t) being a minimum.

2

If E and F are two flags in A, the preceding result yields a unitary U , arbitrarily close to 1, so that the spaces of the flag U F U ∗ are in general position relative to the spaces of E . Dealing with three flags would require the use of two Brownian motions, free from each other and from the flags. In order to obtain flags which are generic for a given intersection of three Schubert cells, this construction would have to be iterated following the inductive procedure of Proposition 5.1.

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References [1] P. Belkale, Local systems on P 1 \ S for S a finite set, Compos. Math. 129 (1) (2001) 67–86. [2] P. Belkale, Geometric proofs of Horn and saturation conjectures, J. Algebraic Geom. 15 (1) (2006) 133–173. [3] H. Bercovici, W.S. Li, Inequalities for eigenvalues of sums in a von Neumann algebra, in: Recent Advances in Operator Theory and Related Topics, Szeged, 1999, in: Oper. Theory Adv. Appl., vol. 127, Birkhäuser, Basel, 2001, pp. 113–126. [4] H. Bercovici, W.S. Li, Eigenvalue inequalities in an embeddable factor, Proc. Amer. Math. Soc. 134 (1) (2006) 75–80. [5] H. Bercovici, W.S. Li, T. Smotzer, Continuous versions of the Littlewood–Richardson rule, selfadjoint operators, and invariant subspaces, J. Operator Theory 54 (1) (2005) 69–92. [6] H. Bercovici, W.S. Li, D. Timotin, The Horn conjecture for sums of compact selfadjoint operators, Amer. J. Math., in press. [7] Ph. 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. [8] A.S. Buch, The saturation conjecture (after A. Knutson and T. Tao), with an appendix by William Fulton, Enseign. Math. (2) 46 (1–2) (2000) 43–60. [9] B. Collins, K. Dykema, A linearization of Connes’ embedding problem, New York J. Math. 14 (2008) 617–641. [10] B. Collins, K. Dykema, On a reduction procedure for Horn inequalities in finite von Neumann algebras, Oper. Matrices 3 (2009) 1–40. [11] A. Connes, Classification of injective factors. Cases II1 , II∞ , IIIλ , λ = 1, Ann. of Math. (2) 104 (1976) 73–115. [12] V.I. Danilov, G.A. Koshevoy, Discrete convexity and Hermitian matrices, Tr. Mat. Inst. Steklova 241 (2003) 68–89. [13] J. Day, W. So, R.C. Thompson, The spectrum of a Hermitian matrix sum, Linear Algebra Appl. 28 (2–3) (1998) 289–332. [14] K. Dykema, Simplicity and the stable rank of some free product C ∗ -algebras, Trans. Amer. Math. Soc. 351 (1999) 1–40. [15] G.A. Elliott, D.E. Handelman, Extending traces to factorial traces, Math. Scand. 49 (1981) 95–98. [16] S. Friedland, Finite and infinite dimensional generalizations of Klyachko’s theorem, Linear Algebra Appl. 319 (1–3) (2000) 3–22. [17] W. Fulton, Eigenvalues of sums of Hermitian matrices (after A. Klyachko), in: Séminaire Bourbaki, vol. 1997/1998, Exp. No. 845, in: Asterisque, vol. 252, 1998, pp. 255–269. [18] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. 37 (2000) 209–249. [19] W. Fulton, Eigenvalues of majorized Hermitian matrices and Littlewood–Richardson coefficients, Linear Algebra Appl. 319 (2000) 23–36. [20] G. Grätzer, General Lattice Theory, second edition, Birkhäuser Verlag, Basel, 1998. [21] U. Haagerup, H. Schultz, Brown measures of unbounded operators affiliated with a finite von Neumann algebra, Math. Scand. 100 (2) (2007) 209–263. [22] U. Haagerup, H. Schultz, Invariant subspaces for operators in a general II1 -factor, Publ. Math. Inst. Hautes Études Sci. 109 (2009) 19–111. [23] A. Horn, Eigenvalues of sums of Hermitian matrices, Pacific J. Math. 12 (1962) 225–241. [24] A.A. Klyachko, Stable bundles, representation theory and Hermitian operators, Selecta Math. (N.S.) 4 (3) (1998) 419–445. [25] A. Knutson, T. Tao, The honeycomb model of GLn (C) tensor products, I. Proof of the saturation conjecture, J. Amer. Math. Soc. 12 (4) (1999) 1055–1090. [26] A. Knutson, T. Tao, Honeycombs and sums of Hermitian matrices, Notices Amer. Math. Soc. 48 (2001) 175–186. [27] A. Knutson, T. Tao, C. Woodward, The honeycomb model of GLn (C) tensor products, II. Puzzles determine facets of the Littlewood–Richardson cone, J. Amer. Math. Soc. 17 (1) (2004) 19–48. [28] R.C. Thompson, L.J. Freede, On the eigenvalues of sums of Hermitian matrices, II, Aequationes Math. 5 (1970) 103–115. [29] R.C. Thompson, L.J. Freede, On the eigenvalues of sums of Hermitian matrices, Linear Algebra Appl. 4 (1971) 369–376. [30] R.C. Thompson, S. Therianos, On a construction of B.P. Zwahlen, Linear Multilinear Algebra 1 (1973/1974) 309– 325. [31] R. Vakil, A geometric Littlewood–Richardson rule, Appendix A written with A. Knutson, Ann. of Math. (2) 164 (2) (2006) 371–421.


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[32] R. Vakil, Schubert induction, Ann. of Math. (2) 164 (2) (2006) 489–512. [33] D. Voiculescu, The analogues of entropy and of Fisher’s information measure in free probability theory, VI. Liberation and mutual free information, Adv. Math. 146 (1999) 101–166. [34] H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte lineare partieller Differentialgleichungen, Math. Ann. 71 (1912) 441–479.


Besov spaces with variable smoothness and integrability Alexandre Almeida a,1 , Peter Hästö b,∗,2 a Department of Mathematics, University of Aveiro, 3810-322 Aveiro, Portugal b Department of Mathematical Sciences, PO Box 3000, FI-90014 University of Oulu, Finland

Received 24 June 2009; accepted 16 September 2009 Available online 22 September 2009 Communicated by N. Kalton

Abstract In this article we introduce Besov spaces with variable smoothness and integrability indices. We prove independence of the choice of basis functions, as well as several other basic properties. We also give Sobolev-type embeddings, and show that our scale contains variable order Hölder–Zygmund spaces as special cases. We provide an alternative characterization of the Besov space using approximations by analytic functions. © 2009 Elsevier Inc. All rights reserved. Keywords: Non-standard growth; Variable exponent; Besov space; Iterated Lebesgue spaces; Hölder–Zygmund space; Sobolev embedding; Approximation

1. Introduction Spaces of variable integrability, also known as variable exponent function spaces, can be traced back to 1931 and W. Orlicz [29], but the modern development started with the paper [24] of Kováˇcik and Rákosník in 1991. Corresponding PDE with non-standard growth have been studied since the same time. For an overview we refer to the surveys [14,21,30,36] and the monograph [13]. Apart from interesting theoretical considerations, the motivation to study such * Corresponding author.

E-mail addresses: [email protected] (A. Almeida), [email protected] (P. Hästö). URL: http://cc.oulu.fi/~phasto/ (P. Hästö). 1 Supported in part by INTAS and the Research Unit Matemática e Aplicações of University of Aveiro. 2 Supported in part by the Academy of Finland, the Emil Aaltonen Foundation and INTAS. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.012

A. Almeida, P. Hästö / Journal of Functional Analysis 258 (2010) 1628–1655

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function spaces comes from applications to fluid dynamics [1,2,34], image processing [11], PDE and the calculus of variation [3,16,18,20,28,35,47]. In a recent effort to complete the picture of the variable exponent Lebesgue and Sobolev spaces, Almeida and Samko [4] and Gurka, Harjulehto and Nekvinda [19] introduced variable exponent Bessel potential spaces Lα,p(·) with constant α ∈ R. As in the classical case, this space coincides with the Lebesgue/Sobolev space for integer α. There was taken a step further by Xu α α [44–46], who considered Besov Bp(·),q and Triebel–Lizorkin Fp(·),q spaces with variable p, but fixed q and α. Along a different line of inquiry, Leopold [25–27] studied pseudo-differential operators with symbols of the type ξ m(x) , and defined related function spaces of Besov-type with variable m(·) smoothness, Bp,p . In fact, Beauzamy [7] had studied similar Ψ DEs already in the beginning of the 70s. Function spaces of variable smoothness have recently been studied by Besov [8–10]: he α(·) generalized Leopold’s work by considering both Triebel–Lizorkin spaces Fp,q and Besov spaces α(·) m(·) Bp,q in Rn . By way of application, Schneider and Schwab [39] used B2,2 (R) in the analysis of certain Black–Scholes equations. For further considerations of Ψ DEs, we refer to Hoh [22] and references therein. Integrating the above mentioned spaces into a single larger scale promises similar gains and simplifications as were seen in the constant exponent case in the 60s and 70s with the advent of the full Besov and Triebel–Lizorkin scales. Most of the advantages of unification do not occur with only one index variable: for instance, traces or Sobolev embeddings cannot be covered in this case, since they involve an interaction between integrability and smoothness. To tackle this, Diening, Hästö and Roudenko [15] introduced Triebel–Lizorkin spaces with all three indices α(·) variable, Fp(·),q(·) and showed that they behaved nicely with respect to trace. Subsequently, Vybíral [43] proved Sobolev (Jawerth) type embeddings in these spaces; they were also studied by Kempka [23]. These studies were all restricted to bounded exponents p and q. α(·) Vybíral [43] and Kempka [23] also considered Besov spaces Bp(·),q —note that only the case of constant q was included. This is quite natural, since the norm in the Besov space is usually defined via the iterated space q (Lp ) so that the space integration in Lp is done first, followed by the sum over frequency scales in q . Therefore, it is not obvious how q could depend on x, which has already been integrated out. It is the purpose of the present paper to propose a method making this dependence possible and thus completing the unification process in the variable α(·) integrability-smoothness case by introducing the Besov space Bp(·),q(·) with all three indices variable. Our space includes the previously mentioned spaces of Besov-type, as well as the Hölder– Zygmund space C α(·) . As in the constant exponent case, it is possible to consider unbounded exponents p and q in the Besov space case, while for the Triebel–Lizorkin space one needs p to be bounded. Another advantage of the Besov space for constant exponent is its simplicity compared to the Triebel–Lizorkin space; for instance, the latter requires vector-valued maximal and multiplier theorems, whereas the simple scalar case suffices in the Besov case. Unfortunately, this is not true for the generalization with variable q (this is to be expected, see Remark 4.2 for a discussion). We will nevertheless see that working in the Besov space is relatively simple once some basic tools have been established for dealing in the “iterated” space q(·) (Lp(·) ) in Sections 3 and 4. α(·) We then define the Besov space Bp(·),q(·) in Section 5 and give several basic properties establishing the soundness of our definition. In Section 6 we prove elementary embeddings between Besov and Triebel–Lizorkin spaces, as well as Sobolev embeddings in the Besov scale. In Sec-

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tion 7 we show that our scale includes the variable order Hölder–Zygmund space as a special α(·) case: B∞,∞ = C α(·) for 0 < α < 1. In Section 8 we give an alternative characterization of the Besov space by means of approximations by analytic functions. Before starting our main presentation with some conventions and results on semimodular and variable exponent spaces, we point out one possible interesting avenue for future research which might be opened by this work: real interpolation. So far, complex interpolation has been considered in the variable exponent context in [13,14]. Real interpolation, however, is more difficult in this setting. Using standard notation, we have, for constant exponents, p L 0 , Lp1 θ,q = Lpθ ,q , where 1/pθ := θ/p0 + (1 − θ )/p1 and Lpθ ,q is the Lorenz space. To obtain interpolation of Lebesgue spaces one simply chooses q = pθ . Although details have not been presented anywhere as best we know, it seems that there are no major difficulties in letting p0 and p1 be variable here, i.e. p (·) p (·) L 0 , L 1 θ,q = Lpθ (·),q , where pθ is defined point-wise by the same formula as before. However, this time we do not obtain an interpolation result in Lebesgue spaces, since we cannot set the constant q equal to the function pθ . In fact, the role of q in the real interpolation method is quite similar to the role of q α . Therefore, we hope that the approach introduced in this paper for Besov in the Besov space Bp,q spaces with variable q will also allow us to generalize real interpolation properly to the variable exponent context. Another interesting challenge is to extend extrapolation [12] to the setting of Besov spaces. 2. Preliminaries In this section we introduce some conventions and notation, and state some basic results. For the latter we refer to [13, Chapters 1–3]. We use c as a generic positive constant, i.e. a constant whose value may change from appearance to appearance. The expression f ≈ g means that 1c g f cg for some suitably independent constant c. By χA we denote the characteristic function of A ⊂ Rn . By supp f we denote the support of the function f , i.e. the closure of its zero set. The notation X → Y denotes continuous embeddings from X to Y . 2.1. Modular spaces The spaces studied in this paper fit into the framework of so-called semimodular spaces. For an exposition of these concepts we refer to the monographs [13,31]. We recall the following definition: Definition 2.1. Let X be a vector space over R or C. A function : X → [0, ∞] is called a semimodular on X if the following properties hold: (1) (0) = 0. (2) (λf ) = (f ) for all f ∈ X and |λ| = 1.


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(3) (λf ) = 0 for all λ > 0 implies f = 0. (4) λ → (λf ) is left-continuous on [0, ∞) for every f ∈ X. A semimodular is called a modular if (5) (f ) = 0 implies f = 0. A semimodular is called continuous if (6) for every f ∈ X the mapping λ → (λf ) is continuous on [0, ∞). A semimodular can be additionally qualified by the term (quasi)convex. This means, as usual, that θf + (1 − θ )g A θ (f ) + (1 − θ ) (g) , for all f, g ∈ X; here A = 1 in the convex case, and A ∈ [1, ∞) in the quasiconvex case. Once we have a semimodular in place, we obtain a normed space in a standard way: Definition 2.2. If is a (semi)modular on X, then X := x ∈ X: ∃λ > 0, (λx) < ∞ is called a (semi)modular space. Theorem 2.3. Let be a (quasi)convex semimodular on X. Then X is a (quasi)normed space with the Luxemburg (quasi)norm given by

1 x := inf λ > 0: x 1 . λ For simplicity we will refer to semimodulars as modulars except when special clarity is needed; similarly, we later drop the word “quasi”. One key method for dealing with the somewhat complicated definition of a norm is the following relationship which follows from the definition and left-continuity: (f ) 1 if and only if f 1. 2.2. Spaces of variable integrability The variable exponents that we consider are always measurable functions on Rn with range (c, ∞] for some c > 0. We denote the set of such functions by P0 . The subset of variable expo+ = ess supA p(x) nents with range [1, ∞] is denoted by P. For A ⊂ Rn and p ∈ P0 we denote pA − + − + − and pA = ess infA p(x); we abbreviate p = pRn and p = pRn . The function ϕp is defined as follows: ⎧ p if p ∈ (0, ∞), ⎨t ϕp (t) = 0 if p = ∞ and t 1, ⎩ ∞ if p = ∞ and t > 1.

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The convention 1∞ = 0 is adopted in order that ϕp be left-continuous. In what follows we write t p instead of ϕp (t), with this convention implied. The variable exponent modular is defined by p(·) (f ) :=

ϕp(x) f (x) dx.

Rn

The variable exponent Lebesgue space Lp(·) and its norm f p(·) are defined by the modular as explained in the previous subsection. The variable exponent Sobolev space W k,p(·) is the subspace of Lp(·) consisting of functions f whose distributional k-th order derivative exists and satisfies |D k f | ∈ Lp(·) with norm f W k,p(·) = f p(·) + D k f p(·) . log

We say that g : Rn → R is locally log-Hölder continuous, abbreviated g ∈ Cloc , if there exists clog > 0 such that g(x) − g(y)

clog log(e + 1/|x − y|)

for all x, y ∈ Rn . We say that g is globally log-Hölder continuous, abbreviated g ∈ C log , if it is locally log-Hölder continuous and there exists g∞ ∈ R such that g(x) − g∞

clog log(e + |x|)

for all x ∈ Rn . The notation P log is used for those variable exponents p ∈ P with

1 p

∈ C log .

log P0

is defined analogously. If p ∈ P log , then convolution with a radially decreasing The class 1 L -function is bounded on Lp(·) : ϕ ∗ f p(·) c ϕ 1 f p(·) . 3. The mixed Lebesgue-sequence space In this section we introduce a generalization of the iterated function space q (Lp(·) ) for the case of variable q, which allows us to define Besov spaces with variable q in Section 5. We give a general but quite strange looking definition for the mixed Lebesgue-sequence space modular. This is not strictly an iterated function space—indeed, it cannot be, since then there would be no space variable left in the outer function space. To motivate our definition, we show that it has several sensible properties (Examples 3.2 and 3.4) and that it concurs with the iterated space when q is constant (Proposition 3.3). Then we show that our modular in fact is a semimodular in the sense defined in the previous section and conclude that it defines a normed space.


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Definition 3.1. Let p, q ∈ P0 . The mixed Lebesgue-sequence space q(·) (Lp(·) ) is defined on sequences of Lp(·) -functions by the modular 1 q(·) (Lp(·) ) (fν )ν := inf λν > 0 p(·) fν /λνq(·) 1 .

ν

Here we use the convention λ1/∞ = 1. The norm is defined from this as usual:

(fν )ν q(·) p(·) := inf μ > 0 q(·) p(·) 1 (fν )ν 1 . (L ) (L ) μ If q + < ∞, then 1 inf λ > 0 p(·) f/λ q(·) 1 = |f |q(·) p(·) . q(·)

Since the right-hand side expression is much simpler, we use this notation to stand for the lefthand side even when q + = ∞. For instance, we often use the notation |fν |q(·) p(·) q(·) p(·) (fν )ν =

(L

)

ν

q(·)

for the modular. The norm in q(·) (Lp(·) ) is usually quite complicated to calculate. Here are some examples where it is possible to simplify its expression. Example 3.2. Suppose that p ≡ ∞. Then 1 inf λν > 0 ∞ fν /λνq(·) 1 . q(·) (L∞ ) (fν )ν =

ν 1

Now ∞ (g) 1 if and only if |g| 1 almost everywhere. Thus |fν |/λνq(·) 1 a.e., hence λν ess supx |fν (x)|q(x) . It follows that q(x) q(·) (L∞ ) (fν )ν = ess supx fν (x) . ν

Note how the case q(x) = ∞ is included by the convention t ∞ = ∞χ(1,∞) (t). Another considerable simplification occurs when q is a constant. In this case q (Lp(·) ) is really an iterated function space in the sense that we take the q -norm of Lp(·) -norms as we now show. This also justifies the notation q(·) (Lp(·) ) even though this is not in general an iterated space. Proposition 3.3. If q ∈ (0, ∞] is constant, then (fν )ν

q (Lp(·) )

= fν p(·) q .

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Proof. Suppose first that q ∈ (0, ∞). Since q is constant, |fν |q p(·) = fν q p(·) q

and thus q q q(·) (Lp(·) ) (fν )ν = fν p(·) = fν p(·) q ν

from which the claim follows. In the case q = ∞, we find inf λν > 0 p(·) fν /λ0ν 1 . ∞ (Lp(·) ) (fν )ν = ν

Here the infimum is zero, unless at least one of the sets over which it is taken is empty, in which case it is infinite. Therefore, the inequality in the definition of the norm,

(fν )ν ∞ p(·) = inf μ > 0 ∞ p(·) (fν )ν 1 , (L ) (L ) μ holds if and only if μ is such that p(·) (fν /μ) 1 for every ν, which means that inf μ = sup fν p(·) = fν p(·) ∞ . 2 Example 3.4. Let us then consider what the norm looks like when (fν ) = (f, 0, 0 . . .). We evaluate the modular:

1 1 1 (fν )ν = inf λ > 0 p(·) f/λ q(·) 1 . q(·) (Lp(·) ) μ μ By the definition of the norm, we need to find the infimum of μ > 0 such that the modular of 1 μ (fν )ν is at most one:

1 1 q(·) inf λ > 0 1 1 . = inf μ > 0 f/λ p(·) q(·) (Lp(·) ) μ

(fν )ν

In order to choose a small μ, we should make λ as big as possible. But the final inequality says that λ 1. Setting λ = 1, we see that

(fν )ν q(·) p(·) = inf μ > 0 p(·) 1 f 1 = f p(·) . (L ) μ Thus we see that the values of q have no influence on the value of (fν )ν q(·) (Lp(·) ) when the sequence has just one non-zero entry, just as in the constant exponent case. So far we have proved various results about the modular q(·) (Lp(·) ) . However, now it is time to investigate properly in what sense it is a modular in terms of Definition 2.1.


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Proposition 3.5. Let p, q ∈ P0 . Then q(·) (Lp(·) ) is a semimodular. Additionally, (a) it is a modular if p + < ∞; and (b) it is continuous if p + , q + < ∞. Proof. We need to check properties (1)–(4) of Definition 2.1 and properties (5)–(6) under the appropriate additional assumptions. Properties (1) and (2) are clear. To prove (3), we suppose that q(·) (Lp(·) ) λ(fν )ν = 0 for all λ > 0. Clearly, q(·) (Lp(·) ) ((0, . . . , 0, λfν0 , 0, . . .)) q(·) (Lp(·) ) (λ(fν )ν ) = 0. Thus it follows from Example 3.4 that fν0 p(·) = 0, and so f = 0. If p is bounded, then the same argument implies (5). To prove the left-continuity we start by noting that μ → q(·) (Lp(·) ) (μ(fν )ν ) in nondecreasing. By relabeling the function if necessary, we see that it suffices to show that q(·) (Lp(·) ) μ(fν )ν q(·) (Lp(·) ) (fν )ν as μ 1. We assume that q(·) (Lp(·) ) (fν )ν < ∞; the other case is similar. We fix ε > 0 and choose N > 0 such that N 1 q(·) (Lp(·) ) (fν )ν − ε < inf λν > 0 p(·) fν /λνq(·) 1 . ν=0

By the left-continuity of μ → p(·) (μf ), we then choose μ∗ < 1 such that N

N 1 1 inf λν > 0 p(·) fν /λνq(·) 1 − ε < inf λν > 0 p(·) μfν /λνq(·) 1

ν=0

ν=0

for all μ ∈ (μ∗ , 1). Then q(·) (Lp(·) ) ((fν )ν ) < q(·) (Lp(·) ) (μ(fν )ν ) + 2ε in the same range, which proves (4). When q + < ∞, a similar argument reduces (6) to the continuity of p(·) , which holds when p + < ∞. 2 Normally, we would have shown that the modular is quasiconvex as part of the previous theorem. Then Theorem 2.3 would immediately imply that the modular in q(·) (Lp(·) ) defines a quasinorm. Unfortunately, we do not know whether the modular is quasiconvex when q + = ∞. Therefore, we prove the quasiconvexity of the norm directly; we do this in two steps, beginning with the true convexity. Notice that our assumption when q is non-constant is not as expected. We also do not know if it is necessary. Theorem 3.6. Let p, q ∈ P. If either is a norm.

1 p

+ q1 1 point-wise, or q is a constant, then · q(·) (Lp(·) )

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Proof. Theorem 2.3 implies all the other claims, except the convexity. If p ∈ P and q ∈ [1, ∞] is a constant, then by Proposition 3.3, the convexity follows directly from the convexity of the modulars in q and Lp(·) . Thus it remains only to consider p1 + q1 1 and to show that (fν )ν + (gν )ν q(·) p(·) (fν )ν q(·) p(·) + (gν )ν q(·) p(·) . (L ) (L ) (L ) Let λ > (fν )ν q(·) (Lp(·) ) and μ > (gν )ν q(·) (Lp(·) ) . Then the claim follows from left-continuity if we show that (fν )ν + (gν )ν q(·) p(·) 1. λ+μ (L ) Moving to the modular, we get the equivalent condition fν + gν q(·) λ + μ p(·) 1, ν q(·)

with our usual convention regarding the case p/q = 0. Since fν q(·) gν q(·) 1 and 1, λ p(·) μ p(·) ν ν q(·)

q(·)

the claim follows provided we show that q(·) fν + gν q(·) λ μ fν q(·) gν λ + μ p(·) λ + μ λ p(·) + λ + μ μ p(·) q(·)

q(·)

q(·)

for every ν. Fix now one ν. Denote the norms on the right-hand side of the previous inequality by σ and τ . Then what we need to show reads

p(x) fν + gν p(x) λσ + μτ − q(x) dx 1. λ+μ λ+μ

(3.7)

Rn

We use Hölder’s inequality (with two-point atomic measure and weights (λ, μ)) as follows: 1

|fν | + |gν | = λσ q(x)

1 |gν |/μ |fν |/λ + μτ q(x) 1/q(x) σ 1/q(x) τ 1 1 1− p(x) − q(x)

(λ + μ)

(λσ + μτ )

1 q(x)

1 |fν |/λ p(x) |gν |/μ p(x) p(x) λ 1/q(x) + μ 1/q(x) . σ τ

With this, we obtain

p(x) fν + gν p(x) λσ + μτ − q(x) |fν |/λ p(x) |gν |/μ p(x) λ μ + . λ+μ λ+μ λ + μ σ 1/q(x) λ + μ τ 1/q(x)


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Integrating the inequality over Rn and taking into account that σ is the norm of fν /λ and τ the norm of gν /μ gives us (3.7), which completes the proof. 2 Then we consider the quasinorm case. Theorem 3.8. If p, q ∈ P0 , then · q(·) (Lp(·) ) is a quasinorm on q(·) (Lp(·) ). Proof. By Theorem 2.3, we only need to cosinder quasiconvexity. Let r ∈ (0, 12 min{p − , q − , 2}] and define p˜ = p/r and q˜ = q/r. Then clearly p1˜ + q1˜ 1. Thus we obtain by the previous theorem that (fν )ν + (gν )ν

q(·) (Lp(·) )

1 r r = (fν )ν + (gν )ν q(·) ˜

˜ ) (Lp(·)

1 |fν |r ν + |gν |r ν rq(·) ˜ (Lp(·) ˜ ) 1 r |gν |r q(·) |fν |r ν q(·) ˜ (Lp(·) ˜ ) + ˜ ) ν ˜ (Lp(·) r r 1 = (fν )ν q(·) (Lp(·) ) + (gν )ν q(·) (Lp(·) ) r 1 2 r −1 (fν )ν q(·) (Lp(·) ) + (gν )ν q(·) (Lp(·) ) , which completes the proof.

2

Surprisingly, the condition p, q 1 is not sufficient to guarantee that the modular q(·) (Lp(·) ) be convex! Although it is not true that the modular q(·) (Lp(·) ) is never convex when q is nonconstant, the following example shows that it may be only quasiconvex for arbitrarily small oscillations of q and for arbitrarily large p − . Note that the example deals only with sequences having a single non-zero entry. In Example 3.4 we saw that the sequence norm · q(·) (Lp(·) ) equals the Lp(·) -norm in this case, so that the triangle inequality holds even though the modular is not convex. We do not know if there exists an example of when · q(·) (Lp(·) ) is not convex and p, q 1. (Recall that the convexity of the modular is sufficient but not necessary for the convexity of the norm.) Example 3.9. Consider (fv ) = (f, 0, 0, . . .) and (gv ) = (g, 0, 0, . . .). Let p ∈ [1, ∞) be a constant. Fix two disjoint unit cubes Q1 and Q2 . Let a, b ∈ (0, ∞) and q1 , q2 ∈ [1, ∞), suppose that q|Q1 = q1 and q|Q2 = q2 , and define f = a 1/q1 χQ1 and g = b1/q2 χQ2 . Since q is constant when f is non-zero, we conclude by Proposition 3.3 that q q(·) (Lp ) (fν )ν = q1 (Lp (Q1 )) (fν )ν = a 1/q1 χQ1 p1 = a. Similarly, q(·) (Lp ) ((gν )ν ) = b. Then we consider the modular of 12 (f + g): q(·) (Lp )

1 1 1 (fν + gν )ν = inf λ > 0 p (f + g)/λ q(·) 1 . 2 2

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The condition in the infimum translates to 1 Rn

f +g 2λ1/q(x)

p dx =

1 2p

p p p qp a 1 b q2 1 a q1 1 b q2 χQ1 + χQ2 dx = p + p . λ λ 2 λ 2 λ

Rn

Since the right-hand side is continuous and decreasing in λ, we see that there exists a unique λ0 > 0 for which equality holds. This number is the value of the modular of 12 (f + g). Therefore the convexity inequality for the modular, q(·) (Lp )

1 1 (fν + gν )ν q(·) (Lp ) (fν )ν + q(·) (Lp ) (gν )ν , 2 2

can be written as a+b λ0 2

where

a λ0

p

q1

b + λ0

p

q2

= 2p .

Let us denote x := a/λ0 and y := b/λ0 . Then the convexity condition becomes 2x+y

p

p

when x q1 + y q2 = 2p .

By monotonicity, we may reformulate this as follows: p

p

x q1 + y q2 2p

when 2 = x + y.

p

(3.10)

p

Thus we need to look for the maximum of x q1 + (2 − x) q2 on [0, 2]. Suppose first that p = 1. Then (3.10) holds with equality at x = y = 1, but this is not a maximum if q1 = q2 . Thus we see that the inequality x 1/q1 + y 1/q2 2 does not hold in this case, which means that the modular is non-convex for arbitrarily small |q1 − q2 |. On the other hand, fix p > 1 and choose q1 = 1. Then we can choose x ∈ (0, 2) so large that 2p − x p/q1 = 1/2. Since y = 2 − x > 0, we can choose q2 so large that y p/q2 > 1/2. Thus we see that there exists q1 and q2 for every p such that (3.10) does not hold. We end the section by explicitly stating the open problem regarding the triangle inequality. Open problem 3.11. Suppose that p, q ∈ P. Is · q(·) (Lp(·) ) a norm on q(·) (Lp(·) )? 4. The maximal operator in the mixed Lebesgue-sequence space Despite its title, this section is actually mostly about how to work around the maximal operator. Recall that the Hardy–Littlewood maximal operator M is defined on L1loc by 1 r>0 |B(x, r)|

Mf (x) = sup

B(x,r)

f (y) dy,


1639

where B(x, r) denotes the ball with center x ∈ Rn and radius r > 0. Although the maximal operator has often proved to be very useful in analysis, it is not well suited to the mixed Lebesguesequence space q(·) (Lp(·) ): Example 4.1. Let us take, for instance, the space q(·) (L2 ). Let q, q1 , q2 , Q1 and Q2 be as in Example 3.9, and let fν := aν χQ1 for constants aν > 0. Then |fν |q(·) 2 = q(·) (L2 ) (fν )ν = aνq1 q(·)

ν

and

|λcaν χQ |q(·) q(·) (L2 ) λ(Mfν )ν 2 ν

2 q(·)

ν

=c

(λaν )q2 . ν

(The constant c depends on the distance between Q1 and Q2 , but is always positive.) If q1 > q2 , then we can choose the sequence such that (aν )ν ∈ q1 \ q2 . But then q(·) (L2 ) (fν )ν < ∞ whereas q(·) (L2 ) λ(Mfν )ν = ∞ for every λ > 0. Thus we see that M : q(·) (L2 ) → q(·) (L2 ). Remark 4.2. This example shows that q(·) (Lp(·) ) does not enjoy one key feature of iterated function spaces, namely inheritance of properties from the constituent spaces. Upon closer reflection, this is not so surprising. In the case q (Lp(·) ), the boundedness of the maximal operator, for instance, is inherited, since the outer norm functions on the inner norm in a global fashion. In the case q(·) (Lp(·) ), this is exactly what we want to avoid, since the global approach would necessarily preclude us from considering q which depends on the local space variable. Thus we see that this undesirable property is a direct consequence of the local character of our function space. The previous example showed that the maximal function is not going to be a good tool in the variable exponent space q(·) (Lp(·) ). Similarly, it was found in [15] that the vector-valued maximal inequality never holds in the iterated function space Lp(·) (q(·) ) when q is non-constant. As in the variable index Triebel–Lizorkin case [15], we use instead so-called η-functions, which have appropriate scaling. The function which we call η is defined on Rn by ην,m (x) :=

2nν (1 + 2ν |x|)m

with ν ∈ N and m > 0. Note that ην,m ∈ L1 when m > n and that ην,m 1 = cm is independent of ν. We next present some useful lemmas from [15]. log

Lemma 4.3. (See [15, Lemma 6.1].) If α ∈ Cloc , then there exists d ∈ (n, ∞) such that if m > d, then 2να(x) ην,2m (x − y) c2να(y) ην,m (x − y) with c > 0 independent of x, y ∈ Rn and ν ∈ N0 .

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The previous lemma allows us to treat the variable smoothness in many cases as if it were not variable at all, namely we can move the term inside the convolution as follows: 2να(x) ην,2m ∗ f (x) cην,m ∗ 2να(·) f (x). Remark 4.4. For most properties of the space, Lemma 4.3 is the only property of the smoothness that we need. In recent years also spaces with constant p and q, but more general smoothness functions βν (x) have been considered, see e.g. [10,23]. These are covered by most of our results provided only the previous lemma holds for them. The next lemma tells us that in most circumstances two convolutions are as good as one. Lemma 4.5. (See [15, Lemma A.3].) For ν0 , ν1 0 and m > n, we have ην0 ,m ∗ ην1 ,m ≈ ηmin{ν0 ,ν1 ,m} with the constant depending only on m and n. The set S denotes the usual Schwartz space of rapidly decreasing complex-valued functions and S denotes the dual space of tempered distributions. We denote the Fourier transform of ϕ by ϕ. ˆ The next lemma often allows us to deal with exponents which are smaller than 1. It is Lemma A.6 in [15]. Lemma 4.6 (“The r-trick"). Let r > 0, ν 0 and m > n. Then there exists c = c(r, m, n) > 0 such that g(x) c ην,m ∗ |g|r (x) 1/r ,

x ∈ Rn ,

for all g ∈ S with supp gˆ ⊂ {ξ : |ξ | 2ν+1 } Let us then prove one more lemma about η-functions which shows that they are well suited also for mixed Lebesgue sequence spaces, and hence Besov spaces as well. Lemma 4.7. Let p, q ∈ P log . For m > n, there exists c > 0 such that (ην,2m ∗ fν )ν q(·) p(·) c(fν )ν q(·) p(·) . (L ) (L ) Proof. By a scaling argument, we see that it suffices to consider the case (fν )ν q(·) (Lp(·) ) = 1 and show that the modular of a constant times the function on the left-hand side is bounded. In particular, we will show that |cην,2m ∗ fν |q(·) p(·) 2 whenever ν

q(·)

|fν |q(·) p(·) = 1. ν

This clearly follows from the inequality |cην,2m ∗ fν |q(·) p(·) |fν |q(·) p(·) + 2−ν =: δ, q(·)

q(·)

q(·)


1641

which we proceed to prove. The claim can be reformulated as showing that −1 δ |cην,2m ∗ fν |q(·) p(·) 1, q(·)

which is equivalent to − 1 δ q(·) cην,2m ∗ fν

p(·)

1. −

1

Since 1/q is log-Hölder continuous and δ ∈ [2−ν , 1 + 2−ν ], we can move δ q(·) inside the convolution by Lemma 4.3: δ −1/q(·) |ην,2m ∗ fν | c|ην,m ∗ (δ −1/q(·) fν )|. Since convolution is bounded in Lp(·) when p ∈ P log , we obtain − 1 δ q(·) cην,2m ∗ fν

Lp(·)

− 1 − 1 cην,m ∗ δ q(·) fν Lp(·) δ q(·) fν Lp(·)

with an appropriate choice of c > 0. Now the right-hand side is less than or equal to one if and only if − 1 |δ q(·) fν |q(·) p(·) 1, q(·)

which follows immediately from the definition of δ.

2

In the previous lemma we required that p, q 1. This restriction can often be circumvented by the r-trick combined with the following identity, which follows directly from the definition: (fν )ν

q(·) (Lp(·) )

1 = |fν |r ν r q(·)

r

(L

p(·) r )

.

5. The definition of the Besov space We use a Fourier approach to the Besov and Triebel–Lizorkin space. For this we need some general definitions, well known from the constant exponent case. Definition 5.1. We say a pair (ϕ, Φ) is admissible if ϕ, Φ ∈ S satisfy ˆ )| c > 0 when 35 |ξ | 53 , • supp ϕˆ ⊆ {ξ ∈ Rn : 12 |ξ | 2} and |ϕ(ξ n ˆ )| c > 0 when |ξ | 5 . • supp Φˆ ⊆ {ξ ∈ R : |ξ | 2} and |Φ(ξ 3 We set ϕν (x) := 2νn ϕ(2ν x) for ν ∈ N and ϕ0 (x) := Φ(x). We always denote by ϕν and ψν admissible functions in the sense of the previous definition. Usually, the Besov space is defined using the functions ϕν ; when this is not the case, it will be ψ explicitly marked, e.g. · α(·) . Bp(·),q(·)

Using the admissible functions (ϕ, Φ) we can define the norms να να α := 2 α := 2 ϕν ∗ f q and f Bp,q ϕν ∗ f p q , f Fp,q p

1642


for constants α ∈ R and p, q ∈ (0, ∞] (excluding p = ∞ for the F -scale). The Triebel–Lizorkin α and the Besov space B α consist of all distributions f ∈ S for which f α < ∞ space Fp,q Fp,q p,q α < ∞, respectively. It is well known that these spaces do not depend on the choice and f Bp,q of the initial system (ϕ, Φ) (up to equivalence of quasinorms). Further details on the classical theory of these spaces can be found in the books of Triebel [40,41]; see also [42] for recent developments. Definition 5.2. Let ϕν be as in Definition 5.1. For α : Rn → R and p, q ∈ P0 , the Besov space α(·) Bp(·),q(·) consists of all distributions f ∈ S such that f

ϕ α(·) Bp(·),q(·)

:= 2να(·) ϕν ∗ f ν q(·) (Lp(·) ) < ∞. α(·)

α(·)

In the case of p = q we use the notation Bp(·) := Bp(·),p(·) . To the Besov space we can also associate the following modular: ϕ α(·) (f ) := q(·) (Lp(·) ) 2να(·) ϕν ∗ f ν , Bp(·),q(·)

which can be used to define the norm. By Proposition 3.3 we directly obtain the following simplification in the case when q is constant: Corollary 5.3. If q is a constant, then f

ϕ α(·) Bp(·),q

= 2να(·) ϕν ∗ f p(·) q .

An important special case of the Besov space is when p = q. In this case we show that the Besov space agrees with the corresponding Triebel–Lizorkin space studied in [15]. This space is defined via the norm ϕ f α(·) := 2να(·) ϕν ∗ f q(·) . Fp(·),q(·)

p(·)

Notice that there is no difficulty with q depending on the space variable x here, since the q(·) norm is inside the Lp(·) -norm. Proposition 5.4. Let p ∈ P0 and α ∈ L∞ . Then Bp(·) = Fp(·) . α(·)

α(·)

Proof. The claim follows from the following calculation: p(·) να(·) ϕ 2να(x) ϕν ∗ f (x)p(x) dx α(·) (f ) = ϕν ∗ f = 2 Bp(·)

1

ν

ν

Rn

p(x) να(x) 2 = ϕν ∗ f (x) dx Rn

=

ν

p(x) να(x) ϕ 2 ϕν ∗ f (x) p(x) dx = α(·) (f ).

Rn

Fp(·)

2


1643

So far we have not considered whether the space given by Definition 5.2 depends on the choice of (ϕ, Φ). Therefore, the previous result has to be understood in the sense that the Besov space defined from a certain (ϕ, Φ) equals the Triebel–Lizorkin space defined by the same ϕ. This is not entirely satisfactory. In [15] it was shown that the Triebel–Lizorkin space is independent of the log basis functions, essentially assuming that p, q, α ∈ P0 ∩ L∞ . We prove now a corresponding log result for the Besov space, but with more general assumptions; namely we allow p, q ∈ P0 to be unbounded, and assume of α ∈ L∞ only local log-Hölder continuity. Theorem 5.5. Let p, q ∈ P0 and α ∈ Cloc ∩ L∞ . Then the space Bp(·),q(·) does not depend on the admissible basis functions ϕν , i.e. different functions yield equivalent quasinorms. log

log

α(·)

Proof. Let (ϕ, Φ) and (ψ, Ψ ) be two pairs of admissible functions. By symmetry, it suffices to prove that f

ϕ α(·) Bp(·),q(·)

c f

ψ . α(·) Bp(·),q(·)

Define K := {−1, 0, 1}. Following classical lines, and using that ϕˆν ψˆ μ = 0 when |μ − ν| > 1, we have ϕν ∗ ψν+k ∗ f. ϕν ∗ f = k∈K

Fix r ∈ (0, min{1, p − }) and m > n large. Since |ϕν | cην,2m/r , with c > 0 independent of ν, we obtain 1/r , |ϕν ∗ ψν+k ∗ f | cην,2m/r ∗ |ψν+k ∗ f | cην,2m/r ∗ ην+k,2m ∗ |ψν+k ∗ f |r where in the second inequality we used the r-trick. By Minkowski’s integral inequality (with exponent 1/r > 1) and Lemma 4.5 we further obtain r 1/r |ϕν ∗ ψν+k ∗ f |r c ην,2m/r ∗ ην+k,2m ∗ |ψν+k ∗ f |r ≈ ην+k,2m ∗ |ψν+k ∗ f |r . This, together with Lemma 4.3 and Lemma 4.7, gives να(·) 2 ϕν ∗ f ν

q(·) (Lp(·) )

1/r = 2να(·)r |ϕν ∗ f |r ν q(·)

c

r

(L

p(·) r )

2να(·)r ην+k,2m ∗ |ψν+k ∗ f |r 1/r ν q(·)

k∈K

c

2να(·)r |ψν+k ∗ f |r 1/r ν q(·)

k∈K

=c

(L

ην+k,m ∗ 2να(·)r |ψν+k ∗ f |r 1/r ν q(·)

k∈K

c

r

r

(L

p(·) r )

2να(·) ψν+k ∗ f q(·) p(·) . ν (L )

k∈K

.

r

p(·) r )

(L

p(·) r )

1644


By the shift invariance of the mixed Lebesgue sequence space, the last sum equals 3 f which completes the proof.

2

ψ , α(·) Bp(·),q(·)

Although one would obviously like to work in the variable index Besov space independent of the choice of basis functions ϕν , the assumptions needed in the previous theorem are quite strong in the sense that many of the later results work under much weaker assumptions. In the interest of clarity, we state those results only with the assumptions actually needed in their proofs. They should then be understood to hold with any particular choice of basis functions. For simplicity, we will not explicitly include the dependence on ϕ, thus omitting ϕ in the notation of the norm and modular. s,s

Remark 5.6. Recently, Schneider [37,38] studied Besov spaces of varying smoothness Bp 0 , were the function x → s(x) determines the smoothness point-wise and s0 is a constant determining the smoothness globally. These spaces are supposed to classify the smoothness behavior of a function in the neighborhood of each point. Nevertheless, they follow a different line of investigation and apparently cannot be included in our scale. s is to replace the Roughly speaking, another way of generalizing the classical scale Bp,q constant smoothness parameter s by appropriate functions or sequences. Function spaces with generalized smoothness have been considered in the literature from different points of view. The paper [17] gives an unified treatment of such spaces following the Fourier analytical approach. In that paper several references can be found including historical remarks on spaces of generalized smoothness. 6. Embeddings The following theorem gives basic embeddings between Besov spaces and Triebel–Lizorkin spaces. Theorem 6.1. Let α, α0 , α1 ∈ L∞ and p, q0 , q1 ∈ P0 . (i) If q0 q1 , then α(·)

α(·)

α (·)

α (·)

Bp(·),q0 (·) → Bp(·),q1 (·) . (ii) If (α0 − α1 )− > 0, then 0 1 → Bp(·),q . Bp(·),q 0 (·) 1 (·)

(iii) If p + , q + < ∞, then α(·)

α(·)

α(·)

Bp(·),min{p(·),q(·)} → Fp(·),q(·) → Bp(·),max{p(·),q(·)} . 1

1

Proof. Assume that q0 q1 . We note that λ q0 (x) λ q1 (x) when λ 1. By the definition it follows that B α(·)

p(·),q0 (·)

for every μ > 0, which implies (i).

(f/μ) B α(·)

p(·),q1 (·)

(f/μ)


1645

By (i), α (·)

α (·)

0 0 → Bp(·),q Bp(·),q + 0 (·) 0

α (·)

α (·)

1 1 and Bp(·),q − → Bp(·),q1 (·) . 1

Therefore, it suffices to prove (ii) for constant exponents q0+ and q1− , which we denote again by q0 , q1 ∈ (0, ∞] for simplicity. Then the proof is similar to the constant exponent situation. Indeed, να1 (·) ϕν ∗ f p(·) q c1 2να0 (·) ϕν ∗ f p(·) ∞ c1 2να0 (·) ϕν ∗ f p(·) q 2

q

with c11 =

1

0

−

2−νq1 (α0 −α1 ) < ∞.

ν0

To prove the first embedding in (iii), let r := min{p, q} and fν (x) := 2να(x) |ϕν ∗ f (x)|. We assume that B α(·) (f ) 1. Then it suffices to show that F α(·) (f ) c. Since r(x) → q(x) , p(·),r(·)

p(·),q(·)

we obtain p(·) fν q(x) p(·) fν r(x) =

Rn

p(x) fνr(x)

r(x)

dx = p(·) r(·)

ν

fνr(·) .

ν

Thus it suffices to show that the right-hand side is bounded by a constant, which follows if the corresponding norm is bounded. Using the triangle inequality, we obtain just this: r(·) f r(·) p(·) = α(·) (f ) 1. f ν ν B p(·) r(·)

ν

ν

r(·)

p(·),r(·)

For the second embedding in (iii), we use a similar derivation, with s = max{p, q}. We assume that F α(·) (f ) 1. Then we estimate the modular in the Besov space with a reverse triangle p(·),q(·)

inequality which holds since p/s 1: s(·) s(·) s(·) fν B α(·) (f ) = fν = fν s(·) p(·) . p(·) p(·) s(·) s(·) p(·),s(·) ν

ν

s(·)

Since p/s is bounded, the right-hand side is bounded if and only if the corresponding modular is bounded. In fact, p(x) s(·) p(·) fν s(·) = fν s(x) dx = F α(·) (f ) 1, s(·)

p(·),q(·)

Rn

so we are done.

2

We next consider embeddings of Sobolev-type which trade smoothness for integrability. For constant exponents it is well known that Bpα00,q → Bpα11,q

(6.2)

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if α0 − pn0 = α1 − pn1 , where 0 < p0 p1 ∞, 0 < q ∞, −∞ < α1 α0 < ∞ (see e.g. [40, Theorem 2.7.1]). This is a consequence of certain Nikolskii inequalities for entire analytic functions of exponential type (cf. [40, p. 18] for constant exponents), which we now generalize to the variable exponent setting. For constant p, the proof of (6.2) can be done via dilation arguments (see [40, Remarks 1 and 4 on pp. 18 and 23]). With the r-trick we overcome the problems with dilations in the variable exponent case. Lemma 6.3. Let p1 , p0 , q ∈ P0 with α − n/p1 and 1/q locally log-Hölder continuous. If p1 p0 , then there exists c > 0 such that ν(α(·)+ p0n(·) − p1n(·) ) q(·) να(·) q(·) g g p1 (·) 2 p0 (·) + 2−ν c2 q(·)

q(·)

for all ν ∈ N0 and g ∈ Lp0 (·) ∩ S with supp gˆ ⊂ {ξ : |ξ | 2ν+1 } such that the norm on the right-hand side is at most one. Proof. Let us denote β := α − n/p1 and ν(β(·)+ p0n(·) ) q(·) g λ := 2 p0 (·) + 2−ν . q(·)

Note that the assumption on the norm implies that λ ∈ [2−ν , 1 + 2−ν ]. Using the r-trick and Lemma 4.3, we get λ

r − q(x) νrβ(x) 2

r − 1 r − r g(x) cλ q(x) 2νrβ(x) ην,2m ∗ |g|r (x) cην,m ∗ λ q(·) 2νβ(·) |g| (x)

for large m. Fix r ∈ (0, p0− ) and set s = p0 /r ∈ P0 . An application of Hölder’s inequality with exponent s yields λ

1 − q(x) νβ(x) 2

νn 1/r − 1 ν(β(·)+ n ) p0 (·) g(x) c2− s(·) ην,m (x − ·)s (·) λ q(·) 2 g p

0 (·)

.

The second norm on the right-hand side is bounded by 1 due to the choice of λ. To show that the first norm is also bounded, we investigate the corresponding modular: νn −ms (y) s (·) 2− s(·) ην,m (x − ·) = 2νn 1 + 2ν |x − y| dy Rn

−m(s )− 1 + |2ν x − z| dz < ∞,

Rn

since m(s )− > n. Now with the appropriate choice of c0 ∈ (0, 1], we find that νβ(x) p (x) 1 − 1 να(x) |g(x)| p1 (x)−p0 (x) − q(x) ν(β(x)+ p n(x) ) g(x) p1 (x) = cp0 (x) c0 2 0 g(x) 0 c0 λ q(x) 2 λ 2 0 −1/q(x) λ − 1 ν(β(x)+ p n(x) ) g(x) p0 (x) . 0 λ q(x) 2


1647

Integrating this inequality over Rn and taking into account the definition of λ gives us the claim. 2 Applying the previous lemma, we obtain the following generalization of (6.2). Theorem 6.4 (Sobolev inequality). Let p0 , p1 , q ∈ P0 and α0 , α1 ∈ L∞ with α0 α1 . If 1/q and α0 (x) −

n n = α1 (x) − p0 (x) p1 (x)

are locally log-Hölder continuous, then α (·)

α (·)

Bp00(·),q(·) → Bp11(·),q(·) . α (·)

Proof. Suppose without loss of generality that the Bp00(·),q(·) -modular of a function is less than 1. Then an application of the previous lemma with α(x) = α1 (x) and g = ϕν ∗ f , shows that the α (·) Bp11(·),q(·) -modular is bounded by a constant. 2 Corollary 6.5. Let p0 , p1 , q0 , q1 ∈ P0 and α0 , α1 ∈ L∞ with α0 α1 . If α0 (x) −

n n = α1 (x) − + ε(x) p0 (x) p1 (x)

is locally log-Hölder continuous and ε − > 0, then α (·)

α (·)

Bp00(·),q0 (·) → Bp11(·),q1 (·) . Proof. By Theorems 6.1(i) and 6.4, α (·)

α (·)

α (·)+ε(·)

Bp00(·),q0 (·) → Bp00(·),∞ → Bp11(·),∞ . α (·)+ε(·)

We combine this with the embedding Bp11(·),∞ the proof. 2

α (·)

→ Bp11(·),q1 (·) from Theorem 6.1(ii) to conclude

Remark 6.6. It suffices to assume uniform continuity in the previous corollary (and hence in Proposition 6.9) instead of log-Hölder continuity. This is achieved by choosing an auxiliary smoothness function α˜ between α0 and α1 with the appropriate continuity modulus. Let Cu be the space of all bounded uniformly continuous functions on Rn equipped with the sup norm. Concerning embeddings into Cu , we have the following result. log

Corollary 6.7. Let α ∈ Cloc , p ∈ P log and q ∈ P0 . If 1 n δ max 1 − ,0 α(x) − p(x) q(x) for some fixed δ > 0 and every x ∈ Rn , then α(·)

Bp(·),q(·) → Cu .

1648


n Proof. Let γ (x) := α(x) − p(x) . By Theorem 6.1(i), we may replace q with the larger exponent log max{1, δ/(δ − γ )} ∈ P . It then follows from Theorem 6.4 that α(·)

γ (·)

Bp(·),q(·) → B∞,q(·) . 0 Since B∞,1 → Cu by classical results (e.g., [40, Proposition 2.5.7]), we will complete the proof by showing that γ (·)

0 . B∞,q(·) → B∞,1

Denote fν := ϕν ∗ f . The remaining embedding can be written, using homogeneity in the usual manner, as ν

sup |fν | c

whenever

x

ν

q(x) sup2νγ (x) fν 1, x

where we used the expression from Example 3.2 for the second modular. We choose xν such that supx |fν | 2|fν (xν )| for each ν. Then it follows from Young’s inequality that ν

sup |fν | ≈ x

νγ (x ) ν f (x )q(xν ) + 2−νγ (xν )q (xν ) 1 + fν (xν ) 2 2−νδ c, ν ν ν

ν

ν

which completes the proof of the remaining embedding.

2

Let Lα,p(·) , α ∈ R, be the Bessel potential space modeled in Lp(·) . It was shown in [15] that = Lα,p(·) when α 0, 1 σ 0. In particular, we have Bp(·),q(·) → Lp(·) if α − > 0 (cf. [4, Corollary 6.2] or [19, Theorem 6.1]). Next we derive a stronger version of this. Let us define

1 −1 and p(x) := max 1, p(x) , x ∈ Rn . σp (x) := n (6.8) min{1, p(x)}

If α − n/p = α − σp − n/p is log-Hölder continuous, p, q ∈ P0 , α ∈ L∞ and (α − σp )− > 0, then by Corollary 6.5 we get α(·)

0 . Bp(·),q(·) → Bp(·),1

We further conclude that f p(·)

ν0

ϕν ∗ f p(·) = f B 0

p(·),1

c f

α(·)−σp(·)

Bp(·),1

.


1649

α(·)

This shows that under the above assumptions the elements from Bp(·),q(·) are regular distributions. A discussion of such results for constant exponents can be found in [40, Section 2.5.3]; see also [33, Section 2.2.4]. In sum, we obtain the following result, without the assumptions p − > 1 and p + < ∞. Proposition 6.9. Assume that p, q ∈ P0 and α ∈ L∞ are such that α − n/p is log-Hölder continuous. Let σp and p be as in (6.8). If (α − σp )− > 0, then α(·)

Bp(·),q(·) → Lp(·) . Let p, q ∈ P0 and α ∈ L∞ . Define α0 := (α − pn )− . Then α α0 + pn =: α1 ∈ L∞ . It is clear that α1 − pn = α0 is log-Hölder continuous. Therefore we obtain by Theorem 6.4 that α(·)

n α1 (·)− p(·)

α (·)

1 → B∞,∞ Bp(·),q(·) → Bp(·),∞

α0 = B∞,∞ → S .

Here the last embedding is just the well-known constant exponent case [40, Theorem 2.3.3]. A similar argument gives the embedding of S into the variable index Besov space. Thus we obtain: Theorem 6.10. If p, q ∈ P0 and α ∈ L∞ , then α(·) S → Bp(·),q(·) → S .

Remark 6.11. As in the classical case (e.g. [40, Theorem 2.3.3]), using the previous theorem one α(·) can prove the completeness of the Besov space Bp(·),q , hence it is a (quasi)Banach space. 7. The Hölder–Zygmund space In this section we show that our scale of Besov spaces includes also the Hölder–Zygmund spaces of continuous functions. This application requires in particular that we include the case of unbounded p and q. We start by generalizing the definition of Hölder–Zygmund spaces to the variable order setting. Such spaces have been considered e.g. in [5,6,32]. Recall that Cu denotes the set of all bounded uniformly continuous functions. Definition 7.1. Let α : Rn → (0, 1]. The Zygmund space C α(·) consists of all f ∈ Cu such that f C α(·) < ∞, where f C α(·) := f ∞ +

|2h f (x)| . α(x) x∈Rn , h∈Rn \{0} |h| sup

For α < 1, the Hölder space C α(·) is defined analogously but with the norm given by f C α(·) := f ∞ +

|1h f (x)| . α(x) x∈Rn ,h∈Rn \{0} |h| sup

1650

A. Almeida, P. Hästö / Journal of Functional Analysis 258 (2010) 1628–1655 j

Here h is the j -th order difference operator (h ∈ Rn , j ∈ N): j +1

1h f (x) = f (x + h) − f (x),

h

j f = 1h h f .

One can easily derive the point-wise inequality sup |h|−α(x) 1h f (x) h

1 −α(x) 2 h f (x), + sup |h| α 2−2 h

x ∈ Rn .

Hence we have C α(·) → C α(·) for α + < 1. In fact, these two spaces coincide for such α, as in the classical case. This is one consequence of the following result. Theorem 7.2. For α locally log-Hölder continuous with α − > 0, α(·) = C α(·) B∞,∞

α(·) (α 1) and B∞,∞ = C α(·)

+ α 0 for p, q ∈ P0 itself.

log

→ C α(·)

log

and α ∈ Cloc . We then move on to the proof of the theorem

Proof of Theorem 7.2. The proof is naturally divided into two parts. First we consider the claim that α(·) C α(·) → B∞,∞

(α 1)

α(·) and C α(·) → B∞,∞

+ α 0.


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Since the Besov space is independent of the choice of admissible ϕ, we may assume without loss of generality that ϕ(−y) = ϕ(y). Then ϕν ∗ f (x) =

1 2

Rn

1 ϕν (h) f (x + h) + f (x − h) dh = 2

ϕν (h)2h f (x − h) dh, Rn

where we used the fact that ϕν (y) dy = ϕν (0) = 0 in the second step. By definition, |2h f (x −h)| f C α(·) |h|α(x−h) . For small h, the log-Hölder continuity implies that |h|α(x−h) c|h|α(x) . Thus we obtain ϕν ∗ f (x) c

ϕν (h)|h|α(x) dh + c

|h| 0 by assumption. Let r ∈ (0, 12 min{p, q, 2}). Using the previous expression and the triangle inequality in the mixed Lebesgue-sequence space (Theorem 3.6), we obtain 1 να(·) −j α(·) (j +ν)α(·) r r 2 2 (f − uν ) ν q(·) (Lp(·) ) = 2 ϕj +ν ∗ f ν q(·)

j 0

r

(L

1 r −j rα(·) (j +ν)rα(·) r 2 2 |ϕj +ν ∗ f | ν q(·)

j 0

2

(j +ν)rα(·) 2 |ϕj +ν ∗ f |r ν

−j rα −

j 0

r

p(·) r )

(L

p(·) r )

1 r

q(·) p(·) r (L r )

c 2να(·) ϕν ∗ f ν q(·) (Lp(·) ) , where the last step follows from the invariance of the norm under shifts in the ν direction. Since u0 p(·) = ϕ0 ∗ f p(·) f B α(·) , we have shown that p(·),q(·)

f u c f B α(·)

.

p(·),q(·)

p(·) be such that f = lim Now we prove the opposite k→∞ uk and inequality. Let (uk )k ∈ U u f < ∞. Then ϕν ∗ f = k−1 ϕν ∗ (uν+k − uν+k−1 ), ν ∈ N0 (with u−1 = 0). Using the r-trick, with r as before, we find that

2να(x) |ϕν ∗ f | 2να(x)

ϕν ∗ (uν+k − uν+k−1 ) k−1

1 ην,m ∗ 2να(·)r |uν+k − uν+k−1 |r r . k−1

−

Since 2να(·) 2(ν+k)α(·) 2−kα , we obtain

1654


να(·) 2 ϕν ∗ f c

ν q(·) (Lp(·) )

1 − 2−kα ην,m ∗ 2(ν+k)α(·)r |uν+k − uν+k−1 |r ν r q(·)

k−1

r

(L

p(·) r )

.

Then we can get rid of the function η by Lemma 4.7. Using |uν+k − uν+k−1 | |f − uν+k | + |f − uν+k−1 |, we find that να(·) − 2 ϕν ∗ f ν q(·) (Lp(·) ) c 2−kα 2(ν+k)α(·) (f − uν+k ) ν q(·) (Lp(·) ) . k−1

Using again the invariance of the sequence space with respect to shifts, we see that the lefthand side can be estimated by a constant times f u . Taking the infimum over u, we obtain f B α(·) c f . 2 p(·),q(·)

Remark 8.3. Compared to the proof given in [40, Theorem 2.5.3] for constant exponents, we used the r-trick to circumvent the use of Fourier multipliers. Consequently, our proof requires only the assumption α − > 0, while the stronger assumption α > σp is needed in [40] even in the constant exponent case. References [1] E. Acerbi, G. Mingione, Regularity results for a class of functionals with nonstandard growth, Arch. Ration. Mech. Anal. 156 (2) (2001) 121–140. [2] E. Acerbi, G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal. 164 (3) (2002) 213–259. [3] E. Acerbi, G. Mingione, Gradient estimates for the p(x)-Laplacian system, J. Reine Angew. Math. 584 (2005) 117–148. [4] A. Almeida, S. Samko, Characterization of Riesz and Bessel potentials on variable Lebesgue spaces, J. Funct. Spaces Appl. 4 (2) (2006) 113–144. [5] A. Almeida, S. Samko, Pointwise inequalities in variable Sobolev spaces and applications, Z. Anal. Anwend. 26 (2) (2007) 179–193. [6] A. Almeida, S. Samko, Embeddings of variable Hajłasz–Sobolev spaces into Hölder spaces of variable order, J. Math. Anal. Appl. 353 (2) (2009) 489–496. [7] A. Beauzamy, Espaces de Sobolev et Besov d’ordre variable définis sur Lp , C. R. Acad. Sci. Paris Ser. A 274 (1972) 1935–1938. [8] O. Besov, On spaces of functions of variable smoothness defined by pseudodifferential operators, Tr. Mat. Inst. Steklova 227 (1999), Issled. po Teor. Differ. Funkts. Mnogikh Perem. i ee Prilozh. 18, 56–74, translation in: Proc. Steklov Inst. Math. 4 (227) (1999) 50–69. [9] O. Besov, Equivalent normings of spaces of functions of variable smoothness, Tr. Mat. Inst. Steklova 243 (2003) (in Russian), Funkts. Prostran., Priblizh., Differ. Uravn., 87–95, translation in: Proc. Steklov Inst. Math. 243 (4) (2003) 80–88. [10] O. Besov, Interpolation, embedding, and extension of spaces of functions of variable smoothness, Tr. Mat. Inst. Steklova 248 (2005) (in Russian), Issled. po Teor. Funkts. i Differ. Uravn., 52–63, translation in: Proc. Steklov Inst. Math. 248 (1) (2005) 47–58. [11] Y. Chen, S. Levine, R. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (4) (2006) 1383–1406. [12] D. Cruz-Uribe, A. Fiorenza, J.M. Martell, C. Pérez, The boundedness of classical operators in variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 13 (2006) 239–264. [13] L. Diening, P. Harjulehto, P. Hästö, M. R˚užiˇcka, Variable Exponent Function Spaces, 2009, book manuscript.


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Sets of finite perimeter and the Hausdorff–Gauss measure on the Wiener space ✩ Masanori Hino Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan Received 25 June 2009; accepted 30 June 2009 Available online 16 July 2009 Communicated by L. Gross

Abstract In Euclidean space, the integration by parts formula for a set of finite perimeter is expressed by the integration with respect to a type of surface measure. According to geometric measure theory, this surface measure is realized by the one-codimensional Hausdorff measure restricted on the reduced boundary and/or the measure-theoretic boundary, which may be strictly smaller than the topological boundary. In this paper, we discuss the counterpart of this measure in the abstract Wiener space, which is a typical infinitedimensional space. We introduce the concept of the measure-theoretic boundary in the Wiener space and provide the integration by parts formula for sets of finite perimeter. The formula is presented in terms of the integration with respect to the one-codimensional Hausdorff–Gauss measure restricted on the measuretheoretic boundary. © 2009 Elsevier Inc. All rights reserved. Keywords: Wiener space; Set of finite perimeter; Hausdorff–Gauss measure; Geometric measure theory

1. Introduction The concept of functions of bounded variation on a domain of Rm is a fundamental concept in geometric measure theory. Let U be a domain of Rm . By definition, a real-valued Lebesgue ✩

This study was supported by a Grant-in-Aid for Young Scientists (B) (18740070, 21740094). E-mail address: [email protected].

0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.06.033

M. Hino / Journal of Functional Analysis 258 (2010) 1656–1681

1657

integrable function f on U has bounded variation if sup

(div G)f dx G ∈ Cc1 U → Rm , G(x)Rm 1 for all x ∈ U < ∞,

U

where Cc1 (U → Rm ) denotes the set of all Rm -valued functions G on U such that G is continuously differentiable and G vanishes outside a certain compact subset of U , and | · |Rm denotes the Euclidean norm on Rm . One of the basic properties of a function f of bounded variation on U is that there exist a positive Radon measure ν on U and a measurable function σ : U → Rm such that |σ (x)|Rm = 1 ν-a.e. x and

(div G)f dx = −

U

G, σ Rm dν

for all G ∈ Cc1 U → Rm ,

(1.1)

U

where ·,·Rm denotes the standard inner product on Rm . This follows directly from the Riesz representation theorem. Roughly speaking, we can say that f has an Rm -valued measure σ dν as the weak gradient. A Lebesgue measurable subset A of U is called a set of finite perimeter or sometimes a Caccioppoli set in U if the indicator function 1A of A has bounded variation on U . Then, Eq. (1.1) is rewritten as

div G dx = −

A

G, σ Rm dν

for all G ∈ Cc1 U → Rm ,

(1.2)

∂A

since the support of ν is proved to be a subset of the topological boundary ∂A of A. When A is a bounded domain with a smooth boundary, Eq. (1.2) is identical to the Gauss–Green formula, and σ and ν are expressed as the unit inner normal vector field on ∂A and the surface measure on ∂A, respectively. Although ∂A is not smooth in general, the deep theorem known as the structure theorem in geometric measure theory guarantees that A has a “measure-theoretical C 1 boundary.” To state this claim more precisely, let us define the reduced boundary ∂ A of A, which is a subset of ∂A, by the set of all points x ∈ Rm such that (i) ν(B(x, r)) > 0 for all r > 0; 1 (ii) limr→0 ν(B(x,r)) B(x,r) σ dν = σ (x); (iii) |σ (x)|Rm = 1. Here, B(x, r) = {y ∈ Rm | |y − x|Rm r}. Further, the measure-theoretic boundary ∂ A of A is defined as Lm (B(x, r) ∩ A) Lm (B(x, r) \ A) > 0 and lim sup > 0 , (1.3) ∂ A = x ∈ Rm lim sup rm rm r→0 r→0 where Lm is the m-dimensional Lebesgue measure. Then, the following theorems hold.

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Theorem 1.1 (Structure theorem). (i) The measure ν is identified by the (m − 1)-dimensional (in other words, one-codimensional) Hausdorff measure Hm−1 restricted ∞ on ∂ A. (ii) ∂ A is decomposed as ∂ A = i=1 Ci ∪N , where ν(N ) = 0 and each Ci is a compact subset of some C 1 -hypersurface Si (i = 1, 2, . . .); moreover, σ |Ci is normal to Si (i = 1, 2, . . .). Theorem 1.2. The following relations hold: ∂ A ⊂ ∂ A ⊂ ∂A and Hm−1 (∂ A \ ∂ A) = 0; in particular, ν is also equal to Hm−1 restricted on ∂ A. In this sense, the measure ν can be regarded as the surface measure on suitable boundaries of A. See, e.g., [3,10] for the proof of these claims. The proof is heavily dependent on the fact that the Lebesgue measure satisfies the volume-doubling property and that the closed balls in Rm are compact; the proof also requires effective use of covering arguments. On the other hand, in [8,9,14,15], a theory for functions of bounded variation on the abstract Wiener space, which is a typical infinite-dimensional space, has been developed in relation to stochastic analysis. In this case, the whole space E is a Banach space equipped with a Gaussian measure μ as an underlying measure, and the tangent space H is a Hilbert space that is continuously and densely embedded in E, as in the framework of the Malliavin calculus. Then, we can define the concepts of functions of bounded variation on E and sets of finite perimeter in a similar manner, and thus, we can obtain integration by parts formulas that are analogous to (1.1) and (1.2). The existence of the measure ν is proved by a version of the Riesz representation theorem in infinite dimensions. This type of Riesz theorem was proved in [7] by utilizing a probabilistic method together with the theory of Dirichlet forms and in [14] by using a purely analytic method. Since the construction of the measure ν is somewhat abstract, the geometric interpretation of ν associated with sets of finite perimeter has been unknown thus far. In this article, we consider Borel sets A of E that have a finite perimeter and prove that the measure ν associated with A as above, which is denoted by A E in this paper, is identified by the one-codimensional Hausdorff–Gauss measure restricted on the measure-theoretic boundary ∂ A of A. This Hausdorff–Gauss measure on the Wiener space has been introduced in [6] (see also [5]) in order to discuss the coarea formula on the Wiener space and the smoothness of Wiener functionals. Further, for the first time, the measure-theoretic boundary ∂ A is introduced in this study as a natural generalization of that in Euclidean space. This identification justifies the heuristic observation that A E can be considered as the surface measure of A. Since Gaussian measures on E do not satisfy the volume-doubling property and closed balls in E are not compact when E is infinite-dimensional, most techniques in geometric measure theory cannot be applied directly. Instead, we adopt the finite-dimensional approximation and utilize some results from geometric measure theory in finite dimensions; this is a reasonable approach since both the Hausdorff–Gauss measure and the measure-theoretic boundary are defined as the limits of the corresponding objects of finite-dimensional sections. The most crucial task in the proof of the main theorem (Theorem 2.11) is to prove that the order of these two limits can be possibly interchanged in a certain sense. Since the limit in the definition of the measure-theoretic boundary is not monotone, this claim is not straightforward and the proof requires rather technical arguments. The representation of A E by the Hausdorff–Gauss measure enables us to take advantage of the general properties of Hausdorff–Gauss measures [6,5]; we can deduce that A E does not charge any sets of zero (r, p)-capacity if p > 1 and rp > 1, where the (r, p)-capacity is defined in the context of the Malliavin calculus. In [14], such a smoothness property was proved by using


1659

a different method, and the similarity between this smoothness property and that of the onecodimensional Hausdorff–Gauss measure was pointed out. Our results clarify this relationship further. Surface measures in infinite dimensions have been studied in various frameworks and approaches, such as in [11,17,16,13,1,2,6]. For example, in the early study by Goodman [11], surface measures and normal vector fields were provided explicitly for what are called H -C 1 surfaces in the Wiener space. In the study by Airault and Malliavin [1], the surface measures on the level sets of smooth and nondegenerate functions are realized by generalized Wiener functionals in the sense of Malliavin calculus. In the paper by Feyel and de La Pradelle [6], the Hausdorff–Gauss measures were introduced to represent the surface measures, which has a great influence on this article. Although these apparently different expressions should be closely related one another, it does not seem evident to derive one formula from another one directly. It would be an interesting problem to clarify such an involved situation. In this study, in contrast to the preceding ones, the smoothness assumption is not explicitly imposed on the boundary of the set under consideration. The author hopes that our study will be useful to develop geometric measure theory in infinite dimensions. This paper is organized as follows. In Section 2, we provide the framework as well as the necessary definitions and propositions and state the main theorem. We provide the proof of this theorem in Section 3. In Section 4, we present some additional results as concluding remarks. 2. Framework and main results Henceforth, we denote the Borel σ -field of X by B(X) for a topological space X. Let (E, H, μ) be an abstract Wiener space. In other words, E is a separable Banach space, H is a separable Hilbert space densely and continuously embedded in E, and μ is a Gaussian measure on (E, B(E)) that satisfies √ exp −1 l(z) μ(dz) = exp −|l|2H /2 , l ∈ E ∗ . E

Here, the topological dual space E ∗ of E is regarded as a subspace of H by the natural inclusion E ∗ ⊂ H ∗ and the identification H ∗ H . The inner product and the norm of H are denoted by ·,·H and | · |H , respectively. We mainly deal with the case in which both E and H are infinitedimensional. However, if necessary, many concepts discussed below can be easily modified such that they are valid even in the finite-dimensional case. Denote by M(E) the completion of B(E) by μ. We define the following function spaces: u(z) = f (h (z), . . . , h (z)), h , . . . , h ∈ E ∗ , 1 n 1 n , F Cb1 (E) = u : E → R f ∈ Cb1 (Rn ) for some n ∈ N

F Cb1 (E → X) = the linear span of u(·)l u ∈ F Cb1 (E), l ∈ X ,

(2.1)

for a Banach space X. Here, Cb1 (Rn ) is the set of all bounded continuous functions on Rn that have continuous bounded derivatives. For a separable Hilbert space X and f ∈ F Cb1 (E → X), the H -derivative of f , denoted by ∇f , is a map from E to H ⊗ X defined by the relation

∇f (z), l

H

= (∂l f )(z)

for all l ∈ E ∗ ⊂ H,

1660


where (∂l f )(z) = lim f (z + εl) − f (z) /ε, ε→0

l ∈ E ∗ ⊂ H ⊂ E.

For each G ∈ F Cb1 (E → E ∗ ), the (formal) adjoint ∇ ∗ G is defined by the following identity:

∗ ∇ G f dμ =

E

G, ∇f H dμ for all f ∈ F Cb1 (E). E

Definition 2.1. (See [9].) We say that a real-valued M(E)-measurable function f on E is of bounded variation (f ∈ BV(E)) if E |f |((log |f |) ∨ 0)1/2 dμ < ∞ and VE (f ) := sup G

∗ ∇ G f dμ

E

is finite, where G is taken over all functions in F Cb1 (E → E ∗ ) such that |G(z)|H 1 for every z ∈ E. A subset A of E is said to have a finite perimeter if the indicator function 1A of A belongs to BV(E). We denote VE (1A ) by VE (A). One of the basic theorems concerning the functions of bounded variation is the following: Theorem 2.2. (See [9, Theorem 3.9].) For each f ∈ BV(E), there exist a finite Borel measure ν on E and an H -valued Borel measurable function σ on E such that |σ |H = 1 ν-a.e. and

∇ ∗ G f dμ =

E

G, σ H dν

for every G ∈ F Cb1 E → E ∗ .

(2.2)

E

Also, ν and σ are uniquely determined in the following sense: if ν and σ are different from ν and σ and also satisfy relation (2.2), then ν = ν and σ (z) = σ (z) for ν-a.e. z. There is no minus sign on the right-hand side of (2.2), in contrast to (1.1); this minus sign is included in the definition of ∇ ∗ . For an A ∈ M(E) that has finite perimeter, the ν and σ associated with f := 1A in the theorem above are denoted by A E and σA , respectively. Then, it is proved that the support of A E is included in the topological boundary ∂A of A in E. In other words, (2.2) is rewritten as follows: for every G ∈ F Cb1 (E → E ∗ ), A

∗ ∇ G dμ =

G, σA H d A E .

(2.3)

∂A

A more detailed assertion has been presented in [9, Theorem 3.15]. In order to state the main theorem in this paper, we introduce the concept of the Hausdorff– Gauss measure on E, essentially following the procedure in [6,5]. We begin with the finitedimensional case. Let F be an m-dimensional subspace of E ∗ (⊂ H ) with m 1. By including


1661

the inner product induced from H in the subspace F , we regard F as an m-dimensional Euclidean space. Let A be a (not necessarily Lebesgue measurable) subset of F . For ε > 0, we set m−1 SF,ε (A) = inf∞

{Bi }i=1

∞

volm−1 (Bi ),

(2.4)

i=1

where {Bi }∞ i=1 is taken over all countable coverings of A such that each Bi is an open ball of diameter less than ε, and

diam(Bi ) volm−1 (Bi ) = Vm−1 · 2

m−1 ,

Vm−1 =

π (m−1)/2 . Γ ((m − 1)/2 + 1)

Note that Vm−1 is equal to the volume of the unit ball in Rm−1 . Then, define m−1 SFm−1 (A) = lim SF,ε (A). ε↓0

SFm−1 is called the (m − 1)-dimensional (or one-codimensional) spherical Hausdorff measure on F . SFm−1 is an outer measure on F and a measure on (F, B(F )). We do not use the standard Hausdorff measure HFm−1 (namely, the measure obtained by removing the restriction that Bi is an open ball in (2.4)) for the reason explained in the remark that follows Proposition 2.4 below. The spherical Hausdorff measure and the Hausdorff measure differ on some pathological Borel sets but coincide on good sets that are considered in this study. For further details on this assertion, we refer to [4, Section 2.10.6, Corollary 2.10.42, Theorem 3.2.26]. The one-codimensional Hausdorff–Gauss measure θFm−1 on F is defined as ∗ θFm−1 (A) =

|x|2 m (2π)−m/2 exp − R SFm−1 (dx), 2

A ⊂ F.

(2.5)

A

∗ Here, denotes the outer integral for the case in which A is not measurable. Note that we adopt a terminology different from [6,5]. θFm−1 is also an outer measure on F and a measure on (F, B(F )). We now consider the infinite-dimensional case. Let F be a finite-dimensional subspace of E ∗ , and let m = dim F . Define a closed subspace F˜ of E by

F˜ = z ∈ E x(z) = 0 for every x ∈ F ⊂ E ∗ . Then, E is decomposed as a direct sum F F˜ , where F is regarded as a subspace of E. The canonical projection operators from E onto F and F˜ are denoted by pF and qF , respectively. In other words, they are given by pF (z) =

m i=1

hi (z)hi ,

qF (z) = z − pF (z),

1662


where {h1 , . . . , hm } ⊂ E ∗ ⊂ H ⊂ E is an orthonormal basis of F in H . Let μF and μF˜ be the image measures of μ by pF and qF , respectively. The measure space (E, μ) can be identified by the product measure space (F, μF ) × (F˜ , μF˜ ). We define M(F ) as the completion of B(F ) by μF , and M(F˜ ) as the completion of B(F˜ ) by μF˜ . For A ⊂ E and y ∈ F˜ , the section Ay is defined as Ay = {x ∈ F | x + y ∈ A}.

(2.6)

Define

DF = A ⊂ E the map F˜ y → θFm−1 (Ay ) ∈ [0, ∞] is M(F˜ )-measurable and θFm−1 (Ay )μF˜ (dy)

ρF (A) =

for A ∈ DF .

(2.7)

F˜

Then, we have the following propositions. Proposition 2.3. (Cf. [6, Proposition 3] or [5, Corollary 2.3].) Every Suslin set of E belongs to DF , and ρF is a measure on (E, B(E)). ∞ ∗ Fix a sequence {li }∞ i=1 ⊂ E (⊂ H ) such that {li }i=1 is a complete orthonormal system of H . For m ∈ N, let Fm be an m-dimensional subspace of E ∗ (⊂ H ⊂ E) defined as

Fm = the linear span of {l1 , . . . , lm }. Set D =

∞

m=1 DFm .

(2.8)

Note that D contains all Suslin sets of E; in particular, D ⊃ B(E).

Proposition 2.4. (See [6, Proposition 6] or [5, Proposition 3.2].) For any A ∈ D, ρFm (A) is nondecreasing in m. The essential part of the proof is contained in [4, Section 2.10.27], where it is explained that such a monotonicity does not hold when we replace SFm−1 with the Hausdorff measure HFm−1 in (2.5). From this proposition, we can define ρ(A) := lim ρFm (A) m→∞

for A ∈ D.

Then, ρ is a (non-σ -finite) measure on (E, B(E)). Denote by Mρ (E) the completion of B(E) by ρ. Proposition 2.5. Mρ (E) ⊂ D, and ρ is a complete measure on (E, Mρ (E)). This proposition is proved in the next section.


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Definition 2.6. (Cf. [6, Definition 8], [5, Definition 3.3].) We call ρ the one-codimensional Hausdorff–Gauss measure on E. Remark 2.7. (i) The measure ρ may depend on the choice of {li }∞ i=1 . In the original studies [6,5], the supremum of ρ(A) is taken over all possible choices of {li }∞ i=1 in order to define the onecodimensional Hausdorff–Gauss measure of A. In this study, such a procedure is not carried out. (ii) Similarly, for each n ∈ N, we can define the n-codimensional Hausdorff–Gauss measure on E. Next, we introduce the concept of the measure-theoretic boundary of a subset of E. Definition 2.8. Let A be a subset of E and let F be a finite-dimensional subspace of E ∗ (⊂ H ). Denote dim F by m and the m-dimensional Lebesgue outer measure on F by Lm . Define Lm (B(pF (z), r) ∩ AqF (z) ) ∂F A := z ∈ E lim sup > 0 and rm r→0 Lm (B(pF (z), r) \ AqF (z) ) lim sup > 0 . rm r→0 Here, B(pF (z), r) is a closed ball in F with center pF (z) and radius r, and AqF (z) is a section of A at qF (z) that is defined as in (2.6). For each y ∈ F˜ , the relation (∂F A)y = ∂ (Ay ) holds, where the left-hand side is the section of at y as in (2.6) and the right-hand side is the measure-theoretic boundary of Ay as in (1.3).

∂F A

Definition 2.9. For A ⊂ E, the measure-theoretic boundary ∂ A of A is defined as ∂ A := lim inf ∂Fm A = m→∞

∞ ∞

∂Fm A.

n=1 m=n

It can be easily seen that ∂ A is a subset of ∂A. In general, the sequence {∂Fm A}∞ m=1 is ; however, in not monotone in m. We also note that ∂ A may depend on the choice of {li }∞ i=1 the case of our study, the difference is negligible, as we infer from the comment that follows Theorem 2.11. Proposition 2.10. If A is a Borel subset of E, then ∂ A is also a Borel set. The proof is left to the next section. The following theorem is the main theorem of this article. Theorem 2.11. Let A be a Borel subset of E that has a finite perimeter. Then, A E coincides with ρ restricted on ∂ A. More precisely,

A E (B) = ρ(B ∩ ∂ A),

B ∈ B(E).

1664


In particular, Eq. (2.3) can be rewritten as

∗ ∇ G dμ =

A

G, σA H dρ.

(2.9)

∂ A

Further, the measure (ρ, Mρ (E)) restricted on ∂ A coincides with the completion of the measure ( A E , B(∂ A)) on ∂ A. As a consequence of this theorem, the symmetric difference of ∂ A and ∂ A is a null set with respect to A E , where ∂ A is the measure-theoretic boundary of A with respect to another complete orthonormal system {li }∞ i=1 . Indeed, by letting B1 = ∂ A \ ∂ A and B2 = ∂ A \ ∂ A, ∞ and denoting the one-codimensional Hausdorff–Gauss measure with respect to {li }i=1 by ρ , we have

A E (B1 ) = ρ(B1 ∩ ∂ A) = 0 and A E (B2 ) = ρ B2 ∩ ∂ A = 0. 3. Proof Proof of Proposition 2.5. Let A ∈ Mρ (E). Then, there exist B, C ∈ B(E) such that B ⊂ A ⊂ C and ρ(C \ B) = 0. Let m ∈ N. From Proposition 2.4, ρFm (C \ B) = 0, which, from Eq. (2.7) and the Fubini theorem, implies that θFm−1 ((C \ B)y ) = 0 for μF˜m -a.e. y ∈ F˜m . For such y, m

θFm−1 (By ) = θFm−1 (Cy ) = θFm−1 (Ay ). Since m m m

(By ) ∈ [0, ∞] F˜m y → θFm−1 m (Ay ) is also M(F˜m )-measurable in y ∈ F˜m . Therefore, we have is M(F˜m )-measurable, θFm−1 m A ∈ DFm and ρFm (A) = ρFm (B). Consequently, we conclude that A ∈ D and ρ(A) = ρ(B). In particular, the measure space (E, Mρ (E), ρ) is the completion of (E, B(E), ρ). 2 Proof of Proposition 2.10. It is sufficient to prove that ∂F A in Definition 2.8 is a Borel set. Let r > 0. Since the map F × F × F˜ (x, w, y) → 1B(x,r) (w)1A (w + y) ∈ R is Borel measurable, from the Fubini theorem, the map F × F˜ (x, y) →

1B(x,r) (w)1A (w + y)Lm (dw) ∈ R F

is Borel measurable. Therefore, Lm (B(pF (z), r) ∩ AqF (z) ) is a Borel measurable function in z ∈ E.


1665

We will prove that

Lm (B(pF (z), r) ∩ AqF (z) ) z ∈ E lim sup >0 rm r→0 Lm (B(pF (z), 2−j ) ∩ AqF (z) ) >0 . = z ∈ E lim sup (2−j )m j ∈N, j →∞

(3.1)

Denote the left-hand side and the right-hand side by B1 and B2 , respectively. The inclusion B1 ⊃ B2 is trivial. When 2−j −1 < r 2−j , Lm (B(x, 2−j ) ∩ Ay ) Lm (B(x, r) ∩ Ay ) 2m · , m r (2−j )m

x ∈ F, y ∈ F˜ .

This implies that lim sup r→0

Lm (B(pF (z), r) ∩ AqF (z) ) Lm (B(pF (z), 2−j ) ∩ AqF (z) ) m 2 lim sup . rm (2−j )m j ∈N, j →∞

Therefore, B1 ⊂ B2 . Hence, (3.1) holds. The Borel measurability of this set results from the expression B2 . Similarly, we can prove that the set {z ∈ E | lim supr→0 r −m Lm (B(pF (z), r) \ AqF (z) ) > 0} is also Borel measurable. 2 The rest of this section is devoted to the proof of Theorem 2.11. We use the same notations as those used in the previous section. In the following discussion, let F be a finite-dimensional subspace of E ∗ or F = E. Let K be a finite-dimensional subspace of F ∩ E ∗ . We regard K as a subspace of H and include the inner product induced from H in K. As a convention, μ is denoted by μF when F = E. When F is finite-dimensional, we define F Cb1 (F → K) (resp. F Cb1 (F˜ → K)), as in (2.1), with respect to the abstract Wiener space (F, F, μF ) (resp. (F˜ , F˜ ∩ H, μF˜ )). In this case, F Cb1 (F → K) is also denoted by Cb1 (F → K). By abuse of notation, the gradient operator and its adjoint operator for both (F, F, μF ) and (F˜ , F˜ ∩ H, μF˜ ) are denoted by the symbols ∇ and ∇ ∗ , respectively, which are the same as those for (E, H, μ). For A ∈ M(F ), we define VF,K (A) = sup

1 ∇ G dμF G ∈ F Cb (F → K), G(x) K 1 for all x ∈ F ( ∞). ∗

A

Proposition 3.1. Suppose VF,K (A) < ∞. Then, there exist a Borel measure A F,K on F and a K-valued Borel measurable function σA,F,K on F such that A F,K (F ) = VF,K (A), |σA,F,K (z)|K = 1 for A F,K -a.e. z, and A

∇ ∗ G dμF =

G, σA,F,K K d A F,K F

for every G ∈ F Cb1 (F → K).

(3.2)

1666


Also, A F,K and σA,F,K are uniquely determined; in other words, if A F,K and σA,F,K are different from A F,K and σA,F,K and satisfy relation (3.2), then A F,K = A F,K and (z) for A F,K -a.e. z. σA,F,K (z) = σA,F,K

Proof. The proof of this proposition is similar to that in [9, Theorem 3.9]; this proof is simpler since K is finite-dimensional. Let k = dim K. Select an orthonormal basis h1 , . . . , hk of K. Let i = 1, . . . , k. Select g ∈ F Cb1 (F ) and let G(·) = g(·)hi ∈ F Cb1 (F → K). Then,

∇ ∗ G (z) = −(∂hi g)(z) + g(z)hi (z).

From [9, Theorem 2.1] and the argument in the first part of the proof of [9, Theorem 3.9], there i on F such that exists a signed Borel measure DA

∗ ∇ G 1A dμF =

F

for all G(·) = g(·)hi ∈ F Cb1 (F → K).

i g dDA

(3.3)

F

i |, where |D i | is the total variation measure of D i . For each i, denote Define A := ki=1 |DA A A i /d by γ . We may assume that γ is Borel measurable. the Radon–Nikodym derivative dDA A i i Define a Borel measure A F,K on F and a K-valued Borel measurable function σA,F,K on F as k

A F,K (dz) = γj (z)2 A (dz),

(3.4)

j =1

σA,F,K (z) =

⎧ k ⎨ i=1 ⎩

γi (z) hi k 2 j =1 γj (z)

0

if if

k

2 j =1 γj (z)

k

= 0, (3.5)

2 j =1 γj (z) = 0.

Then, for any i = 1, . . . , k and g ∈ F Cb1 (F ),

i g dDA =

F

ghi , σA,F,K K d A F,K . F

We obtain (3.2) by combining this equation with (3.3). By construction, |σA,F,K (z)|K = 1 for

A F,K -a.e. z ∈ F . It is evident from (3.2) that the inequality VF,K (A) A F,K (F ) holds. To 1 prove the converse inequality, it is sufficient to select a sequence {Gn }∞ n=1 from F Cb (F → K) such that |Gn (z)|K 1 for all z ∈ F and limn→∞ Gn (z) = σA,F,K (z) for A F,K -a.e. z. The uniqueness is proved in the same manner as in the proof of [9, Theorem 3.9]. 2


1667

Henceforth, let F be a finite-dimensional subspace of E ∗ and K be a subspace of F . As in previous sections, both F and K are regarded as subspaces of H as well as E ∗ . Proposition 3.2. Let A ∈ M(E) with VE,K (A) < ∞. Then, the map F˜ y → VF,K (Ay ) ∈ [0, ∞] is M(F˜ )-measurable, and VF,K (Ay )μF˜ (dy) VE,K (A).

(3.6)

F˜

In particular, VF,K (Ay ) < ∞ for μF˜ -a.e. y ∈ F˜ . Here, Ay is a section of A that is defined in (2.6). Remark 3.3. In fact, equality holds in (3.6). This will be proved in Proposition 3.4(iii). Proof of Proposition 3.2. Let D1,2 (F → K) denote the (1, 2)-Sobolev space of K-valued functions on F ; in other words, it is the completion of Cb1 (F → K) with respect to the norm

f D1,2 := ( F (|∇f |2F ⊗K + |f |2K ) dμF )1/2 . This is a Hilbert space with the inner product 1 f, gD1,2 := F (∇f, ∇gF ⊗K + f, gK ) dμF . Select a sequence {fj }∞ j =1 from Cb (F → K) such that the following hold: • |fj (x)|K 1 for all j ∈ N and x ∈ F . • The set {fj | j ∈ N} is dense in {g ∈ D1,2 (F → K) | |g(x)|K 1 for μF -a.e. x} with the topology of D1,2 (F → K). For any B ∈ M(F ), we have VF,K (B) = sup

j ∈N

∇ ∗ fj (x)μF (dx),

B

since ∇ ∗ extends to a continuous operator from D1,2 (F → K) to L2 (F ). For f ∈ Cb1 (F → K), the map E F × F˜ (x, y) → 1A (x + y)∇ ∗ f (x) ∈ R is M(E)-measurable. By the Fubini theorem, Ay ∈ M(F ) for μF˜ -a.e. y ∈ F˜ , and the map F˜ y →

∇ ∗ f (x)μF (dx) ∈ R

Ay

is M(F˜ )-measurable. Therefore, the map F˜ y → VF,K (Ay ) ∈ [0, ∞] is also M(F˜ )measurable.

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˜ Let ε > 0. We inductively define a sequence {Cj }∞ j =0 of subsets of F as follows: C0 = ∅, j −1 ∗ −1 ˜ \ ∇ fj (x)μF (dx) (1 − ε)VF,K (Ay ) ∧ ε Ci , Cj = y ∈ F Ay ∈ M(F ) and i=0

Ay

j = 1, 2, . . . . Then, Cj ∈ M(F˜ ) for all j and μF˜ (F˜ \ ∞ j =1 Cj ) = 0. n Let n ∈ N. Define Dn = j =1 Cj and gn (x, y) =

n

fj (x)1Cj (y)

for (x, y) ∈ F × F˜ E.

j =1

We also regard gn as an element of L2 (F˜ → D1,2 (F → K)) by the map F˜ y → x → gn (x, y) ∈ D1,2 (F → K). Since F Cb1 (F˜ → D1,2 (F → K)) is dense in L2 (F˜ → D1,2 (F → K)) and Cb1 (F → K) is dense in D1,2 (F → K), F Cb1 (F˜ → Cb1 (F → K)) is dense in L2 (F˜ → D1,2 (F → K)). Therefore, we 1 ˜ 1 can select a sequence {uj }∞ j =1 from F Cb (F → Cb (F → K)) – also considered a subspace of F Cb1 (E → K) – such that • uj → gn in L2 (F˜ → D1,2 (F → K)) and μF˜ -a.e. as j → ∞, • |uj (x, y)|K 1 for all j ∈ N and (x, y) ∈ F × F˜ . For μF˜ -a.e. y ∈ F˜ , we have lim

j →∞ Ay

∇ ∗ uj (·, y) dμF =

∇ ∗ gn (·, y) dμF

Ay

∇ ∗ fm dμF

if y ∈ Cm for some m = 1, . . . , n, if y ∈ / Dn (1 − ε)VF,K (Ay ) ∧ ε −1 · 1Dn (y).

=

Ay

0

Therefore, Dn

(1 − ε)VF,K (Ay ) ∧ ε

−1

μF˜ (dy)

lim

F˜

j →∞ Ay

= lim

j →∞

F˜

Ay

∇ uj (·, y) dμF μF˜ (dy) ∗

∇ uj (·, y) dμF μF˜ (dy) ∗


= lim

j →∞

1669

∇ ∗ uj dμ

A

VE,K (A).

(3.7)

Here, to obtain the equality in the second line, we used the uniform integrability of the sequence { Ay ∇ ∗ (uj (·, y)) dμF }∞ j =1 , which follows from sup

j ∈N

F˜

∇ ∗ uj (·, y) dμF

2

μF˜ (dy) sup

j ∈N

Ay

F˜

∗ 2 ∇ uj (·, y) dμF μF˜ (dy) < ∞.

F

To obtain the equality in the third line in (3.7), we used the identity (∇ ∗ (uj (·, y)))(x) = (∇ ∗ uj )(x, y), which follows from the assumption that K is a subspace of F . By letting ε ↓ 0 and n → ∞ in (3.7), we obtain (3.6). 2 Proposition 3.4. Let A ∈ M(E) with VE,K (A) < ∞. (i) Let f be a bounded Borel measurable function on E. Then, the map F˜ y →

f (x + y) Ay F,K (dx) ∈ R F

is M(F˜ )-measurable, and

f d A E,K =

F˜

E

f (x + y) Ay F,K (dx) μF˜ (dy).

F

In particular, for any B ∈ B(E), the map F˜ y → Ay F,K (By ) ∈ [0, ∞) is M(F˜ )-measurable, and

A E,K (B) =

Ay F,K (By )μF˜ (dy). F˜

(ii) For μF˜ -a.e. y ∈ F˜ , σA,E,K (x + y) = σAy ,F,K (x) (iii) In Eq. (3.6), equality holds.

for Ay F,K -a.e. x ∈ F.

(3.8)

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Proof. We may assume that A is a Borel set. Let k = dim K. Select an orthonormal basis h1 , . . . , hk of K, as in the proof of Proposition 3.1. Let i = 1, . . . , k. Define Ki as a one-dimensional vector space spanned by hi . We denote i and D i for C ∈ M(F ) with VE,Ki and VF,Ki by VE,i and VF,i , respectively. Define DA C VF,i (C) < ∞ so that the relations of the type of Eq. (3.3) in the proof of Proposition 3.1 hold. i |(E) = V i Note that |DA E,i (A) and |DC |(F ) = VF,i (C). From Proposition 3.2, VF,i (Ay )μF˜ (dy) VE,i (A) VE,K (A).

(3.9)

F˜

Let g ∈ F Cb1 (E) and define G(z) = g(z)hi . Then,

i g dDA = E

∇ ∗ G dμ

A

= F˜

∗ 1Ay (x) ∇ G(· + y) (x)μF (dx) μF˜ (dy)

F

=

g(x

F˜

i + y)DA (dx) y

μF˜ (dy).

(3.10)

F

In particular, the map F˜ y →

i g(x + y)DA (dx) ∈ R y

(3.11)

F

is M(F˜ )-measurable. From the domination

i |(dx) sup |g(x + y)||DA z∈E |g(z)| · VF,i (Ay ), y ˜ Eq. (3.9), and the monotone class theorem, Eq. (3.10) and the M(F )-measurability of (3.11) hold for all bounded Borel measurable functions g on E. In particular, for B ∈ B(E), by setting g = 1B , we have F

i (B) = DA

1B (x F˜

=

i + y)DA (dx) y

μF˜ (dy)

F i DA (By )μF˜ (dy), y

(3.12)

F˜ i (B ) ∈ R is M(F˜ )-measurable. and the map F˜ y → DA y y i |(·) = D i (· ∩ S i ) − D i (· \ S i ) (the Hahn decompoSelect a Borel set S i of E such that |DA A A sition). Then,


i (E) = D i S i − D i E \ S i VE,i (A) = DA A A i i i DAy Sy − DA F \ Syi μF˜ (dy) = y

1671

from (3.12)

F˜

i D (F )μ ˜ (dy) Ay F

F˜

=

VF,i (Ay )μF˜ (dy) F˜

VE,i (A)

from (3.9) ,

where Syi is the section of S i , that is, Syi = {x ∈ F | x + y ∈ S i }. Therefore, the inequalities in the above equations can be replaced by equalities. In particular, there exists a μF˜ -null set N˜ i in B(F˜ ) such that for all y ∈ F˜ \ N˜ i , VF,i (Ay ) < ∞ and i D (F ) = D i S i − D i F \ S i , y Ay Ay y Ay i |(·) = D i (· ∩ S i ) − D i (· \ S i ); this provides the Hahn decomposition which implies that |DA y y Ay Ay y i . Then, for any B ∈ B(E), of DA y

i D (B) = D i B ∩ S i − D i B \ S i A A A i i DAy By ∩ Syi − DA By \ Syi μF˜ (dy) = y F˜

=

i D (By )μ ˜ (dy). Ay F

F˜

k i | and i Let N˜ = ki=1 N˜ i . Define A = ki=1 |DA Ay = i=1 |DAy |, which can be defined for y ∈ F˜ \ N˜ . Then, 1 ˜ ˜ (y) · Ay (By ) is Borel measurable in y ∈ F˜ and F \N

A (B) =

Ay (By )μF˜ (dy). F˜

This implies that for any bounded Borel function f on E, 1F˜ \N˜ (y) · F

f (x + y)Ay (dx) is Borel measurable in y ∈ F˜

(3.13)

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and

f dA =

F˜

E

f (x + y)Ay (dx) μF˜ (dy).

(3.14)

F

For z ∈ E, let x = pF (z) ∈ F and y = qF (z) ∈ F˜ . Let B(x, r) = {w ∈ F | |w − x|F r} for r > 0, and define a function ϕ i on E for i = 1, . . . , k by

ϕ i (z) =

⎧ ⎨ lim sup ⎩

n→∞

i (B(x,1/n)) DA y i i (B(x,1/n)\S i ) DAy (B(x,1/n) ∩ Syi )−DA y y

0

if y ∈ F˜ \ N˜ , if y ∈ N˜ ,

where 0/0 = +∞ by definition. Then, from the differentiation theorem (see, e.g., [3, Section 1.6]), for y ∈ F˜ \ N˜ (in particular, for μF˜ -a.e. y), ϕ i (x + y) is equal to the Radon–Nikodym i /d )(x) for -a.e. x ∈ F . derivative (dDA Ay Ay y We will prove that ϕ i (z) is Borel measurable in z ∈ E. Let g be a real-valued, bounded Borel measurable function on F × F × F˜ F × E such that g(x, ·) ∈ F Cb1 (E) for every x ∈ F . Define for (x, w, y) ∈ F × F × F˜ F × E.

Gx (w, y) = g(x, w, y)hi Then, for x ∈ F and y ∈ F˜ \ N˜ ,

i g(x, w, y) dDA (dw) = y

F

∗ ∇ Gx (·, y) (w)μF (dw)

Ay

=

1A (w + y) ∇ ∗ Gx (w, y)μF (dw).

F

From the Fubini theorem, the map F × F˜ (x, y) → 1F˜ \N˜ (y)

i g(x, w, y) dDA (dw) ∈ R y F

is Borel measurable. By the monotone class theorem, this measurability holds for any bounded Borel measurable function g. By letting g(x, w, y) = 1B(x,1/n) (w), g(x, w, y) = 1B(x,1/n) (w)1S i (w + y), g(x, w, y) = 1B(x,1/n) (w)1E\S i (w + y), we show that

and


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i B(x, 1/n) , 1F˜ \N˜ (y)DA y i 1F˜ \N˜ (y)DA B(x, 1/n) ∩ Syi , and y i 1F˜ \N˜ (y)DA B(x, 1/n) \ Syi y are all Borel measurable in (x, y) ∈ F × F˜ . Therefore, ϕ i (z) is Borel measurable in z ∈ E. Now, for any B ∈ B(E), i DA (B) =

from (3.12)

i DA (By )μF˜ (dy) y F˜

=

ϕ (x + y)Ay (dx) μF˜ (dy) i

F˜

=

By

ϕ i dA

from (3.14) .

B i /d . Therefore, ϕ i is equal to the Radon–Nikodym derivative dDA A From the construction of A E,K and Ay F,K by (3.4), we have

k

A E,K (dz) = ϕ j (z)2 A (dz) j =1

and k ϕ j (x + y)2 Ay (dx),

Ay F,K (dx) =

y ∈ F˜ \ N˜ .

j =1

By combining this with (3.13) and (3.14), we prove that claim (i) holds. From expression (3.5), we have

σA,E,K (z) =

σAy ,F,K (x) =

⎧ ⎨ ki=1 ⎩

0

⎧ ⎨ ki=1 ⎩

0

i ϕ (z) hi k j 2 j =1 ϕ (z)

if if

i ϕ (x+y) hi k j 2 j =1 ϕ (x+y)

k

j =1 ϕ

k

j =1 ϕ

if if

j (z)2

= 0,

j (z)2

= 0,

k

j =1 ϕ

k

j =1

j (x

ϕ j (x

+ y)2 = 0, + y)2

=0

Therefore, claim (ii) follows. We obtain (iii) by letting B = E in (3.8).

2

for y ∈ F˜ \ N˜ .

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Proposition 3.5. Let A ∈ M(E) with VE (A) < ∞. Denote the orthogonal projection operator from H to F by πF . Then, σA,E,F (z) A E,F (dz) = πF σA (z) A E (dz). In particular,

A E,F (dz) = πF σ (z)F A E (dz), π σ (z) F A if πF σA (z) = 0, σA,E,F (z) = |πF σA (z)|F 0 if πF σA (z) = 0

for A E,F -a.e. z ∈ E,

and for every B ∈ B(E), A E,Fm (B) increases to A E (B) as m → ∞, where {Fm }∞ m=1 is defined as in (2.8). Proof. Let G ∈ F Cb1 (E → F ). Then, from Proposition 3.4 with K = F ,

G, σA,E,F F d A E,F = E

F˜

F

F˜

F

=

=

G(x + y), σA,E,F (x + y) F Ay F,F (dx) μF˜ (dy)

∗ ∇ G(· + y) (x)1Ay (x)μF (dx) μF˜ (dy)

∇ ∗ G · 1A dμ

E

G, σA H d A E

= E

=

G, πF σA F d A E . E

This proves the assertion.

2

Let m = dim F . For a subset A of F , we define the measure-theoretic boundary ∂ A of A in F by replacing Rm with F in (1.3). Proposition 3.6. Let A ∈ M(F ) satisfy VF,F (A) < ∞. Then, A F,F is equal to the onecodimensional Hausdorff–Gauss measure θFm−1 restricted on ∂ A, and σA,F,F is equal to the σ obtained by replacing U and Rm in (1.2) with F . Proof. Define ξ(x) = (2π)−m/2 exp −|x|2F /2 , Then, for G ∈ Cc1 (F → F ) and f ∈ Cc1 (F ),

x ∈ F.


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(div G)f dL = −

G, ∇f F dLm

m

F

F

=−

G · ξ −1 , ∇f

F

dμF

F

=−

∗ ∇ G · ξ −1 f dμF

F

=−

∗ ∇ G · ξ −1 ξf dLm .

F

Therefore, div G = −(∇ ∗ (G · ξ −1 ))ξ . This implies that A has a locally finite perimeter in F (with respect to the Lebesgue measure) in the following sense: for any bounded domain U in F Rm ,

1 (div G) dL G ∈ Cc (U → F ), G(x) F 1 for all x ∈ U < ∞. m

sup A

For such a set, Theorems 1.1 and 1.2 hold. (See, e.g., Section 5.7.3, Theorem 2 and Section 5.8, Lemma 1 in [3].) In particular, the measure-theoretic boundary ∂ A in F is equal to a countable union of compact subsets of C 1 -surfaces in F , up to an HFm−1 -null set. Here, HFm−1 is the (m − 1)-dimensional Hausdorff measure on F . Thus, any subset B of ∂ A with HFm−1 (B) < ∞ is (HFm−1 , m − 1)-rectifiable in the sense of [4, Section 3.2.14]. From [4, Theorem 3.2.26], HFm−1 (B) = SFm−1 (B). In other words, HFm−1 and SFm−1 coincide as (outer) measures on ∂ A. Then, for G ∈ Cc1 (F → F ),

G, σ F dθFm−1

∂ A

Gξ, σ F dHFm−1

= ∂ A

=− =

div(Gξ ) dLm

from (1.2) and Theorem 1.2

A

∇ ∗ G dμF

because div(Gξ ) = − ∇ ∗ G ξ .

A

Therefore, A F,F is equal to the measure θFm−1 restricted on ∂ A, and σA,F,F = σ .

2

Proof of Theorem 2.11. Let k, m ∈ N with m > k. Denote the linear span of {lk+1 , lk+2 , . . . , lm } by Fm Fk . For ym ∈ F˜m and x ∈ Fm Fk , let (Aym )x = {w ∈ Fk | x + w ∈ Aym } = {w ∈ Fk | x + w + ym ∈ A} .

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Since VFm ,Fm (Aym ) < ∞ for μF˜m -a.e. ym ∈ F˜m , for such ym and any C ∈ B(Fm ), we have

(Ay )x m F

k ,Fk

(Cx )μFm Fk (dx) = Aym Fm ,Fk (C) Aym Fm ,Fm (C),

Fm Fk

by applying Proposition 3.4 to the abstract Wiener space (Fm , Fm , μFm ). By taking C = (E \ ∂Fm A)ym (= Fm \ ∂ (Aym )), from Proposition 3.6, we have

(Ay )x m F

0 = Aym Fm ,Fm (C)

k ,Fk

(Cx )μFm Fk (dx).

Fm Fk

Then, we have 0= F˜m

=

(Ay )x m F

k ,Fk

E \ ∂Fm A y x μFm Fk (dx) μF˜m (dym ) m

Fm Fk

Ayk Fk ,Fk E \ ∂Fm A y μF˜k (dyk ). k

F˜k

Therefore, for μF˜k -a.e. yk ∈ F˜k , ∂ (Ayk ) ⊂ (∂Fm A)yk up to a Ayk Fk ,Fk -null set, where ∂ (Ayk ) is the measure-theoretic boundary of Ayk in Fk . By taking lim infm→∞ , = (∂ A)yk ∂ (Ayk ) ⊂ lim inf ∂Fm A y = lim inf ∂Fm A m→∞

k

m→∞

yk

up to a Ayk Fk ,Fk -null set. Let B ∈ B(E). For μF˜k -a.e. yk ∈ F˜k , we have

Ayk Fk ,Fk (Byk ) = θFk−1 ∂ (Ayk ) ∩ Byk (from Proposition 3.6) k θFk−1 (∂ A)yk ∩ Byk k = θFk−1 (∂ A) ∩ B y . k k

Integrating both sides with respect to μF˜k (dyk ) and applying Proposition 3.4 with K = Fk , we have

A E,Fk (B) ρFk (∂ A) ∩ B .

(3.15)

On the other hand, by applying Proposition 2.4 with (E, H, μ) = (Fm , Fm , μFm ), for μF˜m -a.e. ym ∈ F˜m , we have


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∂ (Aym ) ∩ Bym x μFm Fk (dx) θFm−1 ∂ (Aym ) ∩ Bym θFk−1 m k

Fm Fk

= Aym Fm ,Fm (Bym ). Here, (∂ (Aym ) ∩ Bym )x = {w ∈ Fk | x + w ∈ ∂ (Aym ) ∩ Bym }. Then, ρFk ∂Fm A ∩ B =

θFk−1 k

k

F˜k

∂ μ θFk−1 (A ) ∩ B (dx) μF˜m (dym ) ym ym x Fm Fk k

= Fm Fk

F˜m

∂Fm A ∩ B y μF˜k (dyk )

Aym Fm ,Fm (Bym )μF˜m (dym )

F˜m

= A E,Fm (B)

(from Proposition 3.4)

A E (B). From the Fatou lemma, we obtain ρFk (∂ A) ∩ B lim inf ρFk ∂Fm A ∩ B A E (B). m→∞

(3.16)

From (3.15) and (3.16) and by letting k → ∞, we have

A E (B) ρ (∂ A) ∩ B A E (B) by Proposition 3.5. Therefore, A E (B) = ρ((∂ A) ∩ B) for all B ∈ B(E). The final claim in Theorem 2.11 follows from the standard argument. 2 4. Concluding remarks 4.1. Remarks on A E and σA Let A be a subset of E that has a finite perimeter. To state a further property of A E , we recall the notion of Sobolev spaces and capacities on E in the sense of the Malliavin calculus. Let K be a separable Hilbert space. Let P(E) be the set of all real-valued functions u on E that is expressed as u(z) = g(h1 (z), . . . , hn (z)) for some n ∈ N, h1 , . . . , hn ∈ E ∗ , and some polynomial g on Rn . Denote by P(E → K) the linear span of {u(·)k | u ∈ P(E), k ∈ K}. For r 0 and p > 1, the (r, p)-Sobolev space Dr,p (E → K) on E is defined as the completion of P(E → K) by the (semi-)norm · r,p defined by f r,p = ( E |(I − L)r/2 f |p dμ)1/p , where L = −∇ ∗ ∇ is the Ornstein–Uhlenbeck operator. For f ∈ Dr,p (E → K), f r,p is defined by continuity. We denote Dr,p (E → R) by Dr,p (E). The (r, p)-capacity Cr,p on E is defined as

p Cr,p (U ) = inf f r,p f ∈ Dr,p (E) and f 1 μ-a.e. on U

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when U is an open set of E and

Cr,p (B) = inf Cr,p (U ) U is open and B ⊂ U for a general B ⊂ E. Then, from [5, Theorem 4.4], the n-codimensional Gauss–Hausdorff measure does not charge any set of Cr,p -null set if rp > n. Therefore, by combining this fact with Theorem 2.11, we have the following claims. Proposition 4.1. Suppose that p > 1 and rp > 1. Then, the measure A E does not charge any Cr,p -null set. This proposition has been proved in [14, Proposition 4.6 and Remark 4.7] by using a different method. Such a smoothness property of A E is important for the study of the stochastic analysis on A; refer to [9] for further details on this topic. A K-valued function G on E is called Cr,p -quasicontinuous if for any ε > 0, there exists ˜ μ-a.e. and G ˜ is an open set U ⊂ E such that Cr,p (U ) < ε and G|E\U is continuous. If G = G ˜ is a Cr,p -quasicontinuous modification of G. In the manner Cr,p -quasicontinuous, we say that G similar to the proof of [9, Lemma 4.3], it is not difficult to prove that every G ∈ Dr,p (E → K) ˜ and if a sequence {Gn }∞ converges to G in has a Cr,p -quasicontinuous modification G, n=1 ˜ nk converges to G ˜ pointwise outDr,p (E → K), then there exists some {nk } ↑ ∞ such that G side some Cr,p -null set. Using these facts, we can prove the following corollary. Corollary 4.2. For any p > 1, Eq. (2.9) is valid for any G ∈ D1,p (E → H ) ∩ L∞ (E → H ), where G in the right-hand side of (2.9) should be replaced by the C1,p -quasicontinuous modifi˜ cation G. Proof. From the Meyer equivalence, ∇ ∗ extends to a continuous map from D1,p (E → H ) to Lp (E), and ∇ extends to a continuous map from D1,p (E → H ) to Lp (E → H ⊗ H ); further, p p { E (|f |H + |∇f |H ⊗H ) dμ}1/p provides a norm on D1,p (E → H ) that is equivalent to · 1,p . 1 ∗ From a standard procedure, we can take a sequence {Gn }∞ n=1 from F Cb (E → E ) and a C1,p 1,p ˜ null set N of E such that Gn converges to G in D (E → H ) and Gn (z) converges to G(z) for ˜ all z ∈ E \ N , and sup{Gn (z) | n ∈ N, z ∈ E} ∨ sup{G(z) | z ∈ E \ N } < ∞. Applying (2.9) to Gn and letting n → ∞, we obtain the conclusion. 2 From Proposition 3.5, the H -valued measure σA (z) A E (dz) can be regarded as a kind of projective limit of the H -valued measures associated with finite-dimensional sections of A. From the above fact and the structure theorem (Theorem 1.1), we can say that σA is described as the limit of normal vector fields on finite-dimensional sections of A. The determination of the validity of the infinite-dimensional version of the structure theorem is an open problem, which is stated below. Problem 4.3. Does ∂ A itself have an infinite-dimensional differential structure in a suitable sense, and can σA be interpreted as a normal vector field on ∂ A? If A is given by the set {f > 0} for a nondegenerate function f on E that belongs to some suitable Sobolev space, then the answer is affirmative; see [1,6,5]. In general, it does not seem


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that we can expect this type of a good expression for A. Here, we present the typical examples under consideration. Let d ∈ N and (E, H, μ) be the classical Wiener space on Rd ; in particular,

E = w ∈ C [0, 1] → Rd w(0) = 0 , 1 ˙ 2 d ds < ∞ , H = h ∈ E h is absolutely continuous and h(s) R 0

and μ is the law of the Brownian motion on Rd starting from 0. Let Ω be a domain of Rd that includes 0, and define

A = w ∈ E w(t) ∈ Ω for all t ∈ [0, 1] . We say that Ω satisfies the uniform exterior ball condition if there exists δ > 0 such that for every y in the topological boundary of Ω in Rd , there exists x ∈ Rd \ Ω satisfying B(x, δ) ∩ Ω = {y}, where B(x, δ) is the closed ball with center x and radius δ and Ω is the closure of Ω. For example, bounded domains with boundaries in the C 2 -class and convex domains satisfy this condition. Then, we have the following theorem. Theorem 4.4. (See [15, Theorem 5.1].) Suppose Ω satisfies the uniform exterior ball condition. Then, A is of finite perimeter. Further detailed properties are discussed in [15] in a more general setting. Sets of finite perimeter in the Wiener space appear in a natural manner as presented in [15], and in general, it seems difficult to treat such sets as level sets of smooth and nondegenerate functions. 4.2. Remarks on measure-theoretic boundaries In general, ∂ A is strictly smaller than ∂A. A trivial example is a one point set. It is natural to expect that ∂ A coincides with ∂A when ∂A is smooth in a certain sense. We will state it as a problem as follows: Problem 4.5. Provide sufficient conditions on A such that ∂ A = ∂A. In particular, when A is realized as {f > 0} for some function f on E, what kind of condition on f is sufficient to assure ∂ A = ∂A? As a partial answer, we will provide a simple sufficient condition at which ∂ A = ∂A holds. ◦ In the following discussion, {Fm }∞ m=1 is selected as in (2.8). For A ⊂ E, let A and A denote the interior and the closure of A in E, respectively. Proposition 4.6. Suppose A is a convex set of E with A◦ = ∅. Then, ∂ A = ∂A. For the proof, we state a basic result from convex analysis. Let G be a finite-dimensional affine space of E. For C ⊂ G, let C ◦G , C G , and ∂ G C be the interior, the closure, and the boundary of C with respect to the relative topology of G, respectively. Lemma 4.7. Let A be a convex set of E. If A◦ ∩ G = ∅, then A◦ ∩ G = (A ∩ G)◦G , A ∩ G = A ∩ GG , and (∂A) ∩ G = ∂ G (A ∩ G).

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Proof. Consider y ∈ A◦ ∩ G. We can choose an open ball U with center y that is included in A◦ . First, we prove A◦ ∩ G ⊃ (A ∩ G)◦ . Consider x ∈ (A ∩ G)◦ . There exists s > 0 such that w := 1 s w + 1+s U is an open ball that includes x and is included (1 + s)x − sy ∈ (A ∩ G)◦ . Since 1+s ◦ in A, we conclude that x ∈ A . Since x clearly belongs to G, we conclude that x ∈ A◦ ∩ G. Next, we prove A ∩ G ⊂ A ∩ GG . Consider x ∈ A ∩ G. Then, (1 − t)x + t (U ∩ G) ⊂ (A ∩ G)◦G , t∈(0,1]

and x is an accumulation point on the left-hand side; therefore, we have x ∈ A ∩ GG . Both the converse inclusions are obvious. The last equality in the claim follows from the first two equalities. 2 Proof of Proposition 4.6. It is sufficient to prove ∂ A ⊃ ∂A. Let F∞ = ∞ m=1 Fm , which is a dense subspace of E. Consider z ∈ ∂A. By the assumption A◦ = ∅, we have A◦ ∩ (z + F∞ ) = ∅. Therefore, for sufficiently large m, A◦ ∩ (z + Fm ) = ∅. Denote z + Fm by G. From Lemma 4.7, z ∈ (∂A) ∩ G = ∂ G (A ∩ G). Since A ∩ G is a convex set, it has a Lipschitz boundary in G. (For the proof, see, e.g., [12, Corollary 1.2.2.3].) This implies ∂ G (A ∩ G) = (∂Fm A) ∩ G, therefore z ∈ ∂Fm A. Thus, z belongs to ∂ A. 2 5. Note added in proof Two recent papers [18,19] that are closely relevant to this article were added in the references. References [1] H. Airault, P. Malliavin, Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math. (2) 112 (1988) 3–52. [2] V.I. Bogachev, Smooth measures, the Malliavin calculus and approximations in infinite-dimensional spaces, in: 18th Winter School on Abstract Analysis, Srní, 1990, Acta Univ. Carolin. Math. Phys. 31 (1990) 9–23. [3] L.C. Evans, R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, Boca Raton, 1992. [4] H. Federer, Geometric Measure Theory, Springer, New York, 1969. [5] D. Feyel, Hausdorff–Gauss measures, in: Stochastic Analysis and Related Topics, VII, Kusadasi, 1998, in: Progr. Probab., vol. 48, Birkhäuser Boston, Boston, MA, 2001, pp. 59–76. [6] D. Feyel, A. de La Pradelle, Hausdorff measures on the Wiener space, Potential Anal. 1 (1992) 177–189. [7] M. Fukushima, On semimartingale characterizations of functionals of symmetric Markov processes, Electron. J. Probab. 4 (18) (1999) 1–32. [8] M. Fukushima, BV functions and distorted Ornstein Uhlenbeck processes over the abstract Wiener space, J. Funct. Anal. 174 (2000) 227–249. [9] M. Fukushima, M. Hino, On the space of BV functions and a related stochastic calculus in infinite dimensions, J. Funct. Anal. 183 (2001) 245–268. [10] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monogr. Math., vol. 80, Birkhäuser, 1984. [11] V. Goodman, A divergence theorem for Hilbert space, Trans. Amer. Math. Soc. 164 (1972) 411–426. [12] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Monogr. Stud. Math., vol. 24, Pitman, 1985. [13] 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, Berlin, 1980, pp. 513–531. [14] M. Hino, Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space, in: Stochastic Analysis and Related Topics in Kyoto in Honour of Kiyosi Itô, in: Adv. Stud. Pure Math., vol. 41, 2004, pp. 121–140. [15] M. Hino, H. Uchida, Reflecting Ornstein–Uhlenbeck processes on pinned path spaces, in: Proceedings of RIMS Workshop on Stochastic Analysis and Applications, in: RIMS Kôkyûroku Bessatsu, vol. B6, Res. Inst. Math. Sci. (RIMS), Kyoto, 2008, pp. 111–128.


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[16] H.-H. Kuo, Gaussian Measures in Banach Spaces, Lecture Notes in Math., vol. 463, Springer-Verlag, Berlin, New York, 1975. [17] A.V. Skorohod, Integration in Hilbert Space, Ergeb. Math. Grenzgeb., vol. 79, Springer-Verlag, New York, Heidelberg, 1974, translated from the Russian by Kenneth Wickwire. [18] L. Ambrosio, M. Miranda Jr., S. Maniglia, D. Pallara, Towards a theory of BV functions in abstract Wiener spaces, Phys. D, in press. [19] L. Ambrosio, M. Miranda Jr., S. Maniglia, D. Pallara, BV functions in abstract Wiener spaces, preprint, 2009.


Gradient map of isoparametric polynomial and its application to Ginzburg–Landau system ✩ Jianquan Ge, Yuquan Xie ∗ School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing 100875, China Received 27 June 2009; accepted 2 October 2009 Available online 9 October 2009 Communicated by H. Brezis

Abstract In this note, we study properties of the gradient map of the isoparametric polynomial. For a given isoparametric hypersurface in sphere, we calculate explicitly the gradient map of its isoparametric polynomial which turns out many interesting phenomenons and applications. We find that it should map not only the focal submanifolds to focal submanifolds, isoparametric hypersurfaces to isoparametric hypersurfaces, but also map isoparametric hypersurfaces to focal submanifolds. In particular, it turns out to be a homogeneous polynomial automorphism on certain isoparametric hypersurface. As an immediate consequence, we get the Brouwer degree of the gradient map which was firstly obtained by Peng and Tang with moving frame method. Following Farina’s construction, another immediate consequence is a counterexample of the Brézis question about the symmetry for the Ginzburg–Landau system in dimension 6, which gives a partial answer toward the Open problem 2 raised by Farina. © 2009 Elsevier Inc. All rights reserved. Keywords: Isoparametric hypersurface; Isoparametric polynomial; Ginzburg–Landau system

1. Introduction Let M be a connected oriented isoparametric hypersurface in the unit sphere S n+1 with g distinct principle curvatures. The isoparametric polynomial F of M is a homogeneous polynomial of degree g in the Euclidean space Rn+2 , which is uniquely determined by M and satisfies the ✩

The project is partially supported by the NSFC (Grant Nos. 10531090 and 10701007).

* Corresponding author.

E-mail addresses: [email protected] (J. Ge), [email protected] (Y. Xie). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.001

J. Ge, Y. Xie / Journal of Functional Analysis 258 (2010) 1682–1691

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Cartan–Münzner equations: |∇F |2 = g 2 |x|2g−2 , F =

g2 2

(m2 − m1 )|x|g−2 ,

(1) (2)

where ∇F , F denote the gradient and Laplacian of F in Rn+2 respectively, and m1 , m2 the multiplicities of the maximal and minimal principal curvature of M. Cartan (see [5,6]) considered isoparametric hypersurfaces in spheres and solved the classification problem in the case g ∈ {1, 2, 3}. By using delicate cohomological and algebraic arguments, Münzner (see [16,17]) obtained the splendid result that the number g must be 1, 2, 3, 4 or 6. Deep going study about the geometry and topology of isoparametric hypersurfaces leads to a lot of important results (see [21,20,18,1,9,22,14,7,13], etc.). For a detailed survey of this subject and its generalizations, we’d like to refer to [24]. It is well known that there is a one-to-one correspondence between isoparametric polynomials and families of parallel isoparametric hypersurfaces. Throughout this paper, we identify both isoparametric polynomials and isoparametric hypersurfaces with their congruences under isometries of the Euclidean space. Thus we say an isoparametric polynomial is unique means that it’s unique under congruence. Given an isoparametric hypersurface M, one can construct a function F which turns out to be an isoparametric polynomial with its level hypersurfaces being parallel hypersurfaces of M. Given an isoparametric polynomial F on Rn+2 , let f denote the restriction to S n+1 . Then the level hypersurfaces of f consist of a family of parallel isoparametric hypersurfaces in the sphere. From the Cartan–Münzner equation (1), it is not difficult to show that f must have [−1, 1] as its range. Moreover, the gradient of f on S n+1 can vanish only when f = ±1, and for each s ∈ (−1, 1), the level set Ms = p ∈ S n+1 f (p) = s is a compact connected isoparametric hypersurface, while M1 , M−1 are the focal submanifolds 1 respectively. In other words, the level sets in the sphere with codimension m1 + 1 and m2 + of f give a “singular” foliation of S n+1 as S n+1 = s∈[−1,1] Ms . The gradient map Φ is a map from Rn+2 to Rn+2 defined by Φ = g1 ∇F . Obviously, each component of Φ is a homogeneous polynomial of degree g − 1, and the restriction of Φ to S n+1 provides a homogeneous polynomial map from S n+1 to S n+1 . In [19], Peng and Tang applied the moving frame method successfully to obtain the Brouwer degree of Φ. In this paper, we try to study further properties of the gradient map of the isoparametric polynomial and establish Theorem 1.1. Let F be an isoparametric polynomial of degree g on Rn+2 (g 2), the gradient map Φ = g1 ∇F . Then Φ(Mcos(gτ ) ) = Mcos(g(1−g)τ ) . In particular, (i) Φ maps focal submanifold to focal submanifold; kπ | 1 k < g − 1}, Φ maps isoparametric hypersurface Ms to focal (ii) if s ∈ D = {cos g−1 submanifold; (iii) if s ∈ (−1, 1) − D, Φ maps isoparametric hypersurface Ms to isoparametric hypersurface. In particular, Φ provides a homogeneous polynomial automorphism on certain isoparametric hypersurface.

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As an immediate consequence, we can get the Brouwer degree of the gradient map. In Section 3, we give an application of the gradient maps to the Open problem 2 of Farina [11] about Brézis question on the symmetry for the Ginzburg–Landau system. Question. (See Brézis [4].) Let u : RN → RN be a solution of u = u |u|2 − 1

on RN , N 3,

(3)

with |u(x)| → 1 as |x| → ∞ (possibly with a “good” rate of convergence). Assume deg(u, ∞) = ±1. Does u have the form u(x) =

x h |x| |x|

(4)

(modulo translation and isometry), where h : R+ → R+ is a smooth function, such that h(0) = 0 and h(∞) = 1? In [4], Brézis gave an affirmative answer for the case N = 2 and thus raised the above question. For the case N = 8, Farina [11] gave a negative answer to it. In fact, he constructed a radial x solution which can be written as the form u(x) = G( |x| )h(|x|). Therefore, he formulated his Open problem 2 which is to study Brézis question in dimension N 3 and N = 8. Following Farina’s construction and using the gradient map of some isoparametric polynomial, we give another counterexample. Theorem 1.2. There exists a solution u : R6 → R6 , of the Ginzburg–Landau system (3), satisfying (i) |u(x)| → 1 as |x| → ∞; (ii) deg(u, ∞) = 1, which has not the form (4) (modulo translation and isometry). Furthermore, u is a radial solution, i.e., it can be written in the following way: x h |x| , u(x) = Φ |x|

(5)

where Φ is the gradient map of the isoparametric polynomial in R6 with g = 4, m1 = m2 = 1, and h ∈ C 2 (R+ , R) is the unique solution of ⎧ ⎨

h h + 21 2 = h 1 − h2 , r r ⎩ h(0) = 0, h(∞) = 1. −h − 5

r > 0,

(6)

Remark 1.1. Takagi [20] proved that for the isoparametric polynomial with g = 4, if one of the principal curvatures of M has multiplicity one, then M must be homogeneous. Hence, the isoparametric polynomial in Theorem 1.2 is unique and one can write it as follows 2 2 F = |x|2 + |y|2 − 2 |x|2 − |y|2 + 4 x, y 2 ,


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where (x, y) ∈ R3 × R3 and , denotes the Euclidean inner product in R3 . Thus the map Φ = ∇F 4 in the theorem can be written explicitly. Remark 1.2. In the case g = 6, m1 = m2 = 1, Dorfmeister and Neher [9] showed that the isoparametric hypersurface in S 7 must be homogeneous. In [14], Miyaoka gave an interesting description in this case. She found that a homogeneous hypersurface in S 7 with g = 6 is the inverse image of an isoparametric hypersurface in S 4 with g = 3 under Hopf fibering. On the other hand, Cartan [5,6] determined all isoparametric hypersurfaces with g = 3. In particular, when m1 = m2 = 1, the isoparametric polynomial in R5 can be written as √ √ 3 3 2 3 2 2 2 F (x, y, X, Y, Z) = x − 3xy + x X + Y − 2Z + y X − Y 2 + 3 3XY Z, 2 2 3

2

where (x, y, X, Y, Z) ∈ R5 . Therefore, we can write the isoparametric polynomial with g = 6, m1 = m2 = 1 as F˜ = F ◦ π, where π : R8 → R5 is given by π(u, v) = |u|2 − |v|2 , 2uv¯ , The gradient map Φ = mentioned before.

∇ F˜ 6

u, v ∈ the quaternion field H ∼ = R4 .

: R8 → R8 is exactly the map G in Farina’s counterexample as we

2. Gradient map of isoparametric polynomial In this section, following Münzner [16], firstly we’ll construct the isoparametric polynomial from a given isoparametric hypersurface (see also [8]). Suppose M is a connected oriented isoparametric hypersurface in S n+1 with g distinct principal curvatures λi := cot(θi ), λ1 > · · · > λg . It’s well known that θi = θ1 +

i −1 π, g

i = 1, . . . , g,

and the multiplicity mi of λi satisfies: mi = mi+2 , m2 = mg . Let ξ be the oriented unit normal vector field of M. Consider the normal exponential map φ : M × R → S n+1 defined by φ(x, t) = cos t · x + sin t · ξx .

(7)

We know that φ has rank n + 1, except where cot t is a principal curvature of M. For any regular point (x, t) of φ, there exists an open neighborhood U of (x, t), such that φ is a diffeomorphism of U onto V = φ(U ). Define τ : V → R by τ (p) = θ1 − t,

(8)

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where t is considered as a function on V under the map φ. Note that the function τ is invariant if we take a parallel isoparametric hypersurface of M instead of M in the definition. Then we can obtain a homogeneous function F on the cone of Rn+2 over V by F (rp) = r g cos gτ (p),

p ∈ V , r > 0.

It is well known that the function F is the restriction of a homogeneous polynomial with degree g in Rn+2 , which is uniquely determined by M and satisfies the Cartan–Münzner equations (1) and (2) (see [16,8]). This polynomial is called the isoparametric polynomial of M. Let Φ be the (normalized) gradient map of F , i.e. Φ = ∇F g . Since for g = 1, Φ = ∇F is a constant map. Hence in the following discussion, we assume that g = 2, 3, 4 or 6. Let f denote the restriction of F on S n+1 . For convenience, we write the level set of f as M˜ τ = f −1 (cos gτ ), τ ∈ R. It is not difficult to check that, (i) For any integer j , M˜ 0 = M˜ 2j π , M˜ πg = M˜ (2j +1)π , and S n+1 = g

g

τ ∈Ij

M˜ τ , where Ij =

]; [ jgπ , (j +1)π g ˜ ˜ π (ii) M0 , M g are the focal submanifolds in S n+1 with codimension m1 + 1, m2 + 1, respectively; (iii) For any τ ∈ (0, π ), M˜ τ is an isoparametric hypersurface with the maximal principal curvag

ture cot τ , i.e. θ1 of M˜ τ equals τ . π , then θ1 = Proof of Theorem 1.1. For convenience, let M = f −1 (0) = M˜ 2g π φ(M × { 2g − τ }). It is easily seen that

π 2g

and M˜ τ =

π π − τ · x + cos −τ ·ξ ξ˜ = − sin 2g 2g

(9)

is the oriented unit normal vector field of M˜ τ . π We now calculate the gradient map Φ. At each point p = φ(x, 2g − τ ) ∈ M˜ τ ⊂ S n+1 , τ ∈ π (0, g ), ∇F (p) = ∇S f (p) + p, ∇F p, where ∇, ∇S are the gradient operators in Rn+2 and S n+1 respectively. Since F is a homogeneous polynomial of degree g and f = cos gτ , by Euler’s theorem, ∇F (p) = −g sin gτ · ∇S τ (p) + gF · p.

(10)

On the other hand, by (7) and (8), we have ∇S τ (p) = −ξ˜ . π − τ ), Substituting (11) to (10) implies that for each p = φ(x, 2g

(11)


1 ∇F (p) = cos gτ (p) · p + sin gτ (p) · ξ˜p g π π = cos + (g − 1)τ · x + sin + (g − 1)τ · ξx 2g 2g π = φ x, + (g − 1)τ , 2g

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Φ(p) =

(12)

which follows that Φ(M˜ τ ) = M˜ (1−g)τ .

(13)

By the continuity of the gradient map Φ, equalities (12) and (13) hold for all τ ∈ [0, πg ]. In particular, the focal submanifold M˜ 0 is the fix point set of Φ. If g is odd, Φ maps the other focal submanifold M˜ πg to M˜ (1−g)π = M˜ 0 , and if g is even, Φ maps M˜ πg to M˜ (1−g)π = M˜ πg . Note that g

g

Φ is just the antipodal map when restricted to the focal submanifold M˜ πg . For τ ∈ (0, πg ), it follows from the equality (13), Φ(M˜ τ ) is a focal submanifold if and only if cos(g(1 − g)τ ) = ±1, i.e., τ=

k π, g(g − 1)

1 k < g − 1.

When cos(gτ ) = cos(g(1 − g)τ ), i.e., g = 2, or τ = 2kπ , or τ = g 22kπ , k ∈ Z, Φ maps M˜ τ to g2 −2g itself. In particular, there’s always an isoparametric hypersurface with g 2 on which Φ provides a homogeneous polynomial automorphism. These complete the proof of Theorem 1.1. 2 As a consequence of Theorem 1.1, we can deduce the Brouwer degree of the gradient map n+1 → S n+1 , by counting the number of inverse points, counted with multiplicΦ = ∇F g |S n+1 : S ity ±1 which is the sign of the tangential map of Φ according whether it preserves or reverses the orientation, of any regular value point. Our method here differs from that of [19] where they used the integral definition of Brouwer degree and calculated it by moving frame method (see [3] for different equivalent definitions of Brouwer degree). Corollary 2.1. Let Φ be the gradient map of an isoparametric polynomial with degree g. Then the Brouwer degree of Φ is given by (i) (ii) (iii) (iv)

for g = 2, deg Φ = (−1)m1 +1 ; for g = 3, deg Φ = (−1)m1 +1 + (−1)m1 +m2 +1 ; for g = 4, deg Φ = (−1)m1 +1 + (−1)m1 +m2 +1 + (−1)m2 +1 ; for g = 6, deg Φ = 2 · (−1)m1 +1 + (−1)m1 +m2 +1 + (−1)m2 +1 − 1.

k−1 k π, g(g−1) π), 1 k g − 1. Let Proof. Denote by J = (0, πg ), Jk = ( g(g−1)

M := S n+1 − (M˜ 0 ∪ M˜ πg ) =

τ ∈J

M˜ τ ,

Mk :=

τ ∈Jk

M˜ τ .

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Then M , Mk are open subsets of S n+1 and Theorem 1.1 implies that Φ|Mk : Mk → M is a diffeomorphism for each 1 k g − 1. Thus every point p in M is a regular value point of Φ and its inverse set equals {pk ∈ Mk | Φ(pk ) = p, k = 1, . . . , g − 1} having g − 1 points. Therefore, to calculate the Brouwer degree of Φ, we need only specify the sign of its tangential map Φ∗ at each pk . Assume pk ∈ M˜ τk for some τk ∈ Jk . Then the principal curvatures of M˜ τk are given by λki = cot(τk + i−1 g π) with multiplicities mi satisfying mi = mi+2 , m2 = mg , i = 1, . . . , g. Note that m1 , m2 are determined by the isoparametric polynomial and thus are same for each k. Suppose X is a principal tangent vector of M˜ τk with respect to λki at pk . It is easily seen from formula (12) that Φ∗ (X) =

sin((1 − g)τk + sin(τk +

i−1 g π)

i−1 g π)

X,

(14)

where X in the right side is regarded as the vector at p by parallel translating X from pk to p in Rn+2 . On the other hand, from formulas (9), (11) and (12), we can derive directly Φ∗ (ξ˜pk ) = (1 − g)ξ˜p ,

(15)

where ξ˜pk , ξ˜p are the unit normal vectors of M˜ τk , Φ(M˜ τk ) = M˜ (1−g)τk at pk and p respectively. Notice that the tangent space of S n+1 at pk (resp. p) is spanned by such X’s and ξ˜pk (resp. ξ˜p ), it follows immediately from (14) and (15) that the sign of the tangential map Φ∗ at pk is given by sign Φ∗ |pk =

(−1)

k+1 k−1 2 m1 + 2 m2 +1

for k is odd,

(−1)

k k 2 m1 + 2 m2 +1

for k is even.

(16)

Combining (16) with the following formula for the definition of Brouwer degree deg(Φ) =

g−1

sign Φ∗ |pk ,

k=1

we can conclude the items of Corollary 2.1.

2

When the isoparametric polynomial F is harmonic, i.e. m1 = m2 =: m. According to the tangential map Φ∗ given in (14), (15), one can calculate directly that the tension field B(Φ) := Trace(∇S Φ∗ ) = 0, hence Φ|S n+1 : S n+1 → S n+1 is a harmonic map (see also [10] and [19]). For applications in the next section, we now focus on harmonic isoparametric polynomials. For g = 2, the harmonic isoparametric polynomial is given by F (x, y) = |x|2 − |y|2 , and thus Φ(x, y) = (x, −y), where (x, y) ∈ Rm+1 × Rm+1 . For g = 3, Cartan completely classified the isoparametric polynomials and showed that m1 = m2 = 1, 2, 4, or 8. For g = 4, Abresch [1] showed that harmonic isoparametric polynomials must have m1 = m2 = 1, or 2. These two cases were showed to be unique by Ozeki and Takeuchi [18]. See Remark 1.1 for explicit representation of the one with m = 1. For g = 6, Münzner (see [16,17]) showed that it must have m1 = m2 . Furthermore, Abresch [1] was able to show that the common multiplicity m must be either 1 or 2.


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In the case m = 1, Dorfmeister and Neher showed that it must be homogeneous. See Remark 1.2 for explicit representation for the case of m = 1. Recently, Miyaoka [15] claimed that it is also unique for the case of m = 2. In conclusion, by Corollary 2.1 and discussions above, we have (compare with [23]) Corollary 2.2. Harmonic isoparametric polynomial exists only when (g, m) = (1, m), (2, m), (3, 1), (3, 2), (3, 4), (3, 8), (4, 1), (4, 2), (6, 1), (6, 2), and for each case (except possibly the last one), it’s unique under congruence. Furthermore, its gradient map Φ|S n+1 : S n+1 → S n+1 is a polynomial harmonic map with the Brouwer degree deg Φ (i) (ii) (iii) (iv) (v)

(g, m) = (1, m), deg Φ = 0; (g, m) = (2, m), deg Φ = (−1)m+1 ; (g, m) = (3, 1), deg Φ = 0, (g, m) = (3, 2), (3, 4), (3, 8), deg Φ = −2; (g, m) = (4, 1), deg Φ = 1, (g, m) = (4, 2), deg Φ = −3; (g, m) = (6, 1), deg Φ = 1, (g, m) = (6, 2), deg Φ = −5.

3. Proof of Theorem 1.2 The proof should be translated word by word from Farina [11] once one knows examples of harmonic isoparametric polynomial with Brouwer degree of its gradient map being ±1. For completeness, we state it as follows. x )h(|x|), with a nonBy the results of [2] a function u having the form (5): u(x) = Φ( |x| 2 N −1 N 2 constant Φ ∈ C (S , R ) and a profile h ∈ C (R+ , R) is a solution of the Ginzburg–Landau system (3) if (i) Φ(S N −1 ) ⊂ S N −1 , (ii) there exists a positive integer k such that Φ ∈ (S Hk,N )N (where S Hk,N is the vector space of the spherical harmonics of degree k in RN ), and the profile h satisfies ⎧ h h ⎨ −h − (N − 1) + k(k + N − 2) 2 = h 1 − h2 , r r ⎩ h(0) = 0.

r > 0,

(17)

Therefore, to obtain the desired conclusion it is enough to prove the existence of a map Φ satisfying (i) and (ii) above with N = 6 and k = 3, and a corresponding profile h satisfying (17) with h(∞) = 1. Existence of Φ: Corollary 2.2 implies that the gradient map Φ = ∇F g of a harmonic isoparametric polynomial F has Brouwer degree ±1 if and only if g = 2, or (g, m) = (4, 1), or (g, m) = (6, 1). Obviously, such map Φ satisfies properties (i) and (ii) above. As mentioned before, when g = 2, Φ(x, y) = (x, −y) is congruent to the identity and so is trivial. The map Φ for (g, m) = (6, 1) is given explicitly in Remark 1.2 and has been applied in Farina’s counterexample in dimension N = gm + 2 = 8. The map Φ for (g, m) = (4, 1) is exactly the one we apply

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to construct the counterexample in dimension N = 6. See Remark 1.1 for explicit form of this map Φ. Existence of h: In [12] it is proved that there is a unique solution of the problem ⎧ ⎨ ⎩

h h + k(k + N − 2) 2 = h 1 − h2 , r r h(∞) = 1

−h − (N − 1) h(0) = 0,

r > 0,

(18)

for every integer N 3 and every positive integer k. Furthermore, the profile h is a strictly increasing function. This property implies that u satisfies the condition (i) in Theorem 1.2. On the other hand, owing to the special form of the constructed radial solution u, we have that deg(u, ∞) is equal to the Brouwer degree of the map Φ. This shows that (ii) in Theorem 1.2 is also satisfied. This completes the proof of Theorem 1.2. Acknowledgments It’s our great pleasure to thank Professors Chiakuei Peng and Zizhou Tang for introducing this topic to us and guidance, and also for their careful revision of an earlier version of this paper. References [1] U. Abresch, Isoparametric hypersurfaces with four or six distinct principal curvatures, Math. Ann. 264 (1983) 283– 302. [2] V. Akopian, A. Farina, Sur les solutions radiales de l’équation u = u(1 − |u|2 ) dans R n (N 3), C. R. Acad. Sci. Paris Sér. I Math. 325 (6) (1997) 601–604. [3] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Grad. Texts in Math., vol. 82, Springer-Verlag, New York, 1982. [4] H. Brézis, Symmetry in nonlinear PDE’s, in: Differential Equations, La Pietra, Florence, 1996, in: Proc. Sympos. Pure Math., vol. 65, Amer. Math. Soc., Providence, RI, 1999, pp. 1–12. [5] E. Cartan, Familles de surfaces isoparamétriques dans les espaces à courbure constante, Ann. Mat. 17 (1938) 177– 191. [6] E. Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques, Math. Z. 45 (1939) 335–367. [7] T.E. Cecil, Q.S. Chi, G.R. Jensen, Isoparametric hypersurfaces with four principal curvatures, Ann. of Math. 166 (1) (2007) 1–76. [8] T.E. Cecil, P.T. Ryan, Tight and Taut Immersions of Manifolds, Res. Notes Math., vol. 107, Pitman, London, 1985. [9] J. Dorfmeister, E. Neher, Isoparametric hypersurfaces, case g = 6, m = 1, Comm. Algebra 13 (1985) 2299–2368. [10] J. Eells, A. Ratto, Harmonic Maps and Minimal Immersions with Symmetries, Ann. of Math. Stud., vol. 130, Princeton Univ. Press, 1993. [11] A. Farina, Two results on entire solutions of Ginzburg–Landau system in higher dimensions, J. Funct. Anal. 214 (2004) 386–395. [12] A. Farina, M. Guedda, Qualitative study of radial solutions of the Ginzburg–Landau system in R N (N 3), Appl. Math. Lett. 13 (7) (2000) 59–64. [13] S. Immervoll, On the classification of isoparametric hypersurfaces with four distinct principal curvatures in spheres, Ann. of Math. 168 (3) (2008) 1011–1024. [14] R. Miyaoka, The linear isotropy group of G2 /SO(4), the Hopf fibering and isoparametric hypersurfaces, Osaka J. Math. 30 (1993) 179–202. [15] R. Miyaoka, Isoparametric hypersurfaces with six principal curvatures, preprint, 2008. [16] H.F. Münzner, Isoparametric hyperflächen in sphären, I, Math. Ann. 251 (1980) 57–71. [17] H.F. Münzner, Isoparametric hyperflächen in sphären, II, Math. Ann. 256 (1981) 215–232. [18] H. Ozeki, M. Takeuchi, On some types of isoparametric hypersurfaces in spheres I, Tohoku Math. J. 27 (1975) 515–559;


[19] [20] [21] [22] [23] [24]

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H. Ozeki, M. Takeuchi, On some types of isoparametric hypersurfaces in spheres II, Tohoku Math. J. 28 (1976) 7–55. C.K. Peng, Z.Z. Tang, Brouwer degrees of gradient maps of isoparametric functions, Sci. China Ser. A 39 (1996) 1131–1139. R. Takagi, A class of hypersurfaces with a constant principal curvatures in a sphere, J. Differential Geom. 11 (1976) 225–233. R. Takagi, T. Takahashi, On the principal curvatures of homogeneous hypersurfaces in a sphere, in: Differential Geometry in Honor of K. Yano, Kinokuniya, Tokyo, 1972, pp. 469–481. Z.Z. Tang, Isoparametric hypersurfaces with four distinct principal curvatures, Chinese Sci. Bull. 36 (15) (1991) 1237–1240. Z.Z. Tang, Harmonic Hopf constructions and isoparametric gradient maps, Differential Geom. Appl. 25 (2007) 461–465. G. Thorbergsson, A survey on isoparametric hypersurfaces and their generalizations, in: Handbook of Differential Geometry, vol. I, North-Holland, Amsterdam, 2000, pp. 963–995.


Sofic equivalence relations ✩ Gábor Elek ∗ , Gábor Lippner Mathematical Institute of the Hungarian Academy of Sciences, PO Box 127, Budapest, Hungary Received 29 June 2009; accepted 20 October 2009 Available online 28 October 2009 Communicated by S. Vaes

Abstract We introduce the notion of sofic measurable equivalence relations. Using them we prove that Connes’ Embedding Conjecture as well as the Measurable Determinant Conjecture of Lück, Sauer and Wegner hold for treeable equivalence relations. © 2009 Elsevier Inc. All rights reserved. Keywords: Measurable equivalence relations; Connes Embedding Conjecture; Sofic groups; von Neumann algebras

1. Introduction 1.1. Sofic groups and sofic relations First let us recall the definition of sofic groups. The group Γ is sofic if for any real number 0 < ε < 1 and any finite subset F ⊆ Γ there exists a natural number n and a function ψn : Γ → Sn from Γ into the group of permutations on n elements with the following properties: (a) #fix (φ(e)φ(f )φ(ef )−1 ) (1 − ε)n for any two elements e, f ∈ F , (b) φ(1) = 1, (c) #fix φ(e) εn for any 1 = e ∈ F , ✩

Research sponsored by OTKA Grant Nos. 67867, 69062.


E-mail address: [email protected] (G. Elek). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.013

G. Elek, G. Lippner / Journal of Functional Analysis 258 (2010) 1692–1708

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where #fix π denotes the number of fixed points of the permutation π ∈ Sn . The notion of soficity was introduced by Gromov [8] and Weiss [14] as a common generalization of amenability and residual finiteness. Direct products, subgroups, free products, inverse and direct limits of sofic groups are sofic as well. If N Γ , N is sofic and Γ /N is amenable, then Γ is also sofic. Residually amenable groups are sofic, however there exist finitely generated non-residually amenable sofic groups as well [5]. It is conjectured that there are non-sofic groups, but no example is known yet (see also the survey of Pestov [12]). In our paper we introduce the notion of a sofic measurable equivalence relation (SER). First let us briefly recall some basic definitions from [9]. A countable Borel-equivalence relation is a Borel-subspace E ⊂ X × X, where E is an equivalence relation and all equivalence classes are countable. The space X is a standard Borel-space. Let Γ be a countable group and Γ X be a Borel-action of Γ then it defines a countable Borel-equivalence relation of X and in fact by the theorem of Feldman and Moore any countable Borel-equivalence relation can be obtained by such an action. A probability measure μ is E-invariant if it is invariant under a (and actually under all) Borel-action of a countable group defining the relation E. From now on, let X = {0, 1}N denote the standard Borel-space which we equip with the standard product probability measure μ. For any word w ∈ {0, 1}k Aw ⊂ X is the closed–open set of those points in X which start with w. Let F∞ = γ1 , γ2 , . . . denote the free group on countable generators. For any integer r > 0 let us denoted by Wr ⊂ F∞ the subset of reduced words of length at most r containing only letters γ1 , γ1−1 , γ2 , γ2−1 . . . γr , γr−1 . Clearly, W0 ⊂ W1 ⊂ W2 · · · ∞ and r=0 Wr = F∞ . Suppose θ : F∞ X is a (not necessarily free) Borel group action. Then θ gives rise to a directed graphing (a directed Borel-graph) G ⊂ X × X in a natural way: (x, y) ∈ G if and only if there is an index i such that θ (γi , x) = y. The group action also gives an edgecoloring of this graphing with countable colors such a way that any vertex there is exactly one out-edge and one in-edge of every color. The colors are γ1 , γ1−1 , γ2 , γ2−1 . . . . Since an edge xy might be realized by more than one generator, it will be more convenient to think of G as a multigraphing (i.e. one where multiple edges and loop edges are allowed) and then the action gives us indeed a unique edge-coloring. Also, if xy is colored by γi then yx is colored by γi−1 . Definition 1.1. By an r-neighborhood we mean an r-edge-colored oriented multi-graph. That is the out-edges need to have different colors from the set γ1 , γ1−1 , γ2 , γ2−1 . . . γr , γr−1 and if xy is colored by γi then yx is colored by γi−1 . Also, we have a chosen vertex which is called the root such that any vertex is connected to the root via a path of length at most r. It is obvious that up to colored, rooted isomorphisms there are only finitely many different r-neighborhoods. The set of these will be denoted by U r . Given the group action θ and a point x ∈ X we define its r-neighborhood Br (x) to be the subgraph of G spanned by θ (Wr , x). Its root is x and it inherits the edge-coloring from G. Definition 1.2. By an r-labeled r-neighborhood we mean an r-neighborhood whose vertices are labeled with words taken from {0, 1}r . Again the isomorphism types of such objects form a finite set which we denote by U r,r . Given the group action θ and a point x ∈ X we define its r-labeled r-neighborhood Brr (x) to be the r-neighborhood of x with labeling defined in the following way: any vertex y ∈ Br (x) corresponds to a point y ∈ X. The label of y shall be the unique word w ∈ {0, 1}r for which y ∈ Aw ⊂ X.

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For a fixed action θ and a fix α ∈ U r,r it is easy to see that the set T (θ, α) = {x ∈ X: Brr (x) ≡ α} forms a Borel subset of X. Hence we can take its measure pα (θ ) = μ(T (θ, α)) which is clearly a number between 0 and 1. We can repeat everything for any action θ of F∞ on a finite set Y whose elements are labeled with elements from {0, 1}N . Then pα (θ ) is defined as |T (θ,α)| |Y | . We call such vertex labelled sets X-sets. Definition 1.3. We say that the Borel-action θ is sofic if there is a sequence of actions θn of F∞ on finite X-sets Yn such that for any r 1 and α ∈ U r,r limn→∞ pα (θn ) = pα (θ ). Note this definition is strongly related to the various notions of graph convergence (see e.g. [3]). Remark 1.1. An action θ is sofic if and only if θ r = θ |γ1 ,...,γr , its restriction to the first r generators is sofic. The if part follows from choosing a suitable diagonal sequence from the sequences θnr that prove the soficity of each θ r . For the only-if part one takes the sofic sequence θn and restricts it to the first r generators, thereby obtaining a sequence θnr that is obviously sofic for θ r . We call a countable measured Borel-equivalence relation sofic equivalence relation (SER) if it is defined by a sofic action of F∞ . Obviously, since any countable group is a quotient of F∞ , Borel-equivalence relations can always be defined by F∞ -actions. In Section 2 we shall see that if E is given by actions θ respectively θ and θ is sofic, then θ is sofic as well (Theorem 1). That is soficity is not only a property of groups actions, but the property of measurable equivalence relations. It is quite obvious that if a group Γ has a free sofic action, then Γ is sofic. On the other hand, we do not know whether all free actions of a sofic group are sofic as well. 1.2. Results We shall prove that Connes’ Embedding Conjecture holds for the von Neumann algebra of a sofic equivalence relation (Theorem 2). Also, any sofic relation satisfies the Measure-Theoretic Determinant Conjecture of Lück, Sauer and Wegner (Theorem 3). We also show that treeable equivalence relations are always sofic (Theorem 4). Hence we prove that the two conjectures above hold for free actions of free groups. 2. Orbit equivalence Theorem 1. If θ1 is a sofic action and θ2 is measured orbit equivalent to θ1 then θ2 is also sofic. Proof. By Remark 1.1 it is enough to prove the statement in the special case when θ2 is obtained from θ1 by adding a generator of the free group whose action does not change the orbit structure of the relation. Indeed, from this statement the general case follows easily: to see that the restriction θ2r is sofic add the first r generators of θ2 to θ1 , then restrict to the set of r new generators. Let γ1 , . . . , γd , . . . generate θ1 and let γ denote the new generator in θ2 . Since γ does not change the orbit structure we can find for any point x ∈ X words wx , wx ∈ γ1 , . . . , γd , . . . such


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that θ2 (γ , x) = θ1 (wx , x) and θ2 (γ −1 , x) = θ1 (wx , x). In fact we can do this in a Borel way by taking the shortest and lexicographically smallest wx , wx of all possible choices. Let us fix an ε > 0. For this ε we can find an integer L such that μ(X0 ) < ε/2 where X0 = {x ∈ X: |wx | > L or |wx | > L or either wx or wx contains a generator γi where i > L}. Let us look at X \ X0 . It is partitioned into a finite number of Borel subsets Hi : 1 i K on which wx and wx are constant functions of x. We shall define a sequence of Borel subsets Xi ⊂ X in a recursive way. We start with X0 . Then we take H1 and approximate it by a finite union of standard closed–open subsets of X denoted by H1 so that μ(H1 H1 ) ε/4. (The denotes symmetric difference.) Now let X1 = X0 ∪ (H1 H1 ) and H1 = H1 ∩ H1 . Next we take H2 \ X1 , and approximate it by a H2 which is again a finite union of standard closed–open subsets of X so that μ((H2 \ X1 ) H2 ) ε/8, and set X2 = X1 ∪ ((H2 \ X1 ) H2 ) and H2 = (H2 \ X1 ) ∩ H2 . We continue this process for all Hi ’s. At each step Hi \ Xi−1 is completely disjoint from each Hj : j < i so we can always choose Hi to be disjoint from all Hj : j < i. So at the end we have a partition X = XK ∪ H1 ∪ · · · ∪ HK such that μ(XK ) ε, Hi ⊂ Hi ∩ Hi . During the whole process we considered some large, but finite number of standard closed–open sets. Each such set is defined by fixing the first few digits of x. Let M L denote an integer such that none of the used closed–open sets require fixing more than M digits of x. Now if x ∈ X \ XK then the first M digits of x determine which Hi it is in, and hence which Hi and which Hi it is in. This in turn determines wx and wx . So in fact we have a Borel splitting X = XK ∪ X such that μ(XK ) < ε and for any point x ∈ X the words wx , wx are determined by the first M digits of x. We have the sofic sequence Gn for θ1 . From it we shall construct a sequence Gεn . As a first attempt for each vertex g ∈ Gn we read the first M digits of its label. Then find the corresponding words wx , wx we defined above, and trace these words in Gn starting from g. If they end at h and h respectively then we connect g to h by an oriented edge labeled γ and to h by an oriented edge labeled γ −1 . At this point the graph Gεn might not be the graph of a group action: the γ edge going from g to h might not be matched by a γ −1 edge going from h to g. Let us temporarily call such g vertices “bad”. Let us denote by ε (n) the ratio of bad vertices in Gεn . By the construction of Gεn the badness of a vertex g is determined by its (M, M) neighborhood in Gn . Let us call a neighborhood α ∈ U M,M (θ1 ) “bad” if its root is a bad vertex. Hence ε (n) =

pα (Gn ).

α is bad

Then if x ∈ X has neighborhood α then either x or θ2 (γ , x) has to lie in X1 . Hence

pα (θ1 ) 2ε.

α is bad

This means that lim supn→∞ ε (n) 2ε. Let us complete the construction of Gεn by keeping the γ action for the good vertices, and defining it arbitrarily for the bad vertices to make it a proper action. This can always be done: let us denote the set of good vertices by H . Then γ (H ) is the set of γ -neighbors of the elements of H . Obviously |H | = |γ (H )|, and hence |Gn \ H | = |Gn \ γ (H )|. So there is a bijection between these last two sets. This bijection shall be the action of γ and its inverse the action of γ −1 on Gn \ H and Gn \ γ (H ) respectively. Let us fix r and a neighborhood α ∈ U r,r (θ2 ). Let us suppose for a moment that there are no “bad” vertices at all. Then since each γ edge is at most an M-long path of non-γ edges,

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the r-neighborhood of the θ2 action of any vertex is contained in, and determined by the r · Mneighborhood of the same vertex for the θ1 action. Thus we get a function π : U r·M,r·M (θ1 ) → U r,r (θ2 ). Let Bπ −1 (α). Let H ⊂ Gn denote those vertices x ∈ Gn whose r-neighborhood ε / H then x ∈ T (Gεn , α) if and only if Br (x, x∈ Gn ) contain a “bad” vertices. Then obviously x ∈ β∈B T (Gn , β). In other words T (Gεn , α) ( β∈B T (Gn , β)) ⊂ H . On the other hand if x ∈ Br (y, Gεn ). Hence H is covered x ∈ H since Br (x, Gεn ) contains the “bad” vertex y then also ε by the r-neighborhoods of the “bad” vertices so |pα (Gn ) − β∈B pβ (Gn )| ε (n) · r r . The same holds for X: if X0 happens to be empty then pα (θ2 ) = β∈B p β (θ1 ). However X0 might not be empty, and in this case T (α, θ2 ) is not necessarily the same as β∈B T (β, θ1 ). But if the r-neighborhood (by θ2 ) of a point x ∈ X is disjoint from XK , then it cannot belong to the symmetric difference of the two sets above. Hence pβ (θ1 ) − pβ (Gn ) + μ(XK ) + ε (n) · r r . pα (θ2 ) − pα Gε n β∈B

So letting n → ∞ we get that if α ∈ U r,r then lim suppα Gεn − pα (θ2 ) 3ε · r r . n→∞

Hence letting ε → 0 we can choose a suitable diagonal sequence Gn from the Gεn ’s to get a sofic sequence for θ2 . 2 Corollary 2.1. In the definition of soficity we can take actions of F2 ∗ F2 ∗ · · · = F2(∗∞) instead of F∞ . Proof. By Remark 1.1 it is sufficient to show this on the level of finitely generated actions. Let us take an action θ of Fd on X and consider the underlying simple graphing. It has bounded 2 degree (in fact 2d is a bound), hence it can be properly Borel edge-colored by at most d 2+1 colors (see e.g. [4, Section 5.3]). Hence the same equivalence relation can be generated as an 2 action θ of F2∗d where d = d 2+1 . Then according to Theorem 1 θ is sofic if and only if θ is sofic. 2 3. The von Neumann algebra of a measurable equivalence relation In this section we briefly recall the notion of the von Neumann algebra of an equivalence relation [6,10]. Let R ⊂ X × X be a countable Borel-equivalence relation with an invariant measure μ. Then one has a natural σ -finite measure μˆ on the space R which is μ restricted on X (X ⊂ R is given by the diagonal embedding). The groupoid ring of R; CR is defined as follows. Let L∞ (R, C) be the Banach-space of essentially bounded functions on R with respect to μ. ˆ Then CR := K ∈ L∞ (R, C) there exists wK > 0 such that for almost all x ∈ X: K(x, y) = 0 or K(y, x) = 0 only for wK amount of y’s . The ∗-ring structure and a trace is given by:


• • • •

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(K + L)(x, y) = K(x, y) + L(x, y), KL(x, y) = z∼x K(x, z)L(z, y), K ∗ (x, y) = K(y,

x), trN (R) (f ) = X K(x, x) dμ(x).

The von Neumann algebra is constructed by the GNS-construction. The inner product K, L = trN (R) (L∗ K) defines a pre-Hilbert structure on CR and by K → KL we obtain a representation of CR on the closure H of this pre-Hilbert space. The weak closure of CR in the operator algebra B(H) is the von Neumann algebra N (R). The trace trN (R) extends to N (R) weakly continuously to a finite trace on N (R). In Section 6 we shall study the matrix ring Matd×d N (R) as well. Therefore in our paper we use the following version of the groupoid ring of R. Let Cd R := K ∈ L∞ R, Matd×d (C) there exists wK > 0 such that for almost all x ∈ X: K(x, y) = 0 or K(y, x) = 0 only for wK amount of y’s . Then Cd R is isomorphic to Matd×d (CR). The normalized trace trMatd×d N (R) (K) is defined by trMatd×d N (R) (K) :=

Tr K(x, x) dμ(x), d

X

where Tr is the usual trace on Matd×d (C). Observe that Matd×d N (R) can be obtained via the GNS-construction directly as a weak closure of Cd R. 4. Approximation theorems 4.1. The subalgebra of finite type operators Let R be a sofic equivalence relation on our standard space (X, μ) given by a sofic Borelaction θ : F∞ X. Let θn : F∞ Yn be a sofic approximation as in the Introduction. We define the subalgebra Fθ (the subalgebra of finite type operators) the following way. Call an element K ∈ Cd R r-fine, K ∈ Fθr if for any α ∈ U r,r , K(y1 , x1 ) = K(y2 , x2 ) if x1 , x2 ∈ T (θ, α) and y1 = wx1 , y2 = wx2 for some w ∈ Wr . The following properties are easy to check: • Fθ1 ⊂ Fθ2 ⊂ · · · . max(r,s) • If K ∈ Fθr , L ∈ Fθs then K + L ∈ Fθ , KL ∈ Fθr+s , K ∗ ∈ Fθ2r , Id ∈ Fθ1 . That is Fθ =

∞

r r=1 Fθ

is a unital -subalgebra of Cd R.

Proposition 4.1. Fθ is weakly dense in Cd R. Proof. If K ∈ Cd R then let sK = supx,y K(x, y), where is the usual matrix norm. We say μ → L) if: that {Ln }∞ n=1 ⊂ Cd R converge to L in measure (Ln −

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• there exist bounds w and s such that for any n 1, sLn s, wLn w, • for any ε > 0, limn→∞ μ(Aε (n)) = 0, where Aε (n) := x ∈ X L(y, x) − Ln (y, x) > ε, for some y . μ

Lemma 4.1. If Ln − → L, then {Ln }∞ n=1 weakly converges to L. Proof. We need to prove that for any K ∈ Cd R, trMatd×d N (R) K(Ln − L) → 0. We use the inequality | d1 Tr(AB)| AB. trMat

1 = Tr K(x, z)(Ln − L)(z, x) dμ(x) d×d N (R) K(Ln − L) d x∼z X

1 + εwK sK Tr K(x, z)(L − L)(z, x) dμ(x) n d

Aε (n)

x∼z

μ Aε (n) wK sK (s + sL ) + εwK sK , where s is the bound on the norms of the operators {Ln − L}∞ n=1 .

2

Now for K ∈ Cd R we construct a sequence in Fθ converging to K in measure. First let Kn ∈ Cd R be defined the following way. Let Kn (y, x) = K(y, x) if there exists w ∈ Wn such μ that y = wx, otherwise let Kn (y, x) = 0. Clearly, Kn − → K. Now fix n 1. It is easy to see there exist operators {Kw }w∈Wn ⊂ Cd R such that • Kn (y, x) = w∈Wn Kw (y, x), • Kw (y, x) = 0, if wx = y. Let fw (x) = Kw (wx, x). Then we have an approximating function fw such that 1 , • μ(x ∈ X | fw (x) − fw (x) > n1 ) < n|W n| • fw is constant on the sets T (θ, α), if α ∈ U rw ,rw , where rw is some integer depending on w.

Now let Kn (y, x) = w∈Wn Kw (y, x), where Kw (wx, x) = fw (x) and Kw (y, x) = 0 if y = wx. μ Clearly Kn ∈ Fθ and μ(x ∈ X | Kn (y, x) − K(y, x) > n1 ) < n1 . Therefore Kn − → K. 2 4.2. Norm estimates Let A ∈ Cd R and denote by LA the left-multiplication by A on the groupoid ring Cd R. We give a norm estimate for LA in terms of wA and sA . Proposition 4.2. LA Kd wA sA , where Kd is a constant depending only the dimension d. ∗

For a matrix X ∈ Matd×d (C) M(d) denote the Frobenius norm, that is Tr(Xd X) = M2(d) . We have M(d) kd M and M kd M(d) for some constant kd , where M is the


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usual matrix norm (the l 2 -norm). Now let B ∈ Cd R. Then B2 = trMatd×d N (R) (B ∗ B) = trMatd×d N (R) (BB ∗ ) that is Tr B(x, y)B ∗ (y, x) Tr B(x, y)B(x, y) B = dμ(x) = dμ(x) d d x∼y x∼y 2

X

2 B(x, y) (d) = tx dμ(x), = X

x∼y

X

X

where tx = x∼z B(x, z)2(d) . On the other hand, LA B2 = trMatd×d N (R) (B ∗ A∗ AB) = trMatd×d N (R) (A∗ ABB ∗ ). Hence, Tr A∗ A(x, y)B ∗ B(y, x) dμ(x) LA B = d x∼y 2

X

∗ A A(x, y) X

(d)

∗ BB (y, x)

(d)

.

x∼y

Observe that ∗ BB (y, x) = B(y, z)B(x, z) (d) x∼z

(d)

2 2 B(x, z) 2 + B(y, z) 2 . k kd B(y, z)B(x, z) d x∼z

x∼z

Therefore we have the following inequality: LA B

2

kd4 sA∗ A X

where tˆx =

x∼z B(x, z)

2.

1 2

(tˆx + tˆy ) dμ(x),

x∼y, A∗ A(x,y)=0

Therefore, LA B2 kd6 sA∗ A wA∗ A B2 .

2 , s ∗ s 2 our proposition follows. Since wA∗ A wA A A A The previous proposition can be applied in the case of finite sets as well. Let T be a finite set and K : T × T → Matd×d (C) be matrix-valued kernel function. These kernels form an algebra analogous to Cd R. Again we can define sK := supx,y K(x, y) and the width wK as the supremal number such that for any x ∈ T , KT (x, y) = 0 respectively KT (y, x) = 0 for at most wK y s. The normalized trace Tr (K) is defined as

Tr (K) =

Tr K(x, x) x∈T

d|T |

.

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Again we have the inner product K, L = Tr (L∗ K) and LA (B) = AB. The following lemma is the finite version of Proposition 4.2. Lemma 4.2. LK Kd wK sK . Finally, we prove a simple lemma about convergence in measure. μ

Lemma 4.3. If Ln − → L in Cd R then limn→∞ trMatd×d N (R) (Lin ) = trMatd×d N (R) (Lin ). Proof. The fact that limn→∞ trMatd×d N (R) (Ln ) = trMatd×d N (R) (L) directly follows from the i−1 )L + Li−1 (L − L) a simple induction implies that definition. Since (Lin − Li ) = (Li−1 n n n −L μ i i Ln − →L . 2 4.3. Sofic approximation For K ∈ Fθr and n 1 let Kn : Yn × Yn → C be defined the following way. Let Kn (q, p) := K(y, x) if p ∈ T (θn , α), x ∈ T (θ, α) and wp = q, wx = y for some w ∈ Wr . We call {Kn }∞ n=1 the sofic approximation of K. Proposition 4.3. Let K ∈ Fθr , L ∈ F F T s then: 1. 2. 3. 4. 5. 6.

Kn + Ln − (K + L)n (n) → 0, where A(n) = Tr (A∗ A). Kn Ln − (KL)n (n) → 0. Kn∗ − (K ∗ )n (n) → 0. Idn = Id. There exists CK > 0 such that Kn CK , where A denotes the usual norm. Tr∗ (Kni ) limn→∞ |Y = trMatd×d N (R) K i . n|

Proof. We call a sequence Ln : Yn × Yn → Matd×d (C) negligible if: ∞ • {sLn }∞ n=1 and {wLn }n=1 are bounded above. |Qn | • limn→∞ |Yn | = 1, where

Qn = x ∈ Yn Ln (x, y) = 0, Ln (y, x) = 0 for any y ∈ Yn . It is easy to see that if {Ln }∞ n=1 is negligible then lim Tr∗ (Ln ) = 0 and

n→∞

Tr∗ L∗n Ln = 0.

∞ ∞ ∗ ∗ Observe that {Kn + Ln − (K + L)n }∞ n=1 , {Kn Ln − (KL)n }n=1 and {Kn − (K )n }n=1 are all negligible sequences. Hence 1–3 hold. The fourth statement is trivial and the fifth one immediately follows from Lemma 4.2. Since Tr∗ (Kni − (Kn )i ) → 0 in order to prove 6 one only needs to show that

lim Tr (Kn ) = trMatd×d N (R) (K).

n→∞


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The right-hand side is equal to

μ T (θ, α) c(K, α),

α∈U r,r

where c(K, α) = Tr K(x, x) if x ∈ T (θ, α) and K ∈ Fθr . On the other hand the left-hand side of the equation is equal to T (θn , α) c(K, α). |Yn | r,r

α∈U

Thus by the sofic property 6 follows.

2

5. Connes’ Embedding Conjecture In this section we prove Connes’ Embedding Conjecture for the von Neumann algebras of sofic equivalence relations. First let us very briefly recall the conjecture based on the survey of Pestov [12] (see also [11]). Let R be the hyperfinite factor. Let G be a non-principal ultrafilter on the natural numbers and limG be the corresponding ultralimit. Consider the algebra BR ⊂

∞ ∞ n=1 R, where {ai }i=1 ∈ BR iff i1 ai < ∞. Let J ⊂ BR be the ideal of those elements ∗ a ) = 0, where Tr is the unique finite trace on R. Then R ω := such that lim Tr (a {ai }∞ R i R G i i=1 BR /J is the tracial ultrapower of R, a von Neumann algebra factor with trace ∞ TrG [ai ] i=1 = lim TrR (ai ). G

Conjecture 5.1 (Connes’ Embedding Conjecture). Every separable factor of type II 1 embeds into R ω . We confirm the conjecture in the case of von Neumann algebras of sofic equivalence relations. Theorem 2. Let R be a sofic equivalence relation. Then N (R) embeds into R ω . Proof. By the result of [13] it is enough to prove that the weakly dense ∗-algebra Fθ has a trace preserving ∗-homomorphism into R ω . Therefore it is enough to construct (see [11]) unital maps ψn : Fθ → Matin ×in (C) for some sequence of integers {in }∞ n=1 such that for each K, L ∈ Fθ the following conditions are satisfied: • • • •

limG ψn (K) + ψn (L) − ψn (K + L)(in ) = 0. limG ψn (K)ψn (L) − ψn (KL)(in ) = 0. limG ψn (K ∗ ) − (ψn (K))∗ (in ) = 0. ψn (K) is a bounded sequence.

Now let ψn (K) = Kn as in Section 4. Then by Proposition 4.3 all the conditions above are satisfied. 2

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6. The Measurable Determinant Conjecture The goal of this section is to show that the Measurable Determinant Conjecture of Lück, Sauer and Wegner [10] holds for sofic equivalence relations. Let us recall some basic notions from their paper. Let A ∈ Matd×d (N (R)). Then AA∗ ∈ Matd×d N (R) is a positive, self-adjoint element. Let E(λ) = χ[0,λ] (AA∗ ) ∈ Matd×d N (R) be the spectral projection corresponding to the interval [0, λ] and F (λ) = trMatd×d N (R) E(λ) be the associated spectral distribution function. The Fuglede–Kadison determinant is defined as

det

Matd×d N (R)

∗ AA =

∞ λ dF (λ). 0+

The Measurable Determinant Conjecture states that ∗ det AA 1 Matd×d N (R)

provided that A ∈ Matd×d (ZR), where Zd R ⊂ Cd R is defined by Zd R := K ∈ L∞ R, Matd×d (Z) there exists wK > 0 such that for almost all x ∈ X: K(x, y) = 0 or K(y, x) = 0 only for wK amount of y’s . Theorem 3. If R is a sofic equivalence relation, then the Measurable Determinant Conjecture holds. Proof. First let us suppose that A is an operator of finite type. Then AA∗ ∈ Fθ and we can ∗ ∞ consider the sofic approximations {Ai }∞ i=1 , {Ai Ai }i=1 . Observe that: • det(Ai A∗i ) 1. Indeed Ai A∗i is a positive matrix with integer entries (see e.g. the proof of Theorem 3.1(1) in [10]). • {LAi A∗i }∞ i=1 is uniformly bounded. • limn→∞ Tr ((Ai A∗i )m ) = trMatd×d N (R) ((AA∗ )m ). Then by Lemma 3.2 of [10] detMatd×d N (R) (AA∗ ) 1 holds. μ → AA∗ , where {An A∗n }∞ Now let A be an arbitrary element and An A∗n − n=1 ⊂ Fθ . By the previous observation and Proposition 4.3 the conditions of Lemma 3.2 are satisfied, hence detMatd×d N (R) (AA∗ ) 1. 2 7. Examples of sofic equivalence relations 7.1. The Bernoulli-shift Let Γ be a group. We consider the Bernoulli space {0, 1}Γ = {f : Γ → {0, 1}} The (right) Bernoulli-shift θ : {0, 1}Γ × Γ → {0, 1}Γ is defined by θ (f, γ1 )(γ2 ) = f (γ1 · γ2 ). {0, 1}Γ can be identified with X = {0, 1}N by fixing an enumeration of Γ : {γ1 , γ2 , . . .}. Then a k-digit label is just a function {γ1 , . . . , γk } → {0, 1}.


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Proposition 7.1. The Bernoulli-shift of a sofic group is sofic. Proof. Let Γ be a sofic group generated by s1 , s2 , . . . ∈ Γ . Any element γ ∈ Γ can of course be expressed as a word in these generators, but this expression is usually not unique. For later use let us fix for each element γ ∈ Γ a word wγ that expresses γ in terms of the generators. Let us take a sequence of graphs Gn that prove the soficity of Γ . That is, Gn is a directed graph with each edge being labeled by some si such that each vertex has exactly one in-edge and one out-edge labeled with each generator. We can also think of this as a right action of the free group F∞ = s1 , s2 , . . . on the vertex set of Gn . Furthermore the neighborhood statistics of Gn converge to that of Γ ’s Cayley graph on these generators. We shall label each vertex of Gn with an element of {0, 1}Γ so that the labeled neighborhood statistic of Gn will converge to the labeled neighborhood statistic of θ . To do so we first assign to each vertex of each Gn a random bit. This assignment is simply a random function ω: ∞ n=1 Gn → {0, 1}. Then we take a vertex g ∈ Gn and assign to it a function ωg : Γ → {0, 1} by the formula ωg (γ ) = ω(g · wγ ). Thus now we have an action θn on the {0, 1}Γ -labeled space Gn . We claim that pα (θn ) → pα (θ ) for any labeled neighborhood α for a suitable choice of ω (in fact for almost all ω’s). In order to prove this, we shall first consider {0, 1}-labeled neighborhoods, so let us denote by V r the set of usual r-neighborhoods where each vertex is labeled with 0 or 1, up to labeled isomorphism. For an α ∈ V r and a {0, 1}-labeled graph G the notations T (α, G) and pα (G) extend naturally. In the previous paragraph we described how to obtain a {0, 1}Γ -labeling from an {0, 1}-labeling for the actions θn on Gn . It is clear by that construction that the U r,r neighborhood of a vertex g is determined by the V r+R -neighborhood of the same vertex where R = maxi=1,2,...,r |wγi |. On the other hand there is a natural {0, 1}-labeling on the points of the Bernoulli-shift: just label each f : Γ → {0, 1} by the value of f on the identity element. In this way we can talk about the V r -neighborhoods of points of the Bernoulli-shift, and the U r,r -neighborhoods are again determined by the V r+R -neighborhoods in the exact same fashion. Hence to finish the proof it is enough to show that pα (θn ) → pα (θ ) for all α ∈ V r for almost all ω’s. First let α ∈ V r such that its underlying graph is not isomorphic to the r-neighborhood of the identity of Γ in the Cayley graph. Since the Gn is a sofic sequence for the Cayley graph, it is immediate that pα (θn ) → 0. On the other hand the Bernoulli-shift is essentially free, hence almost all orbits are isomorphic to the Cayley graph of Γ so pα (θ ) = 0. Now let us consider an α ∈ V r whose graph looks like the Cayley graph around the identity. We can think that the vertices of α are indexed by those elements of Γ that have length at most r. Then if f : Γ → {0, 1} is a point in the free part of the Bernoulli-shift then f ∈ T (θ, α) if and only if f (γ ) = α(γ ) for all elements |γ | < r. (Here we α(γ ) denotes the label written on the vertex of α corresponding to γ .) Hence pα (θ ) = 1/2|α| . All we have to prove now is Lemma 7.1. For almost all ω’s pα (Gn ) → 1/2|α| . Proof. Let us say that a vertex g ∈ Gn is normal if its r-neighborhood is isomorphic as a graph to the r-neighborhood of the identity element of the Cayley graph. For any vertex g ∈ Gn let Xg denote a random variable that is 1 if g ∈ T (Gn , α) and 0 otherwise. Obviously P (Xg = 1) = 1/2|α| for any normal vertex g and 0 otherwise, and g∈Gn Xg . pα (Gn ) = |Gn |

1704


If all the Xg ’s were independent, then by the law of large numbers pα (Gn ) would converge to the limit of its expected value with probability 1, and this expected value is simply 1 |{g ∈ Gn normal}| = |α| . E(Xg ) = lim lim E pα (Gn ) = lim n→∞ n→∞ n→∞ 2|α| |Gn | 2 g∈Gn

The Xg ’s are however not independent, but at least they are independent for g’s in different graphs, and also Xg1 , . . . , Xgk are jointly independent if g1 , . . . , gk ∈ Gn are pairwise far from each other, namely d(gi , gj ) > r. Lemma 7.2. There exists a natural number l > 0 (depending on r) and a partition such that if x = y ∈ Bin then the r-neighborhoods of x and y are disjoint.

l

n i=1 Bi

= Gn

Proof. Let Hn be a graph with vertex set V (Gn ). Let (x, y) ∈ E(H )n if and only if Br (x) ∩ Br (y) = ∅. Then deg(x) r r for any x ∈ V (Hn ). Let l = r r + 1 then Hn is vertex-colorable by the colors c1 , c2 , . . . , cl . Let Bin be the vertices coloured by ci . 2 Now for a fix ε > 0 let Bin1 , . . . , Binn be those elements of the partition for which |Binj | ε/ l. q Then since {Xg : g ∈ Binj } are jointly independent, by the previous argument we get lim

Binj ∩ T (Gn , α) |Binj |

= lim E

Binj ∩ T (Gn , α) |Binj |

=

1 2|α|

almost surely for any choice of ij . An easy calculation now shows that setting B = have lim

nq

n j =1 Bij

we

1 B ∩ T (Gn , α) = |α| |B| 2

for the same set of ω’s. Since |Gn \ B| ε, this shows that 1 1 − ε lim inf pα (Gn ) lim sup pα (Gn ) |α| + ε |α| 2 2 almost surely, and finally letting ε → 0 we get the desired almost sure convergence.

2

Thus we have pα (θn ) → pα (θ ) almost surely for all α’s. Hence there exists an ω for which pα (θn ) → pα (θ ), hence the Bernoulli-shift is sofic. 2 Note that the fact that for residually amenable groups the Measurable Determinant Conjecture holds for the Bernoulli-shift has already been proved in [10]. 7.2. Treeable relations Recall [9] that an equivalence relation E ⊂ X × X is called treeable if it has an L-treeing generated by measure-preserving involutions S1 , S2 , . . . . We prove that all treeable equivalence


1705

relations are sofic. The most important examples of such treeable relations are the free actions of free groups. Theorem 4. The action of Γ = γ1 , γ2 , . . . | γi2 = 1 (i = 1, 2, . . .) defined by θ (γi , x) = Si (x) is sofic. Proof. By Remark 1.1 it is again sufficient to work with finitely generated actions. So let us assume Γ is generated by γ1 , . . . , γd . Let us fix a large r. For any α, β ∈ U r,r and any 1 i d let us denote T (θ, α, i, β) = x ∈ T (θ, α): Si (x) ∈ T (θ, β) and it measure (as it is obviously a Borel set) pαiβ (θ ) = μ T (θ, α, i, β) . There numbers together with the pα (θ )’s satisfy certain equations:

pα (θ ) = 1,

α∈U r,r

pαiβ (θ ) = pα (θ )

for any i,

β∈U r,r

pαiβ (θ ) = pβiα (θ )

for any α, i, β.

Let us introduce variables wα : α ∈ U r,r and wαiβ : α, β ∈ U r,r , 1 i d. Then wα = pα (θ ), wαiβ = pαiβ (θ ) is a solution to the following set of linear equations:

wα = 1,

(1)

α∈U r,r

wαiβ = wα

for any i,

(2)

β∈U r,r

wαiβ = wβiα

for any α, i, β.

(3)

Now we use the rational approximation trick of Bowen [2]. Let us fix a small ε > 0. If a set of linear equations with rational coefficients has some solution, then it also has a rational solution in which each variable is at most ε-far from the corresponding value of the initial solution. Further we may also assume that if a variable was 0 in the initial solution then it remains 0 in the new solution. So our set of equations has such a rational solution which will shall simply denote by wα , wαiβ . Since now these numbers are all rational, we may choose a large integer N for which Wαiβ = N · wαiβ is always an even integer. Now take a set Y with N elements and partition it into subsets Yα : α ∈ U r,r with |Yα | = Wα . This can be done because of (1) above. Then fix an index i and do the following: if for a type α the involution Si is fixing the root, then define Si (y) = y for all y ∈ Yα . Otherwise partition Yα into subsets Yαiβ of size Wαiβ . This can be done because of (2) above. Finally define Si to be a random bijection between Yαiβ and Yβiα , or a random matching in Yαiα (this is where we need

1706


that the size of this set is even). This can be done because of (3). Repeat this procedure for each index. Finally for any α ∈ U r,r and any y ∈ Yα look at the label of the root in α. This is a word w ∈ {0, 1}k . Label y with any infinite w ∈ {0, 1}∞ which starts with w. This way we defined an action θ of Γ on the finite labeled set Y . We claim this will be a good approximation to the action θ . To make this precise let us fix an ordering of all possible neighborhood types α1 , α2 , . . . , and for two actions θ , θ let us introduce their statistical distance |pα (θ)−pα (θ )| . It is easy to see that θn is a sofic sequence for θ if and only if ds (θ, θ ) = ∞ i=1 2i ds (θ, θn ) → 0. Lemma 7.3. Let νq denote the ratio of those points in Y through which there is a θ cycle of length at most q. Then for any fixed q we have νq → 0 in probability when N → ∞. Proof. By the construction of Y the probability of the existence of any particular xy edge is at most c/N for some universal constant c depending only on the wαiβ numbers. Hence the probability that a particular cycle of length l exists in θ is at most cl /N l , hence the expected N l l l number of length l cycles is at most c /N · l < c / l! which is a constant. So for fixed q and large N the expected number of points through which there is cycle of length at most q is at most c some constant cq . Then for any fixed ε we have P (νq > ε) εNq so clearly P (νq > ε) → 0 as N → ∞. 2 Then the ratio of those vertices whose r-neighborhood is not a tree is at most d r ν2r since any such neighborhood contains a cycle of length at most 2r, and hence the root of this neighborhood is at most r steps from a vertex in the cycle. For a neighborhood α ∈ U r,r let us denote by α|q ∈ U q,q the subgraph of α spanned by the vertices that are at most q steps from the root and keeping only the first q digits of the labels. The following is easily verified by induction on q: Claim 7.1. If q r and the girth of θ at y ∈ Yα is greater than 2q then Bq (y) ∼ = α|q . Now we can estimate ds (θ, θ ). Let us fix r and let j denote the index of the first αi neighborhood in our listing either whose radius is larger than r or its labels have more than r digits. Let U q,q , Uc = {α ∈ U : α is not a tree}, Ut = U \ Uc . U= qr

If α ∈ Uc then pα (θ ) = 0 since θ is a treeing, and pα (θ ) d r ν2r since at most this many vertices can have cycles in their r-neighborhood. If α ∈ Ut ∩ U q,q then pα (θ ) − pα θ =

β∈U r,r : β|q ∼ =α

β∈U r,r : β|q ∼ =α

pβ (θ ) − pβ θ pβ (θ ) − wβ + d r ν2r ε U r,r + d r ν2r .


1707

The term d r ν2r appears again because pβ (θ ) is not necessarily equal to wβ : the difference comes from exactly those vertices in Yβ whose 2r-neighborhood is not a tree. And finally ∞ d r ν2r ε|U r,r | + d r ν2r 1 |pαi (θ ) − pαi (θ )| + + ds θ, θ = 2i 2i 2i 2i i: αi ∈Uc

i=1

ε U r,r + 2d r ν2r + 1/2j −1 .

i: αi ∈Ut

ij

(4)

So in order to construct a finite action with ds (θ, θ ) < δ first we choose r so large that 1/2j −1 < δ/3 in (4). Then we choose an ε < 3|Uδr,r | . Then we find a rational solution to our

system of Eqs. (1)–(3). Finally we choose N so large, that with positive probability ν2r We pick an action θ satisfying this and hence

δ 6d r .

ds θ, θ ε U r,r + 2d r ν2r + 1/2j −1 < 3 · δ/3 = δ. Hence θ is indeed a sofic action.

2

Note that the previous theorem combined with Theorem 1 shows the all treeable groups are sofic. Recall that a group is treeable if it has a free treeable action (see [7] for examples of such groups). 7.3. Profinite actions The simplest case of sofic action is argueably the case of profinite actions. Let Γ be a countable ∞ residually finite group and Γ ⊃ N1 ⊃ N2 · · · be finite index normal subgroups such that i=1 Ni = {1}. Then G = lim ←− Γ /Ni is the profinite closure with respect to the system {Ni }, a compact group. Then Γ is a dense subgroup of G and so it preserves the Haar-measure ν. It is easy to see that Γ (G, ν) is a sofic action. 8. Conclusion We can conclude that the Connes Embedding Conjecture and the Measurable Determinant Conjecture hold for treeable sofic relations, particularly, for relations induced by free actions of free groups. We end our paper with a question related to Question 10.1 of Aldous and Lyons [1] on unimodular networks. Question 8.1. Does there exist a measurable equivalence relation that is not sofic? References [1] D. Aldous, R. Lyons, Processes on unimodular random networks, Electron. J. Probab. 12 (54) (2007) 1454–1508. [2] L. Bowen, Periodicity and circle packings of the hyperbolic plane, Geom. Dedicata 102 (2003) 213–236. [3] G. Elek, A regularity lemma for bounded degree graphs and its applications: Parameter testing and infinite volume limits, preprint, arXiv:0711.2800. [4] G. Elek, G. Lippner, An analogue of the Szemeredi Regularity Lemma for bounded degree graphs, preprint, arXiv: 0809.2879. [5] G. Elek, E. Szabó, On sofic groups, J. Group Theory 9 (2) (2006) 161–171.

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[6] J. Feldman, C.C. Moore, Ergodic equivalence relations, cohomology, and von Neumann algebras, II, Trans. Amer. Math. Soc. 234 (2) (1977) 325–359. [7] D. Gaboriau, Examples of groups that are measure equivalent to the free group, Ergodic Theory Dynam. Systems 25 (6) (2005) 1809–1827. [8] M. Gromov, Endomorphisms of symbolic algebraic varieties, J. Eur. Math. Soc. (JEMS) 1 (2) (1999) 109–197. [9] A. Kechris, B. Miller, Topics in Orbit Equivalence, Lecture Notes in Math., Springer-Verlag, 1852. [10] W. Lück, R. Sauer, C. Wegner, L2-torsion, the measure theoretic determinant conjecture, and uniform measure equivalence, preprint, arXiv:0903.2925. [11] N. Ozawa, About the QWEP conjecture, Internat. J. Math. 15 (5) (2004) 501–530. [12] V. Pestov, Hyperlinear and sofic groups: A brief guide, Bull. Symbolic Logic 14 (4) (2008) 449–480. [13] C. Pearcy, J.R. Ringrose, Trace-preserving isomorphisms in finite operator algebras, Amer. J. Math. 90 (1968) 444–455. [14] B. Weiss, Sofic groups and dynamical systems, in: Ergodic Theory and Harmonic Analysis, Mumbai, 1999, Sankhya Ser. A 62 (3) (2000) 350–359.


Existence results for semilinear differential equations with nonlocal and impulsive conditions ✩ Zhenbin Fan a,∗ , Gang Li b a Department of Mathematics, Changshu Institute of Technology, Suzhou, Jiangsu 215500, PR China b Department of Mathematics, Yangzhou University, Yangzhou, Jiangsu 225002, PR China

Received 2 July 2009; accepted 27 October 2009 Available online 11 November 2009 Communicated by J. Coron

Abstract This paper is concerned with the existence for impulsive semilinear differential equations with nonlocal conditions. Using the techniques of approximate solutions and fixed point, existence results are obtained, for mild solutions, when the impulsive functions are only continuous and the nonlocal item is Lipschitz in the space of piecewise continuous functions, is not Lipschitz and not compact, is continuous in the space of Bochner integrable functions, respectively. © 2009 Elsevier Inc. All rights reserved. Keywords: Nonlocal conditions; Impulsive differential equations; Fixed point; C0 -semigroup; Mild solutions

1. Introduction In this paper, we are concerned with the existence of mild solutions for the following impulsive differential equation with nonlocal conditions u (t) = Au(t) + f t, u(t) , u(0) = g(u), u(ti ) = Ii u(ti ) , ✩

0 t b, t = ti ,

i = 1, 2, . . . , p, 0 < t1 < t2 < · · · < tp < b,

The work was supported by the NSF of China (10971182).


E-mail addresses: [email protected] (Z. Fan), [email protected] (G. Li). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.023

(1.1)

1710

Z. Fan, G. Li / Journal of Functional Analysis 258 (2010) 1709–1727

where A : D(A) ⊆ X → X is the infinitesimal generator of a strongly continuous semigroup T (t), t 0, X a real Banach space endowed with the norm · , u(ti ) = u(ti+ ) − u(ti− ), u(ti+ ), u(ti− ) denote the right and the left limit of u at ti , respectively. f , g, Ii are appropriate continuous functions to be specified later. About semilinear differential equations, two directions are studied extensively in recent years. One is the semilinear differential equations with nonlocal conditions, i.e., nonlocal problems. In 1990, Byszewski and Lakshmikantham [7] first investigated the nonlocal problems. They studied and obtained the existence and uniqueness of mild solutions for nonlocal differential equations without impulsive conditions. Since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones, differential equations with nonlocal problems have been studied extensively in the literature. And the main difficulty in dealing with the nonlocal problem is how to get the compactness of solution operator at zero, especially when the nonlocal item is Lipschitz continuous or continuous only. Many authors developed different techniques and methods to solve this problem. For more details on this topic we refer to [2,3,5,11–14,18,22, 23,26–28] and references therein. Another direction is the semilinear differential equations with impulsive conditions. Recently, the theory of impulsive differential and partial differential equations has become an important area of investigation because of its wide applicability in control, mechanics, electrical engineering fields and so on. For more details on this theory and on its applications we refer to the monographs Benchohra et al. [6], Lakshmikantham et al. [17], the papers of [1,8–10,15,16,19– 21,25] and references therein. A standard approach in deriving the mild solution of (1.1) is to define the solution operator Q. Then conditions are given such that some fixed point theorem such as Browder’s and Schauder’s fixed point theorems can be applied to get a fixed point for solution operator Q, which gives rise to a mild solution of (1.1). Thus many authors supposed that impulsive functions Ii (1 i p) are compact or Lipschitz continuous. This assumption can be found in many previous papers such as [9,10,15,16,19,20]. However, this property of impulsive functions is not satisfied usually in practical applications (for examples see Section 6). Thus it is nature to ask whether there exists a mild solution when the impulsive functions loss the compactness and Lipschitz continuity. New techniques and methods must be given. In this paper, we mainly apply the techniques of approximate solutions and fixed point to get the mild solution of nonlocal impulsive differential equation (1.1) without the compactness or Lipschitz continuity assumption on impulsive functions. Indeed, we only need to suppose the continuity of them and do not impose any other conditions in the proof. On the other hand, the technique of approximate solutions also solves the difficulty involving by the nonlocal item g, i.e., the compactness problem of solution operator at zero. Therefore, our results essentially generalize and improve many previous ones in this field. The outline of this paper is as follows. In Section 2, we recall some concepts and facts about the strongly continuous semigroups and semilinear differential equations. In Section 3, we give the existence result of Eq. (1.1) when g is Lipschitz in PC(0, b; X). In Section 4, we obtain the existence results of Eq. (1.1) when g is not Lipschitz and not compact. In Section 5, we discuss the existence result of Eq. (1.1) when g is continuous in L1 (0, b; X). And two examples are given to illustrate our abstract results in the last section. 2. Preliminaries Let (X, · ) be a real Banach space. We denote by C(0, b; X) the space of X-valued continuous functions on [0, b] with the norm u = sup{u(t), t ∈ [0, b]} and by L1 (0, b; X) the


1711

b space of X-valued Bochner integrable functions on [0, b] with the norm f L1 = 0 f (t) dt. Let PC(0, b; X) = {u : [0, b] → X: u(t) is continuous at t = ti and left continuous at t = ti and the right limit u(ti+ ) exists for i = 1, 2, . . . , p}. It is easy to check that PC(0, b; X) is a Banach space with the norm uPC = supt∈[0,b] u(t) and C(0, b; X) ⊆ PC(0, b; X) ⊆ L1 (0, b; X). Throughout this work, we suppose that (HA) The linear operator A : D(A) ⊆ X → X generates a compact strongly continuous semigroup {T (t): t 0}, i.e., T (t) is compact for any t > 0. Moreover, there exists a positive number M such that M = supt0 T (t). Let us recall the following definitions and facts. Definition 2.1. A function u ∈ PC(0, b; X) is said to be a mild solution of Eq. (1.1) on [0, b] if it satisfies t u(t) = T (t)g(u) + 0

T (t − s)f s, u(s) ds + T (t − ti )Ii u(ti ) ,

0 t b.

0 0, there exists a function ρl ∈ L1 (0, b; R) such that f (t, x) ρl (t) for a.e. t ∈ [0, b] and all x ∈ Bl ; (Hg1) g : PC(0, b; X) → X is Lipschitz continuous with Lipschitz constant k; (HI) Ii : X → X is continuous for every i = 1, 2, . . . , p. Theorem 3.1. Assume that the conditions (HA), (Hf ), (Hg1) and (HI) are satisfied. Then the nonlocal impulsive problem (1.1) has at least one mild solution on [0, b] provided that p Ii u(ti ) r. (3.1) M g(0) + kr + ρr L1 + sup u∈Wr i=1

To prove the above theorem, we first need some lemmas. For fixed n 1, we consider the following approximate problem: u (t) = Au(t) + f t, u(t) , u(0) = g(u),

1 u(ti ) = T Ii u(ti ) , n

0 t b, t = ti ,

i = 1, 2, . . . , p, 0 < t1 < t2 < · · · < tp < b.

(3.2)


1713

Lemma 3.2. Assume that all the conditions in Theorem 3.1 are satisfied. Then for any n 1, the nonlocal impulsive problem (3.2) has at least one mild solution un ∈ PC(0, b; X). Proof. For fixed n 1, set Qn : PC(0, b; X) → PC(0, b; X) defined by (Qn u)(t) = (Qg u)(t) + (Qf u)(t) + (QIn u)(t) with (Qg u)(t) = T (t)g(u), t (Qf u)(t) =

T (t − s)f s, u(s) ds,

0

(QIn u)(t) =

T (t − ti )T

0 0. ∂ z¯ + ∂ z¯ = |uλ | ∂ z¯ − u ∂ z¯ λ

Due to (5.28), (5.30) we also have 1ε | ∂θ ∂ z¯ + 1 ∂v λ λ 2 ∂ v˜ λ − u ε ∂ z¯ ∂ z¯

∂w λ ∂ z¯ |L2 (G)

C. On the other hand,

λ 2 hλω ∂V 1 ∂v λ ∂ v˜ λ + 1− u − ε ∂ z¯ ∂ z¯ ε ∂ z¯

and the right-hand side converges to zero in measure, as follows from (5.4), (5.14) and the second λ ∂w λ bound in (5.8). Then, using (5.4) and (5.20), we see that 1ε | ∂θ ∂ z¯ + ∂ z¯ | tends to zero in measure λ

λ

∂w p as λ → 1 − 0. Therefore 1ε | ∂θ ∂ z¯ + ∂ z¯ |L (G) → 0 for every 1 p < 2. Finally, we make use of (5.30) to conclude that condition (5.32) is satisfied. Using Lemma 3 we can identify the constant α λ in problem (5.19),

α λ = −2 log γξ λ (∂Ω) + κ λ

|γξ λ | = const on ∂Ω ,

with the remainder κ λ satisfying 1 λ κ → 0 as λ → 1 − 0. ε Next, we identify ϕ˜ λ by the following

(5.33)

L. Berlyand et al. / Journal of Functional Analysis 258 (2010) 1728–1762

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Lemma 4. Let ξ λ be the unique zero of θ λ (cf. Lemma 3), then λ ϕ˜ − ϕξ λ

H 2 (G)

= o(ε) as λ → 1 − 0,

where ϕξ λ is the solution of problem (3.3)–(3.4) with ξ = ξ λ . Proof. Set f λ := ϕ˜ λ − κ λ U, where U is the unique solution of the equation U = 0 in G subject to the boundary conditions λ U = 0 on ∂Ω and U = 1 on ∂ω. Then f λ satisfies − f λ = 2(v λ − v˜ λ ) − eϕ˜ |θ λ + w λ |2 + 1 in G (cf. (5.19), (5.21), (5.22)). Therefore, after simple calculations, we get the following boundary value problem for f λ , ⎧ λ 2 fλ λ ⎪ ⎪ ⎨ − f + |γξ λ | e = 1 + r

in G,

f = 0 on ∂Ω, ⎪ ⎪ ⎩ f λ = −2 log |γ λ | on ∂ω, ξ λ

(5.34)

where 2 λ λ r λ = 2 v λ − v˜ λ + |γξ λ |2 e−κ U − θ λ − ϑ λ eϕ˜ 2 λ − ϑ λ + w λ + 2 θ λ − ϑ λ , ϑ λ + w λ eϕ˜ ,

(5.35)

and ϑ λ is as in Lemma 3. Let us show that L2 -norm of r λ is negligibly small. To this end we use (5.14) and the Sobolev embedding for the first term of (5.35); for the last term we make use of statement (ii) of Lemma 3 and (5.30) in conjunction with the Sobolev embedding; finally, the middle term we represent as −κ λ U 2 log(|θ λ −ϑ λ |/|γ λ |) λ ξ e −1 − e − 1 |γξ λ |2 eϕ˜ and estimate it with the help of the elementary inequality |et − 1| |t|e|t| , Lemma 3, (5.20) and (5.33). As the result we get the following bound λ r

L2 (G)

= o(ε).

(5.36)

This bound allows us to estimate the H 1 -norm of the function f λ −ϕξ λ . We have − (f λ−ϕξ λ )+ λ

|γξ λ |2 ef − |γξ λ |2 e parts,

ϕξ λ

= r λ in G. Multiply this equation by f λ − ϕξ λ to get, after integrating by G

λ ∇ f − ϕξ λ 2 dx

G

λ λ r f − ϕξ λ dx,

1746


where we have used the monotonicity of the operator φ → |γξ λ |2 eφ and the fact that f λ −ϕξ λ = 0 on ∂G. It follows that λ f − ϕξ λ

H 1 (G)

= o(ε).

(5.37)

Next we show that the H 1 -bound (5.37) in conjunction with an L∞ -estimate for f λ (following from (5.20) and (5.33)) yield (f λ − ϕξ λ )L2 (G) = o(ε). By elliptic estimates this will imply that λ f − ϕξ λ

H 2 (G)

= o(ε).

(5.38)

In order to estimate (f λ − ϕξ λ ) we write − f λ − ϕξ λ = r λ − |γξ λ |2

1

λ (1−t)f λ +tϕ λ ξ dt, f − ϕξ λ e

0

to get, using the obvious pointwise inequality |γξ λ | 1 in G, ϕξ λ + f λ

L2 (G)

r λ

L2 (G)

1 +

λ λ f − ϕξ λ e(1−t)f +tϕξ λ

L2 (G)

dt

0

r λ

L2 (G)

+ f λ − ϕξ λ

L4 (G)

e

f λ L∞ (G)

1

2t (ϕ λ −f λ ) 1/2 e ξ 2

L (G)

dt.

0

Thus, in order to accomplish the proof of (5.38), it suffices to show that sup exp 2t ϕξ λ − f λ L2 (G)

remains bounded as λ → 1 − 0.

(5.39)

t∈[0,1]

Indeed, according to (5.36) and (5.37) we have r λ L2 (G) , f λ − ϕξ λ L4 (G) = o(ε), while f λ L∞ (G) ϕ˜ λ L∞ (G) + |κ λ |U L∞ (G) → 0, as follows from (5.20) and (5.33). It is straightforward to verify that for any φ ∈ H 1 (G), φ ≡ 0, and any C1 > 0 exp 2|φ| exp C1 φ2H 1 exp

|φ|2 C1 φ2H 1 (G)

in G.

On the other hand, as shown in [14, Chapter VII], there are C1 , C2 > 0 such that G

exp

|φ|2 dx C2 C1 φ2H 1 (G)

for every φ ∈ H 1 (G), φ ≡ 0.

(5.40)


1747

Therefore, integrating (5.40) over G, we get exp(φ)L2 (G) C exp(C1 φ2H 1 (G) ). Then (5.37)

implies that (5.39) does hold, and thus (5.38) is proved. Finally, since ϕ˜ λ = f λ + κ λ U and κ λ = o(ε), the claim of the lemma follows. 2

Step VI. (Derivation of the lower bound.) Using (5.15), (5.18) and the definition of hλω (hλω is the restriction of curl Aλ to ω), we get 1 − 2

∂ ϕ˜ λ ds = ∂ν

λ

∂V ∂V λ ds = hω |ω| − ds = hλω KG . ∂ν ∂ν

curl A dx

∂ω

∂ω

− hλω

ω

∂ω

Hence, by Lemma 4, λ 2 hω =

1 2 4KG

∂ϕξ λ ∂ν

2 ds

+ o ε2 .

(5.41)

∂ω

It is not hard to show also that, by (5.5), (5.21)–(5.22), (5.30), Lemma 4 and (ii) of Lemma 3, 1−λ ε = 8

2

2 ϕ˜ λ λ 2 e θ + w λ − 1 dx

G

1 − λ = 1 + o(1) 8

2 ϕ |γξ λ |2 e ξ λ − 1 dx.

(5.42)

G

Now substitute (5.41) and (5.42) in (5.31) to get (5.1). Lemma 2 is proved.

2

6. Proof of the key lemma This section is devoted to the Proof of Lemma 3. In the proof we will repeatedly make use of the formula |∇u|2 dx = G

1 2

2 ∂u dx + π u(∂Ω) 2 deg u/|u|, ∂Ω − u(∂ω) 2 deg u/|u|, ∂ω , ∂ z¯

G

(6.1) that is valid for any u ∈ H 1 (G; C) satisfying |u| = const > 0 on ∂Ω and on ∂ω (with possibly another constant). To see (6.1) one integrates the pointwise identity |∇u|2 = 2

∂u ∂ ∂u ∂u ∂u 1 ∂u 2 ∂ u∧ − u∧ + ∧ + = ∂x1 ∂x2 2 ∂ z¯ ∂x1 ∂x2 ∂x2 ∂x1

over G and applies the divergence theorem.

1 ∂u 2 2 ∂ z¯

1748


We first show Lemma 5. We have λ 2 θ → 1 strongly in L2 (G) as λ → 1 − 0

(6.2)

and ∇ θ˜ λ → 0

in Cloc G; C2 .

(6.3)

Proof. Since θ λ satisfies θ λ = 0 in G, we have 2 2 θ λ = 2 ∇θ λ 0 in G. Then, by the maximum principle, |θ λ | max{1, e−α 1 2

λ 2 ∇θ dx = π + 1 4

G

λ /2

(6.4)

} in G. Besides, by (5.28),

λ 2 ∂θ 2 ∂ z¯ dx π + Cε ,

(6.5)

G

where we have used formula (6.1). It follows that θ λ H 1 (G;C) C with a constant C independent of λ. Therefore, up to extracting a subsequence, θ λ → θ weakly in H 1 (G; C), and |θ | = 1 on ∂G. Moreover, in view of (5.28) and (6.5), 2 ∂θ dx = 0, ∂ z¯

G

i.e.

∂θ = 0 in G, ∂ z¯

and 1 2

|∇θ |2 dx π.

(6.6)

G

It follows that θ = const ∈ S1 . Indeed, since ∂θ ∂ z¯ = 0 in G, it suffices to show that |θ | ≡ 1 in G. 2 2 We have |θ | = 2|∇θ | + 2( θ, θ ) = 2|∇θ |2 0 in G and |θ | = 1 on ∂G. If we assume that ∂|θ| |θ | ≡ 1 in ∂G, we obtain, by using Hopf’s lemma, that ∂|θ| ∂ν > 0 on ∂Ω and ∂ν < 0 on ∂ω. ∂|θ| ∂θ ∂θ ∂θ The equation ∂ z¯ = 0 in G implies that θ ∧ ∂τ = ∂ν > 0 on ∂Ω and u ∧ ∂τ = ∂|θ| ∂ν < 0 on ∂ω, consequently deg(θ, ∂Ω) 1 and deg(θ, ∂ω) −1. Hence, by using formula (6.1), we get 1 2

|∇θ |2 dx = π deg(θ, ∂Ω) − π deg(θ, ∂ω) 2π, G

and thus obtain a contradiction with (6.6). We have shown that, up to extracting a subsequence, θ λ → const ∈ S1 weakly in H 1 (G; C) as λ → 1 − 0. The statement of the lemma follows by the Sobolev embedding and elliptic estimates. 2


1749

We next study the pointwise asymptotic behavior of |θ λ | to get the Proof of (i) of Lemma 3. Since deg(θ λ , ∂Ω) = 1 and deg(θ λ /|θ λ |, ∂ω) = 0, θ λ has at least one zero in G. Let ξ λ be a zero of θ λ nearest to ∂Ω then, by (6.2)–(6.3), ξ λ → ∂Ω

as λ → 1 − 0.

Let us prove that ξ λ is the unique zero. To this end we first show that other zeros (if exist) are localized near ξ λ . We use the coarea formula of H. Federer and W.H. Fleming (see, e.g., [12]) to compute

1 − θ λ 2 |∇|θ λ dx =

λ /2)} max{1,exp(−α

1 − t 2 dH1 ,

dt

G

0

{x: |θ λ (x)|=t}

where H1 is 1-dimensional Hausdorff measure on R2 . On the other hand, by the Cauchy– Schwarz inequality, (6.2) and (6.5), we obtain

1 − θ λ 2 |∇|θ λ dx C 1 − θ λ 2 2 → 0 L (G)

as λ → 1 − 0.

G

It follows that there is a regular value t λ ∈ (4/5, 6/7) of |θ λ | such that H1 ({x ∈ G; |θ λ | = t λ }) → 0, as λ → 1 − 0. (Note that by Sard’s lemma almost all t ∈ (0, max{1, exp(−α λ /2)}) are regular values of |θ λ |.) Set T λ := z ∈ G; θ λ < t λ , then, assuming that 1 − λ is sufficiently small, the boundary ∂T λ of T λ consists of a finite collection of k (= k(λ)) closed curves enclosing simply connected subdomains 0λ , . . . , kλ of G, where 0λ is a subdomain containing ξ λ . By the (strong) maximum principle applied to (6.4) we have |θ λ | < t λ in each jλ . This means, in particular, that these domains are disjoint. Moreover, the following lemma shows that for sufficiently small 1 − λ we have λ θ 1/5 in T λ \ λ . 0

(6.7)

Lemma 6. Let be a simply connected domain with a smooth boundary and let v ∈ H 1 (, C) satisfy v = 0 in and |v| 4/5 on ∂ . Then, if |v(y)| 1/5 at a point y ∈ , we have 1 2

|∇v|2 dx

3π . 5

Proof. Since the equation v = 0 and the Dirichlet integral are conformally invariant, we can assume, without loss of generality, that = B1 and |v(0)| 1/5. Then

1750


v = v(0) +

∞

bk zk + ck z¯ k

in B1 (0),

k=1

and the Dirichlet integral is expressed as 1 2

|∇v|2 dx = π

∞ k |bk |2 + |ck |2 , k=1

while 16 1 π 25 2

∞ 2 2 2 |bk | + |ck | . |v| ds = π v(0) +

2

k=1

S1

Therefore 1 2

1 16 − |∇v| dx π 25 25 2

=

3π . 5

2

Proof of (i) of Lemma 3 completed. Lemma 6 in conjunction with (6.5) imply that zero ξ λ lies in 0λ , when λ is sufficiently close to 1. Besides, according to (6.7), λ θ min inf θ λ , inf θ λ 1/5 in G \ λ . 0 T λ \0λ

G\T λ

In order to study θ λ in 0λ we perform the rescaling by means of the conformal map aξ λ , given by (3.6). Prior to that we extend θ λ into ω in order to have θ λ L∞ (Ω;C) , θ λ H 1 (Ω;C) C with C independent of λ (it is possible because of L∞ - and H 1 -bounds already established in the proof of Lemma 5). Set Θ λ (ζ ) := θ λ aξ−1 λ (ζ ) . Thanks to the conformal invariance of the Dirichlet integral we have Θ λ ∈ H 1 (B1 (0); C) and Θ λ H 1 (B1 (0);C) C. Moreover, Θ λ satisfies Θ λ = 0 in aξ λ (G) and Θ λ (0) = θ λ (ξ λ ) = 0. λ

λ

∂Θ Without loss of generality we may assume that ∂Θ ∂ζ (0) is real and ∂ζ (0) 0 (this always can λ be achieved by multiplying θ by a constant with modulus one). We claim that

Θ λ (ζ ) → ζ

weakly in H 1 B1 (0); C as λ → 1 − 0.

Clearly, up to extracting a subsequence, Θ λ converges to some Θ weakly in H 1 (B1 (0); C) as λ → 1 − 0, and |Θ| = 1 on S1 = ∂B1 (0). One easily checks that |aξ (x)| → 1 uniformly on ω¯ as ξ → ∂Ω, therefore for any fixed 0 < r < 1 we have aξ−1 λ (Br (0)) ⊂ G when 1 − λ is sufficiently

small. For such λ, Θ λ satisfies Θ λ = 0 in Br (0), consequently elliptic estimates imply the following convergence result,


Θλ → Θ

in C k Br (0); C for every k > 0.

1751

(6.8)

We have, in particular, ∂Θ λ ∂Θ (0) = lim (0) 0. λ→1−0 ∂ζ ∂ζ

Θ(0) = lim Θ λ (0) = 0 and λ→1−0

Besides, using (6.5) we see that

|∇Θ|2 dζ = lim

r→1−0 λ→1−0 Br (0)

B1 (0)

∇Θ λ 2 dζ

lim

= lim

lim

r→1−0 λ→1−0 a −1 λ (Br (0))

lim

λ→1−0

λ 2 ∇θ dx

ξ

λ 2 ∇θ dx 2π.

(6.9)

G

On the Θ = 0 in B1 (0), as follows from (6.8). Hence Θ can be represented as other hand k + c ζ¯ k ), and we can compute (b ζ Θ= ∞ k k=1 k

|∇Θ| dζ − 2π = 2

B1 (0)

|∇Θ| dζ −

|Θ|2 ds = 2π

2

B1 (0)

∞ (k − 1) |bk |2 + |ck |2 . k=1

S1

Then (6.9) holds only if bk = ck = 0 for k > 1, i.e. Θ = b1 ζ + c1 ζ¯ . Since |b1 |2 + |c1 |2 = 1, b1 = ∂Θ ∂ζ (0) 0 and |c1 |2 =

4 π

∂Θ 2 4 lim ∂ ζ¯ dζ = λ→1−0 π

B1/2 (0)

a −1 λ (B1/2 (0))

λ 2 ∂θ ∂ z¯ dζ = 0

by (5.28) ,

ξ

we conclude that Θ(ζ ) = ζ . λ Now from (6.8) we see that 0λ ⊂ aξ−1 λ (B7/8 (0)) when 1 − λ is sufficiently small (since |θ | = t λ on ∂0λ and t λ ∈ (4/5, 6/7) while → 7/8); min θ λ (x) ; x ∈ ∂aξ−1 λ B7/8 (0) (6.8) also implies that |Θ λ (ζ )| = |ζ |(1 + o(1)) in B7/8 (0) as λ → 1 − 0, or |θ λ | = |aξ λ |(1 + o(1)) ∞ in aξ−1 λ (B7/8 (0)), where o(1) stands for a function whose L -norm vanishes in the limit. On the other hand, by (5.27) and (6.7), we have log(1/5) log |θ λ | max{0, −α λ /2} C in G \ 0λ . Thus ξ λ is the unique zero of θ λ . Moreover C1 |aξ λ | |θ λ | C2 |aξ λ | in G for some constants 0 < C1 < C2 . It remains to note only that |γξ λ | admits the factorization |γξ λ | = |aξ λ | exp(σξ λ ) ¯ when ξ λ → ∂Ω. 2 (see Section 3) and σξ λ → 0 uniformly on G

1752


Let us next introduce ϑ λ satisfying the requirements in (ii) of Lemma 3. Since the unique zero ξ λ of θ λ tends to ∂Ω as λ → 1 − 0, we can assume that aξ−1 λ (B8/9 (0)) ⊂

λ G. Rescaling θ λ as above, Θ λ (ζ ) = θ λ (aξ−1 λ (ζ )), we have Θλ = 0 in B8/9 (0) and Θ (0) = 0.

It follows that Θ λ admits the representation Θ λ (ζ ) =

∞ bk,λ ζ k + ck,λ ζ¯ k

in B8/9 (0).

k=1

We set ϑ˜ λ to be the antiholomorphic part of Θ λ , ϑ˜ λ :=

∞

ck,λ ζ¯ k ,

k=1

and show that λ ϑ˜ (ζ ) Cε|ζ | in B7/8 (0), λ ∇ ϑ˜ Cε in B7/8 (0).

(6.10) (6.11)

Both these bounds follow from the estimate |ck,λ | C(9/8)k ε, where C is independent of k and ε. The latter estimate is verified as follows, π

∞

k(8/9) |ck,λ | = 2k

2

k=1

∂Θ λ 2 dζ ∂ ζ¯ G

B8/9 (0)

λ 2 ∂θ ∂ z¯ dx,

due to (5.28) the right-hand side is bounded by Cε 2 . Now introduce ϑ λ by ϑ λ (z) := σ aξ λ (x) ϑ˜ λ aξ λ (x) , where σ is a smooth cut-off function such that σ (ζ ) =

1 if ζ ∈ B1/4 (0), 0 if ζ ∈ / B1/2 (0).

Lemma 7. We have G

λ 2 ∇ϑ (x) dx Cε 2

λ ϑ (z) Cε γξ λ (z) in G, λ p ∇ϑ (x) dx = o ε p and

(6.12) for every 1 p < 2,

(6.13)

G

∂ λ θ − ϑl = 0 ∂ z¯

in aξ−1 λ B1/4 (0) .

(6.14)


1753

Proof. Bound (6.12) follows from (6.10) and the pointwise inequality |aξ λ | |γξ λ | in G (this inequality can be easily derived from the constructive definition of |γξ λ | given in Section 3); (6.14) is a straightforward consequence of the very definition of ϑ λ . To show the first bound in (6.13) we argue by the conformal invariance of the Dirichlet integral,

λ 2 ∇ϑ (x) dx =

G

∇ σ (ζ )ϑ˜ λ (ζ ) 2 dζ,

B1/2 (0)

and make use of (6.10)–(6.11). Finally the second bound in (6.13) follows from the first one and the fact that the measure of supp(|∇ϑ λ |) tends to zero as λ → 1 − 0. 2 Note that the second bound in (6.13) in conjunction with the fact that ϑ λ = 0 on ∂G imply, by the Sobolev embedding, that ϑ λ Lq (G) = o(ε) for every q 1. In order to complete the proof of (ii), we need to estimate log |s λ |, where s λ := θ λ − ϑ λ /γξ λ . Observe that 0 < C1 |s λ | C2 when 1 − λ is sufficiently small, which follows from (i) and (6.12). We also have |s λ | = 1 on ∂G and |s λ | = const > 0 on ∂ω, moreover deg(s λ , ∂Ω) = deg(s λ /|s λ |, ∂ω) = 0. Thus, we can fix a single-valued branch of log s λ on G, and set S λ :=

1 log s λ . ε

Lemma 8. The real part of S λ converges weakly in H 1 (G) to zero as λ → 1 − 0. Proof. We have G

λ 2 ∇S dx = 1 ε2

G

λ 2 dx C ∇s 2 λ 2 |s | ε

4C ∂s λ 2 4C = 2 dx = 2 ∂ z¯ ε ε G

λ 2 ∇s dx

G

G\a −1 λ (B1/4 (0))

λ 2 ∂s ∂ z¯ dx

ξ

λ 2 λ 2 ∂θ ∂ϑ C1 2 ∂ z¯ + ∂ z¯ dx C2 , ε G

where we successively used the pointwise bound 1/|s λ |2 C, formula (6.1), property (6.14), the bound |γξ λ | |aξ λ | 1/4 in G \ aξ−1 λ (B1/4 (0)), and (5.28) together with the first bound in (6.13). The real part S1λ of S λ satisfies S1λ = 0 on ∂Ω, therefore, after subtracting the mean value S2λ from the imaginary part, we get λ S − i S λ 2

H 1 (G;C)

C.

1754


Thus, up to extracting a subsequence, S λ − i S2λ → S

weakly in H 1 (G; C),

where S ∈ H 1 (G; C) and its real part vanishes on ∂Ω and takes constant values on ∂ω while the imaginary part has zero mean over G. On the other hand λ p ∂S C ∂ z¯ dx ε p

G

G\a −1 λ (B1/4 (0))

λ p λ p ∂θ ∂ϑ dx ∂ z¯ + ∂ z¯ |γ λ |p ξ

ξ

λ p λ p ∂θ C + ∂ϑ dx, p ε ∂ z¯ ∂ z¯ G

and according to (5.32) and the second bound in (6.13) the right-hand side tends to zero as λ → 1 − 0. Thus ∂S = 0 in G. ∂ z¯

(6.15)

It follows that exp(S) : G → C is a holomorphic map, | exp(S)| = 1 on ∂Ω, | exp(S)| = const on ∂ω and deg(exp(S), ∂Ω) = deg(exp(S)/| exp(S)|, ∂ω) = 0 (since the imaginary part of S is a single valued function). Thus, by (6.15), 1 2

G

∇ exp(S) 2 dx = 2

∂ exp(S) 2 ∂ z¯ dx = 0,

(6.16)

G

where we have used formula (6.1). Hence S ≡ 0 in G, because (6.16) implies that S is a constant while the real part of S vanishes on ∂Ω and its imaginary part has zero mean. 2 Lemma 8 implies the convergence of traces, |S λ | → 0 on ∂ω, i.e. log |θ λ | − log |γξ λ | = o(ε), and also, by the Sobolev embedding, the convergence of S λ = 1ε log(|θ λ − ϑ λ |/|γξ λ |) in Lq (G) to zero for every q 1. Lemma 3 is proved. 2 7. Near boundary vortices and δ-like behavior of currents In this section we analyze the behavior of vortices of minimizers as λ → 1 − 0 and describe the effect of δ-like concentration of currents on the outer boundary of G. First we show Lemma 9. For 0 < λ < 1 sufficiently close to 1, uλ has a unique zero (vortex) ξ˜ λ and dist(ξ˜ λ , ξ λ ) = o(dist(ξ λ , ∂G)), where ξ λ is the unique zero of θ λ defined through (5.21)–(5.22). Moreover, there is μ0 > 0 such that


λ u μ 0

in G \ aξ−1 λ B1/2 (0) ,

1755

(7.1)

where aξ λ is the conformal map given by (3.6) with ξ = ξ λ . λ ¯ (cf. (5.20)). It follows Proof. Recall that uλ = eϕ˜ /2 (θ λ + w λ ) and ϕ˜ λ → 0 uniformly on G from (5.30) and (i) of Lemma 3 that for λ → 1 − 0 we have |uλ | C(|γξ λ | − ε) C(|aξ λ | − ε) C(1/2 − ε) in aξ−1 λ (B1/2 (0)), where C is some positive constant independent of λ. This shows (7.1). λ we perform the rescaling x → In order to study local (in aξ−1 λ (B1/2 (0))) behavior of u

λ λ −1 λ λ −1 ζ = aξ λ (x). Set U λ (ζ ) = uλ (aξ−1 λ (ζ )), Θ (ζ ) = θ (aξ λ (ζ )) and W (ζ ) = w (aξ λ (ζ )). Note that (5.23) can be written as

w λ = g1λ

∂θ λ ∂w λ + ∂ z¯ ∂ z¯

+ g2λ curl Aλ + g3λ

with coefficients gkλ whose L∞ -norms are uniformly in λ bounded (this follows from results in Section 5, cf. (5.15), (5.20), (5.21) and the second bound in (5.8)). We will show below that the L∞ -norm of curl Aλ is also uniformly bounded. Thus we get after rescaling the above equation, for λ sufficiently close to 1, λ λ W λ C1 dist ξ λ , ∂Ω ∂Θ + ∂W + C2 dist2 ξ λ , ∂Ω ∂ ζ¯ ¯ ∂ζ

in B3/4 (0),

(7.2)

λ where we have used the obvious bound |∇(aξ−1 λ )| C dist(ξ , ∂Ω) in B3/4 (0). The behavior of

Θ λ when λ → 1 − 0 is already examined in Section 6, and we know that (up to multiplication on a constant with modulus one) Θ λ (ζ ) → ζ in C k (B3/4 (0); C) for every k > 0. By (5.30) we also know that W λ H 1 (B3/4 (0);C) W λ H 1 (a λ (G);C) → 0 as λ → 1 − 0. It follows from (7.2), ξ

2 (B λ by elliptic estimates, that W λ → 0 in Hloc 3/4 (0); C). In particular, W W 1,q (B2/3 (0);C) → 0 for every q 1. Then (7.2) restricted to B2/3 (0) implies that W λ W 2,q (B1/2 (0);C) → 0 (∀q > 1), therefore W λ C 1 (B1/2 (0);C) → 0. Thus Θ λ + W λ has exactly one zero in B1/2 (0) which tends to λ the origin as λ → 1−0, that is uλ = eϕ˜ /2 (θ λ +w λ ) does have a unique zero ξ˜ λ and aξ λ (ξ˜ λ ) → 0. By an explicit computation this yields dist(ξ˜ λ , ξ λ ) = o(dist(ξ λ , ∂G)). 2

Lemma 10. We have (i) hλ L∞ (G) C, (ii) in H 1 (G) as λ → 1 − 0.

∂hλ ∂ν

0 on ∂Ω, (iii) hλ converges to zero weakly

Proof. (i) We assume hereafter that the minimizer (uλ , Aλ ) is in the Coulomb gauge (2.2). Take curl in (2.5) to get the equation − h = 2

2 ∂uλ ∂uλ ∧ − curl uλ Aλ ∂x1 ∂x2

we also have the following boundary conditions

in G,

(7.3)

1756


h=0

on ∂Ω

and h = hλω

on ∂ω.

(7.4)

We can represent hλ as hλ = hˆ λ1 + hˆ λ2 with hˆ λ2 solving hˆ λ2 = curl(|uλ |2 Aλ ) in G subject to the boundary conditions hˆ λ2 = 0 on ∂Ω and hˆ λ2 = hλω on ∂ω. According to (5.3) and bound (5.8) we have |hλω | C and Aλ Lq (G;R2 ) Cq , ∀q 1, where Cq is independent of λ. Therefore, by elliptic estimates, the norm hˆ λ2 W 1,q (G) is uniformly in λ bounded. This in turn implies 1,q (G) ⊂ the uniform boundedness of hˆ λ2 C(G) ¯ , thanks to the compactness of the embedding W ¯ for q > 2. We next consider hˆ λ which satisfies − hˆ λ = 2 ∂uλ ∧ ∂uλ in G and zero boundary C(G) 1

1

∂x1

∂x2

conditions on ∂G. Applying a result from [21] (see also [7]) we have hˆ λ1 H 1 (G) , hˆ λ1 L∞ (G) Cuλ 2H 1 (G;C) , so that the required L∞ -bound follows from the fact that uλ H 1 (G;C) C (cf. Section 5). To demonstrate (iii) we just note that the weak convergence of hλ follows from (5.8), (5.13) and (5.14), since we already know that hλ H 1 (G) is bounded. To prove (ii) it suffices to show that hλ 0 in G (hλ = 0 on ∂Ω). For this purpose we derive from the pointwise equality j λ = −∇ ⊥ h, dividing it by |uλ | and than taking curl, that 1 λ − div ∇h + hλ = 0 in x ∈ G; uλ (x) > 0 . |uλ |2

(7.5)

Let 1 > ρ > 0 be a regular value of |uλ | (by Sard’s lemma almost all ρ ∈ (0, 1) are regular values of |uλ |), and let x0 be a minimum point of hλ on the closure of Gρ = {x ∈ G; |uλ (x)| > ρ}. Assume by contradiction that hλ (x0 ) < 0. Then by the maximum principle applied to (7.5) the point x0 cannot be in the interior of Gρ . It cannot also be on ∂ω, otherwise hλ (x0 ) = hλω < 0 and therefore

∂hλ ds = − ∂ν

∂ω

=

j λ · τ ds = −2π deg uλ , ∂ω +

∂ω

Aλ · τ ds

∂ω

hλ dx = |ω|hλω < 0, ω

thus hλ (x0 ) is not a minimal value of hλ in Gρ . Similar computations show that, if |uλ (x0 )| = ρ then 1 ρ2

|uλ |=ρ

∂hλ ds = −2π + ∂ν

hλ dx,

(7.6)

|uλ | 0. Let Φ be extended into G to have Φ ∈ C 1 (G), −1 Assume that λ is so close to 1 that aξ λ (B1/2 (0)) ⊂ Bρ (ξ ∗ ), then, by Lemma 9, |uλ | μ0 > 0 in G \ Bρ (ξ ∗ ). Multiply (7.5) by Φ to get after integrating over G \ Bρ (ξ ∗ ), −

∂hλ Φ ds = ∂ν

Φj · τ ds = λ

∂Ω

∂Ω

G

1 λ λ ∇Φ · ∇h + h Φ dx. |uλ |2

The right-hand side of this equality tends to zero as λ → 1 − 0, since in

L2 (G; R2 ),

while

hλ

→ 0 weakly in

H 1 (G).

2

1 ∇Φ |uλ |2

→ ∇Φ strongly

Remark 3. A reasoning similar to the proof of Lemma 11 shows that j λ → 0 in D (∂ω) as λ → 1 − 0. (This is due to the fact that uλ has a unique vortex approaching ∂Ω in the limit.) 8. Explicit formula for energy bounds The right-hand side I (ξ, λ) in the upper bound (3.14) can be equivalently rewritten as I (ξ, λ) = π +

1 8KG

∂φξ ds ∂ν

∂ω

2 −

1−λ 8

2 |aξ |2 eφξ − 1 dx,

(8.1)

G

where φ is the unique solution of ⎧ 2 φ ξ in G, ⎪ ⎪ ⎨ − φξ = 1 − aξ (z) e φξ = 0 on ∂Ω, ⎪ ⎪ ⎩ φ = − log a (z) 2 on ∂ω, ξ

(8.2)

ξ

aξ (z) is given by (3.6) with F being a fixed conformal map from Ω onto the unit disk. Indeed, the solution ϕξ of (3.3)–(3.4) and φξ are related by ϕξ = φξ − σξ , where σξ = log |γξ /aξ | ( σξ = 0 in G). Note also that

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∂ϕξ ds = ∂ν

∂ω

∂φξ ds, ∂ν

∂ω

since σξ satisfies (3.7). The following lemma proves the explicit asymptotic formula (3.15). Lemma 12. Let ξ ∈ G and let ξ ∗ = ξ ∗ (ξ ) ∈ ∂Ω to be the nearest point projection of ξ on ∂Ω. Then for sufficiently small δ = dist(ξ, ∂Ω) ∂φ ∗ (i) ∂ω ∂νξ ds = 4πδ ∂V ∂ν (ξ ) + o(δ), where V is the solution of (1.4), (ii) G (|aξ |2 eφξ − 1)2 dx = 8πδ 2 |log δ| + O(δ 2 ). In the proof of both statements of Lemma 12 we will make use of the following formulas, as ξ → ∂Ω G

2 1 − |aξ |2 dx = 8πδ 2 |log δ| + O(1) ,

1 − |aξ |2 dx = O(δ),

(8.3)

(8.4)

G

where δ is the distance from ξ to ∂Ω. We postpone the proof of these formulas and proceed to the Proof of (i) of Lemma 12. We first show that ∂ω

∂φξ ds = ∂ν

∂φ ∗ ξ ∂ν

ds + o(δ)

as δ → 0,

(8.5)

∂ω

where φξ∗ is the unique solution of the auxiliary linear problem ⎧ ∗ ∗ aξ (z) 2 in G, ⎪ − φ + φ = 1 − ⎪ ξ ξ ⎨ φξ∗ = 0 on ∂Ω, ⎪ 2 ⎪ ⎩ ∗ φξ = − log aξ (z) on ∂ω.

(8.6)

We claim that φξ C(G) ¯ < Cδ

as δ → 0.

This implies the bound − φξ − φ ∗ + φξ − φ ∗ 2 = φξ 1 − |aξ |2 + |aξ |2 1 − eφξ + φξ 2 ξ ξ L (G) L (G) 2 2 2 δ 1 − |aξ | L2 (G) + Cδ C1 δ |log δ| + 1 ,

(8.7)


1759

where we have used (8.3). Since φξ = φξ∗ on ∂G, we can easily derive (8.5) by standard elliptic estimates. Proof of claim (8.7). Due to (8.2) we have − (φξ + log |aξ (z)|2 ) = 1 − |aξ (z)|2 eφξ in G \ Br (ξ ) for every r > 0. By applying the maximum principle to this equation, we conclude that φξ − log |aξ (z)|2 in G. The latter inequality implies that 1 − |aξ (z)|2 eφξ 0. Hence, we can apply the maximum principle once more to (8.2) to conclude that φξ 0 in G. Thus, 0 1 − |aξ (z)|2 eφξ 1 − |aξ (z)|2 . On the other hand, a direct computation shows that 2 2 1 − aξ (z) = F (ξ ) − 1

|F (z)|2 − 1 |F (z)F (ξ ) − 1|2

C

δ |z − ξ ∗ |

as δ → 0.

(8.8)

This allows us to estimate Lp -norm of the right-hand side of (8.2) for every 1 < p < 2 and next obtain, by standard elliptic estimates, that φξ W 2,p (G) C(p)δ

as δ → 0.

By using the Sobolev embedding the required result (8.7) follows.

2

Remark 4. In the proof of claim (8.7) we showed that |aξ |eφξ 1 in G, therefore |γξ |eϕξ 1 in G (since ϕξ = φξ − log |γξ /aξ |). Proof of (i) of Lemma 12 completed. Now multiply the equation in (8.6) by V (the unique solution of problem (1.4)) and integrate by parts to get ∂φ ∗ ξ ∂ν

ds =

∂ω

1 − |aξ |2 V −

G

log |aξ |2

∂V ds. ∂ν

(8.9)

∂ω

On the other hand, since log |aξ |2 = 4πδξ (x) in Ω and log |aξ |2 = 0 on ∂Ω, we have 4πV (ξ ) =

log |aξ |2 V dx +

G

log |aξ |2

∂V ds. ∂ν

(8.10)

∂ω

By adding (8.9) to (8.10), we obtain ∂φ ∗ ξ ∂ν

ds = 4π V (ξ ∗ ) − V (ξ ) +

∂ω

log |aξ |2 + 1 − |aξ |2 V dx.

G

Thus, in view of (8.5) it remains only to show that the last term in the above equality is of order o(δ). To this end we split it as G

log |aξ |2 + 1 − |aξ |2 V dx =

G\B√δ (ξ ∗ )

··· + G∩B√δ (ξ ∗ )

· · · =: I1 + I2 .

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According to (8.8) we have |log(1 − 1 + |aξ |2 ) + 1 − |aξ |2 | Cδ 2 /|x − ξ ∗ |2 in G \ B√δ (ξ ∗ ), therefore I1 = O(δ 2 ) (note that |V (x)| C|x − ξ ∗ |); while I2 = O(δ 3/2 |log δ|), that can be verified by using the obvious bound |log |aξ |2 + 1 − |aξ |2 | C(|log |x − ξ || + 1). Statement (i) is proved. 2 Proof of (ii) of Lemma 12. Integrating the identity

|aξ |2 eφξ − 1

2

2 2 = |aξ |2 − 1 + 2|aξ |2 eφξ − 1 |aξ |2 − 1 + |aξ |4 eφξ − 1

over G we use estimates (8.3), (8.4), (8.7) and the Cauchy–Schwarz inequality to establish the following

aξ (z) 2 eϕ − 1 2 dx =

G

a(z, ξ ) 2 − 1 2 dx + O δ 2 = 8πδ 2 |log δ| + O δ 2 .

G

2

Thus Lemma 12 is completely proved.

Calculation of (8.3) and (8.4). For brevity we show only (8.3) (the demonstration of (8.4) follows the same lines). Perform the change of variables ζ = F (x) to get, after simple computations,

2 1 − |aξ |2 dx =

G

2 1 − |mF (ξ ) |2

B1 (0)\F (ω)

=

1 Jac F (ξ )

dζ Jac F (ζ )

2 2 1 − mF (ξ ) (ζ ) dζ + O δ 2 ,

(8.11)

B1 (0)

where mμ (z) = (z − μ)/(μz ¯ − 1) is the classical Möbius conformal map from the unit disk onto itself. The integral in the right-hand side of (8.11) can be calculated explicitly. We obtain, by using the coarea formula twice,

mμ (x) 2 − 1 2 dx =

dt 0

B1 (0)

1

1 = 0

1 =

2 2 t −1

dH1 |∇|mμ (x)||

t

|mμ (x)|=t

2 2 t −1 d

dτ 0

|mμ (x)|=τ

dH1 |∇|mμ (x)||

2 2 . t − 1 d area m−1 μ Bt (0)

0

Note that the inverse conformal map m−1 μ (z) coincides with mμ (z). Moreover, the inverse image m−1 (B (0)) of the disk B (0) is the disk Br (y) with the radius r = t (1 − |μ|2 )/(1 − t 2 |μ|2 ) and t t ξ the center at y = μ(1 − t 2 )/(1 − t 2 |μ|2 ). Therefore we get, after integrating by parts,


mμ (x) 2 − 1 2 dx = 2 1 − |μ|2 2 π

1 0

B1 (0)

1761

(1 − t 2 )t 2 dt 2 , (1 − t 2 |μ|2 )2

and elementary calculations lead to the following asymptotic formula, as |μ| → 1 − 0

2 2 2 1 − mμ (ζ ) dζ = 2π 1 − |μ|2 log 1 − |μ|2 + O(1) .

(8.12)

B1 (0)

Finally, by the conformality of F we have 2 2 1 − F (ξ ) = 4δ 2 Jac F (ξ ) + O δ 3 .

(8.13)

Thus (8.11), (8.12) and (8.13) yield (8.3). Lemma 12 allows us to rewrite the lower bound (5.1) of Lemma 2 in the form

2π 2 2 ∂V λ 2 m(λ) = Fλ uλ , Aλ π + δ ξˆ − πδ 2 (1 − λ)|log δ| 1 + o(1) + o δ 2 , KG ∂ν (8.14) as λ → 1−0, where ξˆ λ is the nearest point projection on ∂Ω of the point ξ λ and δ = dist(ξ λ , ∂Ω) (δ = δ(λ) → 0). Recall that the point ξ λ ∈ G was defined in Section 5 as the unique zero of the auxiliary map θ λ constructed by means of uλ and Aλ . By Lemma 9 we can redefine ξ λ as the unique zero (vortex) of uλ so that (8.14) remains valid. On the other hand, by (3.14), (8.1) and Lemma 12, 2 2π 2 2 ∂V ˆ ˜ ˜ + O δ˜2 + o δ˜2 , (8.15) (ξ ) − π(1 − λ) δ˜2 |log δ| δ m(λ) I (ξ, λ) = π + KG ∂ν where ξˆ is an arbitrary point on ∂Ω and δ˜ > 0 is a small parameter (ξˆ is the nearest point projection on ∂Ω of ξ ∈ G and δ˜ = dist(ξ, ∂Ω)). It follows from (8.14) and (8.15) that, as λ → 1 − 0: −2π | ∂V (ξˆ λ )|2 (1 + o(1))); (a) δ = exp( (1−λ)K G ∂ν ∂V ˆ 2 ˆ ˆλ 2 (b) | ∂V ∂ν (ξ )| → MG , where MG = min{| ∂ν (ξ )| ; ξ ∈ ∂Ω} (> 0).

Thus we have, in particular, that the unique zero (vortex) of uλ converges (up to extracting a 2 subsequence) to a point minimizing | ∂V ∂ν | on ∂Ω. This completes the proof of Theorem 1. Acknowledgments The research of all three authors was partially supported by NSF grant DMS-0708324. This work was initiated when V. Rybalko was visiting The Pennsylvania State University. He is grateful to L. Berlyand and A. Bressan (the Eberly Chair fund) for the support and hospitality received during the visit.

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References [1] L. Almeida, F. Bethuel, Topological methods for the Ginzburg–Landau equations, J. Math. Pures Appl. 77 (1998) 1–49. [2] L. Berlyand, D. Golovaty, V. Rybalko, Nonexistence of Ginzburg–Landau minimizers with prescribed degree on the boundary of a doubly connected domain, C. R. Math. Acad. Sci. Paris 343 (2006) 63–68. [3] L. Berlyand, P. Mironescu, Ginzburg–Landau minimizers with prescribed degrees. Capacity of the domain and emergence of vortices, J. Funct. Anal. 239 (2006) 76–99. [4] L. Berlyand, O. Misiats, V. Rybalko, Minimizers of the magnetic Ginzburg–Landau functional in simply connected domain with prescribed degree on the boundary, Commun. Contemp. Math., in press. [5] L. Berlyand, V. Rybalko, Solutions with vortices of a semi-stiff boundary value problem for the Ginzburg–Landau equation, J. Eur. Math. Soc., in press. [6] F. Bethuel, H. Brezis, F. Hélein, Ginzburg–Landau Vortices, Birkhäuser, 1994. [7] F. Bethuel, J.-M. Ghidaglia, Régularité des solutions de certaines équations elliptiques en dimension deux et formule de la co-aire, in: Équations aux Dérivées Partielles, Exp. 1, Saint-Jean-de-Monts, 1993, École Polytechnique, Palaiseau, 1993, 36 pp. [8] F. Bethuel, T. Rivière, Vortices for a variational problem related to superconductivity, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 243–303. [9] E.B. Bogomol’nyi, The stability of classical solutions, Soviet J. Nuclear Phys. 24 (1976) 449–454. [10] A. Boutet de Monvel-Berthier, V. Georgescu, R. Purice, A boundary value problem related to the Ginzburg–Landau model, Comm. Math. Phys. 142 (1991) 1–23. [11] M. Dos Santos, Local minimizers of the Ginzburg–Landau functional with prescribed degrees, J. Funct. Anal. 25 (2009) 1053–1091. [12] L. Evans, R. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, 1992. [13] A. Fiorenza, C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right-hand side in L1 , Studia Math. 127 (1998) 223–231. [14] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1977. [15] D. Golovaty, L. Berlyand, On uniqueness of vector-valued minimizers of the Ginzburg–Landau functional in annular domains, Calc. Var. Partial Differential Equations 14 (2002) 213–232. [16] P. Mironescu, A. Pisante, A variational problem with lack of compactness for H 1/2 (S 1 ; S 1 ) maps of prescribed degree, J. Funct. Anal. 217 (2004) 249–279. [17] F. Pacard, T. Rivière, Linear and Nonlinear Aspects of Vortices, Birkhäuser, 2000. [18] E. Sandier, S. Serfaty, Vortices in the Magnetic Ginzburg–Landau Model, Birkhäuser, 2007. [19] M. Struwe, On the asymptotic behavior of minimizers of the Ginzburg–Landau model in 2 dimensions, Differential Integral Equations 7 (1994) 1613–1624. [20] C. Taubes, Arbitrary N -vortex solutions to the first order Ginzburg–Landau equations, Comm. Math. Phys. 72 (1980) 277–292. [21] H. Wente, An existence theorem for surfaces of constant mean curvature, J. Math. Anal. Appl. 26 (1969) 318–344.


Stokes formula on the Wiener space and n-dimensional Nourdin–Peccati analysis Hélène Airault a,∗ , Paul Malliavin b , Frederi Viens c a Université de Picardie Jules Verne, Insset, 48 rue Raspail, 02100 Saint-Quentin (Aisne), LAMFA, UMR 6140, 33,

rue Saint-Leu, 80000 Amiens, France b 10, rue Saint-Louis-en-l’Ile, 75004 Paris, France c Dept. Statistics and Dept. Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907-2067, USA

Received 7 July 2009; accepted 7 July 2009 Available online 23 July 2009 Communicated by Paul Malliavin

Abstract Extensions of the Nourdin–Peccati analysis to Rn -valued random variables are obtained by taking conditional expectation on the Wiener space. Several proof techniques are explored, from infinitesimal geometry, to quasi-sure analysis (including a connection to Stein’s lemma), to classical analysis on Wiener space. Partial differential equations for the density of an Rn -valued centered random variable Z = (Z 1 , . . . , Z n ) are obtained. Of particular importance is the function defined by the conditional expectation given Z of the auxiliary random matrix (−DL−1 Z i | DZ j ), i, j = 1, 2, . . . , n, where D and L are respectively the derivative operator and the generator of the Ornstein–Uhlenbeck semigroup on Wiener space. © 2009 Elsevier Inc. All rights reserved. Keywords: Wiener space; Quasi-sure analysis; Density formula; Nourdin–Peccati analysis; Stein’s lemma; Conditional probability

1. Introduction We set up various systems of partial differential equations for the density and other functionals of the distribution of a random variable, extending to the n-dimensional case a formula from [12] * Corresponding author.

E-mail addresses: [email protected] (H. Airault), [email protected] (P. Malliavin), [email protected] (F. Viens). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.07.005

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H. Airault et al. / Journal of Functional Analysis 258 (2010) 1763–1783

based on the Nourdin–Peccati analysis introduced in [8]. The basic tool is the projection (or equivalently the image) of a vector field defined on the Wiener space through a centered nondegenerate map Z, as defined in [7, p. 70]; we connect this projection to [8] and [12] by extending the use of the random variable A = (−DL−1 Z | DZ) defined and employed in those references, where D is the derivative operator and L is the Ornstein–Uhlenbeck generator; these are defined precisely below, also see [7]. For instance, for an Rn -valued random variable Z = (Z 1 , Z 2 , . . . , Z n ) having a density ρ the one-dimensional density formula in [12] is generalized herein to the system of partial differential equations in Rn ∂ 1 ∂ 2 ∂ n β ρ + β ρ + ··· + β ρ = −xj ρ ∂x1 j ∂x2 j ∂xn j

for j = 1, 2, . . . , n,

(0.1)

where βjk is the function on Rn given by the conditional expectation βjk (x) = E Z=x −DL−1 Z j DZ k .

(0.2)

First, we introduce definitions and notations, including those needed for the above expressions. 1.1. Non-degenerate maps We denote by Ω the Wiener space, by μ the standard Wiener measure and by H its Cameron– Martin space. For a function h ∈ H, and ω ∈ Ω, we define the derivative operator D, see [6,7], by F (ω + h) − F (ω) = Dh F (ω) := lim →0

1

Ds F (ω)h (s) ds.

(1.1.1)

0

For a vector field V on the Wiener space and F : Ω → R, we denote DV F := (V | DF ) where (. | .) is the scalar product in the Cameron–Martin space. The divergence operator δ is the dual of the operator D in L2μ . This means that for a vector field A and a random variable Ψ (i.e. Ψ is a real-valued measurable function defined on the Wiener space Ω), respectively in the domain of the operators δ and D, E δ(A)Ψ = E (A | DΨ ) .

(1.1.2)

We define the Ornstein–Uhlenbeck operator L by L := −δD.

(1.1.3)

We have 1 LF = −

Ds F (ω) dω(s). 0

(1.1.4)


1765

The pseudo-inverse L−1 of the operator L plays an important role in our study. For a function F : Ω → R such that E[F ] = 0, we put

L

−1

+∞ =− et L dt.

(1.1.5)

0

Using Wiener chaos, for any F ∈ L2μ (Ω), we can find a sequence of symmetric func∞ 2 n tions ∞ fn ∈ L2 ([0, 1] ), n = 1, 2, . . ., such that F = E[F ] + n=1 In (fn ) and Var[F ] = n=1 n!fn L2 ([0,1]n ) where In (f ) is n! times the iterated Itô integral of f w.r.t. the Wiener process ω: sn−1 1 s2 · · · f (s1 , . . . , sn ) dω(s1 ) · · · dω(sn ). In (f ) = n! 0

0

0

∞ −1 −1 Then it turns out that LF = − ∞ n=1 nIn (fn ) and L F = − n=1 n In (fn ). Details are in Nualart’s book [13, Chapter 1]. Let Z = (Z 1 , Z 2 , . . . , Z n ) be an Rn -valued random variable defined on the Wiener space Ω. Assume that Z ∈ D∞ (Ω), i.e. Z is infinitely differentiable with respect to the operator D. Let p M be the n × n Gram matrix having for coefficients (DZ i | DZ j ); assume that [det(M)]−1 ∈ Lμ for every p: if all the above conditions are fulfilled, we say that the map Z is non-degenerate. It is known from [7] that the law of a non-degenerate map Z has a density ρ relatively to the volume measure of Rn , and that ρ is infinitely differentiable. We denote Z ∗ μ the image of μ through the map Z; thus Z ∗ μ = ρ(x) dx. For f : Rn → R

f Z(ω) dμ(ω) =

f (x)ρ(x) dx

(1.1.6)

where dx = dx1 dx2 · · · dxn is the volume measure on Rn . Let Φ be an R-valued function defined on the Wiener space and consider the measure dν = Φ(ω) dμ(ω); we denote Z ∗ Φ(ω) dμ the image measure of ν by Z. If this image measure has a density with respect to the volume measure dx, we denote this density dZ ∗ (Φ dμ)/dx. The conditional expectation of Φ given Z = (x1 , x2 , . . . , xn ) is E Z=(x1 ,x2 ,...,xn ) [Φ] =

dZ ∗ (Φ dμ)/dx (x1 , x2 , . . . , xn ). dZ ∗ dμ/dx

(1.1.7)

We denote this function by E Z [Φ]. By definition, for any integrable function ψ : Rn → R, E ψ Z(ω) Φ(ω) =

ψ Z(ω) Φ(ω) dμ(ω),

ψ(x)E Z=x [Φ]ρ(x) dx =

ψ Z(ω) Φ(ω) dμ(ω).

(1.1.8) (1.1.9)

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1.2. Inner action of vector fields on differential forms Let v be a vector field on a differentiable manifold M of dimension n, we assume n 2; denote p (M) the vector space of differential forms of degree p on M; for p > 1, the inner product i(v) :

p

(M) →

p−1

(1.2.1)

(M)

is defined through the identity

i(v)(Θ), e1 ∧ · · · ∧ ep−1 = Θ, v ∧ e1 ∧ · · · ∧ ep−1

(1.2.2)

where e1 , . . . , ep−1 are generic vector fields on M. In particular if M = Rn , and θ := dx 1 ∧ · · · ∧ dx n is the canonical volume form of Rn , then i(v)(θ ) ∈ field v as v=

n

βk

k=1

n−1

(1.2.3) (Rn ) and if we represent the vector

∂ ∂x k

(1.2.4)

we get i(v)(θ ) = β 1 × dx 2 ∧ · · · ∧ dx n − β 2 × dx 1 ∧ dx 3 ∧ · · · ∧ dx n + · · · =

n

(−1)j +1 β j × dx k .

(1.2.5)

k =j

j =1

Let d be the classical differential: if u : Rn → R is a function, then du = forms, if

α=

n

∂u k=1 ∂x k

dx k and for

αi1 ···ip dx i1 ∧ dx i2 ∧ · · · ∧ dx ip

i1 x 1 1Z 2 >x 2 ,

j

j = 1, 2.

Z −1 (D(x0 ,+∞))

Then we proceed as in the direct proof of (3.10). We define an approximation of the set function 1D(x,+∞) . For small ε > 0, we define the continuous and almost everywhere differentiable function φε : R2 → R by φε (η1 , η2 ) = 1 if η = (η1 , η2 ) ∈ D(x, +∞) and φε (η1 , η2 ) = 0 / D(x − (ε, ε), +∞). We join these two pieces by planes. For that we put if η = (η1 , η2 ) ∈ φε (η1 , η2 ) = ε −1 (η1 − (x 1 − ε)) if x 1 − ε < η1 < x 1 and η1 − x 1 < η2 − x 2 , then φε (η1 , η2 ) = ε −1 (η2 − (x 2 − ε)) if x 2 − ε < η2 < x 2 and η1 − x 1 η2 − x 2 . We have 1D(x,+∞) = lim φε ε→0

almost everywhere w.r.t. the measure dx

and the derivatives of φε exist almost everywhere. We deduce F j x 1 , x 2 = lim E δAj φε Z 1 (ω), Z 2 (ω) ε→0 1 j 1 ∂φε 2 Z (ω), Z (ω) = lim E A DZ ε→0 ∂η1 ∂φε 1 Z (ω), Z 2 (ω) . + lim E Aj DZ 2 ε→0 ∂η2

(4.8)

The partial derivatives of φε are zero for η ∈ D(x, +∞) and η ∈ / D(x − (ε, ε), +∞). On the strip (of width ε) x 1 − ε < η1 < x 1 and η1 − x 1 < η2 − x 2 , we have ∂φε /∂η1 = ε −1 . We thus obtain the theorem from (4.8). 2 For fixed x0 ∈ R2 , we proved the above theorem approximating the boundary of the domain D(x0 , +∞). However we may define the boundary of the submanifold Z −1 (D(x0 , +∞)) in Wiener space. The boundary ∂D(x0 , +∞) is constituted by the two half-lines l1 , l2 , starting from x0 satisfying dx02 |l1 = 0, and dx01 |l2 = 0, i.e. l1 =

1 2 1 η , x0 , η x01

and l2 =

1 2 2 x0 , η , η x02 .

(4.9)

For k = 1, 2 and j = 1, 2, j = k, we let Lk = Z −1 (lk ), i.e. j Lk = Z k (ω) x0k , Z j (ω) = x0

(4.10)

thus L1 , L2 are two submanifolds of codimension 1 of the Wiener space and the boundary of Z −1 (D(x0 , +∞)) is ∂ Z −1 D(x0 , +∞) = L1 ∪ L2 .

(4.11)

Then with quasi-sure analysis [5,16], we can consider the previous theorem as a projection on R2 of the Stokes formula on the Wiener space. Let A be a vector field on the Wiener space. For a


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differentiable function ψ : Ω → R, it holds ψ(ω)(δ A) dμ(ω) = (A | Dψ) dμ(ω). If ψ is not differentiable but is the set function 1Z −1 (D(x0 ,+∞)) , we have Stokes theorem

(δ A) dμ(ω) =

Z −1 (D(x0 ,+∞))

A

∂[Z −1 (D(x0 ,+∞))]

where we specify the meaning of the boundary integral of the vector field A on the right-hand side by

A= ∂[Z −1 (D(x0 ,+∞))]

A+ L1

A. L2

This interpretation therefore generalizes to the n-dimensional case, as follows. Theorem 4.2. Let Z : Ω → Rn be a non-degenerate map and A be a vector field on Ω. Let x0 = (x 1 , x 2 , . . . , x n ) ∈ Rn . The boundary ∂[Z −1 (D(x0 , +∞))] is Pk ∂ Z −1 D(x0 , +∞) = 1kn

where Pk is the subset of the Wiener space defined by Pk = Z k (ω) = x k , Z j (ω) x j , ∀j = k . It is a submanifold of codimension one in the Wiener space. We have

(δ A) dμ(ω) = Z −1 (D(x0 ,+∞))

A=

∂[Z −1 (D(x0 ,+∞))]

n

A

j =1P

k

with Pk

∂ A=− k ∂η η=x0

A DZ k dμ(ω)

Z −1 (D(η,+∞))

where we put η = (η1 , η2 , . . . , ηn ). Proof. Following [18], one only needs to consider distributions on the Wiener space: we have (δ A) dμ(ω) = Z −1 (D(x0 ,+∞))

n ∂ E A DZ j × 1Z k (ω)x k × j 1Z j (ω)x j . ∂x j =1

k, k =j

2

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5. Approach to n-dimensional density formulae via partial differential equations In dimension n 2, as we have seen that (0.1)–(0.2) are consequence of Section 2, it is possible to find implicit expressions for the density of an Rn -valued random variable Z using partial differential equations as a consequence of Theorem 2.1. 5.1. A system of PDEs employing DZ and LZ If one is able to calculate functions based on DZ and LZ, a system was already known in [1, pp. 355–360]. Let Z : Ω → Rn be an Rn -valued random variable, Z = (Z 1 , Z 2 , . . . , Z n ). We assume that Z has a density ρ(x1 , x2 , . . . , xn ) with respect to the Lebesgue measure dx. We denote and γj = −E Z=x LZ j βij (x) = E Z=x DZ i DZ j then according to [1], the density ρ satisfies the system of partial differential equations (S): ∂ ∂ ∂ (β1j ρ) + (β2j ρ) + · · · + (βnj ρ) = −γj ρ ∂x1 ∂x2 ∂xn

for j = 1, 2, . . . , n.

(S)

Like in Section 2, we can deduce the system (S) from a more general result, as we now see. Proposition 5.1. Let Z = (Z 1 , Z 2 , . . . , Z n ) : Ω → Rn be an Rn -valued random variable with density ρ. Let ψ : Rn → R, and denote and γψ (x) = −E Z=x L(ψ ◦ Z) βψk (x) = E Z=x D(ψ ◦ Z) DZ k then n ∂ k β ρ = −γψ ρ. ∂x k ψ

(5.1.1)

k=1

Proof. For any test function g : Rn → R with suitable boundedness and smoothness assumptions, and μ the Wiener measure, we have by definition of the density and the conditional expectation, first using −L = δD, then integration by parts, and finally the duality relation between δ and D and the chain rule for D,

g(x)γψ (x)ρ(x) dx = −

g(Z)L(ψ ◦ Z) dμ =

g(Z)δD(ψ ◦ Z) dμ

∂g ∂g k (Z) DZ D(ψ ◦ Z) dμ = (x)βψk (x)ρ(x) dx. = ∂xk ∂xk k

k

Integrating again by parts yields (5.1.1). Another proof of this proposition is to apply Theorem 2.1 with f (x) = γψ (x). 2


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Example 5.2. Let Z = (ω(t1 ), ω(t2 )), then βij = ti ∧ tj and Lω(tj ) = ω(tj ), thus γj (x1 , x2 ) = E Z=(x1 ,x2 ) [Lω(tj )] = E Z [ω(tj )] = xj . The system (S) becomes t1

∂ ∂ ρ + t1 ∧ t2 ρ = −x1 ρ, ∂x1 ∂x2

t1 ∧ t2

∂ ∂ ρ + t2 ρ = −x2 ρ. ∂x1 ∂x2

x2

2 −x1 ) The solution is given by ρ(x1 , x2 ) = exp(− 2t11 ) exp(− (x 2(t2 −t1 ) ). 2

As can be seen in the above example, the proposition is easily interpreted when Z is jointly Gaussian: one notes that then the system (S) becomes n i=1

β i,j

∂ρ = −xj ρ(x) ∂xi

for j = 1, 2, . . . , n,

whose solution is evidently the density of Z. An economy of functional parameters can be achieved, and a greater ability to compare the law of an arbitrary random variable Z to a Gaussian law, if one reverts to the use of the matrix h defined in (0.2), see Theorem 2.5. The Gaussian case is equivalent to the case where h is a constant matrix, equal to the covariance matrix of Z. We easily see in this case that the system (S) is identical to the system (0.1)–(0.2). In general, this is not the case. Indeed, while the matrix α in (S) is always symmetric, the matrix h, which coincides with the matrix α only in the Gaussian case, is typically non-symmetric when Z is not Gaussian. On the other hand, from the point of view of PDEs, assume that the hj,k are constants and that the matrix (hj,k ) is invertible; then the system (0.1)–(0.2) has a solution if and only if the matrix (hj,k ) is symmetric. This is proved writing the integrability conditions for the system as follows. Let h−1 be the inverse of the matrix h. We have ∂ log ρ/∂xj = k (h−1 )j,k xk and ∂ 2 log ρ/∂xp ∂xj = (h−1 )j,p . The condition that ∂ 2 log ρ/∂xp ∂xj is symmetric in j , p implies that the matrix (h−1 )j,p is symmetric. 5.2. A general system. Comparison of two random variables The following proposition is also a consequence of Theorem 2.1. It covers both system (S) and system (0.1)–(0.2), see Theorem 2.5. Let Y = (Y 1 , Y 2 , . . . , Y p ) and Z = (Z 1 , Z 2 , . . . , Z n ) be two random variables with values respectively in Rp and in Rn . Let f : Rp → R. In the next proposition, to obtain the system (S) of Section 5.1, we take n = p and Y j = Z j and to obtain (0.1)–(0.2), we take Y j = L−1 Z j . Proposition 5.3. Let ψ : Rp → R. With Y = (Y 1 , Y 2 , . . . , Y p ) and Z = (Z 1 , Z 2 , . . . , Z n ) as above, we denote, for x ∈ Rn , γψ (x) = E Z=x L(ψ ◦ Y )

and βψk (x) = −E Z=x D(ψ ◦ Y ) DZ k .

We assume that the variable Z has a density ρ with respect to the n-dimensional Lebesgue measure. Then ρ satisfies the following system of PDEs for j = 1, 2, . . . , n: n ∂ k β ρ = −γψ ρ. ∂xk ψ k=1

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Proof. The proof is similar to those of Theorem 2.5 or of Proposition 5.1. For a bounded differentiable test function g : Rn → R, we calculate E L(ψ ◦ Y )g Z 1 , Z 2 , . . . , Z n in two different ways: E L(ψ ◦ Y )g Z 1 , Z 2 , . . . , Z n =

g(x1 , x2 , . . . , xn )Z ∗ L(ψ ◦ Y ) dμ

Z ∗ (L(ψ ◦ Y ))dμ Z ∗ dμ Z ∗ dμ = g(x1 , x2 , . . . , xn )E Z L(ψ ◦ Y ) (x1 , x2 , . . . , xn )Z ∗ dμ = g(x)γψ (x)ρ(x) dx.

=

g(x1 , x2 , . . . , xn )

On the other hand E L(ψ ◦ Y )g Z 1 , Z 2 , . . . , Z n = −E δ D(ψ ◦ Y ) g Z 1 , Z 2 , . . . , Z n n ∂g 1 2 =− Z , Z , . . . , Zn E D(ψ ◦ Y ) DZ k ∂xk k=1

=

n

βψk (x)

k=1

=−

n k=1

∂g (x)ρ(x) dx ∂xk

∂ k β ρ (x)g(x) dx ∂xk ψ

2

finishing the proof of the proposition.

The above general proposition gives information when n = 1, p = 1, for calculating conditional expectations for D-differentiable r.v.’s. Indeed, assume Z and V are D-differentiable, and let Y = L−1 V . Then

and β(x) = −E Z=x DL−1 V , DZ .

γ (x) = E[V | Z = x]

Let ρ be the density of Z. The proposition yields (βρ) = −γρ. In particular, ∞ γ (y)ρ(y) dy = E[V 1Z>x ].

β(x)ρ(x) = x

This relation helps to see how β and γ are connected to the issue of how Z and V are correlated. For instance one way to signify that V and Z are from the same distribution but are non-trivially


1781

correlated, is to say that there is some constant K ∈ (0, 1) such that for x > 0, γ (x) = E Z=x [Y ] Kx. Then for x > 0, +∞ zρ(z) dz = Kϕ(x) β(x)ρ(x) K x

where we used the notation ϕ as in Lemma 2.4. Recall from therein that the density formula of Lemma 2.4 is equivalent to ϕ = ρh where h(x) = −E Z=x [ DL−1 Z, DZ] as usual. Therefore on the support of Z, with x > 0 therein, β(x) Kh(x). In other words, the non-trivial correlation of Y and Z can be read off of the above inequality as well. 6. Estimating conditional probabilities Theorem 4.1 is a special case, in dimension 2, of a corollary of the PDE-based Theorem 2.5 (i.e. of the system of PDEs (0.1)–(0.2)), which we now give. Theorem 4.1 and this corollary have the advantage of not referring to the derivatives of ρ. Corollary 6.1. Under the assumptions of Theorem 2.5, recall the distribution-moment-type function F defined in (4.1), i.e. ∞∞ F (x) =

∞ ···

i

x1 x2

yi ρ(y) dyn · · · dy2 dy1 = E[Zi 1Z1 >x1 1Z2 >x2 · · · 1Zn >xn ] xn

for i = 1, 2, . . . , n, and the corresponding matrix h from (4.5). Then for each i, and each x = (x1 , x2 , . . . , xn ) ∈ Rn , F i (x) =

n

I i,j

j =1

where ∞

∞ I i,j :=

··· x1

∞ ···

xj

j · · · dzn , hi,j (z1 , . . . , xj , . . . , zn )ρ(z1 , . . . , xj , . . . , zn ) dz1 · · · dz

xn

where the symbol . means that the corresponding expression is to be omitted. Proof. For each fixed i = 1, 2, . . . , n, if we integrate the corresponding equation in the system of PDEs (0.1)–(0.2), over the orthant D(x, +∞), the expression on the left-hand side of (0.1) becomes precisely the sum − nj=1 I i,j above, while the expression on the right-hand side of (0.1) becomes precisely −F i (x), proving the corollary. 2

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Let us now go back to the special case n = 2. Consider the conditional distribution function of Z 2 given Z 1 , and conversely: with ρZ i the density of Z i , we define ∞ ψ (x) :=

ρ(x1 , z2 ) dz2 = ρZ 1 (x1 )P Z

1

1 =x 1

2 Z > x2 ,

2 =x 2

1 Z > x1 .

x2

∞ ψ (x) :=

ρ(z1 , x2 ) dz1 = ρZ 2 (x2 )P Z

2

x1

Note that F i is an antiderivative with respect to xi of −zi ψ i . Given prior information about the marginal densities ρZ1 and ρZ2 , estimates on H translate into relations on the two functions ψi , as the next proposition shows. Proposition 6.2. With the notation of Theorem 2.5 with n = 2, assume that, for some c ∈ R, for all x ∈ R2 , hi,i (x) 1 for i = 1, 2, and hi,j (x) c when i = j . Then ∞ ψ 1 (z1 , x2 )z1 dz1 ψ 1 (x) + cψ 2 (x), x1

∞ ψ2 (x1 , z2 )z2 dz2 cψ 1 (x) + ψ 2 (x). x2

Similarly, if the inequalities in the assumptions are both reversed, then so are the inequalities in the conclusions. 1 Proof. ∞ 1 This follows immediately from2 Corollary 6.1 (or Theorem 4.1), the fact that F (x1i, x2 ) = x1 ψ (z1 , x2 )z1 dz1 (similarly for F ), the non-negativity of ρ, and the definitions of ψ . 2

References [1] H. Airault, Projection of the infinitesimal generator of a diffusion, J. Funct. Anal. 85 (2) (1989) 353–391. [2] H. Airault, P. Malliavin, Intégration géométrique sur l’espace de Wiener, Bull. Sci. Math. 112 (1988) 3–52. [3] J.-C. Breton, I. Nourdin, G. Peccati, Exact confidence intervals for the Hurst parameter of a fractional Brownian motion, Electron. J. Stat. 3 (2009) 416–425. [4] H. Federer, Geometric Measure Theory, Springer-Verlag, 1996. [5] M. Fukushima, Basic properties of Brownian motion and a capacity on the Wiener space, J. Math. Soc. Japan 36 (1984) 161–176. [6] K. Ito, On Malliavin tensor fields, Comm. Pure Appl. Math. 47 (1994) 377–403. [7] P. Malliavin, Stochastic Analysis, Springer-Verlag, 1997. [8] I. Nourdin, G. Peccati, Stein’s method on Wiener chaos, Probab. Theory Related Fields 145 (1) (2008) 75–118. [9] I. Nourdin, G. Peccati, Stein’s method and exact Berry–Esséen asymptotics for functionals of Gaussian fields, Ann. Probab., in press. [10] I. Nourdin, G. Peccati, Non-central convergence of multiple integrals, Ann. Probab., in press. [11] I. Nourdin, G. Peccati, G. Reinert, Second order Poincaré inequalities and CLTs on Wiener space, J. Funct. Anal. 257 (2009) 593–609. [12] I. Nourdin, F. Viens, Density formula and concentration inequalities with Malliavin calculus, submitted for publication.


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[13] D. Nualart, The Malliavin Calculus and Related Topics, 2nd ed., Springer-Verlag, 2006. [14] D. Nualart, Ll. Quer-Sardayons, Gaussian density estimates for solutions to quasi-linear stochastic partial differential equations, preprint, 2009, http://arxiv.org/abs/0902.1849. [15] Ch. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, in: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. II: Probability Theory, University of California Press, Berkeley, 1972, pp. 583–602. [16] H. Sugita, Positive generalized functions and potential theory over an abstract Wiener space, Osaka J. Math. 25 (1988) 665–696. [17] F. Viens, Stein’s lemma, Malliavin calculus, and tail bounds, with application to polymer fluctuation exponent, Stochastic Process. Appl., published online http://dx.doi.org/10.1016/j.spa.2009.07.002. [18] S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels., Ann. Probab. 15 (1) (1987) 1–39.


Hypercontractivity for log-subharmonic functions Piotr Graczyk a , Todd Kemp b,c,∗,1 , Jean-Jacques Loeb a a Université d’Angers, 2 boulevard Lavoisier, 49045 Angers cedex 01, France b 2-175, MIT 77 Massachusetts Avenue, Cambridge, MA 02139, United States c UCSD, La Jolla, CA, United States

Received 11 January 2008; accepted 28 August 2009 Available online 25 November 2009 Communicated by L. Gross

Abstract We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on Rn and different classes of measures: Gaussian measures on Rn , symmetric Bernoulli and symmetric uniform probability measures on R, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for any symmetric measure on R. A log-Sobolev inequality (LSI) is deduced from the (SHC) for compactly supported measures on Rn , still for log-subharmonic functions. An analogous (LSI) is proved for Gaussian measures on Rn and for other measures for which we know the (SHC) holds. Our log-Sobolev inequality holds in the log-subharmonic category with a constant smaller than the one for Gaussian measure in the classical context. © 2009 Elsevier Inc. All rights reserved. Keywords: Hypercontractivity; Subharmonic; Log-Sobolev inequality

1. Introduction In this paper, we prove some important inequalities – strong hypercontractivity (SHC) and a logarithmic Sobolev inequality – for logarithmically subharmonic functions (cf. Definition 2.1 below). Our paper is inspired by work of Janson [14], in which he began the study of an important property of semigroups called strong hypercontractivity. A rich series of subsequent papers by Janson [15], Carlen [4], Zhao [19], and recently by Gross ([9,10] and a survey [11]) was devoted * Corresponding author at: UCSD, La Jolla, CA, United States.

E-mail addresses: [email protected] (P. Graczyk), [email protected] (T. Kemp), [email protected] (J.-J. Loeb). 1 This work was partially supported by NSF Grant DMS-0701162. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.014

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P. Graczyk et al. / Journal of Functional Analysis 258 (2010) 1785–1805

to this subject on the spaces Cn and, in papers by Gross, on complex manifolds. In contrast to all the aforementioned papers, our results concern the real spaces Rn . In the first part of the paper (Sections 3–4) we prove strong hypercontractivity in the logsubharmonic setting: for 0 < p q < ∞, Tt f Lq (μ) f Lp (μ)

for t

1 q log , 2 p

(SHC)

for the dilation semigroup Tt f (x) = f (e−t x), for any logarithmically subharmonic function f , for different classes of measures μ: including Gaussian measures and some compactly supported measures on R (symmetric Bernoulli and uniform probability measure on [−a, a] for a > 0). We also show that, in numerous important cases, the convolution of two measures satisfying (SHC) also satisfies (SHC). Let us note that in the theory of hypercontractivity for general measures, the semigroup considered is the one associated to the measure by the usual technology of Dirichlet forms. The generator of the semigroup (on a complete Riemannian manifold) takes the form L = − + X where is the Laplace–Beltrami operator and X is a vector field; hence, the semigroup restricted to harmonic functions on the manifold is simply the (backward) flow of X. For Gaussian measure, X = x · ∇, yielding the above flow Tt ; this vector field is often called the Euler operator, denoted E. In a sense, the point of this paper is to show that the strong hypercontractivity theorems about this flow extend beyond harmonic functions to the larger class of logarithmically subharmonic functions. The second part of the paper (Section 5) is devoted to logarithmic Sobolev inequalities (LSI) corresponding to the Strong Hypercontractivity property for log-subharmonic functions. We prove a general implication (SHC) ⇒ (LSI) for compactly supported measures on Rn for log-subharmonic functions. (It is important to note that, while the general technique of this implication – differentiating the inequalities in an appropriate fashion – are well-known, the technical details here involved with regularizing subharmonic functions are quite difficult.) We also show that an analogous log-Sobolev inequality in the log-subharmonic domain holds for Gaussian measures on Rn and for other measures which satisfy the strong hypercontractivity (SHC) considered in the first part. In both cases, the (LSI) we get is stronger than the classical one in the following sense. Let tN (p, q) =

q −1 1 log , 2 p−1

tJ (p, q) =

q 1 log 2 p

denote the Nelson and Janson times (cf. [14,17]), for 1 < p q < ∞ (in fact, tJ makes sense for all positive p q). The classical hypercontractivity for t ctN is equivalent, by Gross’s theorem in [8], to a logarithmic Sobolev inequality with the constant 2c: |f |2 log |f |2 dμ − f 22,μ log f 22,μ 2c f Lf dμ where L is the positive generator of the semigroup. We show that, in the category of logarithmically subharmonic functions, strong hypercontractivity for t ctJ implies (LSI) with constant c: (LSI) |f |2 log |f |2 dμ − f 22,μ log f 22,μ c f Ef dμ


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where E is the Euler operator discussed above. Hence, one cannot obtain this stronger LSI by simply restricting the usual Gaussian LSI to log-subharmonic functions. We call the inequality (LSI) a “strong LSI” both because it corresponds to the strong hypercontractivity and as the constant in the energy integral is smaller than in the classical case (of the Gaussian LSI of [8]). (LSI) could also be appropriately called an Euler type logarithmic Sobolev inequality. We emphasize the fact that the strong (LSI) and the implication (SHC) ⇒ strong (LSI) were never observed before in holomorphic case, in the afore-mentioned papers on strong hypercontractivity. In [9], only the implication classical (LSI) ⇒ (SHC) is proved. The authors of [12] observe and extensively discuss the difficulty in approximating of subharmonic functions. Let us note that the implication (SHC) ⇒ (LSI) in the log-subharmonic case does not follow as easily as in the classical setting. Indeed, if f is log-subharmonic, the functions f |[−N,N ] and f 1|f | 0. (1) The product f g is LSH, as is g p . (2) The sum f + g is LSH. (3) f is subharmonic. Proof. Property (1) is evident. In order to prove (3) (note that non-negativity is built into the definition of LSH functions), we use the fact that if a function ϕ : R → R is increasing and convex and h is a subharmonic function then ϕ(h) is also subharmonic. We apply this fact with ϕ(x) = ex and h = log f when f is LSH. To prove (2), we need the following lemma. Lemma 2.3. Let ϕ : R2 → R be a convex function of two variables, increasing in each variable. If F and G are subharmonic functions then ϕ(F, G) is also subharmonic. Proof. We apply the Jensen inequality in dimension 2 ϕ F (x), G(x) ϕ

– F (x + αy) dα, – G(x + αy) dα ϕ(F, G)(x + αy) dα. O(n)

2

O(n)

It is easy to verify that the function ϕ(x, y) = log(ex + ey ) satisfies the hypotheses of the lemma: to check its convexity, we write log(ex + ey ) = x + log(1 + ex−y ), yielding the result since the function t → ln(1 + et ) is convex. Hence, if f and g are LSH, then f = eF and g = eG for subharmonic functions F, G, and so the lemma yields that ϕ(F, G) = log(f + g) is subharmonic. This ends the proof of the proposition. 2 The next lemma and corollary are based on Proposition 2.2. They are useful in much of the following. Lemma 2.4. Let Ω be a separable metric space, and let μ be a Borel probability measure on Ω. Suppose f : Ω × Rn → R satisfies (1) The function x → f (ω, x) is LSH and continuous for μ-almost every ω ∈ Ω. (2) The function ω → f (ω, x) is bounded and continuous for each x ∈ Rn . (3) For small r > 0, there is a constant Cr > 0 so that, for all ω ∈ Ω and all x ∈ Rn , |f (ω, t)| Cr for t ∈ B(x, r). Then the function f˜(x) =

Ω

f (ω, x) μ(dω) is LSH.


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Proof. By Varadarajan’s theorem (see Theorem 11.4.1 in [5]), there is a sequence of points ωj ∈ Ω such that the probability measures 1 δω j n n

μn =

j =1

converge weakly to μ : μn μ. Note that fñ (x) =

1 f (ωj , x), n n

f (ω, x) μn (dω) =

j =1

Ω

and by Proposition 2.2 part (2), fñ is LSH for each n. Moreover, since f (·, x) ∈ Cb (Ω), weak convergence guarantees that fñ (x) → f˜(x) for each x. Fix > 0; then since fñ and f˜ are nonnegative, fñ + and f˜ + are strictly positive and thus log(fñ (x) + ) → log(f˜(x) + ) for each x. Again using Proposition 2.2, fñ + is LSH and so log(fñ + ) is subharmonic. Let r > 0 be small, and consider – log f˜(t) + dt = – lim log fñ (t) + dt. n→∞

∂B(x,r)

∂B(x,r)

By assumption, |f (ω, t)| Cr for each ω ∈ Ω and t ∈ ∂B(x, r); hence, |fñ (t)| Cr as well. This means there is a uniform bound on log(fñ + ) on ∂B(x, r). We may therefore apply the dominated convergence theorem to find that – log f˜(t) + dt = lim – log fñ (t) + dt n→∞

∂B(x,r)

∂B(x,r)

lim log fñ (x) + = log f˜(x) + , n→∞

where the inequality follows from the fact that log(fñ + ) is subharmonic. Hence, f˜ + is LSH for each > 0. Finally, since f (ω, x) is continuous in x for almost every ω, the boundedness of f in ω shows that f˜ is continuous. Thus the set where f˜ > −∞ is open. Therefore log(f˜(x) + ) is uniformly-bounded in on small enough balls around x, and a simple argument like the one above shows that the limit as ↓ 0 can be performed to show that f˜ is LSH as required. 2 Remark 2.5. It is possible to dispense with the requirement that f (ω, x) is continuous in x by using Fatou’s lemma instead of the dominated convergence theorem; however, the continuity of f (ω, x) in ω is still required for this argument. In all the applications we have planned for Corollary 2.4, f (ω, x) is such that continuity in one variable implies continuity in the other, and so we need not work harder to eliminate this hypothesis. Remark 2.6. In Lemma 2.4, if LSH is replaced with the weaker condition lower-bounded subharmonic (in the premise and conclusion of the statement), then the result follows from Definition 2.1 with a simple application of Fubini’s theorem; moreover, the only assumption needed is that f (·, x) ∈ L1 (Ω, μ) for each x.

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Corollary 2.7. Suppose f : Rn → R is lower-bounded and subharmonic. Then the function f˜(x) =

f (αx) dα

O(n)

is subharmonic. Moreover, if f is also LSH and continuous, then so is f˜. In either case, f˜ depends only on the radial direction: there is a function g : [0, ∞) → [−∞, ∞) with f˜(x) = g(|x|), and g is non-decreasing on [0, ∞). Proof. Suppose f is LSH and continuous. The reader may readily verify that the function (α, x) → f (αx) satisfies all the conditions of Lemma 2.4. (The weaker statement for lowerbounded subharmonic f , not necessarily continuous, follows similarly via Remark 2.6.) Clearly averaging f over rotations makes f˜ radially symmetric. Any radially symmetric subharmonic function is radially non-decreasing, by the maximum principle. 2 3. Hypercontractivity inequalities for the Gaussian measure Let m be a probability measure on Rn . For p 1, we denote the norm on Lp (m) by p,m . p We will denote by LLSH (m) the cone of log-subharmonic functions in Lp (m). Let γ be the standard Gaussian measure on Rn , i.e. γ (dx) = cn exp(−|x|2 /2) dx, where dx is Lebesgue measure and cn = (2π)−n/2 . Given a function f on Rn , and r ∈ [0, 1], we denote by fr the function x → f (rx). The family of operators Sr f = fr , r ∈ [0, 1] is a multiplicative semigroup, whose additive form Tt f (x) = f (e−t x) is considered in connection with holomorphic function spaces in [4,9,14,19] and others (including the second author’s paper [16] in the non-commutative holomorphic category). When f is differentiable, the infinitesimal generator E of (Tt )t0 equals −Ef where E is the Euler operator Ef (x) = x · ∇f. If L is the Ornstein–Uhlenbeck operator L = − + E acting in L2 (Cn , γ ) and f is a holomorphic function then Lf = Ef , so (Tt )t0 and, equivalently, (Sr )r∈[0,1] act on holomorphic functions as the Ornstein–Uhlenbeck semigroup e−tL (cf. [2, p. 22–23]). Before showing the strong hypercontractivity of the semigroup Sr for the Gaussian measure and LSH functions, let us show that the operators Sr are Lp -contractions on non-negative subharmonic functions, for any rotationally invariant probability measure. Proposition 3.1. Let m be a probability measure on Rn which is O(n)-invariant. Then for f 0 subharmonic, r ∈ [0, 1], and p 1, we have fr p,m f p,m . Moreover, this contraction property holds additionally in the regime 0 < p < 1 if f is LSH.


1791

Proof. First consider the case p 1, and assume only that f 0 is subharmonic. Note that, since f 0 and since m is O(n)-invariant, p p fr p,m = f (rx) dm(x) = f (rx)p dm(αx) dα. Rn

O(n) Rn

Changing variables using the linear transformation α in the inside integral and using Fubini’s theorem, we have (replacing α −1 with α in the end) f (rαx)p dα dm(x) = Sr h(x) dm(x), Rn O(n)

Rn

where h(x) = O(n) f (αx)p dα; i.e., with k = f p , h = k˜ in the notation of Corollary 2.7. Since p 1, k is subharmonic, and so by Corollary 2.7 h is also subharmonic and radially increasing. In particular, there is some non-decreasing g : [0, ∞) → R such that h(x) = g(|x|). So Sr h(x) = p g(r|x|) g(|x|) = h(x) for r ∈ [0, 1]. Integrating over Rn we have fr p,m h(x) dx which p equals f p,m by reversing the above argument. This proves the result. If 0 < p < 1, the above argument follows through as well since, if f ∈ LSH then k = f p is LSH by Proposition 2.2. In particular, k is non-negative and subharmonic, and so by Corol˜ The rest of the proof follows verbatim. 2 lary 2.7, so is k. We now show the strong hypercontractivity inequality for Gaussian measure and LSH functions. That is: Tt f q,γ f p,γ whenever f is LSH and t tJ (p, q). This is a generalization (from holomorphic functions to the much larger class of logarithmically-subharmonic functions) of Janson’s original strong hypercontractivity theorem in [14]. Because our test functions f are non-negative and the action of Tt commutes with taking powers of f , this can be reduced to the following simplified form. Theorem 3.2. Let f be a log-subharmonic function. Then for every r ∈ [0, 1], one has fr 1/r 2 ,γ f 1,γ .

(3.1)

Remark 3.3. The inequality (3.1) means that the operators Sr act as contractions between the spaces 1/r 2

Sr : L1LSH (γ ) → LLSH (γ ), or, equivalently, the operator Tt is a contraction between the cones 2t

Tt : L1LSH (γ ) → LeLSH (γ ). In fact, by Proposition 2.2, one gets other hypercontractivity properties. Applying the theorem to the function f p , it follows that the operators Sr are contractions p

p/r 2

Sr : LLSH (γ ) → LLSH (γ ),

1792


and the operators Tt are contractions e2t p

p

Tt : LLSH (γ ) → LLSH (γ ) for any p > 0. Since Tt is an Lq contraction for any q (Proposition 3.1), by the semigroup property the above implies that Tt is a contraction from Lp to Lq for any q e2t p. In other words, Tt is a contraction from Lp to Lq provided that t 12 log(q/p), the Janson time tJ (p, q). This is the strong hypercontractivity theorem proved in [14] for holomorphic functions on Cn ∼ = R2n ; n here we prove it for LSH functions on R . Proof. The case where f = log |g| with g holomorphic on Cn is implicitly proved in [14] but is not given in this form. Using the ideas of Janson, we will prove the general theorem. Nelson’s classical hypercontractivity result plays a crucial role here as in Janson’s paper. Let Pt = e−tN be the Ornstein–Uhlenbeck semigroup. Let us write it in the form Pt f (x) =

(3.2)

Mr (x, y)f (y) γ (dy)

where r = e−t and Mr is the Mehler kernel −n/2 r2 2r 1 + r2 2 2 . exp − |x| + x, y

− |y| Mr (x, y) = 1 − r 2 1 − r2 1 − r2 1 − r2 We can rewrite Eq. (3.2) in terms of Lebesgue measure as Pt f (x) = the modified kernel Kr is given by

(3.3)

Kr (x, y)f (y) dy where

−n/2 |y − rx|2 . Kr (x, y) = 1 − r 2 exp − 1 − r2 Evidently Kr (x, y) is constant in y on spheres around rx. This implies that if f 0 is subharmonic, then for all t > 0 we have Pt f (x) f (e−t x) (indeed, this is at the core of Janson’s proof in [14]). The classical hypercontractivity inequality of Nelson (cf. [17]) is given by: Pt f q(t),γ f p,γ where q(t) = (p − 1)e2t + 1 and p > 1. Hence, for f 0 subharmonic, we have Nelson’s theorem for the dilation semigroup: −t f e x

q(t),γ

f p,γ .

(3.4)

Now take f to be LSH. The function f 1/p is also LSH, so it is positive and subharmonic. Eq. (3.4) applied to f 1/p becomes

1/q(t) 1/p f (x) dγ (x) . fe−t (x)q(t)/p dγ (x)


1793

This implies that fe−t

q(t)/p,γ

f 1,γ .

1 2t −t Observe that limp→∞ q(t) p = e = r 2 where r = e . Applying Fatou’s lemma, we obtain fr r −2 ,γ f 1,γ , the desired result. 2 q−1 is the smallest In the full hypercontractivity theory due to Nelson [17], tN (p, q) = 12 log p−1 time to contraction, for all Lp -functions. The analogous statement holds for Theorem 3.2; the exponent 1/r 2 is optimal in this inequality (with Gaussian measure) over all LSH functions. In fact, it is optimal when restricted just to holomorphic functions on Cn , as is proved (in an analogous non-commutative setting) in [16]; here we present a slightly different proof.

Proposition 3.4. Let r ∈ (0, 1] and C > 0. Assume that for some p > 0, the following inequality holds for every LSH function f : fr p,γ Cf 1,γ .

(3.5)

Then p 1/r 2 and C 1. Remark 3.5. If m is a probability measure then the Lp norm f p,m is a non-decreasing function of p. It follows that if Eq. (3.5) holds for a p > 1 then it also holds for every q ∈ [1, p). Proof. Consider the set of functions f a (x) = eax1 , which are all LSH for a > 0. An easy computation shows that (f a )r p,γ = exp(r 2 a 2 p/2); in particular, (f a )1,γ = exp(a 2 /2). The supposed inequality (3.5) then implies that exp(r 2 a 2 p/2) C exp(a 2 /2) for all a > 0. Set s = r 2 p. Then exp(a 2 (s − 1)/2) C for every real a. Letting a → 0 shows that C 1; letting a → ∞ shows that s 1. 2 Remark 3.6. Hypercontractive inequalities very typically involve actual contractions (i.e. constant C = 1 in Proposition 3.4), since the time constant (tN or tJ in this case) are usually independent of dimension, yielding an infinite-dimensional version of the inequality. Indeed, in Nelson’s original work [17], one main technique was to show that hypercontractivity held in all dimensions up to a fixed (dimension-independent) constant C > 1. The infinite-dimensional version then implies that C = 1 is the best inequality, for if the best constant is > 1 or < 1, a tensor argument shows that in infinite dimensions the constant is ∞ or 0, respectively. We saw that the exponent 1/r 2 is maximal in the (SHC) inequality for Gaussian measures. Below we show that it cannot be bigger for any probability measure with an exponential moment. In the following, |x| refers to the Euclidean norm on Rn . Proposition 3.7. Let μ be a probability measure with a finite exponential moment (i.e. ec|x| is μ-integrable for some c > 0) and such that for a linear form h on Rn

h(x) dμ(x) = 0 and

h(x)2 dμ(x) = 0.

(3.6)

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Fix r ∈ (0, 1). Assume that there exists q(r) > 0 such that fr q(r),μ f 1,μ

(3.7)

for every LSH function f . Then q(r) r −2 . Remark 3.8. Observe that an O(n)-invariant probability measure with an exponential moment and not equal to δ0 satisfies the condition (3.6). Proof. One can assume that the μ-integral of h2 is 1. Take the LSH function f (x) = eh(x) where > 0. The inequality (3.7) implies that

e

rq(r)h(x)

dμ(x)

e

h(x)

q(r) dμ(x) .

(3.8)

Note that the last integral is finite for small enough, because μ has an exponential moment. Put a = rq(r). We use the Taylor expansion ex = 1 + x + x 2 /2 + g(x) where g satisfies: |g(x)| (|x|3 /6)e|x| . We get eah(x) dμ(x) = 1 +

a22 + 2

g(ax) dμ(x).

where the last term is o( 2 ). Similarly, we see that the right-hand side term of (3.8) can be written as 1 + q(r) 2 /2 + o( 2 ). It follows that a 2 q(r), which means that q(r) r −2 . 2 4. Hypercontractivity inequalities for probability measures In this section we study hypercontractivity properties of LSH functions with respect to any probability measure m. We have already seen in Proposition 3.1 that, for rotationally invariant measures m, the semigroup Sr is always an Lp contraction. Theorem 4.1. Fix q > 1 and r ∈ (0, 1]. Suppose that μ1 and μ2 are two probability measures on Rn which verify the hypercontractivity inequality fr q,μ f 1,μ

(4.1)

for any continuous LSH function f . It at least one of μ1 and μ2 is compactly-supported, then the convolved measure μ1 ∗ μ2 also satisfies (4.1). Proof. Let f be a continuous LSH function, and suppose μ1 is compactly-supported. We have

f (rz)q d(μ1 ∗ μ2 )(z) =

f (rx + ry)q dμ1 (x) dμ2 (y)

q f (x + ry) dμ1 (x) dμ2 (y)


1795

since the function x → f (x + ry) is continuous LSH for each fixed y ∈ Rn , and μ1 satisfies (4.1). Let h(y) = f (x + y) dμ1 (x), so that we have proven that

q

fr q,μ1 ∗μ2

q

h(ry)q dμ2 (y) = hr 1,μ2 .

(4.2)

Since f is continuous, the function (x, y) → f (x + y) is continuous in both variables, and also LSH in each. Since supp μ1 is compact and f is continuous, all the conditions of Corollary 2.4 are satisfied, and so h is LSH. Thence, by the assumption of the theorem, the quantity on the q right-hand side of Eq. (4.2) is bounded above by h1,μ2 . By definition, h1,μ2 =

h(y) dμ2 (y) =

f (x + y) dμ1 (x) dμ2 (y) = f 1,μ1 ∗μ2 ,

and this proves that inequality (4.1) also holds for μ1 ∗ μ2 .

2

The Theorem 4.1 suggests the following Conjecture. The convolution property of Theorem 4.1 holds without any assumptions on the measures μ1 , μ2 . It does not however seem easy to prove. This is due to the difficulty of proving that f ∗ μ is upper semi-continuous when f is LSH, without any supplementary conditions on f or μ. In the sequel we will only use Theorem 4.1 as stated, with μ1 equal to a symmetric Bernoulli measure. Most of the following results of this section concern the 1-dimensional case, i.e. log-convex functions on the real line. In that case, one has the following surprisingly general hypercontractivity inequality. Proposition 4.2. For every symmetric probability measure m on R, and for any logarithmically convex function f on R, the following inequality is true for any r ∈ (0, 1]: fr 1/r,m f 1,m . Remark 4.3. Translating this statement into additive language, the dilation semigroup Tt satisfies strong hypercontractivity with time to contraction at most 2 · tJ , for any symmetric probability measure on R, for log-convex functions. As explained above, a simple scaling f → f p yields the comparable result from Lp → Lq for q p > 0. Proof. By the log-convexity of f , for any x ∈ R f (rx) f (0)1−r f (x)r , which implies that f (rx)1/r f (0)1/r−1 f (x). Then by m-integration, f (rx)1/r dm(x) f (0)1/r−1 f 1,m .

1796


Since f is convex, f (0) 12 [f (x) + f (−x)] for all x. Integrating and using the symmetry of m yields f (0) f 1,m . Consequently,

1/r

f (rx)1/r dm(x) f 1,m , and the proposition follows.

2

Remark 4.4. Proposition 4.2 remains true for rotationally invariant measures m and log-convex functions f on Rn . This proof fails, however, for general LSH functions on Rn when n 2. Remark 4.5. Subject to additional regularity on m, the symmetry condition in Proposition 4.2 can be replaced with the much weaker assumption that m is centred: i.e. m has a finite first moment, and x m(dx) = 0. In short, fix a log-convex f , and suppose that m is regular enough that the function η(r) = f (rx) m(dx) is differentiable, so that η (r) = f (rx)x m(dx). (It is easy to see, from convexity of f , that fr ∈ L1 (m) for each r, provided f ∈ L1 (m).) Then η (0) = f (0) x m(dx) = 0, and since f is convex, f is increasing which means that xf (rx) xf (x) for all x, r 0, so η (r) η (0) = 0. Thus, f dm = η(1) η(0) = f (0), and the rest of the above proof follows. For this to work, it is necessary to assume (at minimum) that the functions ∂ 1 ∂r f (rx) = f (rx)x are uniformly bounded in L (m); a convenient way to achieve this is to 1 assume that functions g ∈ L (m) for which x → xg (x) are also in L1 (m) are dense in L1 (m). The kinds of measures for which such a Sobolev-space density is known is a main topic of our subsequent paper [7]. The problem in general is to find, for a fixed measure m, the maximal exponent q such that fr q,m f 1,m for every r ∈ (0, 1] and any log-convex function f on R. For symmetric Bernoulli measures we will show that the optimal exponent q is the same as for Gaussian measures. Proposition 4.6. If m = 12 (δ1 + δ−1 ) then fr 1/r 2 ,m f 1,m

(4.3)

for every r ∈ (0, 1] and any log-convex function f . Remark 4.7. It follows from Proposition 4.6, and a simple rescaling argument, that the same strong hypercontractivity inequality holds for any symmetric Bernoulli measure 12 (δa + δ−a ), a > 0. The optimality of the index 1/r 2 in the inequality (4.3) follows from Proposition 3.7. Proof. Step 1. We justify that it is sufficient to prove the proposition for the two-parameter family of functions h(x) = C exp(ax) with a ∈ R and C > 0. Take f strictly positive. Then there exists h of the form C exp(ax) such that the functions f and h are equal on the set {−1, +1}. Assume now that f is log-convex. Then f h on [−1, 1], and in particular f (r) h(r) and f (−r) h(−r). This implies that

2

f (rx)1/r dm(x)

2

h(rx)1/r dm(x).


1797

If the function h satisfies (4.3), we obtain fr q,m hr q,m h1,m = f 1,m , the last equality following from the fact that f and h coincide on the support of m. This gives the inequality (4.3) for f . Step 2. We show the inequality (4.3) for f (x) = eax (the constant C obviously factors out of the desired inequality). This is essentially a calculus exercise. One has to prove that

r 2 exp(ax) dm(x), exp(ax/r) dm(x)

2

i.e. (cosh( ar ))r cosh a for a real and r ∈ (0, 1]. Put s = 1/r. Then s 1 and the required 2 inequality becomes cosh(sa) (cosh a)s . Taking logarithms and next dividing by s 2 a 2 , we are left to prove that log(cosh(sa)) log(cosh a) . s2a2 a2 In other words, we must prove that the function log(cosh x)/x 2 is decreasing for x 0. Taking the derivative, it is sufficient to see that ρ(x) = x tanh x − 2 log(cosh x) is nonpositive for x 0. Well, ρ(0) = 0, and ρ (x) = x/ cosh2 x − tanh x = x−sinh x2cosh x . This last quotient is noncosh x positive for its numerator is equal to x − (sinh 2x)/2. 2 Remark 4.8. Proposition 4.6 could be obtained from an inequality of A. Bonami [3] similarly to the manner in which Theorem 3.2 was obtained from Nelson’s hypercontractivity theorem for Gaussian measures. She proved that for symmetric Bernoulli measures the same classical hypercontractivity inequalities as for the Gaussian measure hold. In order to prove Proposition 4.3 for a log-convex function f , one compares it to the affine function which takes the same value as f on {−1, 1}. For a function on {−1, 1}, there is a unique affine function on the line which extends it. Thus one can identify the space C{−1, 1} of functions on {−1, 1} and the space of affine functions on the line. We omit the details. Corollary 4.9. The symmetric uniform probability measure λa on [−a, a], a > 0, satisfies the strong hypercontractivity property fr 1/r 2 ,λa f 1,λa for all LSH functions. Proof. Let mx = 12 (δx + δ−x ). It is easy to see that μk := m 1 ∗ m 1 ∗ · · · ∗ m 1 λ1 , 2

4

2k

k → ∞,

where we denote by the convergence in law. By the Proposition 4.6 (and the proceeding Remark 4.7) and Theorem 4.1, the inequality (4.3) holds for the measures μk . The supports of the measures μk and λ1 are compact and included in the segment [−1, 1]. If f is log-convex on R, it 1 1 is continuous and the convergence −1 f dμk → −1 f dλ1 follows from the convergence in law μk λ1 . The statement for all a > 0 now follows from a simple rescaling argument. 2

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Definition 4.1. For convenience, we introduce the notation (SHCc) for the strong hypercontractivity coefficient of a probability measure on Rn : it is the supremum of the positive real numbers c such that for every LSH function f , one has the strong hypercontractivity inequality: fr r −c ,μ f 1,μ ,

r ∈ (0, 1].

We have seen in Proposition 4.2 that for radial measures in dimension 1 the SHCc is at least 1. The following proposition shows that this is not true in higher dimensions. Proposition 4.10. In dimension bigger than one, the SHCc for radial measures can be 0. Proof. For clarity of notation here and in the following, let N (x) = |x| denote the Euclidean norm on Rn . Computing directly, one checks that for n > 1 this function is LSH (the Laplacian of ln N(x) for x nonzero and observe also that ln N (0) = −∞). Then take a probability measure μ with a density s(x) = 0 for N (x) 1 and of the form: DN (x)−(n+2) dx for N (x) > 1, for some constant D > 0. The function N (x) is LSH and integrable for this measure. But it is clear −c that for every positive value of c, the function N (rx)r is not μ-integrable for r near 0. 2 At the end of this Section we study the SHC properties for the probability measures mp (dx) = cp exp −N (x)p dx,

p > 0.

By Theorem 3.2 and Proposition 3.4 we already know that for p = 2, the SHCc is 2 in any Rn . Proposition 3.7 implies that for any p > 0, the SHCc is not greater than 2. Proposition 4.11. (a) In any dimension and for p 1 the SHCc of the probability measure mp is at most p. (b) For p = 1 and in dimension one, the SHCc is 1. Proof. (a) Take a function of the form f (x) := exp(AN (x)p ) dx with 0 < A < 1. As N (x) is convex and positive, the function N (x)p is also convex for p 1 and then also SH. This implies that exp(AN (x)p ) is LSH. Moreover, it is mp integrable. Fix r between 0 and 1. The integrability of the function f (rx)q(r) implies that q(r) r −p , which implies that the SHCc is at most p. (b) For the case p = 1, n = 1, one uses part (a) and the fact that in dimension one, the SHCc is at least one. 2 Open question. Is the SHCc= p for mp when 1 < p < 2 or when p = 1 and the dimension is bigger than 1? 5. Logarithmic Sobolev inequalities for LSH functions Recall that the classical Gaussian logarithmic Sobolev inequality, cf. [2,8], is Ent f 2 =

|f |2 log |f |2 dγ − f 22,γ log f 22,γ 2

f Lf dγ = 2EL (f )

(5.1)


1799

where γ is the standard Gaussian measure, L = − + E is the generator of the Ornstein– Uhlenbeck semigroup and f ∈ A, a standard algebra contained in the domain of the operator L. For the Ornstein–Uhlenbeck semigroup A can be chosen as the space of C ∞ functions with slowly increasing derivatives. The expression Ent(f ) is called the entropy of f and EL (f ) is the Dirichlet form or energy of f , with respect to the generator L of the Ornstein–Uhlenbeck semigroup, cf. [2]. The celebrated theorem of Gross [8] establishes the equivalence between the hypercontractivity property of a semigroup Tt with invariant measure μ and the log-Sobolev inequality relative q−1 , the hyperconto the generator L of Tt . More precisely, recalling the Nelson time tN = 12 ln p−1 tractivity inequalities Tt f q,μ f p,μ for t ctN (p, q) for 1 < p q < ∞ are, together, equivalent to the single log-Sobolev inequality 2 2 2 2 2 Ent f = |f | log |f | dμ − f 2,μ log f 2,μ 2c f Lf dμ = 2cEL (f ). (5.2) In the Gaussian case these inequalities indeed hold with c = 1. In this section we will prove that a strong log-Sobolev inequality 2 2 2 2 2 Ent f = |f | log |f | dμ − f 2,μ log f 2,μ c f Ef dμ = cEE (f )

(5.3)

holds for log-subharmonic functions f and compactly supported measures μ for which a (SHC) property holds. As the Dirichlet form (or energy) on the right-hand side of (5.3) are taken with respect to the generator E of the considered dilation semigroup Tt f (x) = f (e−t x), the inequality (5.3) may also be called an Euler type LSI. Observe that all the above-mentioned inequalities have L1 versions. If in (5.1) we consider √ f > 0 and we substitute f = g, then using the formulas f Lf dγ = (∇f )2 dγ and ∇f = ∇g √ 2 g we get 1 Ent(g) 2

(∇g)2 dγ . g

(5.4)

Let f be LSH, and set g = f 2 in (5.3). Using the fact that Eg = 2f Ef we can write the inequality (5.3) as c Ent(g) = g log g dμ − g1,μ log g1,μ Eg dμ. (5.5) 2 It may seem surprising that the integrals f Ef dμ from (5.3) and, equivalently, Eg dμ from (5.5) are positive when f and g are LSH functions. The following proposition explains this phenomenon, which holds more generally for subharmonic functions. Proposition 5.1. Let m be a probability measure on Rn which is O(n) invariant, and let g ∈ C 1 be a subharmonic function. Then I = Eg(x) dm(x) 0.

1800


Proof. We have

I=

Eg(αx) dα,

dm(x) O(n)

where dα denotes the Haar measure on O(n). Denote by σ the normalized Lebesgue measure on the unit sphere S n−1 . If r = |x|, we have ∂g (ru) σ (du) Eg(αx) dα = (Eg)(ru) σ (du) = r ∂r S n−1

O(n)

=r

∂ ∂r

S n−1

g(ru) σ (du) 0 S n−1

because the function r →

S n−1

g(ru) σ (du) is increasing (cf. Corollary 2.7).

2

5.1. Log-Sobolev inequalities for measures with compact support The following techniques work, in principle, quite generally. However, the usual approximation techniques to guarantee integrability (convolution approximations and cut-offs) are unavailable in the category of subharmonic functions. As such, we include this section which develops the relevant log-Sobolev inequalities in all dimensions, but only for compactly-supported measures (i.e. do the cut-off in the measure rather than the test functions). Extension of these results to a much larger class of measures is the topic of [7]. Theorem 5.2. Let μ be a probability measure on Rn with compact support. Suppose that for some c > 0, the following strong hypercontractivity property holds: for 0 < p q < ∞ and p f ∈ LLSH (μ), fe−t q,μ f p,μ

for t c ·

q 1 log . 2 p

Then for any log-subharmonic function f ∈ C 1 the following logarithmic Sobolev inequality holds: 2 2 2 2 (5.6) f log f dμ − f 2,μ log f 2,μ c f Ef dμ. Remark 5.3. (1) The condition f ∈ C 1 is natural to ensure a good sense of the expression Ef in (5.6). In the classical case in [2] one supposes f ∈ A ⊂ C ∞ and such an LSI inequality is equivalent to the hypercontractivity property [2, Theorem 2.8.2]. (2) In the case of strong hypercontractivity with optimal q = p/r 2 (symmetric Bernoulli measures and their convolutions, symmetric uniform measures on [−a, a]), the constant c is equal to 1. Also Gaussian measures on Rn have the constant c = 1 but evidently they are not covered by the Theorem 5.2. When q = p/r (any symmetric measure on R), the constant c is equal to 2. The time tJ = 12 log pq appearing in Theorem 5.2 is Janson’s time.


1801

(3) Theorem 5.2 is stated and proved here for compactly-supported measures, a class not including the most important Gaussian measures. In the end of this section we will show that the same strong log-Sobolev inequality of Euler type holds for Gaussian measures in all dimensions. Let us reiterate that the following proof applies to a much wider class of measures, but the precise regularity conditions are complicated by the fact that cut-off approximations do not preserve the cone of log-subharmonic functions. This will be covered in [7]. Proof. Let p = 2 and t be the critical time t = c · 12 log pq . Then the variable r = e−t satisfies q(r) = 2r −2/c . The method of proof is classical and consists of differentiating the function α(r) = fr q(r),μ at r = 1. By strong hypercontractivity, α(r) α(1), so α (1) 0 if we prove the existence of this derivative. q(r) = f (rx)q(r) dμ(x) and let β (r) = f (rx)q(r) , so that β(r) = x Define β(r) = α(r) βx (r) dμ(x). Then ∂ q(r) log βx (r) = q (r) log f (rx) + x · ∇f (rx). ∂r f (rx) 2 Since q (r) = − rc q(r), we compute

βx (r) = −

2 q(r) fr (x)q(r) log fr (x)q(r) + fr (x)q(r)−1 (Ef )r (x). rc r

(5.7)

Let 0 < < 1. As f ∈ C 1 , the expression on the right-hand side of (5.7) is bounded for r ∈ (1 − , 1] and x ∈ supp(μ) (which is compact). The Dominated Convergence Theorem then implies that ∂ (5.8) βx (r) dμ(x) = βx (r) dμ(x). β (r) = ∂r Finally, since α(r) = β(r)1/q(r) and β > 0, we have that α is C 1 on (1 − , 1] and a simple calculation shows that

2 α(r) β(r) log β(r) + β (r) . α (r) = q(r)β(r) rc Now, taking r = 1, applying α (1) 0 and the formulas (5.7) and (5.8) we obtain 2 0 β(1) log β(1) + β (1) c 2 2 = f 22,μ log f 22,μ − f 2 log f 2 dμ + 2 f Ef dμ, c c and this is the logarithmic Sobolev inequality (5.6).

2

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P. Graczyk et al. / Journal of Functional Analysis 258 (2010) 1785–1805 p

For p > 0 we define spaces LE (μ) = {f ; f ∈ Lp (μ) and Ef ∈ Lp (μ)} and Lp (μ) log Lp (μ) = {f ; f p | log f p | dμ < ∞} (we think that the notation Lp log Lp is more appropriated than Lp log L that can sometimes be found in the literature). The former is a Sobolev space, the latter an Orlicz space, related to the logarithmic Sobolev inequality 5.6; indeed, in the case p = 2, they are the spaces for which the right- and left-hand sides (respectively) of that inequality are finite. Appealing to the surprising Proposition 4.2, and Theorem 5.2, we have the following. Corollary 5.4. Let μ be a symmetric probability measure on R. Then for any log-subharmonic function f ∈ L2 (μ) log L2 (μ) ∩ L2E (μ) ∩ C 1 the following logarithmic Sobolev inequality holds:

2

f log f

2

dμ − f 2L2 (μ) log f 2L2 (μ)

2

f Ef dμ.

Remark 5.5. In the classical case it is sufficient to suppose only f ∈ L2E (μ); this actually implies that f ∈ L2 (μ) log L2 (μ). The proof of this fact involves approximation by more regular (e.g. compactly supported or bounded) functions, and these tools are unavailable to us here. Proof. By Proposition 4.2 the measure μ as well as the measures μN = μ|[−N,N ] + μ([−N, N ]c )δ0 verify the strong hypercontractivity property for LSH functions with q = p/r and c = 2. Let f verify the hypothesis of the corollary, and set f = f + ; it is easy to check that f also verifies all the conditions of the corollary. By Theorem 5.2, for each N 2 2 2 2 f log f dμN − f 2,μ log f 2,μ 2 f Ef dμN . N

N

When N → ∞, μN μ (weak convergence), and since f ∈ C 1 and is strictly positive, all the functions (f )2 , (f )2 log(f )2 , and f Ef are continuous; hence the integrals in the last formula converge to analogous integrals in terms of f with respect to the measure μ. Finally, we can let ↓ 0 to achieve the result, by the Monotone Convergence Theorem. 2 Corollary 5.6. Let μ be a symmetric probability measure on R. Then for any log-subharmonic function f ∈ L1 (μ) log L1 (μ) ∩ L1E (μ) ∩ C 1 the following logarithmic Sobolev inequality holds:

f log f dμ − f 1,μ log f 1,μ

Ef dμ.

Proof. The proof is similar to the proof of the Corollary 5.4. Note, nevertheless, that Corollary 5.6 does not follow from Corollary 5.4 because the hypothesis Ef ∈ L1 (μ) is weaker than the condition Ef ∈ L2 (μ) supposed in Corollary 5.4 (all other integrability hypotheses are equivalent by the transformation f → f 2 which maps L2 onto L1 ). 2 5.2. Log-Sobolev inequality for Gaussian measures We formulate two versions of the strong logarithmic Sobolev inequality for log-subharmonic functions and Gaussian measures: in the classical context L2 (γ ) (Theorem 5.7) and in the more natural and technically simpler case L1 (γ ) (Theorem 5.8).


1803

Both cases are nearly equivalent since f ∈ L2 (γ ) and log-subharmonic is equivalent to f 2 ∈ L1 (γ ) and log-subharmonic. But the integration hypotheses of the theorems are slightly different, cf. the discussion in the proof of the Corollary 5.6. Theorem 5.7. Let γ be the Gaussian measure with density

2 √ 1 n e−|x| /2 (2π)

on Rn . Then for any

LSH and C 1 function f ∈ L2 (γ ) log L2 (γ ) ∩ L2E (γ ) the following logarithmic Sobolev inequality holds (5.9) f 2 log f 2 dγ − f 22,γ log f 22,γ f Ef dγ . Theorem 5.8. Let γ be as in Theorem 5.7. Then for any LSH and C 1 function g ∈ L1 (γ ) ∩ log L1 (γ ) ∩ L1E (γ ) the following logarithmic Sobolev inequality holds g log g dγ − g1,γ log g1,γ

1 2

Eg dγ .

(5.10)

Note that the method of the proof of Corollary 5.4 cannot be applied because we do not know if the measures γN have the strong hypercontractivity property with Gaussian constant c = 1; by the Theorem 4.2 they have it with c = 2 and we would obtain a weaker inequality with the constant 2 before the energy term EE (f ). Instead, we will use the classical LSI for Gaussian measures. Proof. Let us prove Theorem 5.8; the proof of Theorem 5.7 is similar. It is sufficient to consider the case g = exp(h) with h ∈ C 2 and h 0. It follows that g

(∇g)2 g

which combined with the L1 version of the classical LSI (5.4) gives 1 Ent(g) g dγ . 2 We also have

Eg dγ

g dγ .

Finally Ent(g) which is our strong LSI (5.10).

1 2

Eg dγ

2

Remark 5.9. For the log-subharmonic functions f (x) = eax , a > 0 there is equality in (5.9) and (5.10). Thus the constant c = 1 is optimal in (5.9) and the constant 12 is optimal in (5.10).

1804


Remark 5.10. Let m = 12 (δ1 + δ−1 ) and let μk denote the normalized convolution powers m∗k . By the Central Limit Theorem (CLT), the measures μk converge in law to γ . As Theorem 5.2 applies to the measures μk , one can prove the strong LSI’s for the measure γ on R using a strengthened version of the CLT, cf. [6]. Strong LSI’s are proven in [6] also for Gaussian measures γ on Rn , n 2, using approximation of γ by uniform spherical measures. This approach mirrors, to some extent, Gross’s proof of the Gaussian log-Sobolev inequality in [8]. A direct proof of SHC inequality for γ using Proposition 4.6 is also a corollary of results of [6]. Remark 5.11. In principle, the strong LSI for Gaussian measures or other non-compactly supported measures should follow from the strong hypercontractivity inequalities of Theorem 3.2 via an approach like that in the proof of Theorem 5.2. As we have mentioned, there are challenging regularization issues (due to the nature of logarithmically subharmonic functions) which complicate these techniques. Along the same lines, any measure for which the logarithmic Sobolev inequality holds for LSH functions should also satisfy strong hypercontractive estimates (this was proved in the restricted context of holomorphic functions in [9]). Thus an equivalence SHC

⇐⇒

strong LSI

is a natural conjecture. These issues will be dealt with in a future publication [7]. Other important open problems to be studied are: – proving SHC for semigroups with other generators L; – SHC inequalities for non-symmetric Bernoulli and uniform measures; – a general convolution property, weakening the strong assumptions of Theorem 4.1. Acknowledgments We thank A. Hulanicki for calling the attention of the first and third authors to hypercontractivity problems in the holomorphic category. Thanks also go to L. Gross for many helpful conversations and to an anonymous referee for numerous improvements of the paper. References [1] R. Adamczak, Logarithmic Sobolev inequalities and concentration of measure for convex functions and polynomial chaoses, Bull. Pol. Acad. Sci. Math. 53 (2) (2005) 221–238. [2] C. Ané, Sur les Inégalités de Sobolev Logarithmiques, Panor. Synthèses, vol. 10, Société mathématique de France, 2000. [3] A. Bonami, Etude des coefficients de Fourier de Lp (G), Ann. Inst. Fourier 20 (1971) 335–402. [4] E. Carlen, Some integral identities and inequalities for entire functions and their applications to the coherent state transform, J. Funct. Anal. 97 (1991) 231–249. [5] R. Dudley, Real Analysis and Probability, Cambridge Stud. Adv. Math., vol. 74, Cambridge University Press, Cambridge, 2002, revised reprint of the 1989 original. ˙ [6] P. Graczyk, J.J. Loeb, T. Zak, Strong central limit theorem for isotropic random walks in Rd , submitted for publication. [7] P. Graczyk, T. Kemp, J.J. Loeb, Strong logarithmic Sobolev inequalities for log-subharmonic functions, preprint. [8] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975) 1061–1083. [9] L. Gross, Hypercontractivity over complex manifolds, Acta Math. 182 (2) (2000) 159–206. [10] L. Gross, Strong hypercontractivity and relative subharmonicity, special issue dedicated to the memory of I.E. Segal, J. Funct. Anal. 190 (2002) 38–92.


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[11] L. Gross, Hypercontractivity, logarithmic Sobolev inequalities, and applications: A survey of surveys, in: Diffusion, Quantum Theory, and Radically Elementary Mathematics, in: Math. Notes, vol. 47, Princeton Univ. Press, Princeton, NJ, 2006, pp. 45–73. [12] L. Gross, M. Grothaus, Reverse hypercontractivity for subharmonic functions, Canad. J. Math. 57 (2005) 506–534. [13] L. Hormander, Complex Analysis in Several Variables, North-Holland/Elsevier, 1973. [14] S. Janson, On hypercontractivity for multipliers of orthogonal polynomials, Ark. Mat 21 (1983) 97–110. [15] S. Janson, On complex hypercontractivity, J. Funct. Anal. 151 (1997) 270–280. [16] T. Kemp, Hypercontractivity in non-commutative holomorphic spaces, Comm. Math. Phys. 259 (2005) 615–637. [17] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973) 211–227. [18] A. Sadullaev, R. Madrakhimov, Smoothness of subharmonic functions, Mat. Sb. 181 (2) (1990) 167–182 (in Russian); translation in Math. USSR-Sb. 69 (1) (1991) 179–195. [19] Z. Zhou, The contractivity of the free Hamiltonian semigroup in the Lp space of entire functions, J. Funct. Anal. 96 (1991) 407–425.


Regularity criteria for almost every function in Sobolev spaces A. Fraysse 1 LSS, CNRS, Université Paris Sud, Supélec, 3 rue Joliot Curie, 91192 Gif Sur Yvette, France Received 4 March 2008; accepted 17 November 2009 Available online 1 December 2009 Communicated by J. Bourgain

Abstract In this paper we determine the multifractal nature of almost every function (in the prevalence setting) in a given Sobolev or Besov space according to different regularity exponents. These regularity criteria are based on local Lp regularity or on wavelet coefficients and give a precise information on pointwise behavior. © 2009 Elsevier Inc. All rights reserved. Keywords: Prevalence; Sobolev spaces; Besov spaces; Calderòn–Zygmund regularity criteria; Multifractal analysis; Wavelet bases

1. Introduction The study of regularity, and more precisely of pointwise regularity of signals or functions raised a large amount of interest in scientific communities. This topic allows a better understanding of behavior of functions and it gives also a powerful classification tool in various domains. A recent theory, based on the study of pointwise smoothness is supplied by the multifractal analysis. The multifractal analysis was introduced in order to study the velocity of turbulent flows and was initially applied to understand the behavior of some invariant measures [14,32]. It was then used in several fields, such as signal or image processing [1,2]. But in each case, the criterium of E-mail address: [email protected]. 1 This work was performed while the author was at Laboratoire d’analyse et de mathématiques appliquées, Université

Paris XII, France. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.017

A. Fraysse / Journal of Functional Analysis 258 (2010) 1806–1821

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regularity taken into account is the Hölder exponent, and this exponent is only well defined for locally bounded functions. It would be convenient to define new criteria on more general cases. For instance, the velocity of turbulent fluids is now known not to be bounded near vorticity filaments, see [3]. In the study of turbulent flows, in [27], Leray conjectured that self-similar weak solutions of Navier–Stokes equations with initial value in L2 (R3 ) may develop singularities in a finite time. This problem was then widely studied [26] and different behaviors were produced following the initial value problem involved. Since [8] it is known that the set of singularities of these solutions is of vanishing Hausdorff dimension. In [9], an alternative definition of regularity was supplied which can give better results in elliptic PDEs and especially when viscous solutions occur. It would thus be natural to take this notion, which involves local Lp norms to study irregularities of Navier–Stokes solutions when initial data are supposed in Lp . Furthermore, it would be convenient to establish regularity criteria in image processing, where those properties are widely used. A natural idea would be to determine properties of the characteristic function of sets. But Hölder regularity is not adapted for classification of natural images as it does not take into account the geometry of sets and takes only two values when it is applied to characteristic functions. Furthermore, most natural images, such as clouds images or medical images are discontinuous, see [3] and thus need to be studied in a more general framework. Let us recall the principle of multifractal analysis. The natural notion of regularity used in the study of pointwise behavior is provided by the Hölder exponent, defined as follows. Definition 1. Let α 0; a function f : Rd → R is C α (x0 ) if for each x ∈ Rd such that |x − x0 | 1 there exists a polynomial P of degree less than [α] and a constant C such that, f (x) − P (x − x0 ) C|x − x0 |α .

(1)

The Hölder exponent of f at x0 is hf (x0 ) = sup α: f ∈ C α (x0 ) . In some cases, functions may have an Hölder regularity which changes wildly from point to point. Rather than measure the exact value of the Hölder exponent, one studies the fractal dimension of sets where it takes a given value. The spectrum of singularities, also called multifractal spectrum and denoted d(H ), is the function which gives for each H the Hausdorff dimension of those sets. A function is then called multifractal if the support of its spectrum of singularities is an interval with no empty interior. However, the Hölder exponent has some drawbacks that prevent from using it in any situation. p First, it is only defined for locally bounded functions. If a function f belongs only to Lloc this exponent is no more defined. Furthermore, as pointed by Calderòn and Zygmund in [9], it is not preserved under pseudodifferential operator of order zero, and as stated in [30] cannot thus be characterized with conditions on wavelet coefficients. Another drawback can be emphasized with the example of Raleigh–Taylor instability. This phenomenon occurs when two fluids which are not miscible are placed on top of each other. In this case, thin filaments appear giving to the interface between the two fluids a fractal structure, see [31] for a study. To study geometric properties of this interface, one would be interested on multifractal properties of its characteristic function. Nonetheless as such functions are not continuous and take only two values, their Hölder exponent is not define, and a multifractal approach cannot be carry out.

1808


For all these reasons, it would be convenient to define a new kind of multifractal analysis constructed with more general exponents. Such construction is started in [23,24], where the authors proposed a multifractal formalism based on Calderòn–Zygmund exponents. These exponents p were introduced in [9] as an extension of Hölder exponent to Lloc functions, invariant under pseudodifferential operator of order 0. p

p

Definition 2. Let p ∈ [1, ∞] and u − pd be fixed. A function f ∈ Lloc (Rd ) belongs to Tu (x0 ) if there exist a real R > 0 and a polynomial P , such that deg(P ) 0 such that: ∀ρ R:

1 ρd

f (x) − P (x)p dx

1/p cρ u .

(2)

x−x0 ρ

p

p

The p-exponent of f at x0 is uf (x0 ) = sup{u: f ∈ Tu (x0 )}. p

With this definition, the usual Hölder condition f ∈ C s (x0 ) corresponds to f ∈ Tu (x0 ) where p = ∞. One can also check that the p-exponent is decreasing as a function of p. As it was done for the Hölder exponent one can define for each p the p-spectrum of singularities as the Hausdorff dimension of the set of points where the p-exponent takes a given value. In [23], the authors defined the weak accessibility exponent, given as a parameter of the geometry of the set. Specifically, this weak-scaling exponent deals with the local behavior of the boundary of a set. It is thus well adapted for fractal interfaces that might appear in experimental settings. They showed that this geometrical based exponent coincides with Calderòn–Zygmund exponents of the characteristic function of the boundary of the set. Another regularity criterium, closely related to the previous ones is given by the following definition from [30]. With this exponent we can have a better understanding of the link between Calderòn–Zygmund exponents, Hölder exponent and the pointwise behavior of functions. Definition 3. Let f : Rd → R be a function or a distribution and x0 ∈ Rd be fixed. The weakscaling exponent of f at x0 is the smallest real number β(f, x0 ) satisfying: p

1. β(f, x0 ) uf (x0 ) ∀p 1. ∂f 2. β(f, x0 ) = s ⇔ β( ∂x , x0 ) = s − 1, j = 1, . . . , d. j

Similarly, we define the weak-scaling spectrum, denoted by dws (β) as the Hausdorff dimension of sets of points where β(f, x) takes a given value β. As we will see later, the weak-scaling exponent can be fully characterized by conditions on wavelet coefficients. In practical applications, the classical multifractal spectrum cannot be computed directly, as it takes into account intricate limits. Thus, some formulas, called multifractal formalisms were introduced in purpose to link the spectrum of singularities to some calculable quantities. There are indeed two formalisms based on conditions on wavelet coefficients. Historically the classical multifractal formalism stated in [13] was based directly on wavelet coefficients. Actually, this formula gave unexpected results and was shown to be false in several cases. It is nowadays known that it is the weak-scaling exponent which is involved in this formula. A second multifractal formalism, developed in [22] is based on “wavelet leaders”, which can be seen as the theoretical counterpart of the “Wavelet Transform Modulus Maxima” used in [4]. This “wavelet


1809

leader” based formalism actually gives the spectrum of singularities in term of Hölder exponent. The weak-scaling exponent is thus more appropriated in order to understand the classical multifractal formalism. This exponent is also more stable under the action of differential operators. Furthermore it gives an additional information on the behavior of functions thanks to the following definition. Definition 4. Let f : Rd → R be a function and x0 ∈ Rd . We say that x0 is a cusp singularity for f if β(f, x0 ) = hf (x0 ). If β(f, x0 ) > hf (x0 ), x0 is said to be an oscillating singularity. An example of oscillating function at x0 = 0 is given by f (x) = |x| sin(1/|x|). Here, hf (0) = 1 while β(f, 0) = +∞. And we have a cusp singularity when the behavior of the function at x0 is like |x|α but also like |x|α + |x| sin(1/|x|). Indeed we talk about a cusp singularity when the function does not have oscillations at a point, or if those oscillations are hidden by the Hölder behavior. Many authors have studied generic values of the Hölder exponent in function spaces. In 1931 Banach [5], proved that the pointwise regularity of quasi-all, in a topological sense, continuous functions is zero. Here quasi-all means that this property is true in a countable intersection of dense open sets. Since then, Hunt in [15] showed that the same result is satisfied by measure theoretic almost every continuous functions. Recently, results such as those of [25] and [12] studied Hölder regularity of generic functions in Sobolev spaces in both senses. Whereas a large study of regularity properties for generic sets, there exists no result on genericity of Calderòn– Zygmund exponents or of weak-scaling exponent. Our purpose here is to provide a genericity result of those exponents in given Sobolev and Besov spaces, with the measure-theoretic notion of genericity supplied by prevalence. Prevalence is a measure-theoretic notion of genericity on infinite dimensional spaces. In a finite dimensional space, the notion of genericity in a measure theoretic sense is supplied by the Lebesgue measure. The particular role played by this measure is justified by the fact that this is the only one which is σ -finite and invariant under translation. In a metric infinite dimensional space no measure enjoys these properties. The proposed alternative is to replace conditions on the measure by conditions on sets, see [6,10,17,16] and to take the following definition. Definition 5. Let V be a complete metric vector space. A Borel set B in V is called Haar-null if there exists a probability measure μ with compact support such that μ(B + v) = 0 ∀v ∈ V .

(3)

In this case the measure μ is said to be transverse to B. A subset of V is called Haar-null if it is contained in a Borel Haar-null set. The complement of a Haar-null set is called a prevalent set. With a slight abuse of language we will say that a property is satisfied almost everywhere when it holds on a prevalent set. Let us recall properties of Haar-null sets, see [10,17] and show how they generalize notion of Lebesgue measure zero sets.

1810


Proposition 1. 1. If S is Haar-null, then ∀x ∈ V , x + S is Haar-null. 2. If dim(V ) < ∞, S is Haar-null if and only if meas(S) = 0 (where meas denotes the Lebesgue measure). 3. Prevalent sets are dense. 4. If S is Haar-null and S ⊂ S then S is Haar null. 5. The union of a countable collection of Haar-null sets is Haar null. 6. If dim(V ) = ∞, compact subsets of V are Haar-null. Remarks. Several kinds of measures can be used as transverse measures for a Borel set. Let us give two examples of transverse measure. 1. A finite dimensional space P is called a probe for a set T ⊂ V if the Lebesgue measure on P is transverse to the complement of T . Those measures are not compactly supported probability measures. However one immediately checks that this notion can also be defined in the same way but stated with the Lebesgue measure defined on the unit ball of P . Note that in this case, the support of the measure is included in the unit ball of a finite dimensional subspace. The compactness assumption is therefore fulfilled. 2. If V is a function space, a probability measure on V can be defined by a random process Xt whose sample paths are almost surely in V . The condition μ(f + A) = 0 means that the event Xt − f ∈ A has probability zero. Therefore, a way to check that a property P holds only on a Haar-null set is to exhibit a random process Xt whose sample paths are in V and is such that ∀f ∈ V , a.s.

Xt + f does not satisfy P.

These properties, such as several examples of prevalent results can be found in the survey [16]. 1.1. Statement of main results The purpose of this paper is stated by the two following theorems which give the multifractal properties of almost every functions with regard to exponents defined in the previous section. Theorem 1. Let s0 0 and 1 p0 < ∞ be fixed. 1. For all p 1 such that s0 − pd0 > − pd the p-spectrum of singularities of almost every function in Ls0 ,p0 (Rd ) is given by d dp (u) = p0 (u − s0 ) + d. (4) ∀u ∈ s0 − , s0 p0 2. For almost every function in Ls0 ,q0 (Rd ) the spectrum of singularities for the weak-scaling exponent is given by d dws (β) = p0 (β − s0 ) + d. (5) ∀β ∈ s0 − , s0 p0


1811

This result in Sobolev spaces has an analogous in the Besov setting. Furthermore, Besov spaces are useful when wavelets are involved as it is the case here, those spaces having a simpler characterization. Theorem 2. Let s0 0 and 0 < q, p0 < ∞ be fixed. 1. For all p 1 such that s0 − pd0 > − pd the p-spectrum of singularities of almost every funcs ,q tion in Bp00 (Rd ) is given by d dp (u) = p0 (u − s0 ) + d. ∀u ∈ s0 − , s0 p0

(6)

s ,q

2. For almost every function in Bp00 (Rd ) the spectrum of singularities for the weak-scaling exponent is given by

d ∀β ∈ s0 − , s0 p0

dws (β) = p0 (β − s0 ) + d.

(7)

These theorems seem a bit surprising. Let us compare them with the following proposition from [12]. Proposition 2. • If s − d/p 0, then almost every function in Lp,s is nowhere locally bounded, and therefore its spectrum of singularities is not defined. • If s − d/p > 0, then the Hölder exponent of almost every function f of Lp,s takes values in [s − d/p, s] and ∀H ∈ [s − d/p, s] df (H ) = Hp − sp + d.

(8)

Thus the main change from [12] is given by the fact that here β can take negative values. Indeed, our present theorems give a generic regularity in Sobolev or in Besov spaces that are not imbedded in global Hölder spaces. Even if in such spaces, the classical spectrum of singularities is not define for a prevalent set, we have an idea of the pointwise behavior of almost every distribution. In the other case, when s0 − pd0 > 0 and the spectrum of singularities exists, it coincides with the above spectra for almost every function in Besov spaces. Therefore, in the second case we generalize in this paper the result of [12] to more stable exponents. In [28], it was also proved that in those spaces quasi-all functions, in the Baire’s sense, have no oscillating singularities. Furthermore, presence of oscillating singularities is linked with the failure of the multifractal formalism in [33]. And in [11], it was already proven that almost every function in Besov spaces satisfies the multifractal formalism. The main result of this paper together with Definition 4 show that even if weak-scaling and Hölder exponents do not coincide they share the same spectrum. Thus, in the prevalence setting, oscillating singularities appear as an exceptional behavior in regular Sobolev or Besov spaces. Another remark can be made thanks to the following proposition from [24] and from [34] that gives an upper bound for the p-spectrum.

1812

A. Fraysse / Journal of Functional Analysis 258 (2010) 1806–1821 s ,p0

Proposition 3. Let f ∈ Bp00

(Rd ), where s0 > 0 and let p 1 be such that s0 − pd0 > − pd . Then

d ∀u ∈ s0 − , s0 dp (u) p0 u − s0 p0 + d. p0

(9)

This proposition together with Theorem 2 show that the generic regularity for p criteria is as bad as possible. In Section 2 we will prove Theorems 1 and 2. For the sake of completeness, we first have to define our main tool which is given by wavelet expansions of functions. Wavelets are naturally present in multifractal analysis, see for instance [2]. Furthermore, in our case it allows a characterization of both functional spaces and pointwise regularities. 1.2. Wavelet expansions There exist 2d − 1 oscillating functions (ψ (i) )i∈{1,...,2d −1} in the Schwartz class such that the functions

2dj ψ (i) 2j x − k ,

j ∈ Z, k ∈ Zd

form an orthonormal basis of L2 (Rd ), see [29]. Wavelets are indexed by dyadic cubes λ = [ 2kj ; k+1 [d . Thus, any function f ∈ L2 (Rd ) can be written: 2j f (x) =

(i) cj,k ψ (i) 2j x − k

where

(i)

cj,k = 2dj

f (x)ψ (i) 2j x − k dx.

(Note that we use an L∞ normalization instead of an L2 one, which simplifies the formulas.) If p > 1 and s > 0, Sobolev spaces have thus the following characterization, see [29]:

f ∈ Ls,p Rd

⇐⇒

1/2

|cλ |2 1 + 4j s χλ (x) ∈ Lp R d ,

(10)

λ∈Λ

where χλ (x) denotes the characteristic function of the cube λ and Λ is the set of all dyadics cubes. Homogeneous Besov spaces, which will also be considered, are characterized (for p, q > 0 and s ∈ R) by f

s,q

∈ Bp Rd

⇐⇒

j

q/p |cλ | 2

p (sp−d)j

C

(11)

λ∈Λj

where Λj denotes the set of dyadics cubes at scale j , see [29]. Hölder pointwise regularity can also be expressed in term of wavelet coefficients, see [18].


1813

Proposition 4. Let x be in Rd . If f is in C α (x) then there exists c > 0 such that for each λ: α

|cλ | c2−αj 1 + 2j x − k .

(12)

This proposition is not a characterization. If for any ε > 0, a function does not belong to C ε (Rd ) one cannot express its pointwise Hölder regularity in term of condition on wavelet coefficients. This is an advantage of Calderòn–Zygmund exponent since, as showed in [21], it can be linked to wavelet expansion without global regularity assumption. Definition 6. Let x0 be in Rd and j 0. We denote by λj (x0 ) the unique dyadic cube of width 2−j which contains x0 . And we denote

1 1 3λj (x0 ) = λj (x0 ) + − j , j 2 2

d .

Furthermore, we define the local square function by

1/2 |cλ |2 1λ (x) .

Sf (j, x0 )(x) =

λ⊂3λj (x0 )

Proposition 5. Let p 1 and s 0; if f ∈ T of f satisfy for all j 0

Sf (j, x0 ) Conversely if (13) holds and if s −

d p

p (x ), s− pd 0

Lp

then ∃C > 0 such that wavelet coefficients

c2−j (u+d/p) .

∈ / N then f ∈ T

(13)

p (x ). s− pd 0

As far as we are concerned, we do not need a characterization but a weaker condition which is given by the following proposition from [23]. Proposition 6. Let p 1 and s 0; if f ∈ T

p (x ), s− pd 0

then ∃A, C > 0 such that wavelet coeffi-

cients of f satisfy ∃C ∀j

2j (sp−d)

−sp

|cj,k |p 1 + k − 2j x0 Cj.

(14)

|k−2j x0 |A2j

Furthermore, it is also proved in [23] that the p-exponent can be derived from wavelet coefficients. p

Proposition 7. Let p 1 and f ∈ Lloc . Define p

Σj (s, A) = 2j (sp−d)

|k−2j x0 |A2j

−sp

|cj,k |p 1 + k − 2j x0 ,

(15)

1814


for A > 0 small enough. And denote p log(Σj (s, A)1/p ) ip (x0 ) = sup s: lim inf 0 . −j log 2

(16)

Then the following inequality always holds p

uf (x0 ) ip (x0 ) − δ,p

If furthermore there exists δ > 0 such that f ∈ Bp p

d . p

(17)

then the p-exponent of f satisfies

uf (x0 ) = ip (x0 ) −

d . p

(18)

As seen previously, the p-exponent is also related to the weak-scaling exponent. This one can also be expressed in term of wavelet coefficients, thanks to its relation with two-microlocal spaces, defined in [7]. Definition 7. Let s and s be two real numbers. A distribution f : Rd → R belongs to the two microlocal space C s,s (x0 ) if its wavelet coefficients satisfy that there exists c > 0 such that ∀j, k

−s

|cj,k | c2−sj 1 + 2j x0 − k .

(19)

In [30] the following characterization of the weak-scaling exponent is given. Proposition 8. A tempered distribution f belongs to Γ s (x0 ) if and only if there exists s < 0 such that f belongs to C s,s (x0 ). The weak-scaling exponent of f is β(f, x0 ) = sup s: f ∈ Γ s (x0 ) .

(20)

But we will rather take the following alternative characterization from [23] that gives a simpler condition in term of wavelet coefficients. Proposition 9. Let f be a tempered distribution. The weak-scaling exponent of f at x0 is the supremum of s > 0 such that: ∀ε > 0 ∃c > 0 ∀(j, k) such that 2j x0 − k < 2εj ,

|cj,k | c2−(s−ε)j .

(21)

2. Proofs of Theorems 1 and 2 2.1. The p-spectrum In this section, we only prove the first point of Theorem 2. We will see how this proof can be adapted to Theorem 1 in a second time.


1815

In a first part, the result that we prove is more precise than the one stated. Indeed, we prove that s ,p for each α ∈ (1, ∞) and for each p 1, the p-exponent of almost every function of Bp00 0 (Rd ) is smaller than s−

d d + p αp

(22)

on a set of Hausdorff dimension greater than αd . These fractal sets are closely related to the dyadic approximation of points. Definition 8. Let α ∈ (1, ∞) be fixed. We denote

kn 1 Fα = x: ∃ a sequence (kn , jn ) n∈N x − j αj . 2n 2 n

(23)

This set Fα can also be defined as lim sup i→∞

Fαi,l

l∈Nd

where Fαi,l denotes the cube 2li + [− 21αi ; 21αi ]d . If x ∈ Fα it is said α-approximable by dyadics. The dyadic exponent of x is defined by α(x0 ) = sup{α: x0 is α-approximable by dyadics}. As stated in [20], the Hausdorff dimension of Fα is at least αd . In order to prove our result we show that the set of functions where for α and p 1 given, the p-exponent is larger than (22) at a point of Fα is included in a countable union of Haar-null Borel sets. Let p 1 be given such that s0 − pd0 > − pd . For α 1 fixed we denote s(α) = s0 − pd0 + αpd 0 + s ,p

For ε > 0 fixed, let β = s(α) + ε. We first check that the set of functions in Bp00 0 satisfying (14) with exponent β at a point in Fα is a Haar-null Borel set. This set can be included in a s ,p countable union over A > 0 and c > 0 of sets M(A, c) which are sets of functions in Bp00 0 (Rd ) satisfying d p.

∃x ∈ Fα ∀j

2j (βp−d)

−βp

|cλ |p 1 + k − 2j x c.

|k−2j x|A2j

And for each i ∈ N these sets can be included in the countable union over l ∈ {0, . . . , 2i − 1}d of Mi,l (A, c), defined by the set of f such that ∃x ∈ Fαi,l ∀j

2j (βp−d)

−βp

|cλ |p 1 + k − 2j x c.

|k−2j x|A2j

Each Mi,l (A, c) is a closed set. Indeed, suppose that a sequence (fn ) of elements of Mi,l (A, c) s ,p n the wavelet coefficients of f , for each n ∈ N, and converges to f in Bp00 0 (Rd ). Denote cj,k n cj,k those of f . The mapping giving the wavelet coefficients of a function f in a Besov space is

1816


n converge to c . Furthermore for each n there exists x ∈ F i,l continuous, thus for each j, k cj,k j,k n α such that fn satisfies (14) at xn . Thus

∀j

2j (βp−d)

n p

c 1 + kn − 2j xn −βp c. λ

(24)

|kn −2j xn |A2j

As Fαi,l is a compact set, there exists an accumulation point x ∈ Fαi,l of xn . Furthermore, if kn is such that |kn − 2j xn | A2j for a subsequence xφ(n) such that lim xφ(n) = x, the corresponding kφ(n) converges to k with |k − 2j x| A2j . Thus up to a subsequence, when n tends to infinity, (24) becomes ∀j

2j (βp−d)

−βp

|cλ |p 1 + k − 2j x c.

|k−2j x|A2j

Consequently f belongs to Mi,l (A, c) and M(A, c) is a Borel set. To prove that it is also a Haar-null set, we construct a probe as transverse measure, in this way the compactness assumption is clearly satisfied. This probe is based on a slight modification of the “saturating function” introduced in [20]. Let i ∈ N and l ∈ {0, . . . , 2i − 1}d be fixed. Let n ∈ N be fixed large enough such that N = d dn + 1. Each dyadic cube λ is split into M subcubes of size 2−d(j +n) . For each index 2 > pαε m ∈ {1, . . . , N}, we choose a subcube i(λ) and the wavelet coefficient of gi is given by: dλm where a =

2 p0

=

d

d

1 ( p0 −s0 )j − p0 J 2 ja 2

if m = i(λ), else,

0

(25)

and J j and K ∈ {0, . . . 2J − 1}d are such that k K = J 2j 2 s ,p

is an irreducible form. It is proven in [12] that these functions belong to Bp00 0 . Furthermore, if a point x ∈ (0, 1)d is α-approximable by dyadics, there exists a subsequence (jn , kn ) where jn = [Jn α], Jn and Kn being defined in (23) and kn is such that 2kjnn = 2KJnn . The corresponding wavelet coefficients of all functions gm satisfy that there exists a constant c > 0 such that if (j, k) satisfy |x0 − 2kj | < A: m dj,k

> c(A)

2

( pd −s0 )j − αpd j

2

0

ja

0

.

(26)

s0 ,p0 d Let f = cj,k ψj,k be an arbitrary function in B p0 (R ). Suppose that there exist two points N N m γ1 ∈ R and γ2 ∈ R such that for a = 1, 2, f + m γa g m belong to Mi,l (A, c). By definition there also exist two points x1 and x2 in Fαi,l such that, for a = 1, 2, ∀j

2

j (βp−d)

p N

−βp m m γ a d λ 1 + k − 2 j x c. c λ + j

|k−2j xa |A2

m=1


1817

As β > 0, this condition implies: ∀j

2

j (βp−d)

p N

−βp m m γa dλ 1 + A2j c. c λ + j

|k−2j xa |A2

m=1

But x1 and x2 belong to same dyadic cubes of size j > i. Thus the same k satisfies |k −2j xa | A2j for a = 1, 2 and wavelet coefficients of f1 − f2 are such that for all j > αi

2

j (βp−d)

N p

−βp m m m γ1 − γ2 dλ 1 + A2j 2c. j

|k−2j xa |A2

m=1

It is obvious that −βp m k p m m −βpj −j − x1 2 γ d − γ 2 + λ 1 2 2j m

2j (βp−d) |

k 2j

−x1 |A

2j (˜s p−d)

−˜s p m k p m m −˜s pj −j − x1 2 γ d − γ 2 + . λ 1 2 2j |A m

sup |

k 2j

−x1

Using definition of function gm , if for each j we define j = j + n, at scale j there is only one function gm with nonzero coefficient. And with (26) one finally obtains that there exists a subsequence j such that 2n(βp−d) 2j (βp−d)

−˜s p m k p m m −βpj −j − x1 2 γ d − γ 2 + λ 1 2 2j |A m

sup |

k 2j

−x1

p 1 γ1i − γ2i c˜p pa 2pεj , j where c˜ depends only on n and A. Those two inequalities imply that γ1 − γ2 ∞ 2cc(N )i 1/p0 2−εαpi . p

(27)

Therefore the set of γ such that f + i γ m g m belongs to Mi,l (A, c) is included in a ball of radius less than (2cc(N))N i N/p0 2−εαpN i . Taking the countable union over l, we obtain that for each i0 fixed, the set of γ satisfying ∃x ∈ Fαi0 such that f +

m

is of Lebesgue measure bounded by

γ m g m satisfies (24) at x

1818

A. Fraysse / Journal of Functional Analysis 258 (2010) 1806–1821 ∞

N 2cc(N) i N/p0 2di−εαpN i . i=i0

As N is large enough, this measure tends to zero when i0 tends to infinity. And M(A, c) is then a Haar-null set. As this result does not depend on c or on A, we can take the union over countable cn > 0 s ,p p and An > 0. Then the set of functions in Bp00 0 (Rd ) belonging to Tβ (x) at a point x ∈ Fα is a Haar-null set. Thus, s ,p0

∀p 1, ∀α 1 ∀β > s(α) a.s. in Bp00

∀x ∈ Fα

p

uf (x) β.

Taking ε → 0 it follows by countable intersection that s ,p0

∀p 1, ∀α 1 a.s. in Bp00

∀x ∈ Fα

p

uf (x) s(α).

Therefore, if αn is a dense sequence in (1, ∞), using the same argument, one obtains that s ,p0

∀p 1, a.s. in Bp00

∀n ∈ N ∀x ∈ Fαn

p

uf (x) s(αn ).

(28)

Let f be a function satisfying (28) and α 1 be fixed. Let αφ(n) be a nondecreasing subsequence of αn converging to α. Then the intersection Eα of Fαn contains Fα and for all x ∈ Eα , p and thus for all x ∈ Fα , uf (x) s(α). Furthermore, see [19], there exists a measure mα positive on Fα but such that every set of dimension less than αd is of measure zero. Let us denote by GH the set of points where up (x) < H . According to Proposition 3, this set can be written as a countable union of sets of mα measure zero. Thus, we obtain

mα x: up (x) = H = mα (Fα \GH ) > 0. Which gives us the p spectrum of singularities d ∀u ∈ s0 − , s0 dp (u) = p0 u + d − s0 p0 . p0 This proof does not depend on the choice of q. It can then be extended in the same way for s ,q any Besov space Bp00 for 0 q < ∞. The proof for the Sobolev case is similar. The functions gm defined in (25) also belong s ,1 s ,1 to Bp00 . Since Bp00 → Ls0 ,p0 , the gm belong to Ls0 ,p0 and the remaining of the proof is unchanged. 2.2. Generic values of the weak-scaling spectrum We now prove of the second point of Theorems 1 and 2. As in the previous case, we prove Theorem 2 using the same argument as in the previous part giving the Sobolev case.


1819 s ,q

Proposition 10. Let s0 > 0 and 0 p0 , q < ∞ be fixed. For almost every function in Bp00 the spectrum of singularities for the weak-scaling exponent is given by d ∀β ∈ s0 − , s0 p0

dws (β) = p0 (β − s0 ) + d.

(29)

Proof. Let α 1 be fixed and denote by Fα the set of Definition 8. Let ε > 0 be fixed and define β = s0 − pd0 + pd0 α + ε. According to Proposition 9, we first have to show that for a given c > 0 the set: s ,q Mα,c = f = cλ ψλ ∈ Bp00 : ∃x ∈ Fα ∀ε > 0 ∀(j, k) 2j x − k 2ε j |cλ | c2−(β−ε )j (30) is a Borel Haar-null set. Let us remark that for all i ∈ N, this set is included in the countable union of: s ,q Mα,c (i, l) = f ∈ Bp00 : ∃x ∈ Fαi,l ∀ε > 0 ∀(j, k) 2j x − k 2ε j |cλ | c2−(β−ε )j . (31) One easily checks that Mα,c (i, l) is closed and therefore that Mα,c is a Borel set. To prove that Mα,c is also Haar-null, we use a different transverse measure than in the previous section, by taking the measure induced by a stochastic process. As Mα,c depends only on the dyadic properties of points, we can also restrict the proof to [0, 1]d . Consider the following stochastic process on [0, 1]d : Xx =

∞ j =0 λ∈[0,1]d

εj,k

2

−(s0 − pd )j − pd J 0

ja

2

0

ψ 2j x − k

(32)

where J and a are defined as in (25) and {εj,k }j,k is a Rademacher sequence. That is the εj,k are i.i.d. random variables such that 1 P(εj,k = 1) = P(εj,k = −1) = . 2 s ,q

This process belongs to Bp00 . Furthermore, the measure defined by this stochastic process is supported by the continuous image of a compact set. Thus, (Xx )x∈[0,1]d defines a compactly s ,q supported probability measure on Bp00 . s ,q Let f be an arbitrary function in Bp00 (Rd ). Thanks to Fubini’s theorem, it is sufficient to prove that for all x ∈ Fα , almost surely, condition (21) is not satisfied by f + X. Let x0 ∈ Fα be fixed and suppose that f + X satisfies condition (21) at x0 . Then for all ε > 0 and for all (j, k) such that |k − 2j x0 | 2ε j , −(s0 − pd )j − pd J 0 2 0 cj,k + εj,k 2 c2−(β−ε )j . a j

1820

A. Fraysse / Journal of Functional Analysis 258 (2010) 1806–1821 j

Taking (Jn , Kn ) the sequence of Definition 8, j = [αJn ] and k = K2nJ2n one obtains that there exists a sequence (j, k) such that |2j x0 − k| 1 and the following property holds: εj,k = cj,k j a 2

(s0 − pd + pd α )j 0

0

(s −

+ o 2−(ε−ε )j . d

+

d

)j

Taking ε = 2ε , one obtains that εj,k ∼ cj,k j a 2 0 p0 p0 α n when jn → ∞. Since the cj,k are deterministic, this result implies that there exists an infinite sequence of independent stochastic variables which are deterministic. This event is of probability zero and Mα,c is a Haar-null set. Therefore, taking countable unions over c > 0 and ε → 0, it follows that for all α 1, the s ,q set of functions in Bp00 with a weak-scaling exponent greater than s0 − pd0 + pd0 α at some point of F α is a Haar-null set. Let (αn )n∈N be a dense sequence in (1, ∞) and take a countable union over αn . We finally obtain s ,q d

R

a.s. in Bp00

∀n ∈ N ∀x ∈ Fαn

β(f, x) s0 −

d d + . p0 p0 αn

With a similar argument as in Section 2.1, one can prove that: s ,q d

a.s. in Bp00

R

∀α 1 ∀x ∈ Fα

β(f, x) s0 −

d d + . p0 p0 α

(33)

Furthermore, we saw in Section 2.1 that there exists a measure mα which is positive on Fα and such that d d > 0. mα x; up (x) = s0 − + p0 p0 α And by definition, ∀p 1, β(f, x) up (x), thus mα

d d x; β(f, x) = s0 − > 0. + p0 p0 α

Which states that the spectrum of singularities for the weak-scaling exponent of almost every s ,q function in Bp00 (Rd ) is given by d ∀β ∈ s0 − , s0 p0

dws (β) = p0 β + d − s0 p0 .

2

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Homotopy of unitaries in simple C ∗ -algebras with tracial rank one ✩ Huaxin Lin a,b a Department of Mathematics, East China Normal University, Shanghai 200062, China b Department of Mathenmatics, Univerisity of Oregon, Eugene, OR 97403, USA

Received 2 July 2008; accepted 24 November 2009 Available online 16 December 2009 Communicated by D. Voiculescu

Abstract Let > 0 be a positive number. Is there a number δ > 0 satisfying the following? Given any pair of unitaries u and v in a unital simple C ∗ -algebra A with [v] = 0 in K1 (A) for which uv − vu < δ, there is a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ A such that v(0) = v,

v(1) = 1 and

uv(t) − v(t)u <

for all t ∈ [0, 1].

An answer is given to this question when A is assumed to be a unital simple C ∗ -algebra with tracial rank no more than one. Let C be a unital separable amenable simple C ∗ -algebra with tracial rank no more than one which also satisfies the UCT. Suppose that φ : C → A is a unital monomorphism and suppose that v ∈ A is a unitary with [v] = 0 in K1 (A) such that v almost commutes with φ. It is shown that there is a continuous path of unitaries {v(t): t ∈ [0, 1]} in A with v(0) = v and v(1) = 1 such that the entire path v(t) almost commutes with φ, provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented. © 2009 Elsevier Inc. All rights reserved. Keywords: Homotopy; Unitary; Simple C ∗ -algebras

✩ This work is partially supported by Chang-Jiang Lecturer from East China Normal University, the 111 Project and a grant from NSF (DMS074813). E-mail address: [email protected].


H. Lin / Journal of Functional Analysis 258 (2010) 1822–1882

1823

1. Introduction Fix a positive number > 0. Can one find a positive number δ such that, for any pair of unitary matrices u and v (K1 (Mn ) = {0} for any integer n 1) with uv − vu < δ, there exists a continuous path of unitary matrices {v(t): t ∈ [0, 1]} for which v(0) = v, v(1) = 1 and uv(t) − v(t)u < for all t ∈ [0, 1]? The answer is negative in general. A Bott element associated with the pair of unitary matrices may appear. The hidden topological obstruction can be detected in a limit process. This was first found by Dan Voiculescu [29]. On the other hand, it has been proved that there is such a path of unitary matrices if an additional condition, bott1 (u, v) = 0, is provided (see, for example, [2] and also 3.12 of [13]). It was recognized by Bratteli, Elliott, Evans and Kishimoto [2] that the presence of such continuous path of unitaries in general simple C ∗ -algebras played an important role in the study of classification of simple C ∗ -algebras and perhaps plays important roles in some other areas such as the study of automorphism groups (see, for example, [12,24,21]). They proved what they called the Basic Homotopy Lemma: For any > 0, there exists δ > 0 satisfying the following: For any pair of unitaries u and v in A with sp(u) δ-dense in T and [v] = 0 in K1 (A) for which uv − vu < δ

and bott1 (u, v) = 0,

there exists a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ A such that v(0) = v,

v(1) = 1A

and v(t)u − uv(t) <

for all t ∈ [0, 1], where A is a unital purely infinite simple C ∗ -algebra or a unital simple C ∗ algebra with real rank zero and stable rank one. Define φ : C(T) → A by φ(f ) = f (u) for all f ∈ C(T). Instead of considering a pair of unitaries, one may consider a unital homomorphism from C(T) into A and a unitary v ∈ A for which v almost commutes with φ. In the study of asymptotic unitary equivalence of homomorphisms from an AH-algebra to a unital simple C ∗ -algebra, as well as the study of homotopy theory in simple C ∗ -algebras, one considers the following problem: Suppose that X is a compact metric space and φ is a unital homomorphism from C(X) into a unital simple C ∗ -algebra A. Suppose that there is a unitary u ∈ A with [u] = 0 in K1 (A) and u almost commutes with φ. When can one find a continuous path of unitaries {u(t): t ∈ [0, 1]} ⊂ A with u(0) = u and u(1) = 1 such that u(t) almost commutes with φ for all t ∈ [0, 1]? Let C be a unital AH-algebra and let A be a unital simple C ∗ -algebra. Suppose that φ, ψ : C → A are two unital monomorphisms. Let us consider the question when φ and ψ are asymptotically unitarily equivalent, i.e., when there is a continuous path of unitaries {w(t): t ∈ [0, ∞)} ⊂ A such that lim w(t)∗ φ(c)w(t) = ψ(c)

t→∞

for all c ∈ C.

When A is assumed to have tracial rank zero, it was proved in [14] that they are asymptotically unitarily equivalent if and only if [φ] = [ψ] in KK(C, A), τ ◦ φ = τ ◦ ψ for all tracial states τ of A and a rotation map associated with φ and ψ is zero. Apart from some direct applications, this result plays crucial roles in the study of the problem to embed crossed products into unital simple AF-algebras and in the classification of simple amenable C ∗ -algebras which do not have finite tracial rank (see [30,21,20]). One of the key tools in the study of the above mentioned

1824


asymptotic unitary equivalence is the so-called Basic Homotopy Lemma concerning a unital monomorphism φ and a unitary u which almost commutes with φ. In this paper, we study the case that A is no longer assumed to have real rank zero, or tracial rank zero. The result of W. Winter in [30] provides the possible classification of simple finite C ∗ -algebras far beyond the cases of finite tracial rank. However, it requires to understand much more about asymptotic unitary equivalence in those unital separable simple C ∗ -algebras which have been classified. An immediate problem is to give a classification of monomorphisms (up to asymptotic unitary equivalence) from a unital separable simple AH-algebra into a unital separable simple C ∗ -algebra with tracial rank one. For that goal, it is paramount to study the Basic Homotopy Lemmas in a simple separable C ∗ -algebras with tracial rank one. This is the main purpose of this paper. A number of problems occur when one replaces C ∗ -algebras of tracial rank zero by those of tracial rank one. First, one has to deal with contractive completely positive linear maps from C(X) into a unital C ∗ -algebra C with the form C([0, 1], Mn ) which are not homomorphisms but almost multiplicative. Such problem is already difficult when C = Mn but it has been proved that these above mentioned maps are close to homomorphisms if the associated K-theoretical data of these maps are consistent with those of homomorphisms. It is problematic when one tries to replace Mn by C([0, 1], Mn ). In addition to the usual K-theory and trace information, one also has to handle the maps from U (C)/CU(C) to U (A)/CU(A), where CU(C) and CU(A) are the closure of the subgroups of U (C) and U (A) generated by commutators, respectively. Other problems occur because of lack of projections in C ∗ -algebras which are not of real rank zero. The main theorem is stated as follows: Let C be a unital separable simple amenable C ∗ algebra with tracial rank one which satisfies the Universal Coefficient Theorem. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, a finite subset G ⊂ C and a finite subset P ⊂ K(C) satisfying the following: Suppose that A is a unital simple C ∗ -algebra with tracial rank no more than one, suppose that φ : C → A is a unital homomorphism and u ∈ U (A) such that φ(c), u < δ

for all c ∈ G

and Bott(φ, u)|P = 0.

(1.1)

Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} ⊂ A such that u(0) = u,

u(1) = 1 and φ(c), u(t) <

for all c ∈ F

(1.2)

and for all t ∈ [0, 1]. We also give the following Basic Homotopy Lemma in simple C ∗ -algebras with tracial rank one (see 6.3 below): Let > 0 and let : (0, 1) → (0, 1) be a non-decreasing map. We show that there exist δ > 0 and η > 0 (which does not depend on ) satisfying the following: Given any pair of unitaries u and v in a unital simple C ∗ -algebra A with tracial rank no more than one such that [v] = 0 in K1 (A), [u, v] < δ,

bott1 (u, v) = 0 and μτ ◦ı (Ia ) (a)

for all open arcs Ia with length a η, there exists a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ A such that


v(0) = v,

v(1) = 1 and u, v(t) <

1825

for all t ∈ [0, 1],

where ı : C(T) → A is the homomorphism defined by ı(f ) = f (u) for all f ∈ C(T) and μτ ◦ı is the Borel probability measure induced by the state τ ◦ ı. It should be noted that, unlike the case that A has real rank zero, the length of {v(t)} cannot be controlled. In fact, it could be as long as one wishes. In a subsequent paper [23], we use the main homotopy result (Theorem 8.4) of this paper and the results in [22] to establish a K-theoretical necessary and sufficient condition for homomorphisms from unital simple AH-algebras into a unital separable simple C ∗ -algebra with tracial rank no more than one to be asymptotically unitarily equivalent which, in turn, combining with a result of W. Winter, provides a classification theorem for a class of unital separable simple amenable C ∗ -algebras which properly contains all unital separable simple amenable C ∗ algebras with tracial rank no more than one which satisfy the UCT as well as some projectionless C ∗ -algebras such as the Jiang–Su algebra. 2. Preliminaries and notation 2.1. Let A be a unital C ∗ -algebra. Denote by T(A) the tracial state space of A and denote by Aff(T(A)) the set of affine continuous functions on T(A). Let C = C(X) for some compact metric space X and let L : C → A be a unital positive linear map. Denote by μτ ◦L the Borel probability measure induced by the state τ ◦ L, where τ ∈ T(A). 2.2. Let a and b be two elements in a C ∗ -algebra A and let > 0 be a positive number. We write a ≈ b if a − b < . Let L1 , L2 : A → C be two maps from A to another C ∗ -algebra C and let F ⊂ A be a subset. We write L1 ≈ L2

on F ,

if L1 (a) ≈ L2 (a) for all a ∈ F . Suppose that B ⊂ A. We write a ∈ B if there is an element b ∈ B such that a − b < . Let G ⊂ A be a subset. We say L is -G-multiplicative if, for any a, b ∈ G, L(ab) ≈ L(a)L(b) for all a, b ∈ G. 2.3. Let A be a unital C ∗ -algebra. Denote by U (A) the unitary group of A. Denote by U0 (A) the normal subgroup of U (A) consisting of those unitaries in the path connected component of U (A) containing the identity. Let u ∈ U0 (A). Define celA (u) = inf length u(t) : u(t) ∈ C [0, 1], U0 (A) , u(0) = u and u(1) = 1A . We use cel(u) if the C ∗ -algebra A is not in question. 2.4. Denote by CU(A) the closure of the subgroup generated by the commutators of U (A). For u ∈ U (A), we will use u¯ for the image of u in U (A)/CU(A). If u, ¯ v¯ ∈ U (A)/CU(A), define

1826


dist(u, ¯ v) ¯ = inf x − y: x, y ∈ U (A) such that x¯ = u, ¯ y¯ = v¯ . If u, v ∈ U (A), then dist(u, ¯ v) ¯ = inf uv ∗ − x : x ∈ CU(A) . 2.5. Let A and B be two unital C ∗ -algebras and let φ : A → B be a unital homomorphism. It is easy to check that φ maps CU(A) to CU(B). Denote by φ ‡ the homomorphism from U (A)/CU(A) into U (B)/CU(B) induced by φ. We also use φ ‡ for the homomorphism from U (Mk (A))/CU(Mk (A)) into U (Mk (B))/CU(Mk (B)) (k = 1, 2, . . .). 2.6. Let A and C be two unital C ∗ -algebras and let F ⊂ U (C) be a subgroup of U (C). Suppose that L : F → U (A) is a homomorphism for which L(F ∩ CU(C)) ⊂ CU(A). We will use L‡ : F /CU(C) → U (A)/CU(A) for the induced map. 2.7. Let A and B be as in 2.6, let 1 > > 0 and let G ⊂ A be a subset. Suppose that L is a -Gmultiplicative unital completely positive linear map. Suppose that u, u∗ ∈ G. Define L (u) = L(u)L(u∗ u)−1/2 . Definition 2.8. Let A and B be two unital C ∗ -algebras. Let h : A → B be a homomorphism and let v ∈ U (B) such that h(g)v = vh(g)

for all g ∈ A.

¯ ⊗ g) = h(f )g(v) for f ∈ A and Thus we obtain a homomorphism h¯ : A ⊗ C(S 1 ) → B by h(f 1 g ∈ C(S ). From the following splitting exact sequence: 0 → SA → A ⊗ C S 1 A → 0

(2.3)

and the isomorphisms Ki (A) → K1−i (SA) (i = 0, 1) given by Bott periodicity, one obtains two injective homomorphisms: β (0) : K0 (A) → K1 A ⊗ C S 1 , β (1) : K1 (A) → K0 A ⊗ C S 1 .

(2.4) (2.5)

Note, in this way, one can write Ki (A ⊗ C(S 1 )) = Ki (A) ⊕ β (1−i) (K1−i (A)). We use (i) : K (A ⊗ C(S 1 )) → β (1−i) (K β i 1−i (A)) for the projection to the summand β (1−i) (K1−i (A)). For each integer k 2, one also obtains the following injective homomorphisms: (i) βk : Ki (A, Z/kZ) → K1−i A ⊗ C S 1 , Z/kZ ,

i = 0, 1.

(2.6)

Thus we write (i) K1−i A ⊗ C S 1 , Z/kZ = K1−i (A, Z/kZ) ⊕ βk Ki (A, Z/kZ) ,

i = 0, 1.

(2.7)


1827

(i) (1−i) (i) , i = 1, 2. If Denote by βk : Ki (A ⊗ C(S 1 ), Z/kZ) → βk (K1−i (A, Z/kZ)) similarly to β (i) (i) x ∈ K(A), we use β(x) for β (x) if x ∈ Ki (A) and for βk (x) if x ∈ Ki (A, Z/kZ). Thus we β : K(A ⊗ C(S 1 )) → β(K(A)). Thus one have a map β : K(A) → K(A ⊗ C(S 1 )) as well as

may write K(A ⊗ C(S 1 )) = K(A) ⊕ β(K(A)). On the other hand h¯ induces homomorphisms h¯ ∗i,k : Ki (A ⊗ C(S 1 ), Z/kZ) → Ki (B, Z/kZ), (i) k = 0, 2, . . . , and i = 0, 1. We use Bott(h, v) for all homomorphisms h¯ ∗i,k ◦ βk . We write Bott(h, v) = 0, if h¯ ∗i,k ◦ βk(i) = 0 for all k 1 and i = 0, 1. We will use bott1 (h, v) for the homomorphism h¯ 1,0 ◦ β (1) : K1 (A) → K0 (B), and bott0 (h, u) for the homomorphism h¯ 0,0 ◦ β (0) : K0 (A) → K1 (B). Since A is unital, if bott0 (h, v) = 0, then [v] = 0 in K1 (B). For a fixed finite subset P ⊂ K(A), there exist δ > 0 and a finite subset G ⊂ A such that, if v ∈ B is a unitary for which h(a)v − vh(a) < δ

for all a ∈ G,

then Bott(h, v)|P is well defined. In what follows, whenever we write Bott(h, v)|P , we mean that δ is sufficiently small and G is sufficiently large so it is well defined. Now suppose that Ki (A) is finitely generated (i = 0, 1). For example, A = C(X), where X is a finite CW complex. When Ki (A) is finitely generated, Bott(h, v)|P0 defines Bott(h, v) for some sufficiently large finite subset P0 . In what follows such P0 may be denoted by PA . Suppose that P ⊂ K(A) is a larger finite subset, and G ⊃ G0 and 0 < δ < δ0 . A fact that we be used in this paper is that, Bott(h, v)|P defines the same map Bott(h, v) as Bott(h, v)|P0 defines, if h(a)v − vh(a) < δ

for all a ∈ G,

when Ki (A) is finitely generated. In what follows, in the case that Ki (A) is finitely generated, whenever we write Bott(h, v), we always assume that δ is smaller than δ0 and G is larger than G0 so that Bott(h, v) is well defined (see 2.10 of [13] for more details). 2.9. In the case that A = C(S 1 ), there is a concrete way to visualize bott1 (h, v). It is perhaps helpful to describe it here. The map bott1 (h, v) is determined by bott1 (h, v)([z]), where z is the identity map on the unit circle. Denote u = h(z) and define 1 − 2t, if 0 t 1/2, f e2πit = −1 + 2t, if 1/2 < t 1, (f (e2πit ) − f (e2πit )2 )1/2 , if 0 t 1/2, g e2πit = 0, if 1/2 < t 1 and

1828


0, h e2πit = (f (e2πit ) − f (e2πit )2 )1/2 ,

if 0 t 1/2, if 1/2 < t 1.

These are non-negative continuous functions defined on the unit circle. Suppose that uv = vu. Define b(u, v) =

f (v) g(v) + uh(v)

g(v) + h(v)u∗ 1 − f (v)

(2.8)

.

Then b(u, v) is a projection. There is δ0 > 0 (independent of unitaries u, v and A) such that if [u, v] < δ0 , the spectrum of the positive element p(u, v) has a gap at 1/2. The Bott element of u and v is an element in K0 (A) as defined in [9,8] which may be represented by bott1 (u, v) = χ[1/2,∞) b(u, v) −

1 0 0 0

(2.9)

.

Note that χ[1/2,∞) is a continuous function on sp(b(u, v)). Suppose that sp(b(u, v)) ⊂ (−∞, a] ∪ [1 − a, ∞) for some 0 < a < 1/2. Then χ[1/2,∞) can be replaced by any other positive continuous function F for which F (t) = 0 if t a and F (t) = 1 if t 1/2. Definition 2.10. Let A and C be two unital C ∗ -algebras. Let N : C+ \ {0} → N and K : C+ \ {0} → R+ \ {0} be two maps. Define T = N × K : C+ \ {0} → N × R+ \ {0} by T (c) = (N (c), K(c)) for c ∈ C+ \ {0}. Let L : C → A be a unital positive linear map. We say L is T -full if for any c ∈ C+ \ {0}, there are x1 , x2 , . . . , xN (c) ∈ A with xi K(c) such that N (c)

xi∗ L(c)xi = 1A .

i=1

Let H ⊂ C+ \ {0}. We say that L is T -H-full if N (c)

xi∗ L(c)xi = 1A

i=1

for all c ∈ H. Definition 2.11. Denote by I the class of unital C ∗ -algebras with the form where Xi = [0, 1] or Xi is one point.

m

i=1 C(Xi , Mn(i) ),

Definition 2.12. Let k 0 be an integer. Denote by Ik the class of all C ∗ -algebras B with the form B = P Mm (C(X))P , where X is a finite CW complex with dimension no more than k, P is a projection in Mm (C(X)). Recall that a unital simple C ∗ -algebra A is said to have tracial rank no more than k (write TR(A) k) if the following holds: For any > 0, any positive element a ∈ A+ \ {0} and any finite subset F ⊂ A, there exist a non-zero projection p ∈ A and a C ∗ -subalgebra B ∈ Ik with 1B = p such that


1829

(1) xp − px < for all x ∈ F ; (2) pxp ∈ B for all x ∈ F ; and (3) 1 − p is von Neumann equivalent to a projection in aAa. If TR(A) k and TR(A) = k − 1, we say A has tracial rank k and write TR(A) = k. It has been shown that if TR(A) = 1, then, in the above definition, one can replace B by a C ∗ -algebra in I (see [15]). All unital simple AH-algebra with slow dimension growth and real rank zero have tracial rank zero (see [5] and also [17]) and all unital simple AH-algebras with no dimension growth have tracial rank no more than one (see [10], or, Theorem 2.5 of [19]). Note that all AH-algebras satisfy the Universal Coefficient Theorem. There is unital separable simple C ∗ algebra A with TR(A) = 0 (and TR(A) = 1) which is not amenable. 3. Unitary groups The following is taken from an argument of N.C. Phillips [25]. Lemma 3.1. Let H > 0 be a positive number and let N 2 be an integer. Then, for any unital C ∗ -algebra A, any projection e ∈ A and any u ∈ U0 (eAe) with celeAe (u) < H , dist u + (1 − e), 1¯ < H /N,

(3.10)

if there are mutually orthogonal and mutually equivalent projections e1 , e2 , . . . , e2N ∈ (1 − e)A(1 − e) such that e1 is also equivalent to e. Proof. Since celeAe (u) < H , there are unitaries u0 , u1 , . . . , uN ∈ eAe such that u0 = u,

uN = 1 and ui − ui−1 < H /N,

We will use the fact that

v v 0 = 0 0 v∗ In particular,

v

0 0 v∗

0 1

0 1

1 0

v∗ 0

0 1

i = 1, 2, . . . , N.

(3.11)

0 1 . 1 0

is a commutator. Note that

u ⊕ u∗ ⊕ u1 ⊕ u∗ ⊕ · · · ⊕ u∗ ⊕ uN − u ⊕ u∗ ⊕ u1 ⊕ u∗ ⊕ · · · ⊕ u∗ ⊕ uN < H /N. N 1 2 1 N −1 (3.12) Since uN = 1, u ⊕ u∗ ⊕ u1 ⊕ u∗1 ⊕ · · · ⊕ u∗N −1 ⊕ uN is a commutator. Now we write u ⊕ e1 ⊕ · · · ⊕ e2N = u ⊕ u∗1 ⊕ u1 ⊕ · · · ⊕ u∗N ⊕ uN e ⊕ u1 ⊕ u∗1 ⊕ · · · ⊕ uN ⊕ u∗N . We obtain z ∈ CU((e +

2N

i=1 ei )A(e +

2N

i=1 ei ))

such that

u ⊕ e1 ⊕ · · · ⊕ e2N − z < H /N.

1830


It follows that dist u + (1 − e), 1¯ < H /N.

2

Definition 3.2. Let C = P Mk (C(X))P , where X is a compact metric space and P ∈ Mk (C(X)) is a projection. Let u ∈ U (C). Recall (see [27]) that Dc (u) = inf a: a ∈ Cs.a. such that det exp(ia) · u (x) = 1 for all x ∈ X . If no self-adjoint element a ∈ As.a. exists for which det(exp(ia) · u)(x) = 1 for all x ∈ X, define Dc (u) = ∞. Lemma 3.3. Let K 1 be an integer. Let A be a unital simple C ∗ -algebra with TR(A) 1, let e ∈ A be a projection and let u ∈ U0 (eAe). Suppose that w = u + (1 − e) and suppose η > 0. Suppose also that [1 − e] K[e]

in K0 (A)

¯ < η. and dist(w, ¯ 1)

(3.13)

Then, if η < 2, celeAe (u) <

Kπ + 1/16 η + 8π 2

and dist(u, ¯ e) ¯ < (K + 1/8)η,

and if η = 2, celeAe (u)
1 such that max{L/R1 , 2/R1 , ηπ/R1 } < min{η/64, 1/16π}. η > > 0 with + η < 2, since TR(A) 1, there exist a projection p ∈ A For any 32K(K+1)π ∗ and a C -subalgebra D ∈ I with 1D = p such that (1) [p, x] < for x ∈ {u, w, e, (1 − e)}; (2) pwp, pup, pep, p(1 − e)p ∈ D; (3) there is a projection q ∈ D and a unitary z1 ∈ qDq such that q − pep < , z1 − quq < , z1 ⊕ (p − q) − pwp < and z1 ⊕ (p − q) − c1 < + η;


1831

(4) there is a projection q0 ∈ (1 − p)A(1 − p) and a unitary z0 ∈ q0 Aq0 such that q0 − (1 − p)e(1 − p) < , z0 − (1 − p)u(1 − p) < , z0 ⊕ (1 − p − q0 ) − (1 − p)w(1 − p) < , z0 ⊕ (1 − p − q0 ) − c0 < + η; (5) [p − q] = K[q] in K0 (D), [(1 − p) − q0 ] = K[q0 ] in K0 (A); (6) 2(K + 1)R1 [1 − p] < [p] in K0 (A); (7) cel(1−p)A(1−p) (z0 ⊕ (1 − p − q0 )) L + , where c1 ∈ CU(D) and c0 ∈ CU((1 − p)A(1 − p)). Note that DD (c1 ) = 0 (see 3.2). Since + η < 2, there is h ∈ Ds.a. with h 2 arcsin( +η 2 ) such that (by (3) above) z1 ⊕ (p − q) exp(ih) = c1 .

(3.15)

DD z1 ⊕ (p − q) exp(ih) = 0.

(3.16)

It follows that

By (5) above and applying 3.3 of [27], one obtains that

DqDq (z1 ) K2 arcsin + η . 2

(3.17)

If 2K arcsin( +η 2 ) π , then 2K

+η π π. 2 2

It follows that K( + η) 2 dist(z1 , q).

(3.18)

Since those unitaries in D with det(u) = 1 (for all points) are in CU(D) (see, for example, 3.5 of [7]), from (3.17), one computes that, when 2K arcsin( +η 2 ) < π,

+η K( + η). dist(z1 , q) < 2 sin K arcsin 2

(3.19)

By combining both (3.18) and (3.19), one obtains that dist(z1 , q) K( + η) Kη +

η . 32(K + 1)π

(3.20)

By (3.17), it follows from 3.4 of [27] that π +η celqDq (z1 ) 2K arcsin + 6π K( + η) + 6π 2 2

π 1 K + η + 6π. 2 64(K + 1)

(3.21)

1832


By (5) and (6) above, (K + 1)[q] = [p − q] + [q] = [p] > 2(K + 1)R1 [1 − p]. Since K0 (A) is weakly unperforated, one has 2R1 [1 − p] < [q].

(3.22)

v ∗ (1 − p − q0 )v q.

(3.23)

v1∗ z0 ⊕ (1 − p − q0 ) v1 = z0 ⊕ v ∗ (1 − p − q0 )v.

(3.24)

z0 ⊕ v ∗ (1 − p − q0 )v v ∗ c∗ v1 − q0 ⊕ v ∗ (1 − p − q0 )v < + η.

(3.25)

There is a unitary v ∈ A such that

Put v1 = q0 ⊕ (1 − p − q0 )v. Then

Note that 1 0

Moreover, by (7) above, cel z0 ⊕ v ∗ (1 − p − q0 )v L + .

(3.26)

It follows from (3.22) and Lemma 6.4 of [19] that cel(q0 +q)A(q0 +q) (z0 ⊕ q) 2π + (L + )/R1 .

(3.27)

Therefore, combining (3.21), cel(q0 +q)A(q0 +q) (z0 + z1 )

1 π η + 6π. 2π + (L + )/R1 + K + 2 64(K + 1)

(3.28)

By (3.26), (3.22) and 3.1, in U0 ((q0 + q)A(q0 + q))/CU((q0 + q)A(q0 + q)), (L + ) . R1

(3.29)

(L + ) η < (K + 1/16)η. + Kη + R1 32(K + 1)π

(3.30)

and u − (z0 + z1 ) < 2.

(3.31)

dist(z0 + q, q0 + q) < Therefore, by (3.19) and (3.29), dist(z0 ⊕ z1 , q0 + q) < We note that e − (q0 + q) < 2


1833

It follows that dist(u, ¯ e) ¯ < 4 + (K + 1/16)η < (K + 1/8)η.

(3.32)

Similarly, by (3.28),

1 π η + 6π celeAe (u) 4π + 2π + (L + )/R1 + K + 2 64(K + 1)

π < K + 1/16 η + 8π. 2 This proves the case that η < 2. Now suppose that η = 2. Define R = [cel(w) + 1]. Note that e ∈ MR+1 (A) such that

cel(w) R

(3.33) (3.34)

< 1. There is a projection

(1 − e) + e = (K + RK)[e]. It follows from 3.1 that cel(w) . dist w ⊕ e , 1A + e < R+1

(3.35)

Put K1 = K(R + 1). To simplify notation, without loss of generality, we may now assume that [1 − e] = K1 [e]

¯ < and dist(w, ¯ 1)

cel(w) . R+1

(3.36)

It follows from the first part of the lemma that celeAe (u) <

1 cel(w) K1 π + + 8π 2 16 R + 1

Kπcel(w) 1 + + 8π. 2 16

(3.37) 2

(3.38)

Theorem 3.4. Let A be a unital simple C ∗ -algebra with TR(A) 1 and let e ∈ A be a non-zero projection. Then the map u → u + (1 − e) induces an isomorphism j from U (eAe)/CU(eAe) onto U (A)/CU(A). Proof. It was shown in Theorem 6.7 of [19] that j is a surjective homomorphism. So it remains to show that it is also injective. To do this, fix a unitary u ∈ eAe so that u¯ ∈ ker j . We will show that u ∈ CU(eAe). There is an integer K 1 such that K[e] [1 − e] in K0 (A). Let 1 > > 0. Put v = u + (1 − e). Since u¯ ∈ ker j , v ∈ CU(A). In particular,

1834


¯ < /(Kπ/2 + 1). dist(v, ¯ 1) It follows from Lemma 3.3 that dist(u, ¯ e) ¯ <

Kπ + 1/16 /(Kπ/2 + 1) < . 2

It then follows that u ∈ CU(eAe).

2

Corollary 3.5. Let A be a unital simple C ∗ -algebra with TR(A) 1. Then the map m j : a → diag(a, 1, 1, . . . , 1) from A to Mn (A) induces an isomorphism from U (A)/CU(A) onto U (Mn (A))/CU(Mn (A)) for any integer n 1. 4. Full spectrum One should compare the following with Theorem 3.1 of [28]. Lemma 4.1. Let X be a path connected finite CW complex, let C = C(X) and let A = C([0, 1], Mn ) for some integer n 1. For any unital homomorphism φ : C → A, any finite subset F ⊂ C and any > 0, there exists a unital homomorphism ψ : C → B such that φ(c) − ψ(c) <

for all c ∈ F

(4.39)

and ⎛ ψ(f )(t) = W (t)∗ ⎝

⎞

f (s1 (t)) ..

⎠ W (t),

.

(4.40)

f (sn ((t))) where W ∈ U (A), sj ∈ C([0, 1], X), j = 1, 2, . . . , n, and t ∈ [0, 1]. Proof. To simplify the notation, without loss of generality, we may assume that F is in the unit ball of C. Since X is also locally path connected, choose δ1 > 0 such that, for any point x ∈ X, B(x, δ1 ) is path connected. Put d = 2π/n. Let δ2 > 0 (in place of δ) be as required by Lemma 2.6.11 of [16] for /2. We will also apply Corollary 2.3 of [28]. By Corollary 2.3 of [28], there exists a finite subset H of positive functions in C(X) and δ3 > 0 satisfying the following: For any pair of points {xi }ni=1 and {yi }ni=1 , if {h(xi )}ni=1 and {h(yi )}ni=1 can be paired to within δ3 one by one, in increasing order, counting multiplicity, for all h ∈ H, then {xi }ni=1 and {yi }ni=1 can be paired to within δ3 /2, one by one. Put 1 = min{/16, δ1 /16, δ2 /4, δ3 /4}. There exists η > 0 such that f (t) − f t < 1 /2 for all f ∈ φ(F ∪ H)

(4.41)


1835

provided that |t − t | < η. Choose a partition of the interval: 0 = t0 < t1 < · · · < t N = 1 such that |ti − ti−1 | < η, i = 1, 2, . . . , N . Then φ(f )(ti ) − φ(f )(ti−1 ) < 1

for all f ∈ F ∪ H,

(4.42)

i = 1, 2, . . . , N . There are unitaries Ui ∈ Mn and {xi,j }nj=1 , i = 0, 1, 2, . . . , N , such that ⎛ f (x ) i,1 φ(f )(ti ) = Ui∗ ⎝

⎞ ..

⎠ Ui .

.

(4.43)

f (xi,n ) By the Weyl spectral variation inequality (see [1]), the eigenvalues of {h(xi,j )}nj=1 and {h(xi−1,j )}nj=1 can be paired to within δ3 , one by one, counting multiplicity, in decreasing order. It follows from Corollary 2.3 of [28] that {xi,j }nj=1 and {xi−1,j }nj=1 can be paired within δ3 /2. We may assume that, (4.44)

dist(xi,σi (j ) , xi−1,j ) < δ3 /2,

where σi : {1, 2, . . . , n} → {1, 2, . . . , n} is a permutation. By the choice of δ3 , there is a continuous path {xi−1,j (t): t ∈ [ti−1 , (ti + ti−1 )/2]} ⊂ B(xi−1 , δ3 /2) such that xi−1,j (ti−1 ) = xi−1,j

and xi−1,j (ti−1 + ti )/2 = xi,σi (j ) ,

(4.45)

j = 1, 2, . . . , n. Put ⎛ f (x (t)) i,1 ∗ ⎝ ψ(f )(t) = Ui−1

⎞ ..

⎠ Ui−1

.

(4.46)

f (xi,n (t)) for t ∈ [ti−1 , (ti−1 + ti )/2] and for f ∈ C(X). In particular, ⎛ f (x

i,σi (1) ) ti−1 + ti ∗ ⎝ = Ui−1 ψ(f ) 2

⎞ ..

⎠ Ui−1

.

(4.47)

f (xi,σi (n) )

for f ∈ C(X). Note that φ(f )(ti−1 ) − ψ(f )(t) < δ2 /4

and ψ(f )(t) − φ(f )(ti ) < δ2 /4 + 1 /2 < δ2 /2 (4.48)

for all f ∈ F and t ∈ [ti−1 , ti−12+ti ]. There exists a unitary Wi ∈ Mn such that

1836


ti−1 + ti Wi = φ(f )(ti ) Wi∗ ψ(f ) 2

(4.49)

for all f ∈ C(X). It follows from (4.48) and (4.49) that

Wi ψ(f ) ti−1 + ti − ψ(f ) ti−1 + ti Wi < δ2 2 2

(4.50)

for all f ∈ F . By the √ choice of δ2 and by applying Lemma 2.6.11 of [16], we obtain hi ∈ Mn such that Wi = exp( −1hi ) and

hi ψ(f ) ti−1 + ti − ψ(f ) ti−1 + ti hi < /4 2 2

(4.51)

√ √ exp( −1thi )ψ(f ) ti−1 + ti − ψ(f ) ti−1 + ti exp( −1thi ) < /4 2 2

(4.52)

and

for all f ∈ F and t ∈ [0, 1]. From this we obtain a continuous path of unitaries {Wi (t): t ∈ [ ti−12+ti , ti ]} ⊂ Mn such that Wi

ti−1 + ti 2

= 1,

Wi (ti ) = Wi

(4.53)

and

Wi (t)ψ(f ) ti−1 + ti − ψ(f ) ti−1 + ti Wi (t) < /4 2 2

(4.54)

for all f ∈ F and t ∈ [ ti−12+ti , ti ]. Define ψ(f )(t) = Wi∗ (t)ψ( ti−12+ti )Wi (t) for t ∈ [ ti−12+ti , ti ], i = 1, 2, . . . , N . Note that ψ : C(X) → A. We conclude that φ(f ) − ψ(f ) <

for all F .

(4.55)

Define U (t) = U0

t1 , for t ∈ 0, 2

U (t) = U0 W1 (t)

for t ∈

ti + ti+1 U (t) = U (ti ) for t ∈ ti , , 2 ti + ti+1 , ti+1 , U (t) = U (ti )Wi+1 (t) for t ∈ 2 i = 1, 2, . . . , N − 1 and define

t1 , t2 , 2

(4.56)

(4.57)


t1 t1 for t ∈ , t2 , 2 2

ti + ti+1 sj (t) = xi,σi (j ) (t) for t ∈ ti , , 2

ti + ti+1 ti + ti+1 for t ∈ , ti+1 , sj (t) = sj 2 2

sj (t) = x0,j (t)

t1 , for t ∈ 0, 2

sj (t) = sj

1837

(4.58)

(4.59)

i = 1, 2, . . . , N − 1. Thus U (t) ∈ A and, by (4.45), sj (t) ∈ C([0, 1], X). One then checks that ψ has the form: ⎛

⎞

f (s1 (t))

ψ(f ) = U (t)∗ ⎝

..

⎠ U (t)

.

(4.60)

f (sn (t)) for f ∈ C(X). In fact, for t ∈ [0, t1 ], it is clear that (4.60) holds. Suppose that (4.60) holds for t ∈ [0, ti ]. Then, by (4.49), for f ∈ C(X), ⎛ f (x ψ(f )(ti ) = U (ti )∗ ⎝

⎞

i,σi (1) )

..

⎛ f (x ) i,1 ∗⎝ = Ui

⎠ U (ti )

. f (xi,σi (n) ) ⎞

..

⎠ Ui .

.

(4.61)

f (xi,n ) Therefore, for t ∈ [ti , ti +t2i+1 ], ⎛ f (x (t)) i,1 ∗⎝ ψ(f )(t) = Ui ⎛ f (x = U (ti )∗ ⎝ ⎛ = U (t)∗ ⎝

⎞ ..

⎠ Ui

.

(4.62)

f (xi,n (t))

⎞

i,σi (1) (t))

..

⎠ U (ti )

.

(4.63)

f (xi,σi (n) (t)) ⎞

f (s1 (t)) ..

⎠ U (t).

.

(4.64)

f (sn (t)) For t ∈ [ ti +t2i+1 , ti+1 ],

ti + ti+1 Wi+1 (t) 2 ⎛ f (s1 ( ti +t2i+1 )) ⎜ = Wi+1 (t)∗ U (ti )∗ ⎝

ψ(f )(t) = Wi+1 (t)∗ ψ

(4.65) ⎞ ..

.

f (sn ( ti +t2i+1 ))

⎟ ⎠ U (ti )Wi+1 (t)

(4.66)

1838


⎛

⎞

f (s1 (t))

= U (t)∗ ⎝

..

⎠ U (t).

.

(4.67)

f (sn (t)) This verifies (4.60).

2

Lemma 4.2. Let X be a finite CW complex and let A ∈ I. Suppose that φ : C(X) ⊗ C(T) → A is a unital homomorphism. Then, for any > 0 and any finite subset F ⊂ C(X), there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in A such that u(0) = φ(1 ⊗ z),

u(1) = 1 and φ(f ⊗ 1), u(t) <

(4.68)

for f ∈ F and t ∈ [0, 1]. Proof. It is clear that the general case can be reduced to the case that A = C([0, 1], Mn ). Let q1 , q2 , . . . , qm be projections of C(X) corresponding to each path connected component of X. Since φ(qi )Aφ(qi ) ∼ = C([0, 1], Mni ) for some 1 ni n, i = 1, 2, . . . , we may reduce the general case to the case that X is path connected and A = C([0, 1], Mn ). Note that we use z for the identity function on the unit circle. For any > 0 and any finite subset F ⊂ C(X), by applying 4.1, one obtains a unital homomorphism ψ : C(X) ⊗ C(T) → A such that φ(g) − ψ(g) <

for all g ∈ {f ⊗ 1: f ∈ F } ∪ {1 ⊗ z}

(4.69)

and ⎛

⎞

f (s1 (t))

ψ(f )(t) = U (t)∗ ⎝

..

⎠ U (t),

.

(4.70)

f (sn (t)) for all f ∈ C(X × T), where U (t) ∈ U (C([0, 1], Mn )), sj : [0, 1] → X × T is a continuous map, j = 1, 2, . . . , n, and for all t ∈ [0, 1]. There are continuous paths of unitaries {uj (r): r ∈ [0, 1]} ⊂ C([0, 1]) such that uj (0)(t) = (1 ⊗ z) sj (t) ,

uj (1) = 1,

j = 1, 2, . . . , n.

(4.71)

Define ⎛ u (r)(t) j ∗⎝ u(r)(t) = U (t)

⎞ ..

⎠ U (t).

. un (r)(t)

Then u(r)ψ(f ⊗ 1) = ψ(f ⊗ 1)u(r) It follows that

for all r ∈ [0, 1].

(4.72)


φ(f ⊗ 1), u(r) <

for all r ∈ [0, 1] and for all f ∈ F .

1839

2

Definition 4.3. Let X be a compact metric space. We say that X satisfies property (H) if the following holds: For any > 0, any finite subsets F ⊂ C(X) and any non-decreasing map : (0, 1) → (0, 1), there exists η > 0 (which depends on and F but not ), δ > 0, a finite subset G ⊂ C(X) and a finite subset P ⊂ K(C(X)) satisfying the following: Suppose that φ : C(X) → C([0, 1], Mn ) is a unital δ-G-multiplicative contractive completely positive linear map for which μτ ◦φ (Oa ) (a)

(4.73)

for any open ball Oa with radius a η and for all tracial states τ of C([0, 1], Mn ), and [φ]|P = [Φ]|P ,

(4.74)

where Φ is a point-evaluation. Then there exists a unital homomorphism h : C(X) → C([0, 1], Mn ) such that φ(f ) − h(f ) <

(4.75)

for all f ∈ F . It is a restricted version of some relatively weakly semi-projectivity property. It has been shown in [22] that any k-dimensional torus has the property (H). So do those finite CW complexes X with torsion free K0 (C(X)) and torsion K1 (C(X)), any finite CW complexes with form Y × T where Y is contractive and all one-dimensional finite CW complexes. Theorem 4.4. Let X be a finite CW complex for which X × T has the property (H). Let C = C(X) and let : (0, 1) → (0, 1) be a non-decreasing map. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, η > 0 and there exists a finite subset G ⊂ C satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, φ : C → A is a unital homomorphism and u ∈ A is a unitary and suppose that φ(c), u < δ

for all c ∈ G

and Bott(φ, u) = {0}.

(4.76)

Suppose also that there exists a unital contractive completely positive linear map L : C ⊗ C(T) → A such that (with z the identity function on the unit circle) L(c ⊗ 1) − φ(c) < δ,

L(c ⊗ z) − φ(c)u < δ

for all c ∈ G

(4.77)

and μτ ◦L (Oa ) (a)

for all τ ∈ T (A)

and

(4.78)

for all open balls Oa of X × T with radius 1 > a η, where μτ ◦L is the Borel probability measure defined by L. Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in A such that

1840


u(0) = u,

u(1) = 1

and φ(c), u(t) <

(4.79)

for all c ∈ F and for all t ∈ [0, 1]. Proof. Let 1 (a) = (a)/2. Denote by z ∈ C(T) the identity map on the unit circle. Let B = C ⊗ C(T) = C(X × T). Put Y = X × T. Without loss of generality, we may assume that F is in the unit ball of C. Let F1 = {c ⊗ 1: c ∈ F } ∪ {1 ⊗ z}. Let η1 > 0 (in place of η), δ1 > 0 (in place of δ), G1 ⊂ C(Y ) (in place of G) be a finite subset, P1 ⊂ K(C(Y )) (in place of P) and U1 ⊂ U (C(Y )) be as required by Theorem 10.8 of [22] corresponding to /16 (in place of ), F1 and 1 (in place of ) above. Without loss of generality, we may assume that U1 = ζ1 ⊗ 1, . . . , ζK1 ⊗ 1, p1 ⊗ z ⊕ (1 − p1 ⊗ 1), . . . , pK2 ⊗ z ⊕ (1 − pK2 ⊗ 1) , (4.80) where ζk ∈ U (C), k = 1, 2, . . . , K1 and pj ∈ C is a projection, j = 1, 2, . . . , K2 . Denote zi = pi ⊗ z ⊕ (1 − pi ⊗ 1), i = 1, 2, . . . , K2 . We may also assume that U1 ⊂ U (Mk (C(Y ))). For any contractive completely positive linear map L from C(Y ), we will also use L for L ⊗ idMk . Fix a finite subset G2 ⊂ C(Y ) which contains G1 . Choose a small δ1 > 0. We choose G2 so large and δ1 so small that, for any δ1 -G2 -multiplicative map L form C(Y ) to a unital C ∗ and u , u , . . . , u in M (B ) such that algebra B , there are unitaries w1 , w2 , . . . , wK k K2 1 2 1 L (ζi ) − w < δ1 /16 and L (zj ) − u < δ1 /16, i j

(4.81)

i = 1, 2, . . . , K1 and j = 1, 2, . . . , K2 . Let η2 > 0 (in place of η), δ2 > 0 (in place of δ), G3 ⊂ C(Y ) (in place of G) be required by 10.7 of [22] for min{δ1 /16, δ1 /16, 1 (η1 )/16, /16} (in place of ), G1 ∪ F1 and 1 (in place of ) above. We may assume that G3 ⊃ G2 ∪ G1 ∪ F1 and G3 is in the unit ball of C(Y ). Moreover, we may further assume that G3 = {c ⊗ 1: c ∈ F2 } ∪ {z ⊗ 1, 1 ⊗ z, 1 ⊗ 1} for some finite subset F2 . Suppose that G ⊂ A is a finite subset which contains at least F2 . We may assume that δ2 < δ1 . δ

δ1 Let δ = min{ 16 , 161 , 14(η) }. Let A be a unital simple C ∗ -algebra with TR(A) 1, let φ : C(X) → A and u ∈ U0 (A) be such that

φ(c), u < δ

for all c ∈ G


(4.82)

We may assume that (4.77) holds for η = η1 /2. We also assume that there is a δ-G3 -multiplicative contractive completely positive linear map L : C(Y ) → A such that L(f ⊗ 1) − φ(f ) < δ, and

L(1 ⊗ z) − u < δ

(4.83)


μτ ◦L (Oa ) 21 (a)

for all a η,

1841

(4.84)

for all τ ∈ T(A) and for all f ∈ F2 . We will continue to use L for L ⊗ idMk . By (4.81), one may also assume that there are unitaries w1 , w2 , . . . , wK1 and unitaries u1 , u2 , . . . , uK2 such that L(ζi ⊗ 1) − wi < δ1 /16 and L(zj ) − uj < δ1 /16,

(4.85)

i = 1, 2, . . . , K1 and j = 1, 2, . . . , K2 . We note that, by (4.76), [ui ] = 0 in K1 (A). Put H = max cel(ui ): 1 i K2 . Let N 1 be an integer such that max{1, π, H + δ1 + δ} < δ/4 N

and

1 < 1 (η)/4. N

(4.86)

For each i, there are self-adjoint elements ai,1 , ai,2 , . . . , ai,L(i) ∈ A such that ui =

L(i)

√ exp( −1ai,j )

L(i)

and

j =1

ai,j H + δ/64,

(4.87)

i=1

i = 1, 2, . . . , K2 . Put Λ = max{L(i): 1 i K2 }. Let 0 > 0 such that if p a − ap < 0 for any self-adjoint element a and projection p , p exp(ia) − exp(ia)p < δ1 /16Λ. By applying Corollary 10.7 of [22], we obtain mutually orthogonal projections P0 , P1 , P2 ∈ A with P0 + P1 + P2 = 1 and a C ∗ -subalgebra D = sj =1 C(Xj , Mr(j ) ), where Xj = [0, 1] or Xj is a point, with 1D = P1 , a finite dimensional C ∗ -subalgebra D0 ⊂ A with 1D0 = P2 , a unital contractive completely positive linear map L0 : C(X) → D0 and there exists a unital homomorphism Φ : C(Y ) → D such that L(g) − P0 L(g)P0 + L0 (g) + Φ(g) < min δ1 , δ1 , 1 (η1 ) , for all g ∈ G2 16 16 4 16 (4.88) and (2N + 1)τ (P0 + P2 ) < τ (P1 )

for all τ ∈ T(A).

(4.89)

Moreover, [x, P0 ] < min 0 , δ1 /16, δ /16, 1 (η1 )/4, /16 1

for all x ∈ {ai,j : 1 j L(i), 1 i K2 } ∪ {L(G2 )}.

(4.90)

1842


There exists a unitary uj ∈ Mk (P0 AP0 ) and uj ∈ Mk (D0 ) such that u − P0 L(zj )P0 < δ1 /16 j

and uj − L0 (zj ) < δ1 /16

(4.91)

k

where P0 = diag(P0 , P0 , . . . , P0 ). It follows from (4.90) and (4.87) that [uj ] = 0 in K1 (A) and cel uj H + δ1 /16 + δ/64,

j = 1, 2, . . . , K2

(4.92)

(in Mk (P0 AP0 )). Note also, since uj ∈ Mk (D0 ), cel(uj ) π,

j = 1, 2, . . . , K2 .

By applying 4.2, there exists a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ D such that v(0) = Φ(1 ⊗ z),

v(1) = P1

and Φ(f ⊗ 1), v(t) < /4

(4.93)

for all t ∈ [0, 1]. Define a contractive completely positive linear map L1 : C(Y ) = C(X) ⊗ C(T) → A by L1 (f ⊗ 1) = P0 L(f ⊗ 1)P0 + L0 (f ⊗ 1) + Φ(f ⊗ 1)

(4.94)

L1 (1 ⊗ g) = g(1) · P0 + g(1) · P2 + Φ 1 ⊗ g(z)

(4.95)

and

for all f ∈ C(X) and g ∈ C(T). We compute (by choosing large G2 ) that μτ ◦L1 (Oa ) 1 (a)

for all a η

(4.96)

and τ ◦ L1 (g) − τ ◦ L(g) < δ

for all g ∈ G1

(4.97)

and (by the fact that Bott(φ, u) = {0}) [L]|P1 = [L1 ]|P1 .

(4.98)

We also have (by (4.83) and (4.88)) dist L‡ (ζi ⊗ 1), L‡1 (ζi ⊗ 1) < δ1 /16,

i = 1, 2, . . . , K1 .

(4.99)


1843

Moreover, for j = 1, 2, . . . , K2 , ∗ dist L‡ (zj ), L‡1 (zj ) < δ1 /16 + dist uj + uj + Φ(zj ) P0 + P2 + Φ(zj ) , 1¯ ∗ = δ1 /16 + dist uj + uj + P1 , 1¯ max{π, H + δ1 /16 + δ}/64 N < δ1 /16 + δ/2 < δ1 , < δ1 /16 +

(4.100) (4.101) (4.102) (4.103)

where the third inequality follows from (4.92) and 3.1. From (4.96), (4.97), (4.98), and (4.102), by applying Theorem 10.8 of [22], one obtains a unitary W ∈ A such that ad W ◦ L1 (g) − L(g) < /16

for all g ∈ {c ⊗ 1: c ∈ F ⊗ 1} ∪ {1 ⊗ z}.

(4.104)

Define u (t) = W ∗ P0 ⊕ v(t) W,

t ∈ [0, 1].

(4.105)

Then u (0) = W ∗ (P0 ⊕ Φ(1 ⊗ z))W and u (1) = 1. It follows from (4.93) and (4.104) that φ(c), u (t) < /2 for all c ∈ F

(4.106)

u (0) − u < /8.

(4.107)

and for t ∈ [0, 1]. Note that

One then obtains a continuous path {u(t): t ∈ [0, 1]} ⊂ A by connecting u (0) with u by a path with length no more than /2. The theorem follows. 2 Corollary 4.5. Let C = C(X, Mn ) where X = [0, 1] or X = T and : (0, 1) → (0, 1) be a nondecreasing map. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, η > 0 and there exists a finite subset G ⊂ C satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, φ : C → A is a unital monomorphism and u ∈ A is a unitary and suppose that φ(c), u < δ

for all c ∈ G,

(4.108)

bott0 (φ, u) = {0} and bott1 (φ, u) = {0}.

(4.109)

Suppose also that there exists a unital contractive completely positive linear map L : C ⊗ C(T) → A such that (with z the identity function on the unit circle) L(c ⊗ 1) − φ(c) < δ, and

L(c ⊗ z) − φ(c)u < δ

for all c ∈ G

1844


μτ ◦L (Oa ) (a)

(4.110)

for all open balls Oa of [0, 1] × T with radius 1 > a η, where μτ ◦L is the Borel probability measure defined by restricting L on the center of C ⊗ C(T). Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} such that u(0) = u,

u(1) = 1

and φ(c), u(t) <

(4.111)

for all c ∈ F and for all t ∈ [0, 1]. Corollary 4.6. Let C = C([0, 1], Mn ) and let T = N × K : (C ⊗ C(T))+ \ {0} → N × R+ \ {0} be a map. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, a finite subset H ⊂ (C ⊗ C(T))+ \ {0} and there exists a finite subset G ⊂ C satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, φ : C → A is a unital monomorphism and u ∈ A is a unitary and suppose that φ(c), u < δ

for all c ∈ G

(4.112)

and bott0 (φ, u) = {0}.

(4.113)

Suppose also that there exists a unital contractive completely positive linear map L : C ⊗ C(T) → A which is T -H-full such that (with z the identity function on the unit circle) L(c ⊗ 1) − φ(c) < δ

and L(c ⊗ z) − φ(c)u < δ

for all c ∈ G.

(4.114)

Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in A such that u(0) = u,

u(1) = 1

and φ(c), u(t) <

(4.115)

for all c ∈ F and for all t ∈ [0, 1]. Proof. Fix T = N × K : N × R+ \ {0}. Let : (0, 1) → (0, 1) be the non-decreasing map associated with T as in Proposition 11.2 of [22]. Let G ⊂ C, δ > 0 and η > 0 be as required by 4.5 for and F given and the above . It follows from 11.2 of [22] that there exists a finite subset H ⊂ (C ⊗ C(T))+ \ {0} such that for any unital contractive completely positive linear map L : C ⊗ C(T) → A which is T -H-full, one has that μτ ◦L (Oa ) (a) for all open balls Oa of X × T with radius a η. The corollary then follows immediately from 4.5.

2

(4.116)


1845

The following is an easy and known fact. Lemma 4.7. Let C = Mn . Then, for any > 0 and any finite subset F , there exist δ > 0 and a finite subset G ⊂ C satisfying the following: For any unital C ∗ -algebra A with K1 (A) = U (A)/U0 (A) and any unital homomorphism φ : C → A and any unitary u ∈ A if φ(c), u < δ

and bott0 (φ, u) = {0},

(4.117)

then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} ⊂ A such that u(0) = u,

u(1) = 1 and φ(c), u(t) <

(4.118)

for all c ∈ F and t ∈ [0, 1]. Proof. First consider the case that φ(c) commutes with u for all c ∈ C. Then one has a unital homomorphism Φ : Mn ⊗ C(T) → A defined by Φ(c ⊗ g) = φ(c)g(u) for all c ∈ C and g ∈ C(T). Let {ei,j } be a matrix unit for Mn . Let uj = ej,j ⊗ z, j = 1, 2, . . . , n. The assumption bott0 (φ, u) = {0} implies that Φ∗1 = {0}. It follows that uj ∈ U0 (A), j = 1, 2, . . . , n. One then obtains a continuous path of unitaries {u(t): t ∈ [0, 1]} ⊂ A such that u(0) = u,

u(1) = 1

and φ(c), u(t) = 0

for all c ∈ C(T) and t ∈ [0, 1]. The general case follows from the fact that C ⊗ C(T) is weakly semi-projective.

2

Remark 4.8. Let X be a compact metric space and let A be a unital simple C ∗ -algebra. Suppose that φ : C(X) → A is a unital injective completely positive linear map. Then it is easy to check (see 7.2 of [13], for example) that there exists a non-decreasing map : (0, 1) → (0, 1) such that μτ ◦φ (Oa ) (a) for all a ∈ (0, 1) and for all τ ∈ T(A). 5. Changing the spectrum Lemma 5.1. Let n 64 be an integer. Let > 0 and 1/2 > 1 > 0. There exist 2n > δ > 0 and a finite subset G ⊂ D ∼ M satisfying the following: = n Suppose that A is a unital C ∗ -algebra with T(A) = ∅, D ⊂ A is a C ∗ -subalgebra with 1D = 1A , suppose that F ⊂ A is a finite subset and suppose that u ∈ U (A) such that

[f, x] < δ

for all f ∈ F and x ∈ G,

(5.119)

and [u, x] < δ

for all x ∈ G.

(5.120)

1846


Then, there exist a unitary v ∈ D and a continuous path of unitaries {w(t): t ∈ [0, 1]} ⊂ D such that f, w(t) < nδ < /2

u, w(t) < nδ < ,

(5.121)

for all f ∈ F and for all t ∈ [0, 1], w(0) = 1,

w(1) = v

and μτ ◦ı (Ia )

(5.122) 2 3n2

(5.123)

for all open arcs Ia of T with length a 4π/n and for all τ ∈ T(A), where ı : C(T) → A is defined by ı(f ) = f (vu) for all f ∈ C(T). Moreover, length w(t) π.

(5.124)

If, in addition, π > b1 > b2 > · · · > bm > 0 and 1 = d0 > d1 > d2 · · · > dm > 0 are given so that μτ ◦ı0 (Ibi ) di

for all τ ∈ T(A), i = 1, 2, . . . , m,

(5.125)

where ı0 : C(T) → A is defined by ı0 (f ) = f (u) for all f ∈ C(T), then one also has that μτ ◦ı (Ici ) (1 − 1 )di

for all τ ∈ T(A),

(5.126)

where Ibi and Ici are any open arcs with length bi and ci , respectively, and where ci = bi + 1 , i = 1, 2, . . . , m. Proof. Let 1 d i 0 < δ0 < min : 1 i m . 16n2 Let {ei,j } be a matrix unit for D and let G = {ei,j }. Define v=

n

√

e2

−1j π/n

(5.127)

ej,j .

j =1 √

√

2 −1π/n | Let f1 ∈ C(T) with f1 (t) = 1 for |t − e2 −1π/n √ | < π/n and f1 (t) = 0 if |t − e 2 −1j π/n 2π/n and 1 f1 (t) 0. Define fj +1 (t) = f1 (e t), j = 1, 2, . . . , n − 1. Note that

√ fi e2 −1j π/n t = fi+j (t) where i, j ∈ Z/nZ.

for all t ∈ T

(5.128)


1847

Fix a finite subset F0 ⊂ C(T)+ which contains fi , i = 1, 2, . . . , n. Choose δ so small that the following hold: √

(1) there exists a unitary ui ∈ ei,i Aei,i such that e2 −1iπ/n ei,i uei,i − ui < δ02 /16n2 , i = 1, 2, . . . , n; (2) ei,j f (u) − f (u)ei,j < δ02 /16n2 for all f ∈ F0 ; √ (3) ei,i f (vu) − ei,i f (e2 −1iπ/n u) < δ02 /16n2 for all f ∈ F0 ; and ∗ f (u)e 2 2 (4) ei,j i,j − ej,j f (u)ej,j < δ0 /16n for all f ∈ F0 . Fix k. For each τ ∈ T(A), by (2), (3) and (4) above, there is at least one i such that τ ej,j fi (u) 1/n2 − δ02 /16n2 .

(5.129)

Choose j so that k + j = i mod (n). Then, τ fk (vu) τ ej,j fk (vu)

(5.130)

√ δ2 τ ej,j fk e2 −1j π/n u − 0 2 16n 2 δ 2δ02 1 . = τ ej,j fi (u) − 0 2 2 − 16n n 16n2

(5.131) (5.132)

It follows that √ 2δ02 1 μτ ◦ı B e2 −1kπ/n , π/n 2 − n 16n2

for all τ ∈ T(A)

(5.133)

and for k = 1, 2, . . . , n. It is then easy to compute that μτ ◦ı (Ia ) 2/3n2

for all τ ∈ T(A)

(5.134)

and for any open arc with length a 2(2π/n) = 4π/n. Note that if [x, ei,i ] < δ, then ! n λi ei,i < nδ < /2 x, i=1

! n and u, λi ei,i < nδ < /2 i=1

for any λi ∈ T. Thus, one obtains a continuous path {w(t): t ∈ [0, 1]} ⊂ D with length({w(t)}) π and with w(0) = 1 and w(1) = v so that (5.121) holds. Let {x1 , x2 , . . . , xK } be an 1 /64-dense set of T. Let Ii,j be an open arc with center xj and length bi , j = 1, 2, . . . , K and i = 1, 2, . . . , m. For each j and i, there is a positive function gj,i ∈ C(T)+ with 0 gj,i 1 and gj,i (t) = 1 if |t − xj | < di and√ gj,i (t) = 0 if |t − xj | di + 1 /64, j = 1, 2, . . . , K, i = 1, 2, . . . , m. Put gi,j,k (t) = gj,i (e2 −1kπ/n · t) for all t ∈ T, k = 1, 2, . . . , n. Suppose that F0 contains all gj,i and gj,i,k . We have, by (2), (3) and (4) above,

1848


di τ gj,i (u)el,l , τ gj,i,k (u)el,l − δ 2 /16n2 n


(5.135)

l = 1, 2, . . . , n, j = 1, 2, . . . , K and i = 1, 2, . . . , m. Thus √ δ2 τ ek,k gj,i (vu) τ ek,k gj,i e2 −1kπ/n u − n 0 2 16n

δ2 di − 0 n 8n

(5.136)


(5.137)

k = 1, 2, . . . , n, j = 1, 2, . . . , K and i = 1, 2, . . . , m. Therefore δ2 τ gj,i (vu) di − 0 (1 − 1 )di 8n


(5.138)

j = 1, 2, . . . , K and i = 1, 2, . . . , m. It follows that μτ ◦ı (Ii,j ) (1 − 1 )di


(5.139)

j = 1, 2, . . . , K and i = 1, 2, . . . , m. Since {x1 , x2 , . . . , xK } is 1 /64-dense in T, it follows that μτ ◦ı (Ici ) (1 − 1 )di

for all τ ∈ T(A), i = 1, 2, . . . , m.

2

(5.140)

Remark 5.2. If the assumption that [f, x] < δ for all f ∈ F and for all x ∈ G is replaced by for all x ∈ D with x 1, then the conclusion can also be strengthened to [f, w(t)] < δ for all f ∈ F and t ∈ [0, 1]. The proof of the following is similar to that of 5.1. Lemma 5.3. Let n 64 be an integer. Let > 0 and 1/2 > 1 > 0. There exist 2n > δ > 0 and a ∼ finite subset G ⊂ D = Mn satisfying the following: Suppose that X is a compact metric space, F ⊂ C(X) is a finite subset and 1 > b > 0. Then there exists a finite subset F1 ⊂ C(X) satisfying the following: Suppose that A is a unital C ∗ -algebra with T(A) = ∅, D ⊂ A is a C ∗ -subalgebra with 1D = 1A , φ : C(X) → A is a unital homomorphism and suppose that u ∈ U (A) such that

[x, u] < δ

and x, φ(f ) < δ

for all x ∈ G and f ∈ F1 .

(5.141)

Suppose also that, for some σ > 0, τ φ(f ) σ

for all τ ∈ T(A)

and

(5.142)

for all f ∈ C(X) with 0 f 1 whose support contains an open ball of X with radius b. Then, there exist a unitary v ∈ D and a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ D such that


1849

φ(f ), v(t) < nδ <

(5.143)

u, v(t) < nδ < ,

for all f ∈ F and t ∈ [0, 1], v(0) = 1,

(5.144)

v(1) = v

(5.145)

and 2σ τ φ(f )g(vu) 2 3n

for all τ ∈ T(A)

(5.146)

for any pair of f ∈ C(X) with 0 f 1 whose support contains an open ball with radius 2b and g ∈ C(T) with 0 g 1 whose support contains an open arc of T with length at least 8π/n. Moreover, length v(t) π.

(5.147)

If, in addition, 1 > b1 > b2 > · · · > bk > 0, 1 > d1 d2 · · · dk > 0 are given and τ φ f g (u) di

for all τ ∈ T(A)

(5.148)

for any functions f ∈ C(X) with 0 f 1 whose support contains an open ball of X with radius bi /2 and g ∈ C(T) with 0 g 1 whose support contains an arc with length bi , then one also has that τ φ f g (vu) (1 − 1 )di


(5.149)

where f ∈ C(X) with 0 f 1 whose support contains an open ball of radius ci and g ∈ C(T) with 0 g 1 whose support contains an arc with length 2ci with ci = bi + 1 , i = 1, 2, . . . , k. 1 di Proof. Let 0 < δ0 = min{ 16n 2 : i = 1, 2, . . . , k}. Let {ei,j } be a matrix unit for D and let G = {ei,j }. Define

v=

n

√

e2

−1j π/n

(5.150)

ej,j .

j =1 √

√

Let gj ∈ C(T) with gj (t) = 1 for |t − e2 −1j π/n | < π/n and gj (t) = 0 if |t − e2 −1j π/n | 2π/n and 1 gj (t) 0, j = 1, 2, . . . , n. As in the proof of 5.1, we may also assume that √ gi e2 −1j π/n t = gi+j (t)

for all t ∈ T

(5.151)

where i, j ∈ Z/nZ. Let {x1 , x2 , . . . , xm } be a b/2-dense subset of X. Define fi ∈ C(X) with fi (x) = 1 for x ∈ / B(xi , 2b) and 0 fi 1, i = 1, 2, . . . , m. B(xi , b) and fi (x) = 0 if x ∈

1850


Note that τ φ(fi ) σ

for all τ ∈ T(A), i = 1, 2, . . . , m.

(5.152)

Fix a finite subset F0 ⊂ C(T) which at least contains {g1 , g2 , . . . , gn } and a finite subset F1 ⊂ C(X) which at least contains F and {f1 , f2 , . . . , fm }. Choose δ so small that the following hold: √

(1) there exists a unitary ui ∈ ei,i Aei,i such that e2 −1iπ/n ei,i uei,i − ui < δ02 /16n4 , i = 1, 2, . . . , n; (2) ei,j g(u) − g(u)ei,j < δ02 /16n4 , ei,j φ(f ) − φ(f )ei,j < δ02 /16n4 , for f ∈ F1 and g ∈ F0 , j, k = 1, 2, . . . , n and√s = 1, 2, . . . , m; (3) ei,i g(vu) − ei,i g(e2 −1iπ/n u) < δ02 /16n4 for all g ∈ F0 ; and ∗ g(u)e 2 ∗ 2 4 4 (4) ei,j i,j − ej,j g(u)ej,j < δ0 /16n , ei,j φ(f )ei,j − ej,j φ(f )ej,j < δ0 /16n for all f ∈ F1 and g ∈ F0 , j, k = 1, 2, . . . , n and s = 1, 2, . . . , m. It follows from (4) that, for any k0 ∈ {1, 2, . . . , m}, τ φ(fk0 )ej,j σ/n − nδ02 /16n4 .

(5.153)

Fix k0 and k. For each τ ∈ T(A), there is at least one i such that τ φ(fk0 )ej,j gi (u) σ/n2 − δ02 /16n4 .

(5.154)

Choose j so that k + j = i mod (n). Then, √ δ2 τ φ(fk0 )gk (vu) τ φ(fk0 )ej,j gk e2 −1j π/n u − 0 4 16n δ2 = τ φ(fk0 )ej,j gi (u) − 0 4 16n

2δ02 σ − n2 16n4


(5.155) (5.156) (5.157)

It is then easy to compute that 2σ τ φ(f )g(vu) 2 3n

for all τ ∈ T(A)

(5.158)

and for any pair of f ∈ C(X) with 0 f 1 whose support contains an open ball with radius 2b and g ∈ C(T) with 0 g 1 whose support contains an open arc of length at least 8π/n. Note that if [φ(f ), ei,i ] < δ, then ! n λi ei,i < nδ < φ(f ), i=1


1851

for any λi ∈ T and f ∈ F1 . We then also require that δ < /2n. Thus, one obtains a continuous path {v(t): t ∈ [0, 1]} ⊂ D with length({v(t)}) π and with v(0) = 1 and v(1) = v so that the second part of (5.143) holds. Now we consider the last part of the lemma. Note also that, if f ∈ F1 and g ∈ F0 with 0 f, g 1, n δ2 τ φ(f )g(vu) τ φ(f )ej,j g(vu) − 0 3 16n

(5.159)

j =1

n δ2 τ φ(f )ej,j g (j ) (vu) − 0 2 16n


(5.160)

j =1

√

where g (j ) (t) = g(e2 −1j π/n · t) for t ∈ T. If the support of f contains an open ball with radius bi /2 and that of g contains open arcs with length at least bi , so does that of g (j ) . So, if F0 and F1 are sufficiently large, by the assumptions of the last part of the lemma, as in the proof of 5.1, we have δ2 τ φ(f )g(vu) di − 0 2 16n

for all τ ∈ T(A)

(5.161)

for all τ ∈ T(A). As in the proof of 5.1, this lemma follows when we choose F0 and F1 large enough to begin with. 2 Lemma 5.4. Let C be a unital separable simple C ∗ -algebra with TR(C) 1 and let n 1 be an integer. For any > 0, η > 0, any finite subset F ⊂ C, there exist δ > 0, a projection p ∈ A and a C ∗ -subalgebra D ∼ = Mn with 1D = p such that [x, p] <

[pxp, y] <

for all x ∈ F ;

for all x ∈ F and y ∈ D with y 1

(5.162) (5.163)

and τ (1 − p) < η

for all τ ∈ T (C).

(5.164)

Proof. Choose an integer N 1 such that 1/N < η/2n and N 2n. It follows from (the proof of) Theorem 5.4 of [19] that there is a projection q ∈ C and there exists a C ∗ -subalgebra B of C L with 1B = q and B ∼ = i=1 MKi with Ki N such that [x, q] < η/4

for all x ∈ F ; [qxq, y] < /4 for all x ∈ F and y ∈ B with y 1

(5.165)

τ (1 − q) < η/2n for all τ ∈ T (C).

(5.167)

(5.166)

and

1852


Write Ki = ki n + ri with ki 1 and 0 ri < n for some integers ki and ri , i = 1, 2, . . . , L. Let p ∈ B be a projection such that the rank of p is ki in each summand MKi of B. Take D1 = pBp. We have [x, p] < /2 for all x ∈ F ; [pxp, y] < for all x ∈ F , y ∈ D1 with y 1

(5.168) (5.169)

and τ (1 − p) < η/2n + n/N < η/2n + η/2 < η


Note that there is a unital C ∗ -subalgebra D ⊂ D1 such that D ∼ = Mn .

(5.170)

2

Lemma 5.5. Let n 1 be an integer with n 64. Let > 0 and 1/2 > 1 > 0. Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, suppose that F ⊂ A is a finite subset and suppose that u ∈ U (A). Then, for any > 0, there exist a unitary v ∈ A and a continuous path of unitaries {w(t): t ∈ [0, 1]} ⊂ A such that x, w(t) <

for all f ∈ F and for all t ∈ [0, 1],

w(0) = 1,

(5.171)

w(1) = v

(5.172)

15 24n2

(5.173)

and μτ ◦ı (Ia )

for all open arcs Ia of T with length a 4π/n and for all τ ∈ T(A), where ı : C(T) → A is defined by ı(f ) = f (vu). Moreover, length w(t) π.

(5.174)

If, in addition, π > b1 > b2 > · · · > bm > 0 and 1 = d0 > d1 > d2 · · · > dm > 0 are given so that μτ ◦ı0 (Ibi ) di

for all τ ∈ T(A), i = 1, 2, . . . , m,

(5.175)

where ı0 : C(T) → A is defined by ı0 (f ) = f (u) for all f ∈ C(T), then one also has that μτ ◦ı (Ici ) (1 − 1 )di


(5.176)

where Ibi and Ici are any open arcs with length bi and ci , respectively, and where ci = bi + 1 , i = 1, 2, . . . , m.


1853

Proof. Let > 0, and let n 64 be an integer. Put 2 = min{1 /16, 1/64n2 }. Let F ⊂ A be a finite subset and let u ∈ U (A). Let δ1 > 0 (in place of δ) be as in 5.1 for , 2 (in place of 1 ) and let G = {ei,j } ⊂ D ∼ = Mn be as required by 5.1. Put δ = δ1 /16. By applying 5.4, there is a projection p ∈ A and a C ∗ -subalgebra D ∼ = Mn with 1D = p such that [x, p] < δ

[pxp, y] < δ

for all x ∈ F ;

(5.177)

for all x ∈ F and y ∈ D with y 1

(5.178)

and τ (1 − p) < 2


(5.179)

There is a unitary u0 ∈ (1 − p)A(1 − p) and a unitary u1 ∈ pAp. Put A1 = pAp and F1 = {pxp: x ∈ F }. We apply 5.1 to A1 , F1 and u1 . We check that the lemma follows. 2 The proof of the following lemma follows the same argument using 5.4 as in that of 5.5 but one applies 5.3 instead of 5.1. Lemma 5.6. Let n 64 be an integer. Let > 0 and 1/2 > 1 > 0. Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, X is a compact metric space, φ : C(X) → A is a unital homomorphism, F ⊂ C(X) is a finite subset and suppose that u ∈ U (A). Suppose also that, for some σ > 0 and 1 > b > 0, τ φ(f ) σ

for all τ ∈ T(A)

and

(5.180)

for all f ∈ C(T) with 0 f 1 whose supports contain an open ball with radius at least b. Then, there exist a unitary v ∈ A and a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ A such that v(0) = 1, v(1) = v, φ(f ), v(t) <

and u, v(t) <

15σ τ φ(f )g(vu) 24n2

for all f ∈ F and t ∈ [0, 1],

for all τ ∈ T(A)

(5.181) (5.182)

for any f ∈ C(X) with 0 f 1 whose support contains an open ball of radius at least 2b and any g ∈ C(T) with 0 g 1 whose support contains an open arc of T with length a 8π/n. Moreover, length v(t) π.

(5.183)

If, in addition, 1 > b1 > b2 > · · · > bk > 0, 1 > d1 d2 · · · dk > 0 are given and τ φ f g (u) di

for all τ ∈ T(A)

(5.184)

for any functions f ∈ C(X) with 0 f 1 whose support contains an open ball with radius bi /2 and any function g ∈ C(T) with 0 g 1 whose support contains an arc with length bi , then one also has that

1854


τ φ f g (vu) (1 − 1 )di


(5.185)

where f ∈ C(X) with 0 f 1 whose support contains an open ball with radius ci and g ∈ C(T) with 0 g whose support contains an arc with length 2ci , where ci = bi + 1 , i = 1, 2, . . . , k. 5.7. Define 00 (r) =

1 2(n + 1)2

if 0
0 and F ⊂ C be as described. Put F1 = φ(F ). The corollary follows from 5.8 by taking u(t) = w(t)u. 2 The proof of the following lemma follows from the same argument used in that of 5.8 by applying 5.6 instead. Lemma 5.10. Let : (0, 1) → (0, 1) be a non-decreasing map, let η > 0, let X be a compact metric space and let F ⊂ C(X) be a finite subset. Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, suppose that φ : C(X) → A is a unital homomorphism and suppose that u ∈ U (A) such that μτ ◦φ (Oa ) (a)

for all τ ∈ T(A)

(5.204)


1857

for any open ball with radius a η. For any > 0, there exist a unitary v ∈ U0 (A) and a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ U0 (A) such that φ(f ), v(t) < ,

v(0) = 1, v(1) = v, u, v(t) < for all f ∈ F and t ∈ [0, 1]

(5.205) (5.206)

and τ φ(f )g(vu) D0 ()(a)

for all τ ∈ T(A)

(5.207)

for any f ∈ C(X) with 0 f 1 whose support contains an open ball with radius a 4η and any g ∈ C(T) with 0 g 1 whose support contains an open arc with length a 4η, where D0 () is defined in 5.7. 6. The Basic Homotopy Lemma for C(X) In this section we will prove Theorem 6.2 below. We will apply the results of the previous section to produce the map L which was required in Theorem 4.5 by using a continuous path of unitaries. Lemma 6.1. Let X be a compact metric space, let : (0, 1) → (0, 1) be a non-decreasing map, let > 0, let η > 0 and let F ⊂ C(X) be a finite subset. There exist δ > 0 and a finite subset G ⊂ C(X) satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, suppose that φ : C(X) → A and suppose that u ∈ U (A) such that φ(f ), u < δ

for all f ∈ G

(6.208)

and μτ ◦φ (Ob ) (a)

for all τ ∈ T(A)

(6.209)

for any open balls Ob with radius b η/2. There exist a unitary v ∈ U0 (A), a unital completely positive linear map L : C(X × T) → A and a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ U0 (A) such that φ(f ), v(t) < for all f ∈ F and t ∈ [0, 1], v(0) = u, v(1) = v, L(f ⊗ 1) − φ(f ) < for all f ∈ F L(f ⊗ z) − φ(f )v < ,

(6.210) (6.211)

and μτ ◦L (Oa ) (2/3)D0 ()(a/2)

for all τ ∈ T(A)

for any open balls Oa of X × T with radius a 5η, where D0 () is defined in 5.7.

(6.212)

1858


Proof. Fix > 0, η > 0 and a finite subset F ⊂ C(X). Let F1 ⊂ C(X) be a finite subset containing F . Let 0 = min{/2, D0 ()(η)/4}. Let G ⊂ C(X) be a finite subset containing F , 1C(X) and z. There is δ0 > 0 such that there is a unital completely positive linear map L : C(X × T) → B (for unital C ∗ -algebra B) satisfying the following: L (f ⊗ z) − φ (f )u < 0

for all f ∈ F1

(6.213)

for any unital homomorphism φ : C(X) → B and any unitary u ∈ B whenever φ (g), u < δ0

for all g ∈ G.

(6.214)

Let 0 < δ < min{δ0 /2, /2, 0 /2} and suppose that φ(g), u < δ

for all g ∈ G.

(6.215)

It follows from 5.10 that there is a continuous path of unitaries {z(t): t ∈ [0, 1]} ⊂ U0 (A) such that z(0) = 1, z(1) = v1 , u, z(t) < δ/2 φ(f ), z(t) < δ/2,

(6.216) for all t ∈ [0, 1]

(6.217)

and τ φ(f )g(v1 u) D0 ()(a)

(6.218)

for any f ∈ C(X) with 0 f 1 whose support contains an open ball with radius 4η and g ∈ C(T) with 0 g 1 whose support contains open arcs with length a 4η. Put v = v1 u. Then we obtain a unital completely positive linear map L : C(X × T) → A such that L(f ⊗ z) − φ(f )v < 0

and L(f ⊗ 1) − φ(f ) < 0

for all f ∈ F1 .

(6.219)

If F1 is sufficiently large (depending on η only), we may also assume that μτ ◦L (Ba × Ja ) (2/3)D0 ()(a/2) for any open ball Ba with radius a and open arcs with length a, where a 5η.

(6.220) 2

Theorem 6.2. Let X be a finite CW complex so that X × T has the property (H) (see 4.3). Let C = P C(X, Mn )P for some projection P ∈ C(X, Mn ) and let : (0, 1) → (0, 1) be a nondecreasing map. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, η > 0 and there exists a finite subset G ⊂ C satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, φ : C → A is a unital homomorphism and u ∈ A is a unitary and suppose that φ(c), u < δ

for all c ∈ G


(6.221)


1859

Suppose also that μτ ◦φ (Oa ) (a)

(6.222)

for all open balls Oa of X with radius 1 > a η, where μτ ◦φ is the Borel probability measure defined by restricting φ on the center of C. Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in A such that u(0) = u,

u(1) = 1 and φ(c), u(t) <

(6.223)

for all c ∈ F and for all t ∈ [0, 1]. Proof. First it is easy to see that the general case can be reduced to the case that C = C(X, Mn ). It is then easy to see that this case can be further reduced to the case that C = C(X). Then the theorem follows from the combination of 4.4 and 6.1. 2 Corollary 6.3. Let k 1 be an integer, let > 0 and let : (0, 1) → (0, 1) be any nondecreasing map. There exist δ > 0 and η > 0 (η does not depend on ) satisfying the following: For any k mutually commutative unitaries u1 , u2 , . . . , uk and a unitary v ∈ U (A) in a unital separable simple C ∗ -algebra A with tracial rank no more than one for which [ui , v] < δ,

bottj (ui , v) = 0,

j = 0, 1, i = 1, 2, . . . , k,

and μτ ◦φ (Oa ) (a)


for any open ball Oa with radius a η, where φ : C(Tk ) → A is the homomorphism defined by φ(f ) = f (u1 , u2 , . . . , uk ) for all f ∈ C(Tk ), there exists a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ A such that v(0) = v, v(1) = 1 and ui , v(t) <

for all t ∈ [0, 1], i = 1, 2, . . . , k.

Remark 6.4. In 6.3, if k = 1, the condition that bott0 (u1 , v) = 0 is the same as v ∈ U0 (A). Note that in Theorem 6.2, the constant δ depends not only on and the finite subset F but also depends on the measure distribution . As in Section 9 of [13], in general, δ cannot be chosen independent of . Unlike the Basic Homotopy Lemma in simple C ∗ -algebras of real rank zero, in Theorem 6.2 as well as in 6.3, the length of {u(t)} (or {vt }) cannot be possibly controlled. To see this, one notes that it is known (see [26]) that cel(A) = ∞ for some simple AH-algebras with no dimension growth. It is proved (see [10], or Theorem 2.5 of [19]) that all of these C ∗ -algebras A have tracial rank one. For those simple C ∗ -algebras, let k = 1. For any number L > π , choose u = v and v ∈ U0 (A) with cel(v) > L. This gives an example that the length of {vt } is longer than L. This shows that, in general, the length of {vt } could be as long as one wishes. However, we can always assume that the path {u(t): t ∈ [0, 1]} is piece-wise smooth. For example, suppose that {u(t): t ∈ [0, 1]} satisfies the conclusion of 6.2 for /2. There are 0 = t0 < t1 < · · · < tn = 1 such that

1860


u(ti ) − u(ti−1 ) < /32,

i = 1, 2, . . . , n.

There is a selfadjoint element hi ∈ A with hi /8 such that √ u(ti ) = u(ti−1 ) exp( −1hi ),

i = 1, 2, . . . , n.

Define

√ t − ti−1 hi for all t ∈ [ti−1 , ti ), −1 w(t) = u(ti−1 ) exp ti − ti−1 i = 1, 2, . . . , n. Note that φ(c), w(t) <

for all t ∈ [0, 1].

On the other hand, it is easy to see that w(t) is continuous and piece-wise smooth. 7. An approximate unitary equivalence result The following is a variation of some results in [11]. We refer to [11] for the terminologies used in the following statement. Theorem 7.1. (Cf. Theorem 1.1 of [11].) Let C be a unital separable amenable C ∗ -algebra satisfying the UCT. Let b 1, let T : N2 → N, L : U (M∞ (C)) → R+ , E : R+ × N → R+ and T1 = N × K : C+ \ {0} → N × R+ \ {0} be four maps. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, a finite subset G ⊂ C, a finite subset H ⊂ C+ \ {0}, a finite subset P ⊂ K(C), a finite subset U ⊂ U (M∞ (C)), an integer l > 0 and an integer k > 0 satisfying the following: For any unital C ∗ -algebra A with stable rank one, K0 -divisible rank T , exponential length divisible rank E and cer(Mm (A)) b (for all m), if φ, ψ : C → A are two unital δ-Gmultiplicative contractive completely positive linear maps with [φ]|P = [ψ]|P

and

cel φ (u)∗ ψ (u) L(u)

(7.224)

for all u ∈ U , then for any unital δ-G-multiplicative contractive completely positive linear map θ : C → Ml (A) which is also T -H-full, there exists a unitary u ∈ Mlk+1 (A) such that k

k

∗ u diag φ(a), θ (a), θ (a), . . . , θ (a) u − diag ψ(a), θ (a), θ (a), . . . , θ (a) <

(7.225)

for all a ∈ F . Proof. Suppose that the theorem is false. Then there exist 0 > 0 and a finite subset F ⊂ C such that there are a sequence of positive numbers {δn } with δn 0, an increasing sequence of finite subsets {Gn } whose union is dense in C, an increasing sequence of finite subsets "{Hn } ⊂ C+ \ {0} whose union is dense in C+ , a sequence of finite subsets {Pn } of K(C) with ∞ n=1 Pn = K(C),


1861

a sequence of finite subsets {Un } ⊂ U (M∞ (C)), two sequences of {l(n)} and {k(n)} of integers (with limn→∞ l(n) = ∞), a sequence of unital C ∗ -algebra An with stable rank one, K0 -divisible rank T , exponential length divisible rank E and cer(Mm (An )) b (for all m) and sequences {φn }, {ψn } of Gn -δn -multiplicative contractive completely positive linear maps from C into An with [φn ]|P = [ψn ]|P

and cel φ (u)ψ u∗ L(u)

(7.226)

u ∈ Un satisfying the following: inf sup v ∗ diag φn (a), Sn (a) v − diag ψn (a), Sn (a) : a ∈ F 0

(7.227)

where the infimum is taken among all unital T1 -Hn -full and δn -Gn -multiplicative contractive completely positive linear maps σn : C → Ml(n) (An ) and where k(n)

Sn (a) = diag σn (a), σn (a), . . . , σn (a) , (An ). and among allunitaries v in M #l(n)k(n)+1 ∞ A , B = B , Q(B) = B/B0 and π : B → Q(B) be the quotient map. Let B0 = ∞ n=1 n n=1 n Define Φ, Ψ : C → B by Φ(a) = {φn (a)} and Ψ (a) = {ψn (a)} for a ∈ C. Note that π ◦ Φ and π ◦ Ψ are homomorphism. For any u ∈ Um , since An has stable rank one, when n m, ∗ φn (u) ψn (u) ∈ U0 (An )

∗ and cel φn (u) ψn (u) L(u).

(7.228)

It follows #that, for all n m (by Lemma 1.1 of [11] for example), there is a continuous path {U (t) ∈ ∞ n=m An : t ∈ [0, 1]} such that U (0) = φn (u) nm

and U (1) = ψn (u) nm .

Since this holds for each m, it follows that (π ◦ Φ)∗1 = (π ◦ Ψ )∗1 .

(7.229)

It follows from (2) of Corollary 2.1 of [11] that K0 (B) =

K0 (Bn )

$ and K0 Q(B) = K0 (Bn )/ K0 (Bn ).

b

b

(7.230)

n

Then, by (7.226) and by using the fact that each Bn has stable rank one again, one concludes that (π ◦ Φ)∗0 = (π ◦ Ψ )∗0 .

(7.231)

Moreover, with the same argument, by (7.226) and by applying (2) of Corollary 2.1 of [11], [π ◦ Φ]|Ki (C,Z/kZ) = [π ◦ Ψ ]|Ki (C,Z/kZ) ,

k = 2, 3, . . . , and i = 0, 1.

(7.232)

1862


Since C satisfies the UCT, by [4], [π ◦ Φ] = [π ◦ Ψ ]

in KL C, Q(B) .

(7.233)

On the other hand, since each σn is δn -Gn -multiplicative and T1 -Hn -full, we conclude that π ◦ Σ is a full homomorphism, where Σ : C → B is defined by Σ(c) = {σn (c)} for c ∈ C. It follows from Theorem 3.9 of [18] that there exist an integer N and a unitary W¯ ∈ Q(B) such that N

∗ W¯ diag π ◦ Φ(c), π ◦ Σ(c), . . . , π ◦ Σ(c) W¯

(7.234)

N

− diag π ◦ Ψ (c), π ◦ π ◦ Σ(c), . . . , π ◦ Σ(c) < 0 /2

(7.235)

for all c ∈ F . There exists a unitary un ∈ An for each n such that π({un }) = W¯ . Therefore, by (7.236), for some large n0 0, N

∗ u diag φn (c), σn (c), . . . , σn (c) un n

(7.236)

N

− diag ψn (c), σn (c), . . . , σn (c) < 0 for all c ∈ F . This contradicts (7.227).

(7.237)

2

Remark 7.2. Suppose that U (C)/U0 (C) = K1 (C). Then, from the proof, one sees that it is enough to consider U ⊂ U (C). Theorem 7.3. Let C be a unital separable simple amenable C ∗ -algebra with TR(C) 1 satisfying the UCT and let D = C ⊗ C(T). Let T = N × K : D+ \ {0} → N+ × R+ \ {0}. Then, for any > 0 and any finite subset F ⊂ D, there exist δ > 0, a finite subset G ⊂ D, a finite subset H ⊂ D+ \ {0}, a finite subset P ⊂ K(C) and a finite subset U ⊂ U (D) satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1 and φ, ψ : D → A are two unital δ-G-multiplicative contractive completely positive linear maps such that φ, ψ are T -H-full, τ ◦ φ(g) − τ ◦ ψ(g) < δ

for all g ∈ G

(7.238)

for all τ ∈ T(A), [φ]|P = [ψ]|P

(7.239)

dist φ ‡ (w), ¯ ψ ‡ (w) ¯ 0 and a finite subset F ⊂ D. Fix T = N × K as given. Let T1 = N × 2K. Let 1 > δ1 > 0, G1 ⊂ D, H1 ⊂ D+ \ {0}, P1 ⊂ K(D), U1 ⊂ U (M∞ (D)), integer l and k as required by 7.1 for /8, F and T as well as for b = 2, T (n, m) = 1, L(u) = 2cel(u) + 8π + 1, E(l, k) = 8π + l/k. We may assume that δ1 < min{/8, 1/8π} and k 2. Without loss of generality, we may assume that G1 = {g ⊗ 1, g ∈ G0 } ∪ {1 ⊗ z}, where G0 ∈ C and z is the identity function on T, the unit circle. Note that K1 (D) = K1 (A) ⊕ K0 (A). It is clear that K1 (D) is generated by u ⊗ 1 and (p ⊗ z) + (1 − p) ⊗ 1 for u ∈ U (A) and projections p ∈ A. In particular, K1 (D) = U (D)/U0 (D). Thus (see Remark 7.2), we may assume that U1 ⊂ U (A). Since TR(C) 1, for any δ2 > 0, there exist a projection e ∈ C and a C ∗ -subalgebra C0 ∈ I with 1C0 = e and a contractive completely positive linear map j1 : C → C0 such that (1) [x, e] < δ2 for x ∈ G0 ; (2) dist(exe, j1 (x)) < δ2 /4 for x ∈ G0 ; and (3) (2kl + 1)τ (1 − e) < τ (e) and τ (1 − e) < δ2 /(2kl + 1) for all τ ∈ T (C). Put z0 = (1 − e) ⊗ z, z1 = e ⊗ z and j0 (c) = (1 − e)c(1 − e) for c ∈ C. We may also assume that δ2 < δ1 /4. Put G00 = j1 (G0 ). Thus dist(exe, G00 ) < δ/4

for all x ∈ G0 .

(7.242)

Let D0 = C0 ⊗ C(T). Let T = T |(D0 )+ \{0} . Let δ3 > 0 (in place δ), let G2 (in place of G) be a finite subset of D0 , let H2 ⊂ (D0 )+ \ {0}, let P2 (in place of P) be a finite subset of K(D0 ) and let U2 (in place of U ) be a finite subset of U (M∞ (D0 )) required by Theorem 11.5 of [22] for δ1 /4 (in place of ), G00 ∪ {z1 } (in place of F ) (and D0 in place of C). Here we identify e with 1D0 . Let J = j1 ⊗ idC(T) : D0 → D be the obvious embedding and J0 = j0 ⊗ idC(T) . Let P2 ∈ K(D) be the image of P2 under [J ]. Now let δ = min{δ2 /(8kl + 1), δ3 /(8kl + 1)}, G = G1 ∪ G2 ∪ {e, (1 − e)}. Here we also view G2 as a subset of D. Let H = H1 ∪ H2 , let P = P1 ∪ P2 and U = U1 ∪ {(e + j0 (u)): u ∈ U1 } ∪ {(1 − e) + v: v ∈ U2 }. Suppose that φ and ψ satisfy the assumptions of the theorem for the above G, H, P and U . Let φ = φ ◦ J , ψ = ψ ◦ J . There is a unitary u0 ∈ A such that u∗0 ψ (e)u0 = φ (e) = e0 ∈ A. Put A1 = e0 Ae0 . We have [ad u0 ◦ ψ ]|P2 = [φ ]|P2 and, for g ∈ G, t ◦ ad u0 ◦ ψ (g) − t ◦ φ (g)
0. Since TR(A ) 1, by Let G00 00 1 4 1 Lemma 5.5 of [19], there are mutually orthogonal projections q0 , q1 , q2 , . . . , q8kl+4 with [q0 ] multiplicative contractive [q1 ] and [q1 ] = [qi ], i = 1, 2, . . . , 8kl + 4, and there are unital δ4 -G00 completely positive linear maps L0 : D0 → q0 Aq0 and Li : D0 → qi Aqi (i = 1, 2, . . . , 8kl + 4) such that

φ ≈δ4 L0 ⊕ L1 ⊕ L2 ⊕ · · · ⊕ L8kl+4

on G00 ,

(7.246)

and there exists a unitary Wi ∈ (q1 + qi )A(q1 + qi ) such that ad Wi ◦ Li = L1 ,

i = 1, 2, . . . , 8kl.

(7.247)

we may also asSince φ is T -H-full, with sufficiently small δ4 and sufficiently large G00 8i+4 sume that each Li ◦ j1 is also T1 -H1 -full and δ4 < δ/4. Define Qi = j =4+8(i−1) qi , Q0 = 4 8i+4 i=1 qi and Φi = j =4+8(i−1) Li , i = 1, 2, . . . , kl. Note by (7.247), Φi are unitarily equivalent to Φ1 . Since K0 (A) is weakly unperforated (see Theorem 6.11 of [15]), we check that

[p0 + q0 + Q0 ] [Qi ] and 2[p0 + q0 + Q0 ] [Qi ], Put φ0 = φ ◦ J0 ⊕ L0 ◦ J ⊕ compute that

4

i=1 Li

i = 1, 2, . . . , kl.

◦ J and ψ0 = ψ ◦ J0 ⊕ L0 ◦ J ⊕

¯ ψ0‡ (w) ¯ < δ1 dist φ0‡ (w),

4

i=1 Li

for all w ∈ U.

(7.248)

◦ J . By 3.3, we

(7.249)

It follows from Lemma 6.9 of [19] that cel φ0 (u)ψ0 (u)∗ < 8π + 1

for all u ∈ U.

(7.250)

We also have [φ0 ]|P1 = [ψ0 ]|P1 .

(7.251)

Since Φ1 ◦ j is T1 -H1 -full, by applying 7.1, we obtain a unitary w ∈ U (A), kl

∗ w diag ψ0 (c), Φ1 ◦ J (c), . . . , Φ1 ◦ J (c) w kl

− diag φ0 (c), Φ1 ◦ J (c), . . . , Φ1 ◦ J (c) < /8

(7.252)


1865

for all c ∈ F . Since Φi ◦ j1 is unitarily equivalent to Φ1 ◦ j1 , there is a unitary w ∈ U (A) such that ∗ w diag ψ0 (c), Φ1 ◦ J (c), . . . , Φkl ◦ J (c) w − diag φ0 (c), Φ1 ◦ J (c), . . . , Φkl ◦ J (c) < /8

(7.253) (7.254)

for all c ∈ F . It follows that ∗ w diag ψ ◦ J0 (c), L0 ◦ J (c), φ (c) w − diag φ ◦ J0 (c), L0 ◦ J (c), φ (c) < /4

(7.255)

for all c ∈ F . Let u = ((1 − e0 ) ⊕ e0 u0 u1 )w . Then, by (7.245), we have ∗ u diag ψ ◦ J0 (c), ψ (c) u − diag φ ◦ J0 (c), φ (c) < /2

(7.256)

for all c ∈ F . It follows that ad u ◦ ψ ≈ φ

on F .

2

(7.257)

Corollary 7.4. Let C be a unital separable amenable simple C ∗ -algebra with TR(C) 1 which satisfies the UCT, let D = C ⊗ C(T) and let A be a unital simple C ∗ -algebra with TR(A) 1. Suppose that φ, ψ : D → A are two unital monomorphisms. Then φ and ψ are approximately unitarily equivalent, i.e., there exists a sequence of unitaries {un } ⊂ A such that lim ad un ◦ ψ(d) = φ(d)

n→∞

for all d ∈ D,

if and only if [φ] = [ψ] τ ◦φ =τ ◦ψ

in KL(D, A),

for all τ ∈ T(A)

and ψ ‡ = φ ‡ .

8. The Main Basic Homotopy Lemma Lemma 8.1. Let C be a unital separable simple C ∗ -algebra with TR(C) 1 and let : (0, 1) → (0, 1) be a non-decreasing map. There exists a map T = N × K : D+ \ {0} → N+ × R+ \ {0}, where D = C ⊗ C(T), satisfying the following: For any > 0, any finite subset F ⊂ C and any finite subset H ⊂ D+ \ {0}, there exist δ > 0, η > 0 and a finite subset G ⊂ C satisfying the following: for any unital separable unital simple C ∗ -algebra A, any unital homomorphism φ : C → A and any unitary u ∈ A such that φ(c), u < δ and

for all c ∈ G

(8.258)

1866


μτ ◦ı (Oa ) (a)

for all τ ∈ T(A)

(8.259)

and for all open balls Oa with radius a η, where ı : C(T) → A is defined by ı(f ) = f (u), there is a unital completely positive linear map L : D → A such that L(c ⊗ 1) − φ(c) < L(c ⊗ z) − φ(c)u <

for all c ∈ F

(8.260)

and L is T -H-full. Proof. We identify D with C(T, C). Let f ∈ D+ \ {0}. There is positive number b 1, g ∈ D+ with 0 g b · 1 and f1 ∈ D+ \ {0} with 0 f1 1 such that gf gf1 = f1 .

(8.261)

There is a point t0 ∈ T such that f1 (t0 ) = 0. There is r > 0 such that τ f1 (t) τ f1 (t0 ) /2 for all τ ∈ T (C) and for all t with dist(t, t0 ) < r. Define 0 (f ) = inf{τ (f1 (t0 ))/4: τ ∈ T (C)} · (r). There is an integer n 1 such that n · 0 (f ) > 1.

(8.262)

Define T (f ) = (n, b). Put η = inf 0 (f ): f ∈ H /2 and 1 = min{, η}. We claim that there exists an 1 -F ∪ H-multiplicative contractive completely positive linear map L : D → A such that L(c ⊗ 1) − φ(c) <

for all c ∈ F ,

L(1 ⊗ z) − u <

(8.263)

and % τ ◦ L(f1 ) − τ φ f1 (s) dμτ ◦ı (s) < η

for all τ ∈ T(A)

(8.264)

T

and for all f ∈ H. Otherwise, there exists a sequence of unitaries {un } ⊂ U (A) for which μτ ◦ın (Oa ) (a) for all τ ∈ T(A) and for any open balls Oa with radius a an with an → 0, and for which lim φ(c), un = 0

n→∞

(8.265)

for all c ∈ C and suppose for any sequence of contractive completely positive linear maps Ln : D → A with


lim Ln (ab) − Ln (a)Ln (b) = 0 for all a, b ∈ D, n→∞ lim Ln (c ⊗ f ) − φ(c)f (un ) = 0, n→∞

1867

(8.266) (8.267)

for all c ∈ C, f ∈ C(T) and % lim inf max τ ◦ Ln (f1 ) − τ φ f1 (s) dμτ ◦ın (s): f ∈ H η n

(8.268)

T

for some τ ∈ T(A), where ın : C(T) → D is defined by ın (f ) = f (un ) for f ∈ C(T) (or no contractive completely positive linear maps# Ln exists so that (8.266), #(8.267) and (8.267)). Put An = A, n = 1, 2, . . . , and Q(A) =# n An / n An . Let π : n An → Q(A) be the quotient map. Define a linear map L : D → n An by L(c ⊗ 1) = {φ(c)} and L (1 ⊗ z) = {un }. Then π ◦ L : D → Q(A) is a unital homomorphism. It follows from a theorem # of Effros and Choi [3] that there exists a contractive completely positive linear map L : D → n An such that π ◦ L = π ◦ L . Write L = {Ln }, where Ln : D → An is a contractive completely positive linear map. Note that lim Ln (a)Ln (b) − Ln (ab) = 0 for all ab ∈ D.

n→∞

# Fix τ ∈ T(A), # define tn : n An → C by t n ({dn }) = τ (dn ). Let t be a limit point of {tn }. Then t gives a state on n An . Note that if {dn } ∈ n An , then tm ({dn }) → 0. It follows that t gives a state t¯ on Q(A). Note that (by (8.267)) t¯ π ◦ L(c ⊗ 1) = τ φ(c) for all c ∈ C. It follows that t¯ π ◦ L(f ) =

%

t¯ π ◦ L f (s) ⊗ 1 dμt¯◦π◦L|1⊗C(T)

T

%

τ φ f (s) dμt¯◦π◦L|1⊗C(T)

=

(8.269)

T

for all f ∈ C(T, C). Therefore, for a subsequence {n(k)}, % τ ◦ Ln(k) (f1 ) − τ φ f1 (s) dμτ ◦ı (s) < η/2 n(k)

(8.270)

T

for all f ∈ H. This contradicts with (8.268). Moreover, from this, it is easy to compute that μt¯◦π◦L|1⊗C(T) (Oa ) (a) for all open balls Oa of t with radius 1 > a. This proves the claim.

1868


Note that %

τ ◦ φ f1 (s) dμτ ◦ı (s) τ φ f1 (t0 )/2 · (r)

T

for all τ ∈ T(A). It follows that τ L(f1 ) inf t f1 (t0 ) /2: t ∈ T (C) − η/2 (4/3)0 (f )

(8.271)

for all f ∈ H. By Corollary 9.4 of [22], there exists a projection e ∈ L(f1 )AL(f1 ) such that τ (e) 0 (f )


(8.272)

It follows from (8.262) that there exists a partial isometry w ∈ Mn (A) such that n

w ∗ diag( e, e, . . . , e )w 1A . Thus there x1 , x2 , . . . , xn ∈ A with xi 1 such that n

xi∗ exi 1.

(8.273)

xi∗ gf gxi 1.

(8.274)

i=1

Hence n i=1

It then follows that there are y1 , y2 , . . . , yn ∈ A with yi b such that n

yi∗ fyi = 1.

(8.275)

i=1

Therefore L is T -H-full.

2

Lemma 8.2. Let C be a unital separable amenable simple C ∗ -algebra with TR(C) 1 satisfying the UCT. For 1/2 > σ > 0, any finite subset G0 and any projections p1 , p2 , . . . , pm ∈ C. There is δ0 > 0, a finite subset G ⊂ C and a finite subset of projections P0 ⊂ C satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, φ : C → A is a unital homomorphism and u ∈ U0 (A) is a unitary such that φ(c), u < δ < δ0

for all c ∈ G ∪ G0

and bott0 (φ, u)|P0 = {0},

(8.276)

where P0 is the image of P0 in K0 (C). Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in A with u(0) = u and u(1) = w such that


φ(c), u(t) < 3δ

for all c ∈ G ∪ G0

1869

(8.277)

and wj ⊕ 1 − φ(pj ) ∈ CU(A),

(8.278)

where wj ∈ U0 (φ(pj )Aφ(pj )) and wj − φ(pj )wφ(pj ) < σ,

(8.279)

j = 1, 2, . . . , m. Moreover, cel wj ⊕ 1 − φ(pj ) 8π + 1/4,

j = 1, 2, . . . , m.

(8.280)

Proof. It follows from the combination of Theorem 4.8 and Theorem 4.9 of [7] and Theorem 10.10 of [19] that one may write C = limn→∞ (Cn , ψn ), where each Cn =

R(n) $

Pn,j C(Xn,j , Mr(n,j ) )Pn,j

j =1

and where Pn,i ∈ C(Xn,i , Mr(n,i) ) is a projection and Xn,i is a connected finite CW complex of dimension no more than two with torsion free K1 (C(Xn,i )) and K0 (C(Xn,j )) = Z ⊕ Z/s(j )Z (s(i) 1) and with positive cone {(0, 0) ∪ (m, x): m 1} (when s(j ) = 1, we mean K0 (C(Xn,j )) = Z), or Xn,i is a connected finite CW complex of dimension three with K0 (C(Xn,i )) = Z and torsion K1 (C(Xn,i )). Let d(j ) be the rank of Pn,j . It is known that one #R(n) may assume that d(j ) j =1 s(j ) + 6, j = 1, 2, . . . , R(n). This can be seen, for example, from Lemmas 2.2, 2.3 (and the proof of Theorem 2.1) of [6]. Without loss of generality, we may assume that G0 ⊂ ψn,∞ (Cn ) and that there are projections pi,0 ∈ Cn such that ψn,∞ (pi,0 ) = pi , i = 1, 2, . . . , m. Choose, for each j , mutually orthogonal (0) (0) rank one projections qj,0 , qj,1 ∈ Pn,j (C(Xn,j , Mr(n,j ) ))Pn,j such that (0) qj,0 = (1, 0)

and

(0) ¯ ∈ Z ⊕ Z/s(j )Z, qj,1 = (1, 1)

(0) (0) =ψ or qj,1 = 0, if K0 (C(Xn,j )) = Z, j = 1, 2, . . . , R(n). Put qj,i n,∞ (qj,i ) and qj,i = φ(qj,i ), ∗ i = 0, 1 and j = 1, 2, . . . , R(n). Clearly, in C, pk may be written as Wj Qj Wj , where Qj is a finite orthogonal sum of qj,0 and qj,1 , j = 1, 2, . . . , R(n). By choosing a sufficiently large G which contains G0 (and which contains Qj , qj,i as well as Wj , among other elements) and sufficiently small δ0 > 0, one sees that it suffices to show the case that {p1 , p2 , . . . , pm } ⊂ {qj,0 , qj,1 : j = 1, 2, . . . , R(n)}. Thus we obtain a finite subset G and δ0 so that when G ⊃ G and δ0 < δ0 one can make the assumption that {p1 , p2 , . . . , pm } ⊂ {qj,i , i = 0, 1, j = 1, 2, . . . , R(n)}. In particular, {qj,i , i = 0, 1, j = 1, 2, . . . , R(n)} ⊂ G . , q : j = 1, 2, . . . , R(n)}. Fix 0 < η < min{σ/4, δ /2, 1/16}. Note that Let G0 = G0 ∪ {qj,0 j,1 0 Pn,j is locally trivial in C(Xn,j , Mr(n,j ) ). Since TR(C) 1, it has (SP) (see [15]). It is then easy

1870


to find a projection ej ∈ ψn,∞ (Pn,j )Cψn,∞ (Pn,j ) and Bj ∼ = Md(j ) ⊂ ψn,∞ (Pn,j )Cψn,∞ (Pn,j ) with 1Bj = ej such that [x, ej ] < η dist(ej xej , Bj ) < η

for all x

for all x ∈ G0 ,

∈ G0

and

(8.281)

ej qj,1 ej , ej qj,0 ej

= 0,

(8.282)

∈ B with rank j = 1, 2, . . . , R(n). Furthermore, one may require that there is a projection q¯j,i j one in Bj such that

q¯ − ej q ej < 2η, j,i j,i

i = 0, 1, j = 1, 2, . . . , R(n).

(8.283)

by one of its nearby projections, we may assume To simplify notation further, by replacing qj,i = q¯ + (q − q¯ ) and q q¯ , i = 0, 1 and j = 1, 2, . . . , R(n). Since s(j )[q ] = that qj,i j,i j,i j,i j,i j,i j,1 ], there is a unitary Y ∈ P C(X , M s(j )[qj,0 j n,j n,j r(n,j ) )Pn,j such that s(j )

s(j )+3

Yj∗ diag qj,1 Yj = diag qj,0 . , qj,1 , . . . , qj,1 , qj,0 , qj,0 , qj,0 , qj,0 , . . . , qj,0 (Note that d(j )

#R(n) j =1

(j )

(8.284)

has rank one.) s(j ) + 6 and each qj,i

Let {ei,k } be a matrix unit for Bj , j = 1, 2, . . . , R(n). We choose a finite subset G which , q¯ and {Y , Y ∗ }, j = 1, 2, . . . , R(n). Suppose that v contains G0 as well as {ei,k }, q¯j,0 j j,0 ∈ j j,1 (j )

(j )

(j )

U0 (φ(e1,1 )Aφ(e1,1 )) and d(j )

vj = diag( vj,0 , vj,0 , . . . , vj,0 ),

j = 1, 2, . . . , R(n).

(8.285)

for all x ∈ Bj , j = 1, 2, . . . , R(n).

(8.286)

Then φ(x)vj = vj φ(x) Choose P0 =

(j ) qj,0 , qj,1 , ei,i , q¯j,0 , q¯j,1 , j = 1, 2, . . . , R(n) .

), i = 0, 1 and j = 1, 2, . . . , R(n). We choose δ > 0 such that bott (φ, u)| Put q¯j,i = φ(q¯i,j 0 P0 is 0 well defined which is zero and there is a unitary uj,i ∈ U0 (q¯j,i Aq¯j,i ) such that

u − q¯j,i uq¯j,i < 2δ , j,i 0

i = 0, 1, j = 1, 2, . . . , R(n),

whenever, [φ(c), u] < δ0 for all c ∈ G. Let δ0 = {1/32, δ0 /4, δ0 /4, σ/8}. Suppose that (8.276) holds for the above G, P0 and 0 < δ < δ0 . One obtains a unitary uj,i ∈ U0 (q¯j,i Aq¯j,i ) and a unitary uj,i ∈ U0 ((qj,i − q¯j,i )A(qj,i − q¯j,i )) such that


uj,i − qj,i uqj,i < 2δ,

1871

(8.287)

where uj,i = uj,i + uj,i , i = 0, 1 and j = 1, 2, . . . , R(n). It follows 3.4 (see also Theorem 6.6 (j )

(j )

of [19]) that there is vj,0 ∈ U0 (φ(e1,1 )Aφ(e1,1 )) such that &

'' d(j ) (j ) d(j ) vj,0 + 1 − = u∗j,0 , φ ei,i &

in U0 (A)/CU(A), j = 1, 2, . . . , R(n).

i=2

(8.288) Put vj as in (8.285). It follows from (8.288) that ¯ vj ⊕ 1 − φ(ej ) uj,0 ⊕ (1 − qj,0 ) = 1,

(8.289)

j = 1, 2, . . . , R(n). Since vj,0 ∈ U0 (φ(ej )Aφ(ej )), one has a continuous path of unitaries (j ) {vj,0 (t): t ∈ [0, 1]} such that vj,0 (0) = φ(e1,1 ) and vj,0 (1) = vj,0 , j = 1, 2, . . . , R(n). Put d(j )

vj (t) = diag vj,0 (t), vj,0 (t), . . . , vj,0 (t) ,

j = 1, 2, . . . , R(n).

It follows that φ(x)vj (t) = vj (t)φ(x)

for all x ∈ Bj

(8.290)

and t ∈ [0, 1], j = 1, 2, . . . , R(n). Put u(t) =

&R(n)

& vj (t) + 1 −

j =1

R(n)

'' φ(ej )

u for t ∈ [0, 1].

j =1

Note that, u(0) = u and, if η is sufficiently small, φ(c), u(t) < 2(δ + η) < 3δ

for all c ∈ G.

(8.291)

Put w = u(1),

wj,0 = vj ⊕ 1 − φ(ej ) uj,0 ⊕ (1 − qj,0 )

(8.292)

and wj,1 = vj ⊕ 1 − φ(ej ) uj,1 ⊕ (1 − qj,1 ) ,

wj,i = vj uj,i ,

(8.293)

i = 0, 1, and j = 1, 2, . . . , R(n). Define w¯ j = vj ⊕ 1 − φ(ej ) uj,0 ⊕ uj,1 ⊕ (1 − qj,0 − qj,1 ) .

(8.294)

1872


We have that w¯ j qj,i = wj,i qj,i = vj uj,i = wj,i = qj,i wj,i ,

i = 0, 1 and j = 1, 2, . . . , R(n). Note that, by (8.290), (8.281) and (8.283), wj,i qj,i − qj,i wqj,i < σ

(8.295)

i = 0, 1, j = 1, 2, . . . , R(n). By (8.289), wj,0 ∈ CU(A),

j = 1, 2, . . . , R(n).

(8.296)

It follows from Lemma 6.9 of [19] that cel(wj,0 ) 8π + 1/4.

(8.297)

Put s(j )

Ej = 1 − φ Yj∗ diag qj,1 Yj . , qj,1 , . . . , qj,1 , qj,0 , qj,0 , qj,0 It follows from (8.284) that in U0 (A)/CU(A), s(j )

3 = diag(w wj,1 wj,0 ¯ j qj,1 , w¯ j qj,1 , . . . , w¯ j qj,1 , wj,0 , wj,0 , wj,0 ) ⊕ Ej

(8.298)

s(j )

∗

= φ Yj diag(w¯ j qj,1 , w¯ j qj,1 , . . . , w¯ j qj,1 , w¯ j qj,0 , w¯ j qj,0 , w¯ j qj,0 )φ(Yj ) ⊕ Ej s(j )

(8.299) ¯ = diag(wj,0 , wj,0 , . . . , wj,0 ) ⊕ Ej = 1,

(8.300)

s(j )+3

where j = 1, 2, . . . , R(n). By (8.296), the above implies that s(j )

wj,1 = 1,

j = 1, 2, . . . , R(n).

(8.301)

It follows from Theorem 6.11 that wj,1 ∈ CU(A),

j = 1, 2, . . . , R(n).

(8.302)

It follows from Lemma 6.9 of [19] that cel(wj,1 ) 8π + 1/4,

j = 1, 2, . . . , R(n).

2


1873

Lemma 8.3. Let C be a unital separable simple amenable C ∗ -algebra with TR(C) 1 satisfying the UCT. Let : (0, 1) → (0, 1) be a non-decreasing map. Then, for any > 0 and any finite subset F ⊂ C, there exist δ > 0, η > 0, a finite subset G ⊂ C and a finite subset P ⊂ K(C) satisfying the following: For any unital simple C ∗ -algebra A with TR(A) 1, any unital homomorphism φ : C → A and any unitary u ∈ U (A) with φ(f ), u < δ,

Bott(φ, u)|P = {0}

(8.303)

and μτ ◦ı (Oa ) (a)

for all a η,

(8.304)

where ı : C(T) → A is defined by ı(f ) = f (u) for all f ∈ C(T), there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} ⊂ A such that u(0) = u,

u(1) = 1 and φ(f ), u(t) <

(8.305)

for all f ∈ F and t ∈ [0, 1]. Proof. Let 1 : (0, 1) → (0, 1) be defined by 1 (a) = (a)/2 for all a ∈ (0, 1). Put D = C ⊗ C(T). Let T = N × K : D+ \ {0} → N × R+ \ {0} be associated with as in 8.1 and T = N × K : D+ \ {0} → N × R+ \ {0} be associated with 1 as in 8.1. Let N1 = max{N, N } and K1 = max{K, K }. Define T0 (h) = N1 (h) × K1 (h) for h ∈ D+ \ {0}. Let > 0 and F ⊂ C be a finite subset. Let F1 = {f ⊗ g: f ∈ F ∪ {1C }, g ∈ {z, 1C(T) }}. Let δ1 > 0 (in place of δ), G1 ⊂ D (in place of G), H0 ⊂ D+ \ {0}, P1 ⊂ K(D) (in place of P) and U ⊂ U (D) be as required by 7.3 for /256 (in place of ), F1 and T0 (in place of T ). We may assume that δ1 < /256. To simplify notation, without loss of generality, we may assume that H0 is in the unit ball of D and G1 = {c ⊗ g: c ∈ G1 and g = 1C(T) , g = z}, where 1C ∈ G1 is a finite subset of C. Without loss of generality, we may assume that U = U1 ∪ {z1 , z2 , . . . , zn }, where U1 ⊂ {w ⊗ 1C(T) : w ∈ U (C)} is a finite subset and zi = qi ⊗ z ⊕ (1 − qi ) ⊗ 1C(T) , i = 1, 2, . . . , n and {q1 , q2 , . . . , qn } ⊂ C is a set of projections. We write K(D) = K(C) ⊕ β(K(C)) (see 2.8). Without loss of generality, we may also assume that P1 = P0 ∪ β(P2 ), where P0 , P2 ∈ K(C) are finite subsets. Furthermore, we assume that qj ∈ G1 and [qj ] ∈ P2 , j = 1, 2, . . . , n. Let δ0 > 0 and let G0 ⊂ C be finite subset such that there is a unital completely positive linear map L : D → A such that L (c ⊗ g) − φ(c)g(u) < δ1 /2

for all c ∈ G1 and g = 1 or g = z,

(8.306)

whenever there is a unitary u ∈ A such that [φ(c), u] < δ0 for all c ∈ G0 . By applying 8.1, we may assume that, L is T -H0 -full if, in addition, μτ ◦ı (Oa ) 1 (a) for all open balls Oa of T with radius a η0 for some η0 > 0 and for all τ ∈ T(A), where ı : C(T) → A is defined by ı(g) = g(u) for all g ∈ C(T).

1874


We may assume that L P = L P 0

(8.307)

0

for any pair of unital completely positive linear maps L , L : C ⊗ C(T) → A for which (8.306) holds for both L and L and L ≈δ1 L

on G1 .

(8.308)

Choose an integer K0 1 such that [ K0δ−1 ] 128/δ1 . In particular, (8π + 1)/[ K0δ−1 ] < δ1 . 1 1 Since TR(C) 1, there is a projection p ∈ C and a C ∗ -subalgebra B = kj =1 C(Xj , Mr(j ) ), where Xj = [0, 1], or Xj is a point, with 1B = p such that [x, p] < min{/256, δ0 /4, δ1 /16} for all x ∈ G ∪ G0 , 1

(8.309)

dist(pxp, B) < min{/256, δ0 /4, δ1 /16} for all x ∈ G1 ∪ G0

(8.310)

and τ (1 − p) < min δ1 /K0 , (η0 )/4, δ0 /4


(8.311)

We may also assume that there are projections q1 , q2 , . . . , qn ∈ (1 − p)C(1 − p) such that q − (1 − p)qi (1 − p) < min{/16, δ0 /4, δ1 /16}, i

i = 1, 2, . . . , n.

(8.312)

To simplify notation, without loss of generality, we may assume that p commutes with G ∪ G0 . Moreover, we may assume that there is a unital completely positive linear map L00 : C → pCp → B (first sending c to pcp then to B) such that x − (1 − p)x(1 − p) + L00 (x) < min{/16, δ0 /2, δ1 /4} for all x ∈ G1 .

(8.313)

Put L0 (c) = (1 − p)c(1 − p) and L0 (c) = L0 (c) + L00 (pcp) for all c ∈ C. We may further assume that [L00 ](P2 ) and [L0 ](P2 ) are well defined and [L0 ]|P0 ∪P2 = [idC ]|P0 ∪P2 .

(8.314)

Put P3 = [L0 ](P2 ) ∪ {[qi ]: 1 i n} ∪ P0 and P4 = [L00 ](P2 ). From the above, x = [L0 ](x) + [L00 ](x) for x ∈ P2 . We also assume that L β(P ∪P ∪∩P ) = L β(P ∪P ∪∩P ) 2 3 4 2 3 4 for any pair of unital completely positive linear maps from C ⊗ C(T) → A such that

(8.315)


L1 ≈δ2 L2

on G2

1875

(8.316)

and items in (8.315) are well defined for some δ2 > 0 and a finite subset G2 . Let δ2 > 0 (in place of δ0 ), G2 ⊂ (1 − p)C(1 − p) and P0 ⊂ (1 − p)C(1 − p) be as required by 8.2 for C = (1 − p)C(1 − p), σ = δ1 /16, G1 ∪ G0 (in place of G0 ) and q1 , q2 , . . . , qn (in place of p1 , p2 , . . . , pm ). Note that we may assume that P0 ⊂ G2 . Put P3 = [L0 ](P2 ) ∪ {[q]: q ∈ P0 }. Note again that elements in P3 are represented by elements in (1 − p)C(1 − p). We may assume that Bott(φ, u)|P = Bott φ, u P 3

(8.317)

3

for any pair of unitaries u and u in A for which φ(c), u < min{δ1 , δ0 },

φ(c), u < 2 min{δ1 , δ0 }

and for which there exists a continuous path of unitaries {W (t): t ∈ [0, 1]} ⊂ (1 − φ(p))A × (1 − φ(p)) with W (0) − 1 − φ(p) u 1 − φ(p) < min{δ1 , δ0 }

(8.318)

W (1) − 1 − φ(p) u 1 − φ(p) < min{δ1 , δ0 },

(8.319)

and

and φ(c), W (t) < min{δ1 , δ0 } for all c ∈ G2 and t ∈ [0, 1]. Write pi = 1C(Xi ,Mr(i) ) ∈ B, i = 1, 2, . . . , k. Let F0,i = {pi xpi : x ∈ F }, i = 1, 2, . . . , k. We may assume that Xj = [0, 1], j = 1, 2, . . . , k0 k and Xj is a point for i = k0 + 1, k0 + 2, . . . , k. Put Dj = C(Xj , Mr(j ) ) ⊗ C(T). Define Ti = N |(Dj )+ \{0} × 2R|(Dj )+ \{0} , j = 1, 2, . . . , k0 . Let δ0,i > 0 (in place of δ), Hi ⊂ (Di )+ \ {0} and G0,i ⊂ C(Xi , Mr(i) ) be required by 4.6 for /256k and F0,i and Ti , i = 1, 2, . . . , k0 . Let δ0,i > 0 (in place of δ), G0,i ⊂ Mr(i) be required by 4.7 for /256k and F0,i , i = k0 + 1, k0 + 2, . . . , k. (i) } a matrix unit for Mr(i) , i = 1, 2, . . . , k. Put Denote by {es,j R¯ = max N (h)R(h): h ∈ Hi , i = 1, 2, . . . , k0 . Let δ3 = min{/512, δ2 /2, δ2 /2, δ1 /16, δ0,1 /2, δ0,2 , . . . , δ0,k /2}. Let G3 = G2 ∪ G1 ∪ G2 ∪ "k0 "k0 {1 − p, p} i=1 G0,i . Let H = H0 ∪ {php: h ∈ H0 } i=1 Hi and let P = P2 ∪ P3 ∪ P4 ∪ (i) {[1 − p], [p], [ej,j ], [pi ], i = 1, 2, . . . , k}. It follows from 8.1 that there exist δ4 > 0, η > 0 and a finite subset G ⊂ C satisfying the following: there exists a contractive completely positive linear map L : D → A which is T -Hfull such that

1876


L(c ⊗ 1) − φ(c) < δ3 /16k R¯

and L(c ⊗ z) − φ(c)w < δ3 /16k R¯

for all c ∈ G3 (8.320)

and [L]|P1 ∪β(P2 ) is well defined, provided that w ∈ A is a unitary with φ(b), w < 3δ4

for all b ∈ G

(8.321)

and μτ ◦ı (Oa ) (a)

(8.322)

for all open balls Oa of T with radius a η for all τ ∈ T(A), where ı : C(T) → A is defined by ı(f ) = f (w) for f ∈ C(T). We may assume that η < η0 and δ4 < /256. Note that, for h ∈ Hi , L(h) L hpi hL(pi ),

i = 1, 2, . . . , k.

(8.323)

Therefore, we may assume that (with a smaller δ4 ), L(h) − φ(pi )L(h)φ(pi ) < δ3 /2k R¯

(8.324)

for any h ∈ Hi , i = 1, 2, . . . , k0 . We may also assume that φ(pi )L(c ⊗ z)φ(pi ) − φ(c)w < δ3 /16k R¯ i

for all c ∈ pi G3 pi ,

(8.325)

provided that wi ∈ U (φ(pi )Aφ(pi )) such that w − φ(pi )uφ(pi ) < 3δ4 , i

i = 1, 2, . . . , k.

(8.326)

For any function g ∈ C(T) with 0 g 1 and for any unitary u ∈ U (A), τ g(u) = τ φ(p)g(u)φ(p) + τ 1 − φ(p) g(u) 1 − φ(p)

(8.327)

τ φ(p)g(u)φ(p) τ g(u) − τ 1 − φ(p)

(8.328)

and for all τ ∈ T(A).

Thus, we may assume (by choosing smaller δ4 ) that μτ ◦ı (Oa ) (a)/2

(8.329)

for all a η and τ ∈ T(A), where ı : C(T) → A is defined by ı (f ) = f (w) (for f ∈ C(T)) for any w ∈ U (A) for which w = w0 ⊕ w1 , where w0 ∈ U ((1 − φ(p))A(1 − φ(p))) and w1 ∈ U (φ(p)Aφ(p)), such that w1 − φ(p)uφ(p) < 2δ4 ,

(8.330)


1877

where u and φ satisfy (8.320) and (8.321). Put δ = min{δ4 /12, δ3 /12} and G4 = G ∪ G3 . Put (0) G = G4 ∪ {(1 − p)g(1 − p): g ∈ G3 } ∪ {ei,s , [qj,0 ]}. Now suppose that φ and u ∈ A satisfy the assumptions of the lemma for the above δ, η, G and P. In particular, u ∈ U0 (A). To simplify notation, without loss of generality, we may assume that all elements in G and in H have norm no more than 1. By applying 8.2, one obtains a continuous path of unitaries {w0 (t): t ∈ [0, 1]} ⊂ (1 − φ(p))A(1 − φ(p)) and unitaries wj ∈ U0 (φ(qj )Aφ(qj )) such that φ(c), w0 (t) < 3δ

for all c ∈ pGp

and

(8.331)

w0 (0) − 1 − φ(p) u 1 − φ(p) < δ1 /16, w − φ q w0 (1)φ q < δ1 /16 j j j

(8.332) (8.333)

wj ⊕ 1 − φ(p) − φ qj ∈ CU 1 − φ(p) A 1 − φ(p) ,

(8.334)

for all t ∈ [0, 1],

and

j = 1, 2, . . . , n. Define w = w0 (1) ⊕ w1 for some unitary w1 for which (8.330) holds. We compute (by (8.303), (8.330) and (8.317)) that Bott(φ, w)|P = {0}.

(8.335)

By (8.330), one also has that μτ ◦ı (Oa ) 1 (a)

for all τ ∈ T(A)

(8.336)

and for any open balls Oa of T with radius a η, where ı : C(T) → A is defined by ı (g) = g(w) for all g ∈ C(T). Let L : D → A be a unital completely positive linear map which satisfies (8.320). We may also assume that [L]|P is well defined [L]|P0 = [φ]|P0

and [L]|β(P ) = {0}

by (8.335) .

(8.337)

There is a unital completely positive linear map Φ : (1 − p)C(1 − p) ⊗ C(T) → (1 − φ(p))A(1 − φ(p)) such that Φ c ⊗ g(z) − φ(c)g w0 (1) < δ1 /2

(8.338)

for all c ∈ G1 ∪ G0 and g = 1C(T) and g = z. Define L1 , L2 : C ⊗ C(T) → A as follows: L1 c ⊗ g(z) = Φ (1 − p)c(1 − p) ⊗ g ⊕ φ(p)L(c ⊗ g)φ(p) and

(8.339)

1878


L2 (c ⊗ g) = φ (1 − p)c(1 − p) g(1) ⊕ φ(p)L(c ⊗ g)φ(p)

(8.340)

for all c ∈ C and g ∈ C(T). By (8.307), we compute that, [L]|P0 = [L1 ]|P0 = [L2 ]|P0 .

(8.341)

φ(1 − p)L2 φ(1 − p) β(P ) = {0}.

(8.342)

It is easy to see that

2

One also has, by (8.308), [L2 ]|β([L00 ](P2 )) = [L ◦ L00 ]|β(P2 )

(8.343)

= [L]|β([L00 ](P2 )) = Bott(φ, u)|[L00 ](P2 ) = {0}.

(8.344)

Combining (8.342) and (8.344), one obtains that [L2 ]|β(P2 ) = {0}.

(8.345)

[L1 ]|β(P2 ) = [L]|β(P2 ) = Bott(φ, u)|P2 = {0}.

(8.346)

[L1 ]|P1 = [L]|P1 .

(8.347)

From (8.317), one computes that

It follows that

It is routine to check that τ ◦ L(g) − τ ◦ L1 (g) < δ1

for all g, ∈ G1 , for all τ ∈ T(A).

(8.348)

If v ∈ U1 , since φ(v) − L1 (v ⊗ 1) < δ1 /2 and φ(v) − L2 (v ⊗ 1) < δ1 /2, it follows that ¯ L‡2 (v) ¯ < δ1 . dist L‡1 (v),

(8.349)

If ζj = qj ⊗ z, j = 1, 2, . . . , n, by (8.332), (8.333) and (8.334), by the choice of K0 and by applying 3.1, one has that dist L‡1 (ζ¯j ), L‡2 (ζ¯j ) < δ1 .

(8.350)

Note also that, by (8.336) and by 8.1, both L1 and L2 are T0 -H0 -full. It then follows from (8.344), (8.345), (8.349), (8.350) and 7.3 that there exists a unitary W ∈ U (A) such that ad W ◦ L2 ≈/256 L1

on F1 .

(8.351)


1879

We may assume that ui − φ(pi )uφ(pi ) < 2δ

and w1 =

k

ui

(8.352)

i=1

for some ui ∈ U (φ(pi )Aφ(pi )), i = 1, 2, . . . , R(n) and ui ∈ U0 φ(pi )Aφ(pi ) ,

i = 1, 2, . . . , k

(8.353)

(since Bott(φ, u)|P = {0}). There is a positive element ai ∈ φ(pi )Aφ(pi ) such that ¯ ai L(pi )ai = φ(pi ) and ai − φ(pi ) < δ3 /8k R,

i = 1, 2, . . . , k.

(8.354)

Let Ψi : Di → φ(pi )Aφ(pi ) be defined by Ψi (a) = ai φ(pi )L(a)φ(pi )ai for all a ∈ Di , i = 1, 2, . . . , k. Thus Ψi (h) − φ(pi )L(h)φ(pi ) < δ3 /4k R¯

(8.355)

for all h ∈ Hi , i = 1, 2, . . . , k. Note also that (by (8.325)) Ψi (c ⊗ 1) − φ(c) < δ + δ3 /4k R¯

and Li (c ⊗ z) − φ(c)ui < δ3 /4k R¯

(8.356)

for all c ∈ G0,i , i = 1, 2, . . . , k. Note also that bott0 (φ|C(Xi ,Mr(i) ) , ui ) = {0},

i = 1, 2, . . . , k.

(8.357)

Furthermore, for each h ∈ Hi , there exist x1 (h), x2 (h), . . . , xN (h) (h) with and xj R(h), j = 1, 2, . . . , N(h) such that N (h)

xj (h)∗ L(h)xj (h) = 1A .

(8.358)

j =1

It follows from (8.355) that N (h)

δ3 ∗ < δ3 /4k. xj (h) Ψi (h)xj (h) − 1A < N(h)R(h) 4k R¯

(8.359)

j =1

Therefore that there exists y(h) ∈ A+ with y(h) N (h)

√ 2 such that

∗ y(h) xj (h) Φi (h) xj (h) y(h) = φ(pi ).

j =1

It follows that Φi is Ti -Hi -full, i = 1, 2, . . . , k.

(8.360)

1880


It follows from 4.6 and 4.7 that there is a continuous path of unitaries {ui (t): t ∈ [0, 1]} ⊂ φ(pi )Aφ(pi ) such that ui (0) = ui ,

ui (1) = pi

(8.361)

and Ψi (c), ui (t) < /k256 for all c ∈ F0,i

(8.362)

and for all t ∈ [0, 1], i = 1, 2, . . . , k. Define a continuous path of unitaries {z(t): t ∈ [0, 1]} ⊂ A by k z(t) = 1 − φ(p) ⊕ ui (t)

for all t ∈ [0, 1].

i=1

Then z(0) = (1 − φ(p)) +

k

i=1 ui

and z(1) = 1A . By (8.362), (8.355) and (8.320),

φ(c), z(t) < /128 for all c ∈ F .

(8.363)

Define u (t) = (w0 (t)w0 (1)∗ ⊕ (1 − φ(p)))W ∗ z(t)W for t ∈ [0, 1]. Then u (1) = 1A and we estimate by (8.330), (8.320), (8.351), (8.338) and (8.330) again that u (0) ≈ 2δ4 +δ3 /2R¯ w0 (0)w0 (1)∗ ⊕ 1 − φ(p) W ∗ L2 (1 ⊗ z)W ≈ /256 w0 (0)w0 (1)∗ ⊕ 1 − φ(p) L1 (1 ⊗ z) ≈ δ1 /2+δ3 /2R¯ w0 (0) ⊕ 1 − φ(p) 1 − φ(p) ⊕ w1 ≈ δ1 /16+2δ4 1 − φ(p) u 1 − φ(u) ⊕ φ(p)uφ(u).

(8.367)

u (0) − u < /8.

(8.368)

(8.364) (8.365) (8.366)

It follows that

Moreover, by (8.351), W ∗ φ(c)W ≈/256 φ(c) for all c ∈ F . It follows that φ(c), u (t) < /2

for all c ∈ F and t ∈ [0, 1].

(8.369)

The lemma follows when one connects u with u (0) with a continuous path of length no more than (/8)π . 2 Theorem 8.4. Let C be a unital separable amenable simple C ∗ -algebra with TR(C) 1 which satisfies the UCT. For any > 0 and any finite subset F ⊂ C, there exist δ > 0, a finite subset G ⊂ C and a finite subset P ⊂ K(C) satisfying the following: Suppose that A is a unital simple C ∗ -algebra with TR(A) 1, suppose that φ : C → A is a unital homomorphism and u ∈ U (A) such that


φ(c), u < δ

for all c ∈ G

and Bott(φ, u)|P = 0.

1881

(8.370)

Then there exists a continuous and piece-wise smooth path of unitaries {u(t): t ∈ [0, 1]} such that u(0) = u,

u(1) = 1 and φ(c), u(t) <

for all c ∈ F

(8.371)

and for all t ∈ [0, 1]. Proof. Fix > 0 and a finite subset F ⊂ C. Let δ1 > 0 (in place of δ), η > 0, G1 ⊂ C (in place of G) be a finite subset and P ⊂ K(C) be finite subset as required by 8.3 for , F and = 00 . We may assume that δ1 < . Let δ = δ1 /2. Suppose that φ and u satisfy the conditions in the theorem for the above δ, G and P. It follows from 5.9 that there is a continuous path of unitaries {v(t): t ∈ [0, 1]} ⊂ U (A) such that v(0) = u,

v(1) = u1

and φ(c), v(t) < δ1

(8.372)

for all c ∈ G1 and for all t ∈ [0, 1], and μτ ◦ı (Oa ) (a)

for all τ ∈ T(A)

(8.373)

and for all open balls of radius a η. By applying 8.3, there is a continuous path of unitaries {w(t): t ∈ [0, 1]} ⊂ A such that w(0) = u1 ,

w(1) = 1 and φ(c), w(t) <

(8.374)

for all c ∈ F and t ∈ [0, 1]. Put u(t) = v(2t) for all t ∈ [0, 1/2)

and u(t) = w(2t − 1/2) for all t ∈ [1/2, 1].

Remark 6.4 shows that we can actually require, in addition, the path is piece-wise smooth.

2

References [1] R. Bhatia, Perturbation Bounds for Matrix Eigenvalues, Pitman Res. Notes Math., vol. 162, Longman, London, 1987. [2] O. Bratteli, G.A. Elliott, D.E. Evans, A. Kishimoto, Homotopy of a pair of approximately commuting unitaries in a simple C ∗ -algebra, J. Funct. Anal. 160 (1998) 466–523. [3] M.-D. Choi, E. Effros, The completely positive lifting problem for C ∗ -algebras, Ann. of Math. 104 (1976) 585–609. [4] M. D˘ad˘arlat, T. Loring, A universal multicoefficient theorem for the Kasparov groups, Duke Math. J. 84 (1996) 355–377. [5] G.A. Elliott, G. Gong, On the classification of C ∗ -algebras of real rank zero. II, Ann. of Math. 144 (1996) 497–610. [6] G.A. Elliott, G. Gong, L. Li, Approximate divisibility of simple inductive limit C ∗ -algebras, in: Operator Algebras and Operator Theory, Shanghai, 1997, in: Contemp. Math., vol. 228, Amer. Math. Soc., Providence, RI, 1998, pp. 87–97. [7] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C ∗ -algebras. II. The isomorphism theorem, Invent. Math. 168 (2007) 249–320. [8] R. Exel, T. Loring, Almost commuting unitary matrices, Proc. Amer. Math. Soc. 106 (1989) 913–915.

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[9] R. Exel, T. Loring, Invariants of almost commuting unitaries, J. Funct. Anal. 95 (1991) 364–376. [10] G. Gong, On the classification of simple inductive limit C ∗ -algebras. I. The reduction theorem, Doc. Math. 7 (2002) 255–461 (electronic). [11] G. Gong, H. Lin, Almost multiplicative morphisms and K-theory, Internat. J. Math. 11 (2000) 983–1000. [12] A. Kishimoto, A. Kumjian, The Ext class of an approximately inner automorphism. II, J. Operator Theory 46 (2001) 99–122. [13] H. Lin, Approximate homotopy of homomorphisms from C(X) into a simple C ∗ -algebra, Mem. Amer. Math. Soc., in press, arXiv:math/0612125. [14] H. Lin, Asymptotically unitarily equivalence and asymptotically inner automorphisms, Amer. J. Math., in press, arXiv:math/0703610. [15] H. Lin, Tracial topological ranks of C ∗ -algebras, Proc. London Math. Soc. 83 (2001) 199–234. [16] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras, World Scientific Publishing, River Edge, NJ, ISBN 981-02-4680-3, 2001, xii+320 pp. [17] H. Lin, Simple AH-algebras of real rank zero, Proc. Amer. Math. Soc. 131 (2003) 3813–3819. [18] H. Lin, An approximate universal coefficient theorem, Trans. Amer. Math. Soc. 357 (2005) 3375–3405. [19] H. Lin, Simple nuclear C ∗ -algebras of tracial topological rank one, J. Funct. Anal. 251 (2007) 601–679. [20] H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras – an appendix, preprint, arXiv:math/ 0709.1654, 2007. [21] H. Lin, AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups, Int. Math. Res. Pap. 2008 (2008), doi:10.1093/imrp/rpn007, Article ID rpn007, 67 pages. [22] H. Lin, Approximate unitary equivalence in simple C ∗ -algebras of tracial rank one, preprint, arXiv:0801.2929, 2008. [23] H. Lin, Asymptotically unitary equivalence and classification of simple amenable C ∗ -algebras, preprint, arXiv: 0806.0636, 2008. [24] H. Matui, AF embeddability of crossed products of AT algebras by the integers and its application, J. Funct. Anal. 192 (2002) 562–580. [25] N.C. Phillips, Approximation by unitaries with finite spectrum in purely infinite C ∗ -algebras, J. Funct. Anal. 120 (1994) 98–106. [26] N.C. Phillips, A survey of exponential rank, in: C ∗ -Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 352–399. [27] N.C. Phillips, Reduction of exponential rank in direct limits of C ∗ -algebras, Canad. J. Math. 46 (1994) 818–853. [28] H. Su, On the classification of C ∗ -algebras of real rank zero: Inductive limits of matrix algebras over non-Hausdorff graphs, Mem. Amer. Math. Soc. 114 (547) (1995). [29] D. Voiculescu, Asymptotically commuting finite rank unitary operators without commuting approximates, Acta Sci. Math. (Szeged) 45 (1983) 429–431. [30] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, preprint, arXiv:0708.0283, 2007.


Ornstein–Uhlenbeck semi-groups on stratified groups Françoise Lust-Piquard Université de Cergy-Pontoise, UMR CNRS 8088, Département de Mathématiques, F-95000 Cergy-Pontoise, France Received 10 September 2008; accepted 16 November 2009 Available online 1 December 2009 Communicated by K. Ball

Abstract We consider, in the setting of stratified groups G, two analogues of the Ornstein–Uhlenbeck semi-group, namely Markovian diffusion semi-groups acting on Lq (pdγ ), whose invariant density p is a heat kernel at time 1 on G. Both generators have the same “carré du champ”. The first semi-group is symmetric on L2 (pdγ ), with generator ni=1 Xi∗ Xi , where (Xi )ni=1 is a basis of the first layer of the Lie algebra of G. q 2 The second one is compact n on ∗L (pdγ ), 1 < q < ∞, non-symmetric on L (pdγ ) and the formal real part of its generator N is i=1 Xi Xi . The spectrum of N is the set of non-negative integers if polynomials are dense in L2 (pdγ ), i.e. if G has at most 4 layers; we determine in this case the eigenspaces. When G is step two, we give another description of these eigenspaces, similar to the classical definition of Hermite polynomials by their generating function. © 2009 Elsevier Inc. All rights reserved. Keywords: Stratified groups; Sub-Laplacian; Heat kernel measure; Ornstein–Uhlenbeck semi-groups

1. Introduction and notation Let G be a stratified Lie group equipped with its (biinvariant) Haar measure dg and dilations (δt )t0 . Let Q be the homogeneous dimension of G. We denote by D(G) the space of C ∞ compactly supported functions on G, by S(G) the space of Schwartz functions, by S (G) its dual, and Lq (ϕdg) = Lq (G, ϕdg) for a measurable non-negative function ϕ. E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.012

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F. Lust-Piquard / Journal of Functional Analysis 258 (2010) 1883–1908

As usual, elements Z of the Lie algebra G are identified with left invariant vector fields by (Zf )(g) =

d f (g exp tZ). dt t=0

Let (Xi )1in be a linear basis of the first layer of G, defining a first layer (or horizontal) gradient and a sub-Laplacian L ∇ = (X1 , . . . , Xn ),

L=−

n

Xi2 .

(1)

1

L commutes with left translations and satisfies δt −1 Lδt = t 2 L,

t > 0.

(2)

The following facts are well known [10, Propositions 1.68, 1.70, 1.74]: − L2 generates a strongly t continuous semi-group e− 2 L of convolution operators which are contractions on Lq (dg), 1 t q ∞. The kernel pt of e− 2 L is a positive function such that pt (g) = pt (g −1 ), it lies in S(G) and has norm one in L1 (dg). Denoting p1 = p, Q

pt (g) = t − 2 p ◦ δ √1 (g). t

Equivalently, for f ∈ Lq (dg), e

− 2t L

(f )(γ ) = f ∗ pt (γ ) =

f γ g −1 pt (g) dg =

G

f γ δ√t g −1 p(g) dg.

(3)

G

The aim of this paper is to generalize the Ornstein–Uhlenbeck semi-group in the setting of stratified groups, namely to consider diffusion Markovian semi-groups acting on Lq (pdγ ), 1 q ∞, for which pdγ is an invariant measure. Besides its own interest, we hope this study might throw some light on properties of the heat kernel p, in the spirit of [1]. Actually, there are two natural candidates, each one keeping only some of the nice properties of the classical O–U semigroup. One is, by definition, symmetric on L2 (pdγ ); the other one is not, but is defined by an analogue of Mehler formula. The difference between the two generators being a (antisymmetric) first order differential operator, they have the same “carré du champ”, and actually the same as L. The classical Ornstein–Uhlenbeck semi-group is defined on S(Rn ) by Mehler formula e−tN0 (f )(x) =

f e−t x + 1 − e−2t y p(y) dy,

t 0,

Rn n

1

where the gaussian density p(y) = (2π)− 2 e− 2 |y| is the kernel of e− 2 , and is the (positive) Laplacian on Rn . The O–U semi-group is Markovian, hence contracting on Lq (Rn , pdx), 2


1885

1 q ∞, compact if 1 < q < ∞, but not compact on L1 (R, pdx) [8, Theorem 4.3.5], symmetric on L2 (Rn , pdx), and pdx is an invariant measure. The generator −N0 satisfies

n n ∂p n ∂ ∗ ∂ ∂ ∂xj ∂ N0 = =− =+ xj =+A ∂xj ∂xj p ∂xj ∂xj j =1

j =1

j =1

where ( ∂x∂ j )∗ denotes the adjoint on L2 (Rn , pdx) and A is the generator of dilations on Rn . On Lq (Rn , pdx), 1 < q < ∞, the spectrum of N0 is the set of integers N, and the Hermite polynomials on Rn form an orthogonal basis of eigenvectors of e−tN0 in L2 (Rn , pdx). The generator N0 has a fruitful generalization in (commutative or non-commutative) analysis on deformed or q-Fock spaces, namely the number operator N , i.e. the second differential quantization of identity. −N generates a symmetric completely positive semi-group (e−tN )t>0 , defined by a substitute of Mehler formula, and this semi-group is the compression of a one parameter group of unitary dilations, see e.g. [17]. This paper is an attempt to exploit Mehler formula in another setting. Results and organization of the paper In Section 2 we recall some properties of the semi-group whose generator is −∇ ∗ ∇ = − ni=1 Xi∗ Xi , Xi∗ being the formal adjoint of Xi with respect to L2 (pdγ ). We give a simple proof of the known Poincaré inequality in this space. In the main Section 3 we consider another generalization, the Mehler semi-group, which is defined for t 0 by (Theorem 3) Tt (f )(γ ) = f (δe−t γ δ√ −2t g)p(g) dg = e−tN (f )(γ ). 1−e

G

Some properties are described in 3.2: invariant measure, non-symmetry, relation between the generator −N and −∇ ∗ ∇; in particular N = L + A where A is the generator of the group (δes )s∈R of dilations, studied in 3.3. We show in 3.4 that every Tt , t > 0, is compact on Lq (pdγ ), 1 < q < ∞, (Proposition 6), with common spectrum e−tN ∪ {0} on the closed subspace spanned by polynomials (Theorem 7), which coincides with the whole space only if the number of layers of G is 4 (Proposition 8). We describe the eigenspaces in this case. In 3.5 we give another description of these eigenspaces if G is step two, similar to the usual definition of one variable Hermite polynomials by their generating function. More notation We denote G =V1 ⊕ · · · ⊕ Vk , where V1 , . . . , Vk are the layers of the Lie algebra G of G, Vk = Z being the central layer, so that [10, p. 5] [Vj , Vh ] ⊂ Vj +h ,

[V1 , Vh ] = Vh+1 ,

The homogeneous dimension of G is Q=

k j =1

j dim Vj .

1 h < k.

1886


Elements of the layers are denoted respectively by X, Y, . . . , U , and respective basis of the layers are denoted by (X1 , . . . , Xn ), (Y1 , . . . , Ym ), . . . , (U1 , . . . , Uk ). Such a basis is also denoted by (Zj )1j N . We denote accordingly

g = exp(X + Y + · · · + U ) = exp yi Yi + · · · + ui Ui xi X i + = (x, y, . . . , u) = exp

N

zj Zj

j =1

= (zj )N j =1 ,

N since the mapping (zj )N j =1 → g is a diffeomorphism: R → G. We denote by P the space of polynomials on G [10, Chapter I–C] for the fixed basis (Zj )N j =1 : N they are polynomials w.r. to the coordinates (zj )j =1 . The dilations δt , t 0, are defined on G and G by

δt (X + Y + · · · + U ) = tX + t 2 Y + · · · + t k U,

δt (exp Z) = exp δt (Z),

Z ∈ G.

For a function f on G, δt (f ) = f ◦ δt . The generator A of the one parameter group (δes )s∈R of dilations on G satisfies: for f ∈ S(G) and s > 0 d d d f ◦ δt = A(f ) = −t A+1 t −A (f ) = −tδt (f ◦ δ 1 ). (4) t dt t=1 dt dt 2. The semi-group e−t∇

∗∇

on L2 (pdg)

This semi-group has already been introduced in [3], in connection with some “natural OU processes” on Lie groups. We use instead an analytic point of view as in [19]. We consider this semi-group firstly because it is a natural generalization of the classical O–U semi-group, secondly because its generator ∇ ∗ ∇ is closely related to the generator N studied in 3.2. 2.1. Definition and some properties We consider the (closed) accretive sesquilinear form a(f, h) =

(∇f.∇h)p dg = G

n

Xi f Xi hp dg

G i=1

whose (dense) domain in L2 (pdg) is the Hilbert space

H 1 (p) = f ∈ L2 (pdg) Xi f ∈ L2 (pdg), 1 i n


1887

equipped with the norm f 2H 1 (p) = f 2L2 (p) + |∇f | 2L2 (p) ; this form is continuous on H 1 (p) × H 1 (p). Hence [19, Proposition 1.51, Theorem 1.53] it defines an operator, which we denote by ∇ ∗ ∇, such that −∇ ∗ ∇ is the generator of a strongly continuous semi-group of self-adjoint contractions on L2 (pdg). Obviously, on S(G), ∗

∇ ∇=

n

Xi∗ Xi

=L−

i=1

n Xi p i=1

p

Xi = L − B.

(5)

Since Xi is a derivation, the chain rule holds, hence Xi (f + ) = (Xi f )1{f >0} by the same proof as for usual derivations on RN [19, Proposition 4.4], and a(f + , f − ) = 0; since the ∗ form a also preserves real-valued functions, the semi-group (e−t∇ ∇ )t>0 is positivity preserving [19, Theorem 2.6]. Since it is unital, it is thus contracting on L∞ (pdg). Since it is self-adjoint on L2 (pdg), it is measure preserving, i.e. e

−t∇ ∗ ∇

(f )p dg =

G

t > 0,

fp dg, G

so it extends as a contraction semi-group on L1 (pdg) hence on Lq (pdg), 1 < q < ∞, by interpolation. 2.2. Poincaré inequality in L2 (pdg) Poincaré inequality [9, Theorem 4.2] means that the spectrum of ∇ ∗ ∇ on L2 (pdg) lies in {0} ∪ [C −1 , ∞[: there exists C > 0 such that, for f ∈ S(G), 2 f − fp dg 2 G

L (pdg)

C

|∇f |2 p dg = C

G

f ∇ ∗ ∇f p dg.

(6)

G

(6) follows from the inequality (used for q = 2) [9, Theorem 4.1] −tL q ∇ e f Cq e−tL |∇f |q ,

1 < q < ∞,

(7)

which was proved first for H1 , then for nilpotent groups (see T. Melcher’s thesis), using Malliavin calculus. See also [3] for some extensions. We shall show in Proposition 1 that (7) also follows easily from gaussian estimates of p and ∇p. Using the explicit formula for the Carnot–Caratheodory distance, H.Q. Li [16, Corollary 1.2] obtained (7) for q = 1, on the 3-dimensional Heisenberg group G = H1 . As well known [1, Théorème 5.4.7], this implies Log-Sobolev inequality for the measure pdg on H1 and (6). Another proof of this Log-Sobolev inequality for H1 , hence for Hk , is given in [13, Theorem 7.3]. Note: After this paper was sent to the referee, we were aware of [2, Theorem 4.1], where a proof of (7) is given for H1 and q 1. The proof for q > 1 follows the same line as ours. Many consequences for q = 1 are mentioned.

1888


Proposition 1. (See [9].) Let G be a stratified group. Then (7) and Poincaré inequality (6) hold true. Proof. By [9, Theorem 4.2, Proposition 2.6, Lemma 2.3] it is enough to prove (7) for t = 12 , at γ = 0. Hence, it is enough to prove, for an element X of the basis of V1 , and f ∈ S(G), −1L X e 2 f (0) = X(f ∗ p)(0) = (Xf )p dg Cq,X |∇f | q ; L (pdg) G

here [10, p. 22 and Proposition 1.29] d )(g) = f (exp tX)g , (Xf dt t=0

=X+ X

QX,j Zj

j >n

where (Zj )N j =1 is a basis of G respecting the layers and QX,j is a polynomial (with homogeneous degree h − 1 if Zj ∈ Vh , 2 h k). Since [V1 , Vh−1 ] = Vh , 2 h k, we may choose Zj ∈ Vh such that Zj = [Y, A], where Y is an element of the basis of V1 and A ∈ Vh−1 . Then Zj f QX,j p dg Yf A(QX,j p) dg + Af Y (QX,j p) dg . G

G

G

)p dg| is finally less than a finite number Iterating for A ∈ V1 + · · · + Vk−1 and soon, | G (Xf (which does not depend on f ) of terms | G Yf Z(Qp) dg| where Y is an element of the basis of V1 , Z ∈ G, and Q is a polynomial. Each term can be estimated by Zp Yf Z(Qp) dg |∇f | q

ZQ Lq (pdg) + Q L (pdg) p 1 q

+

1 q

Lq (pdg)

G

where

= 1. Here ZQ Lq (pdg) is finite since ZQ is a polynomial and p ∈ S(G). The

main point is that Q Zp p Lq (pdg) is finite. Indeed, denoting d(g) = d(0, g) where d is the Carnot distance on G, one uses [5, Theorem IV.4.2 and Comments on Chapter IV]: for 0 < ε < 1,

1

Cε e− 2−2ε d

2 (g)

1

p(g) Kε e− 2+2ε d

2 (g)

,

(8)

and, for Z ∈ G, 1

(Zp)(g) Kε,Z e− 2+2ε d

2 (g)

(9)

.

r Hence Q Zp p lies in L (pdg), 1 r < ∞, which ends the proof.

2


1889

3. Definition and properties of the Mehler semi-group 3.1. Preliminaries The next proposition extends a classical property of independant gaussian variables and will imply the semi-group property of our family of operators. Proposition 2. Let γ , g be independant G-valued random variables with law pdg. Then the r.v. 0θ

δcos θ γ δsin θ g,

π 2

has the same law, i.e. for any bounded borelian function f on G,

f (δcos θ γ δsin θ g)p(γ )p(g) dγ dg =

f (g)p(g) dg. G

G2

More generally, if g1 , . . . , gn are G-valued i.i.d r.v. with law pdg and j =n (aj 0), the law of j =1 δaj gj is pdg.

2 1j n aj

= 1,

Proof. By two changes of variables, denoting C = sin θ cos θ , f (δcos θ γ δsin θ g)p(g)p(γ ) dγ dg =

1 CQ

G2

f γ g p δ

1 cos θ

γ p δ

1 sin θ

g dγ dg

G2

1 = Q C

f (g)p δ

1 cos θ

γ p δ

1 sin θ

−1 γ g dγ dg

G2

f (g)(pcos2 θ ∗ psin2 θ )(g) dg

= G

=

f (g)p(g) dg. G

The second assertion follows by iteration.

2

Remark 1. A central limit theorem for i.i.d. centered random variables with values in a stratified group G and law μ with order 2 moments is proved in [6, Theorem 3.1]. The density p of the limit law is the kernel at time 1 of a diffusion semi-group whose generator satisfies (2). Remark 2. If X, Y are i.i.d. standard gaussian vectors with values in Rn , the couple (X cos θ + d (X cos θ + Y sin θ )) has the same joint law as (X, Y ). It follows that cosN0 θ is the Y sin θ , dθ compression of the isometry Rθ of L2 (Rn × Rn , p(x)p(y) dx dy) defined by Rθ (F )(x, y) = F (x cos θ + y sin θ, −x sin θ + y cos θ )

1890


and (Rθ )θ∈R is a one parameter group preserving the measure p(x)p(y) dx dy. This point of view was exploited e.g. in [20, Theorem 2.2]. See also [17] in the q-Fock setting. In the stratified setting we were not able to exhibit explicit unitary dilations for the Mehler operators Tt defined below. 3.2. The Mehler semi-group We now define the Mehler semi-group on Lq (G, pdg). Theorem 3. Let L, defined by (1), be a sub-Laplacian on a stratified group G, and let p be the L kernel of e− 2 . Let A be the generator of dilations. a) The family of operators (Tt )t0 defined on S(G) by Tt (f )(γ ) =

f (δe−t γ δ√

L

1−e−2t

g)p(g) dg = e− 2 (1−e

−2t )

(f )(δe−t γ )

(10)

G

is a semi-group whose generator −N is defined on S(G) by N = L + A.

(11)

b) The probability measure pdγ is invariant by (Tt )t0 i.e.

Tt (f )p dγ = G

fp dγ

(12)

G

and, for f ∈ S(G), G (Nf )p dg = 0. c) (Tt )t0 extends as a Markovian semi-group of contractions on Lq (G, pdγ ), 1 q ∞, strongly continuous if q = ∞. d) If f ∈ Lq (pdγ ), 1 q < ∞, Tt (f ) − fp dg G

→ 0.

Lq (pdγ )

t→∞

e) (i) (Tt )t>0 is not self-adjoint on L2 (G, pdγ ) as soon as G is not abelian. (ii) Formally ∇ ∗ ∇ is the real part of N , i.e., for f, h ∈ S(G), Nf, hL2 (pdγ ) = ∇ ∗ ∇ + iC f, h L2 (pdγ ) where C is non-zero and symmetric. In particular, for real-valued f ∈ S(G),

(Nf )fp dγ =

G

|∇f |2 p dγ =

G

G

∗ ∇ ∇f fp dγ .


1891

(iii) The “carré du champ” operators satisfy, for f, h ∈ S(G), ΓN (f, h) = Γ∇ ∗ ∇ (f, h) = ΓL (f, h) =

n (Xi f )(Xi h). i=1

We recall [1, Definition 2.5.1] that, for a differential operator D, 2ΓD (f, h) = −D(f h) + f Dh + (Df )h. By the change of notation e−t = cos θ , < θ < π2 , (10) can be rewritten as cosN θ (f )(γ ) =

1

f (δcos θ γ δsin θ g)p(g) dg = δcos θ ◦ e− 2 sin

2 θL

(f )(γ ).

(13)

G

Proof. a) Let ϕ(g ) = Tt (f )(g ); then Ts (ϕ)(γ ) =

ϕ(δe−s γ δ√

1−e−2s

h)p(h) dh

G

=

f δe−t [δe−s γ δ√

1−e−2s

h]δ√

1−e−2t

g p(g)p(h) dg dh

G2

=

f (δe−(t+s) γ δ√

1−e−2(s+t)

k)p(k) dk = Ts+t (f )(γ )

G

where the third equality comes from Proposition 2 applied to (h, g). By the chain rule applied to (10), Nf = −

d Tt (f ) = Lf + Af. dt t=0

b) Proposition 2 gives (12). Differentiating (12) at t = 0 for f ∈ S(G) implies (Nf )p dg = 0. G

Another proof will be given in Remark 3. c) Tt is contracting both on L1 (G, pdγ ), since it is positivity and measure preserving, and on L∞ (G, pdγ ), since it is positivity preserving and Tt (1) = 1. Hence Tt is contracting on Lq (G, pdγ ), 1 q ∞, by interpolation. Since D(G) is norm dense in Lq (G), it is norm dense in Lq (pdγ ), 1 q < ∞, because p is bounded. Writing e−t = cos θ , one has, for f ∈ D(G),

1892


Tt (f ) − f q q L

q f (δcos θ γ δsin θ g) − f (γ ) p(g) dg = (pdγ )

Lq (pdγ )

G

f (δcos θ γ δsin θ g) − f (γ )q p(γ )p(g) dγ dg,

G2

which converges to 0 as θ → 0 by the dominated convergence theorem. Since Tt is contracting, the strong continuity on Lq (pdγ ) follows by density. d) Similarly, if f is bounded and continuous on G, f (δe−t γ δ√

1−e−2t

g) → f (g); t→∞

by dominated convergence theorem Tt (f ) →t→∞ G fp dg pointwise and in the norm of Lq (pdγ ). The claim follows by density. e) (i) By (11), (5) and Lemma 4 below, for f ∈ S(G), Xj p Xj f = N − ∇ ∗ ∇ f = Af + bj Zj f = iCf = (A + B)f p 1j n

1j N

where the functions bj are not zero if Zj belongs to the second layer. (ii) For f, h ∈ S(G), ∗ N − ∇ ∇ (f )hp dg = − f bj (Zj h)p + hZj (bj p) dg. G

G

1j N

By b), the left-hand side is zero for h = 1, hence 1j N Zj (bj p) = 0. Since Tt preserves real-valued functions, so does N , hence N − ∇ ∗ ∇ (f )hp dg = − f N − ∇ ∗ ∇ (h)p dg = − f N − ∇ ∗ ∇ (h)p dg, G

G

G

which proves (iC)∗ = −iC and the last assertion. (iii) The “carré du champ” of a first order differential operator is zero and L, N , ∇ ∗ ∇ only differ by such an operator. Remark 3. We now give another instructive proof of G (Nf )p dg = 0, f ∈ S(G), hence of (12). We claim that, for f, h ∈ S(G), d (Nf )h dg = f L(h) − Qh + h ◦ δ 1 dg = f (L − QI d − A)(h) dg. t dt t=1 G

G

G

Indeed, N = L + A, L is formally selfadjoint on L2 (dg) and the claim follows by differentiating at t = 1 the right-hand side of f (δt γ )h(γ ) dγ = t −Q f γ h(δ 1 γ ) dγ . t

G

G


1893

1 By (4) and [18, Lemma 2], p may be precisely defined as the unique solution in L (G), satisfying p(g) dg = 1, of G

(L − QI d − A)(p) = Lp − Qp + tδt

d (p ◦ δ 1 ) = 0. t dt

2

Remark 4. We do not know if the following Log-Sobolev inequality for real-valued f ∈ S(G) holds on general stratified groups:

2 2 2 f ln f − ln f p dg p dg c |∇f |2 p dg. G

G

G

By e) (ii) and [1, Theorem 2.8.2], it holds if and only if the diffusion Markovian semi-groups ∗ (e−tN )t>0 and (e−t∇ ∇ )t>0 (with invariant measure pdg) are both hypercontractive on G, and these semi-groups are simultaneously hypercontractive or not. As already mentioned in Section 2.2, Log-Sobolev inequality holds for Hk . 3.3. The generator of dilations We may identify G with a group of finite matrices [23, Theorem 3.6.6]. The derivation formula for an exponential of a matrix-valued function, see e.g. [11, Theorem 69], applied to a smooth function Z : R → G, gives exp Z(t + h) − exp Z(t) d exp Z(t) = lim h→0 dt h exp(Z(t) + hZ (t)) − exp Z(t) = lim h→0 h = exp Z(t) V Z(t) ,

(14)

where, if G has k layers, k−1 l (−1)l AdZ(t) Z (t) . V Z(t) = (d exp)Z(t) Z (t) = Z (t) + (l + 1)!

(15)

l=1

Hence

exp Z(t + h) = exp Z(t) exp h V Z(t) + o(1) ,

which entails for f ∈ C ∞ (G) d f exp Z(t) = V Z(t) (f ) exp Z(t) . dt Lemma 4. Let A be the generator of the group of dilations (δes )s∈R . Then Af = aj Zj f 1j N

where the functions aj are polynomials, and are not zero if Zj lies in the second layer of G.

(16)

1894


Proof. Assume that G has k layers, k 2. Let δt g = exp tX + t 2 Y + · · · + t k U = exp Z(t). By (16) A = V (Z(1)). Noting that Z − Z ∈ V2 + · · · + Vk , we get (AdZ(1))l (Z (1)) ∈ V3 + · · · + Vk , l 1. So A − (X + 2Y ) lies in V3 + · · · + Vk . 2 Notation. We denote by Pn the (finite dimensional) space of homogeneous polynomials on G with homogeneous degree n, n ∈ N, i.e. satisfying δt (P ) = t n P ,

P ∈ Pn ,

equivalently, Pn is the eigensubspace of A on P associated to n. The finite dimensional subspaces Bn = P0 + · · · + Pn are stable under L and dilations, hence tL under e− 2 and cosN θ by (13), these operators being naturally extended on S (G). In particular L − L2 e 2 is well defined on Bn and is the inverse of e , which is thus one to one on every Bn hence on P = n0 Bn . The next lemma is the key for the computation of the spectrum of cosN θ . It will be exploited again in Section 3.5. Lemma 5. a) The generator A of dilations satisfies [L, A] = 2L on C ∞ (G). L L b) e− 2 ◦ cosN θ = δcos θ ◦ e− 2 on S (G). L c) The set of polynomials e 2 (Pn ) is a space of eigenvectors of cosN θ associated to the eigenvalue cosn θ , n 0. Proof. a) follows by differentiating at s = 0 formula (2) rewritten as LesA = e2s esA L,

s ∈ R.

b) By (3), on S(G), hence on S (G), for t > 0, t2

L

e − 2 L = δ 1 ◦ e − 2 ◦ δt .

(17)

t

Hence, on S (G)), by (13) and (17) applied to t = cos θ , L

L

e− 2 ◦ cosN θ = e− 2 ◦ δcos θ ◦ e−

sin2 θ 2

L

L

= δcos θ ◦ e− 2 .

L

c) Since e− 2 is invertible on P, which is stable under cos N θ , b) implies on P L

L

cosN θ ◦ e 2 = e 2 ◦ δcos θ . Applying this to Pn proves the result.

2


1895

3.4. Compacity and spectrum of cosN θ on Lq (pdγ ) Proposition 6. Let cosN θ be defined by (13). Then a) cosN θ is a Hilbert–Schmidt operator on L2 (pdγ ). b) cosN θ is compact on Lq (pdγ ), 1 < q < ∞. Its non-zero eigenvalues and corresponding eigenspaces are the same on L2 (pdγ ) and Lq (pdγ ). In particular its spectrum σ (cosN θ ) does not depend on q and σ cosN θ = (cos θ )σ (N ) ∪ {0}. Actually, cosN θ is a trace class operator on L2 (pdγ ) by a) and the semi-group property of

(e−tN )t>0 .

Proof. a) We must show that the kernel of cosN θ lies in L2 (G × G, pdγ ⊗ pdg). For fixed γ and θ, 0 < θ < π2 , f (δcos θ γ δsin θ g)p(g) dg = G

1

Q

sin θ

f (z)p δ cos θ γ −1 δ sin θ

1 sin θ

z dz,

G

so we must prove the convergence of the integral I (θ ) =

p 2 δ cos θ γ −1 δ sin θ

1 sin θ

z

p(γ ) p(z)

dz dγ .

G2

By the gaussian estimates (8) 2 d 2 (δ cos θ γ −1 δ 1 z)

p(γ ) d (z) d 2 (γ ) Cε 2 sin θ sin θ −1 exp − − = exp β. δ p δ cos θ γ 1 z sin θ sin θ p(z) 2 − 2ε 2 + 2ε 1+ε Kε3 The Carnot distance d satisfies d(g) +d(γ ) + d γ −1 g

and d(δt g) = td(g).

Hence

2 d 2 (z) 1 cos θ d 2 (γ ) (1 + ε)β − d(z) − d(γ ) − 2 sin θ sin θ 2(1 − ε)2

1 cos θ − cos2 θ 1 − cos θ 1 2 − d 2 (z) (γ ) + d − . 2 − 4ε 2 sin2 θ sin2 θ

1896


θ The coefficient of d 2 (γ ) is strictly negative; since 1−cos > sin2 θ 2 d (z) for small enough ε > 0. Hence, for some c, C > 0,

I (θ ) C

e

−c(d 2 (z)+d 2 (γ ))

1 2

on ]0, π2 ], so is the coefficient of

dz dγ = C

e

−cd 2 (z)

2 dz

.

G

G2

By the left-hand side of (8), for small ε, Cε

e−cd

2 (z)

dz

G

p 2c(1−ε) (z) dz, G

and the last integral is finite since p ∈ S(G). This proves a). b) By interpolation, since cos N θ is compact on L2 (p(g)dg) and bounded on L∞ (pdg) and L1 (pdg), it is compact on Lq (pdg), 1 < q < ∞, with the same spectrum and the same eigenspaces associated to non-zero eigenvalues [8, Theorems 1.6.1 and 1.6.2]. By the compacity on Lq (pdg), the set of these eigenvalues is {cosλ θ | λ ∈ σq (N )} where σq (N ) denotes the spectrum of N on Lq (pdg) [15, Chapter 34.5, Theorem 13]. Hence σq (N ) = σ2 (N ) is discrete and lies in {λ ∈ C | Reλ 0} since cos N θ is contracting on L2 (pdg) (or since ReNf, f 0). 2 Theorem 7. Let G be a step k stratified group. 1) If k 4, a) the spectrum of cosN θ on L2 (pdg) is σ (cosN θ ) = (cos θ )N ∪ {0} and σ (N ) = N; b) the corresponding eigenspaces En , n 0, (which are not pairwise orthogonal in L2 (pdg)) are L

En = e 2 (Pn ). 2) If k > 4, assertions a), b) remain true for the restriction of cosN θ to the closed subspace L2P (pdg) spanned by polynomials. If k = 1, polynomials in En are the Hermite polynomials with degree n. Proof. 1) follows from 2) and Proposition 8 below. L 2) We first define En by En = e 2 (Pn ). By Lemma 5, En lies in the eigenspace of cosN θ associated to the eigenvalue cosn θ . By Proposition 6, cosN θ is compact on L2P (pdg). The claim then follows from the following facts: Let T : E → E be a compact operator on an infinite dimensional Banach space E; let Λ be a set of eigenvalues of T and let Eλ , λ ∈ Λ, be eigensubspaces whose union is total in E. Then a) the spectrum of T is Λ ∪ {0}, b) Eλ is the whole eigenspace associated to λ ∈ Λ.


1897

Indeed, assume that T has an eigenvalue λ0 ∈ / Λ. Then T − λ0 I has a closed range with non-zero finite codimension (see e.g. [15, Chapter 21.1, Theorems 3, 4]). But this range contains the linear span of the Eλ ’s, λ ∈ Λ, hence is the whole of E. This contradiction proves a). Let λ0 ∈ Λ; since Eλ0 is stable under T , T acts on the quotient space E/Eλ0 and is still compact. The Eλ ’s, λ ∈ Λ \ {λ0 } span a dense subspace of E/Eλ0 . Applying a) to E/Eλ0 , λ0 cannot belong to the spectrum of T on the quotient space, which proves b). 2 The proof of the next proposition is essentially due to W. Hebisch (private communication). Proposition 8. Let G be a stratified group. Then the polynomials are dense in L2 (pdg) if and only if G is step k with k 4. Proof. 1) We recall why polynomials are dense in L2 (R, e−c|x| dx) if and only if α 12 : obviously, this does not depend on c and is equivalent to the density of polynomials in α L2 (R+ , e−x dx). If 0 < α < 12 , [21, Part III, Problem 153] produces a non-zero bounded funcα tion gα which is orthogonal to polynomials in L2 (R+ , e− cos(απ)x dx). If α 12 , the result α follows from the trick of [12, pp. 197–198]. Indeed, if ψ ∈ L2 (R+ , e−x dx) and α 12 , the function √ α 2α F (z) = ψ(x)e xz e−x dx = ψ y 2 eyz e−y y dy α

R+

R+

is bounded and holomorphic on {Re z < β} for some β > 0, by Cauchy–Schwarz inequality. √ Expanding z → e xz in power series, one gets F (−z) = −F (z) if ψ is orthogonal to polyα nomials in L2 (R+ , e−x dx). Thus F extends as a bounded entire function, which must be zero 2α by Liouville theorem since F (0) = 0. Hence the Fourier transform of y → ψ(y 2 )e−y y is zero, i.e. ψ = 0 a.s. 2) We identify g = exp Z ∈ G with the coordinates (x, y, . . . , w) of Z w.r. to a basis respecting the layers and denote η(g) =

il

|xi |2 +

2

|yi | 2 + · · · +

im

Obviously η(δs g) = s 2 η(g), in particular η(g) = d 2 (g)η(δ

2

|wi | k .

ir

1 d(g)

g), d denoting the Carnot distance.

Since η is strictly positive and bounded on the d-unit sphere of G, there exist constants c , C > 0 such that c η(g) d 2 (g) C η(g). By (8) there exist constants c, C > 0 such that the following embeddings L2 e−Cη(g) dg → L2 (pdg) → L2 e−cη(g) dg are continuous, with dense ranges since D(G) is dense in the three spaces.

1898


3) The algebraic tensor product E=

2 2 k L2 e−Cxi dxi ⊗ · · · ⊗ip L2 e−C|wi | dwi ,

il

is dense in L2 (e−Cη(g) dg). For k 4, one variable polynomials are dense in every factor of E by step 1), hence polynomials are dense in L2 (e−Cη(g) dg) and in L2 (pdg). 2

Let k 5. By 1) there exists a non-zero function g ∈ L2 (e−c|wr | k dwr ) which is orthogonal to polynomials w.r. to wr . Then 1 ⊗ · · · ⊗ 1 ⊗ g ∈ L2 (e−cη(g) dg) is orthogonal to all polynomials, so polynomials are neither dense in L2 (e−cη(g) dg), nor in L2 (pdg). 2 3.5. Generating functions of eigenvectors of N The usual Hermite polynomials on R, denoted by Hn , n ∈ N, are the eigenvectors of the Ornstein–Uhlenbeck operator N0 , and have the generating function 1 2

eixt+ 2 t =

(it)n n!

n0

t2 Hn (x) = e 2 ◦ δt eix = e 2 ◦ δt eix ,

noting that x → eix is a bounded eigenvector of . In particular t2 d n i Hn (x) = n e 2 δt eix . dt t=0 n

We shall verify that a similar formula gives polynomial eigenvectors of N (Proposition 11). When G is step two, these vectors are total in Lq (pdg), 1 q < ∞, see Theorem 12 below. More precisely we give in 3.5.1 a technical lemma producing eigenvectors of N out of eigenvectors of L. In 3.5.3 we apply this lemma to eigenvectors of L which are also coefficient functions of a representation of G (Proposition 11). We first gather in 3.5.2 known facts about these functions. 3.5.1. Candidates for generating functions of eigenvectors of N We gather in the next lemma technical assumptions ensuring the validity of the computation L of some eigenvectors of N . Using Lemma 5(b), the point is to define “e 2 ϕ” for suitable functions ϕ: in Lemma 5(c), we choose ϕ ∈ P, here we choose eigenvectors of L. Lemma 9. Let G be a stratified group and let ϕ ∈ S (G) ∩ C ∞ (G) be such that Lϕ = λϕ. We assume that, for n 1, (i) (ii)

dn dt n |t=0 δt (ϕ) is a polynomial on G; dn dn −1 −1 dt n t=0 G δt (ϕ)(γ g )p(g) dg = G dt n t=0 δt (ϕ)(γ g )p(g) dg.

Let ft = e

t2λ 2

δt (ϕ),

t > 0;

d n hn = n ft . dt t=0


1899

Then hn is a polynomial on G and cosN θ (hn ) = cosn θ hn . Proof. Since ϕ ∈ C ∞ (G), t → ft is C ∞ on R+ . By (2) L ◦ δt (ϕ) = t 2 λδt (ϕ), so that δt (ϕ) = L e− 2 ft . By Lemma 5(b) L

L

L

e− 2 cosN θ (ft ) = δcos θ e− 2 ft = δcos θ δt (ϕ) = δt cos θ (ϕ) = e− 2 ft cos θ .

(18)

We claim that

n L d n d − L2 N − L2 N = e− 2 cosN θ (hn ). e cos θ (f ) = e cos θ | f t t=0 t n n dt t=0 dt L

(19)

L

d − − In particular, applying (19) with θ = 0, dt n |t=0 e 2 (ft ) = e 2 (hn ). Hence, by (19) and (18), n

L

e− 2 cosN θ (hn ) =

L L d n e− 2 ft cos θ = e− 2 cosn θ hn . n dt t=0

(20)

By Leibnitz rule, it is enough to prove the claim for δt (ϕ) instead of ft . By Lemma 5(b) we may L L replace e− 2 cosN θ in the claim by δcos θ e− 2 . The claim now follows from assumption (ii). By Leibnitz rule and assumption (i), hn is a polynomial. So is cosN θ (hn ) and the result L follows from (20) since e− 2 is one to one on P. 2 Remark 5. ϕ and ϕ ◦ δβ , β > 0, give colinear hn ’s, since n 1 2 2 1 2 d n t β λ n d 2 e δtβ (ϕ) = β e 2 t λ δt (ϕ) = β n hn . dt n t=0 dt n t=0 3.5.2. A total set of eigenvectors of L in Lq (pdg), 1 q < ∞ Let Π : G → B(L2 (Rk , dξ )) be a non-trivial unitary irreducible representation of G. By definition, F ∈ L2 (Rk ) is a C ∞ vector for Π if the vector-valued function: g → Π(g)(F ) is C ∞ on G. We still denote by Π the associated differential representation, defined for a C ∞ vector F and X ∈ G by d XΠ(g)(F ) = Π(g exp tX)(F ) = Π(g)Π(X)(F ), dt t=0

g ∈ G,

(21)

and Π(X m ) = Π(X)m , see e.g. [4, p. 227]; by definition, Π(X m )(F ) still lies in L2 (Rk ) and is still a C ∞ vector for Π . Π extends as a representation of the convolution algebra M(G) by Π(μ) =

Π(g) dμ(g). G

1900


In particular (Π(pt dg))t0 is a semigroup of operators on L2 (Rk ), whose generator is −Π(L). Indeed, for a C ∞ vector F , by (21), d − dt

Π(g)(F )pt (g) dg = G

Π(g)(F )(Lpt )(g) dg =

G

L ◦ Π(g)(F )pt (g) dg G

Π(g) ◦ Π(L)(F )pt (g) dg →t→0+ Π(L)(F ).

= G 1

Since p ∈ S(G), Π(pdg) = e− 2 Π(L) is a trace class operator [4, Theorem 4.2.1]; in particular 1 its non-zero eigenvalues are e− 2 λ , where λ runs through the eigenvalues of Π(L) on L2 (Rk ). Moreover, for F ∈ L2 (Rk ), the function Π(pdg)(F ) is a C ∞ vector for Π [4, Theorem A.2.7 p. 241]. Let U be a set of non-trivial unitary irreducible representations of G whose equivalence classes support the Plancherel measure for G. By Kirillov theory, there exists an integer k, which does not depend on Π ∈ U , such that Π : G → B(L2 (Rk )), see more details in 3.5.4 below. Proposition 10. Let G be a stratified group and let F be the set of coefficient functions

F = ϕ Π,μ,μ = Π(.)(Fμ ), Fμ Π ∈ U, Fμ , Fμ ∈ BΠ ⊂ L∞ (dg) 1

where BΠ is an orthogonal basis of L2 (Rk ) chosen among eigenvectors of e− 2 Π(L) . Then F , which lies in C ∞ (G), is a set of eigenvectors of L which is total in Lq (p(g)dg), 1 q < ∞. For fixed Π, μ the functions {ϕ Π,μ,μ | Fμ ∈ BΠ } are independent and belong to the same eigenspace of L. 1

1

Proof. a) Since Π(pdg)(Fμ ) = e− 2 Π(L) (Fμ ) = e− 2 λμ Fμ , Fμ is a C ∞ vector for Π , hence ϕ Π,μ,μ ∈ C ∞ (G). By (21) ϕ Π,μ,μ is an eigenvector of L with eigenvalue λμ . Since Π is irreducible, the closed invariant subspace

F ∈ L2 Rk ∀g ∈ G Π(g)(Fμ ), F = 0

is reduced to {0}, which implies the independence of the ϕ Π,μ,μ ’s. (In the Heisenberg case, see [22, pp. 19, 51].) b) Let ψ ∈ Lq (pdg), q1 + q1 = 1, be orthogonal to F , i.e. for Π ∈ U , 0=

Π(g)(Fμ ), Fμ ψ(g)p(g) dg =

G

Π(g)ψ(g)p(g) dg (Fμ ), Fμ .

G

Equivalently Π(ψp) = 0 for Π ∈ U . Then Plancherel formula for G (see e.g. [4, Theorem 4.3.10]) implies that ψp = 0 dg a.s. Indeed, this is clear if ψp ∈ L2 (dg), in particular if q 2. In general, ψp ∈ L1 (dg), (ψp) ∗ pt ∈ L2 (dg) and (ψp) ∗ pt − ψp L1 (dg) →t→0 0; moreover (ψp) ∗ pt = 0 a.s. since, for every Π ∈ U , Π (ψp) ∗ pt = Π(ψp)Π(pt ) = 0. 2


1901

3.5.3. Polynomial eigenvectors of N built from coefficients of representations 1 2 We now consider the functions e 2 t λμ ϕ Π,μ,μ ◦δt as generating functions of polynomial eigenvectors of N .

Proposition 11. Let ϕ Π,μ,μ ∈ F be as in Proposition 10. For n 1, let 1 2 d n = e 2 t λμ ϕ Π,μ,μ ◦ δt . hΠ,μ,μ n n dt t=0 Π,μ,μ

Then hn

is a polynomial eigenvector of cosN θ with eigenvalue cosn θ .

Proof. By Proposition 10 and Lemma 9, it is enough to prove assumptions (i) and (ii) in Lemma 9. We claim the existence of a polynomial ψn , n 1, which does not depend on t, such that, for 0 t 1 and n 0, n d Π,μ,μ ϕ ◦ δ t ψn . dt n Since g → ψn (γ g −1 ) is still a polynomial, it lies in L1 (pdg) for every γ ∈ G, and this will prove assumption (ii). We now verify the claim. Case 1: The computation of derivatives being easier if G is step two, we first consider this setting. By Schur lemma, the restriction of Π to the center exp Z of G is given by a character u→ eil,u where l is some linear form on Z, see e.g. [4, p. 184]. If g = (x, u) and X = nj=1 xj Xj ∈ V1 ,

ϕ Π,μ,μ (δt g) = eit

2 l,u

2 Π,μ,μ Π(exp tX)(Fμ ), Fμ = eit l,u Φt (x)

and, by (21), d m Π,μ,μ Φt (x) = Π(exp tX)Π(X)m (Fμ ), Fμ . m dt Since Π(X)m (Fμ ) lies in L2 (Rk ), Π(X)m (Fμ ), Fμ and Π(X)m (Fμ ) L2 (Rk ) are polynodm Π,μ,μ ) is a polynomial w.r. to x, u. The claim follows since mials w.r. to x, and dt m |t=0 δt (ϕ m

Π,μ,μ

d (x)| Π(X)m (Fμ ) L2 (Rk ) Fμ L2 (Rk ) . This proves (i) and (ii) in this case. | dt m Φt General case: As in (14) and (15), for g = exp Z = exp(X + Y + · · · + U ) and t > 0, since V (Π(δt Z)) = Π(V (δt Z)),

d Π,μ,μ d ϕ exp Π(δt Z)(Fμ ), Fμ = Π V (δt Z) (Fμ ), exp −Π(δt Z)(Fμ ) . (δt g) = dt dt At t = 0 this reduces to the polynomial Π(X)(Fμ ), Fμ . Since Π(V (δt Z) has polynomial coefficients w.r. to t and the coordinates of g, so does Π(V (δt Z))(Fμ ) L2 (Rk ) . Hence there is a polynomial ψ1 w.r. to the coordinates of g such that sup0t1 Π V (δt Z) (Fμ )L2 (Rk ) ψ1 .

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This proves the claim for n = 1. Clearly this can be iterated for upper derivatives, which proves (i) and (ii). 2 3.5.4. The step two setting: generalized Hermite polynomials The key facts are now the extension of the explicit functions ϕ Π,μ,μ ∈ F as entire functions on the complexification of G and the explicit expression of p. Theorem 12 gives another proof of Theorem 7(a) in this setting, with another description of the eigenspaces of N , by generating functions. Theorem 12. Let G be a step two stratified group. Then

a) every ϕ Π,μ,μ ∈ F lies in the closed subspace of Lq (pdg), 1 q < ∞, spanned by constants Π,μ,μ , n 1} defined in Proposition 11. and the polynomials {hn b) The set of generalized Hermite polynomials

∪ϕ Π,μ,μ ∈F hΠ,μ,μ , n1 n together with the constants is a set of eigenvectors of N which is total in Lq (pdg), 1 q < ∞. Π,μ,μ } spans the eigenspace of N associated to n in Lq (pdg), c) For fixed n 1, ∪ϕ Π,μ,μ ∈F {hn 1 < q < ∞. In contrast, if G has more than 4 layers, assertion b) is false by Proposition 8, hence a) is false for some ϕ Π,μ,μ ∈ F , by Proposition 10. If G has 3 or 4 layers, we do not know if the conclusions of Theorem 12 hold true. Proof of Theorem 12. a) implies b) by Propositions 10 and 11. b) and Proposition 6 imply c), as recalled in the proof of Theorem 7. a) The proof is given in three steps. In step 1 we give two standard sufficient conditions ensuring the statement; in step 2 we verify these conditions when G is a Heisenberg group; in step 3 we show how the general step 2 case mimicks the Heisenberg case. Step 1: Let ϕ Π,μ,μ ∈ F and assume that (i) for every g ∈ G, the function t → ϕ Π,μ,μ (δt g) extends as a holomorphic function Π,μ,μ z → ϕz (g) on C. (ii) for some connected neighborhood Ω of the real axis, for every compact K ⊂ Ω, there exists wK ∈ Lq (pdg), 1 q < ∞, such that Π,μ,μ ϕ wK , z

z ∈ K. Π,μ,μ

We claim that ϕ Π,μ,μ = ϕ then lies in the closed subspace of Lq (pdg) spanned by hn n 1, and the constants. Indeed, let ψ ∈ Lq (pdg), q1 + q1 = 1, and let m(t) =

ϕ(δt g)ψ(g)p(g) dg. G

,


1903

By the assumptions, m extends as a holomorphic function on Ω and m

d dn m= ϕz ψp dg, m 0. dzn dzm G

By Proposition 10, L(ϕ) = λϕ for some λ. By Leibnitz rule n 1 2 1 2 d d n z λ z λ 2 2 e m= e ϕz ψp dg = hΠ,μ,μ ψp dg, n n n dz z=0 dz z=0 G

n 1.

G

Π,μ,μ

1 2

If ψ is orthogonal to {hn , n 1} and the constants, these derivatives are zero, hence e 2 z λ m is zero on Ω. In particular m(1) = 0, i.e. ψ is orthogonal to ϕ, which proves the claim. Step 2: The Heisenberg groups Hk . A basis of the first layer of the Lie algebra is X1 , Y1 , . . . , Xk , Yk where [Xj , Yj ] = −4U , U spans the center, and the other commutators are zero. By the Campbell–Hausdorff formula, g = exp

k

xj Xj + yj Yj + uU

k

= exp uU

j =1

exp(−2xj yj U ) exp yj Yj exp xj Xj .

j =1

We first consider the Schrödinger (unitary irreducible) representation ΠS : Hk → B(L2 (Rk )), defined on the Lie algebra by i ∂ 1 ∂ , ΠS (Yj ) = iξj , ΠS (U ) = − , iξj = − I. ΠS (Xj ) = ∂ξj 4 ∂ξj 4 For F ∈ L2 (Rk ), this implies u

i

ΠS (g)(F )(ξ ) = e−i 4 e 2

k

j =1 xj yj

ei

k

j =1 yj ξj

F (ξ + x),

and ΠS (L) = H =

k j =1

∂2 − 2 + ξj2 ∂ξj

is the harmonic oscillator. If k = 1, an o.n. basis of eigenvectors of H in L2 (R) is the sequence of Hermite functions Fμ , μ ∈ N. The so-called special Hermite functions [22, pp. 18–19] are, for μ, μ ∈ N and εμ,μ = sgn(μ − μ),

ΠS (x, y, 0)(Fμ ), Fμ = Φμ,μ (x, y) =

e R

iyξ

Fμ

x x ξ+ Fμ ξ − dξ 2 2

1 2 2 = rμ,μ x 2 + y 2 e− 2 (x +y ) (x + iεμ,μ y)|μ−μ | , where rμ,μ = rμ ,μ is a one variable polynomial with real coefficients.

(22)

1904


An o.n basis of eigenvectors of H in L2 (Rk ) is the sequence ( kj =1 Fμj (ξj ))μ∈Nk , which gives, for μ, μ ∈ Nk and g = (x, y, u), k u ϕ ΠS ,μ,μ (g) = Π(g)(Fμ ), Fμ = e−i 4 Φμj ,μj (xj , yj ). j =1

By (22) the function z → ϕ ΠS ,μ,μ (zx, zy, z2 u) is holomorphic on C. Let

Ra,δ = α + iβ |α| < a, |β| < δ ⊂ C. For some constant Ca,δ , and z ∈ Ra,δ , k Π ,μ,μ 2 2 2 1 ϕ S zx, zy, z2 u Ca,δ e 2 aδ|u| eδ (xj +yj ) . j =1

We now look for conditions on a, δ ensuring that the r.h.s. lies in Lq (pdg). We recall [14] that p(x, y, u) =

eiλu Q(x, y, λ) dλ = ck

R

Noting that Q(x, y, λ) = 1 2

eiλu

j =1

R

k

j =1 Q1 (xj , yj , λ)

q

e 2 aδ|u| p(x, y, u) du R

R

k λ 2λ − th2λ (xj2 +yj2 ) e dλ. sh2λ

is even w.r. to λ, we get, for q 1,

q q . ch aδu p(x, y, u) du = Q x, y, iaδ 2 2

We need the convergence of k R2k

e

qδ 2 (xj2 +yj2 )

j =1

which holds for qaδ of step 1 on

Q1

k qaδ q (qδ 2 − 12 tgqaδ )(xj2 +yj2 ) xj , yj , i aδ dx dy = c e dxj dyj , 2 j =1 2 R

π 4

and a > 2δ. Thus, taking a = N ∈ N, ϕ ΠS ,μ,μ satisfies the assumptions Ω=

N 2

π . RN, 4qN

Plancherel formula for Hk (see e.g. [22, Theorem 1.3.1] or [4, p. 154]) involves the representations ρh (x, y, u) = e− 4 hu ΠS (x, hy, 0). i

By the Stone–von Neumann theorem [22, Theorem 1.2.1] every irreducible unitary representai tion Π of Hk satisfying Π(0, 0, u) = e− 4 hu for a real h = 0 is unitarily equivalent to ρh . Hence


1905

ρβ 2 is equivalent to ΠS ◦ δβ , and ρ−β 2 is equivalent to ΠS ◦ σ ◦ δβ , β > 0, where σ is the automorphism of Hk defined by σ (x, y, u) = (x, −y, −u)). Since ΠS (L) = ΠS ◦ σ (L), we get ϕ ΠS ◦σ,μ,μ = ϕ ΠS ,μ,μ ◦ σ = ϕ Π,μ,μ , hence

F = ϕ ΠS ,μ,μ ◦ δβ , ϕ ΠS ,μ,μ ◦ δβ , β > 0, μ, μ ∈ Nk .

The conditions of step 1 are satisfied by ϕ Π,μ,μ ◦ δβ , replacing Ra,δ by Rβa,βδ , which ends the Π ,μ,μ

Π ,μ,μ

, hn S } proof of Theorem 12 for Hk . Taking Remark 5 into account, the set ∪μ,μ ,n {hn S q together with the constants is total in L (Hk , pdg), 1 q < ∞. Step 3. We first recall some more facts on representations and compute the set F for step 2 stratified groups. We shall follow Cygan’s scheme [7]. Let l ∈ G ∗ . Among the Lie subalgebras M ⊂ G satisfying l, [X, Y ] = 0 for every X, Y ∈ M, some have minimal codimension ml and are denoted by Ml . Then the map Z ∈ Ml → eil,Z

(23)

is a representation of the subgroup exp Ml and induces an irreducible unitary representation of ml G as follows [4, Theorems 1.3.3, 2.2.1 and p. 41]. One chooses independent vectors (Xj )i=1 ml such that G = Ml + span{(Xj )i=1 }. For (g, ξ ) ∈ G ×Rml there exist (ξ , M) ∈ Rml × Ml such that exp

ml i=1

ξi Xi .g = exp M. exp

m

ξi Xi .

i=1

Then, for F ∈ L2 (Rml ), Πl (g)(F )(ξ ) = eil,M F ξ .

(24)

The set of C ∞ vectors for Πl is S(Rk ) [4, Corollary 4.1.2]. Every irreducible unitary representation of G is equivalent to a representation constructed in this way; different Ml , Ml and different l, l in the same coadjoint orbit induce equivalent representations [4, Theorems 2.2.2, 2.2.3, 2.2.4]. By Kirillov theory there is an integer k and a set U0 ⊂ G ∗ of “generic” orbits with maximal dimension 2k, such that ml = k for l ∈ U0 . The Plancherel measure is supported by U0 [4, Theorem 4.3.10]. We now compute such a Πl when G is step 2. Let U1 , . . . , Ud be a basis of the central layer Z and let χ1 , . . . , χn be a basis of the first layer V1 of G. Let l ∈ G ∗ and let λ = dj =1 λj Uj∗ be its central part, identified with a vector λ ∈ Rd . Let Aλ be the n × n matrix with coefficients λ, [χj , χh ]. By Campbell–Hausdorff formula, for Y ∈ G, g = exp(X + U ) where X ∈ V1 , U ∈ Z, exp Adg(Y ) = g exp Y g −1 = e[X,Y ] exp Y = exp Y + [X, Y ] , hence the coadjoint orbit of l, i.e. {l ◦ Adg, g ∈ G} ⊂ G ∗ , is l + range Aλ .

1906


We now assume that l lies in U0 , so that the range of Aλ has dimension 2k. There exists an orthogonal matrix Ωλ such that Aλ = Ωλ Aλ Ωλ∗ where Aλ is block diagonal, the non-zero blocks having the form νj (λ)

0 1 , −1 0

νj (λ) > 0.

(25)

The new basis of V1 (defined by the columns of Ωλ ) is denoted by X1 , Y1 , . . . , Xk , Yk , S1 , . . . , Sn−2k , so that λ, [Xj , Xh ] = 0 = λ, [Yj , Yh ] ,

λ, [Xj , Yh ] = νj (λ)δj h ,

1 j, h k.

(26)

We denote t = Ωλ (x, y, s) ∈ Rn , i.e. n

tj χj =

j =1

k

xj Xj + yj Yj +

j =1

n−2k

sh Sh = X + Y + S ∈ V1 .

h=1

Choosing Ml = Z + span{Yj , Sh }1j k, 1hn−2k , let us compute Πl . By definition Πl (exp uj Uj ) = eiuj λj . For g = exp(X + Z) and Z = Y + S, exp

k

ξj Xj g = exp

j =1

k

ξj Xj , X + Z g exp

j =1

= exp

k

ξj X j

j =1

k j =1

k 1 ξj Xj , X + Z + [X, Z] exp Z exp X exp ξj X j 2

= exp M exp

j =1

k (ξj + xj )Xj . j =1

Hence, by (24) and (26), for F ∈ L2 (Rk ), Πl (g)(F )(ξ ) = eil,M F (ξ + x) = ei

k

1 j =1 νj yj (ξj + 2 xj )

eil,Y +S F (ξ + x).

(27)

Since we may replace l by l in the orbit of l, we may suppose l, Yj = 0, 1 j k. In particular, by (27), Πl (Xj ) =

∂ , ∂ξj

Πl (Yj ) = iνj ξj ,

1 j k,

Πl (Sh ) = il, Sh I,

1 h n − 2k.


Since Ωλ is orthogonal, −L =

k

2 j =1 (Xj

Πl (L) =

k

−

j =1

+ Yj2 ) +

n−2k h=1

h=1

Sh2 , which entails

n−2k ∂2 2 2 + ν ξ + l, Sh 2 I. j j 2 ∂ξj h=1

A basis of eigenvectors of Πl (L) is thus (x, y, s, u), ϕ Πl ,μ,μ (g) = eiλ,u ei

n−2k

1907

k

√

j =1 Fμj (

sh l,Sh

νj ξj ) μ∈Nk . By (27) and (22), for g =

k 1 √ √ √ Φμj, μj ( νj xj , νj yj ). νj

j =1

Hence, for z ∈ Ra,δ and some constant Ca,δ , with t = Ωλ (x, y, s), k n−2k Π ,μ,μ 1 δ 2 νj (xj2 +yj2 ) ϕ l zt, z2 u Ca,δ e2aδ|λ,u| eδ h=1 |sh l,Sh | √ e νj j =1

=e

2aδ|λ,u|

wa,δ,l (x, y, s).

By [7, Corollary 5.5] the heat kernel p(t, u) is the Fourier transform of CQ(t, λ) w.r. to the central variables, where Q(t, λ) =

n−2k

1 2

e − 2 sh

k j =1

h=1

νj = Q(t, −λ). Q1 xj , yj , 4

Again, we need the convergence of Rn

q wa,δ,l (x, y, s)

n−2k

e

− 12 sh2

h=1

k j =1

Q1

iqaδνj xj , yj , 2

dx dy ds,

which holds if qaδ max νj π4 and a > 2δ. This ends the proof of Theorem 12. 2 Acknowledgment We thank W. Hebisch who gave us the idea of the proof of Proposition 8. References [1] C. Ané, et al., Sur les Inégalités de Sobolev Logarithmiques, Panor. Synthèses, vol. 10, SMF, 2000. [2] D. Bakry, F. Baudoin, M. Bonnefont, D. Chafaï, On gradient bounds for the heat kernelon the Heisenberg group, J. Funct. Anal. 255 (2008) 1905–1938. [3] F. Baudoin, M. Hairer, J. Teichmann, Ornstein–Uhlenbeck processes on Lie groups, J. Funct. Anal. 255 (2008) 877–890. [4] L. Corwin, F.P. Greenleaf, Representations of nilpotent Lie groups and their applications Part 1: Basic theory and examples, Cambridge Stud. Adv. Math., vol. 18, 2004.

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[5] T. Coulhon, L. Saloff-Coste, N.Th. Varopoulos, Analysis and Geometry on Groups, Cambridge University Press, 1992. [6] P. Crepel, A. Raugi, Théorème central limite sur les groupes nilpotents, Ann. Inst. H. Poincaré Sect. B 14 (2) (1978) 145–164. [7] J. Cygan, Heat kernels for class 2 nilpotent groups, Studia Math. LXIV (1979) 227–238. [8] E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, 1989. [9] B. Driver, T. Melcher, Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal. 221 (2005) 340–365. [10] G.B. Folland, E. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, 1982. [11] B.C. Hall, Lie Groups, Lie Algebras and Representations: An Elementary Introduction, Grad. Texts in Math., vol. 222, Springer-Verlag, 2003. [12] H. Hamburger, Zur Konvergenztheorie der Stieltjesschen Kettenbrüche, Math. Z. 4 (1919) 209–211. [13] W. Hebisch, B. Zegarlinski, Coercive inequalities on metric measure spaces, preprint. [14] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic hypoellipticity of certain subelliptic operators on the Heisenberg group, Studia Math. 56 (1976) 165–173. [15] P. Lax, Functional Analysis, Wiley, 2002. [16] H.-Q. Li, Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg, J. Funct. Anal. 236 (2006) 369–394. [17] F. Lust-Piquard, Riesz transforms on deformed Fock spaces, Comm. Math. Phys. 205 (1999) 519–549. [18] F. Lust-Piquard, A simple minded computation of heat kernels on Heisenberg groups, Colloq. Math. 97 (2003) 233–249. [19] E.M. Ouhabaz, Analysis of Heat Kernels on Domains, London Math. Soc. Monogr. Ser., vol. 31, Princeton University Press, 2006. [20] G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis, Varenna, Italy, 1985, in: Lecture Notes in Math., vol. 1206, Springer-Verlag, 1986, pp. 167–241. [21] G. Polya, G. Szegö, Problems and Theorems in Analysis, Springer-Verlag, 1978. [22] S. Thangavelu, Harmonic Analysis on the Heisenberg Group, Progr. Math., vol. 159, Birkhäuser, 1998. [23] V.S. Varadarajan, Lie Groups, Lie Algebras and Their Representations, Springer-Verlag, 1984.


Inductive limits of subhomogeneous C ∗ -algebras with Hausdorff spectrum Huaxin Lin a,b,∗ a Department of Mathematics, East China Normal University, Shanghai, China b Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

Received 12 January 2009; accepted 23 June 2009 Available online 9 December 2009 Communicated by S. Vaes

Abstract We consider unital simple inductive limits of generalized dimension drop C ∗ -algebras. They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop C ∗ -algebras. Suppose that A is one of these C ∗ -algebras. We show that A ⊗ Q has tracial rank no more than one, where Q is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let A and B be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then A∼ = B if and only if they have the same Elliott invariant. © 2009 Published by Elsevier Inc. Keywords: Simple C ∗ -algebras; Classification

1. Introduction A unital AH-algebra is an inductive limit of finite direct sums of C ∗ -algebras with the form P C(X, Mk )P , where X is a finite dimensional compact metric space, k 1 is an integer and P ∈ C(X, Mk ) is a projection. One of the successful stories in the program of classification of amenable * Address for correspondence: Department of Mathematics, University of Oregon, Eugene, OR 97403, USA.

E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.06.036

1910


C ∗ -algebras, otherwise known as the Elliott program is the classification of unital simple AHalgebras with no dimension growth (see [3]) by their K-theoretical invariant, otherwise called the Elliott invariant. Other classes of inductive limits of building blocks studied in the program include so-called simple inductive limits of dimension drop interval (or circle) algebras (see [14, 2,15,4,13], for example). One reason to study the simple inductive limits of dimension drop interval algebras is to construct unital simple C ∗ -algebras with the Elliott invariant which cannot be realized by unital simple AH-algebras (see also [4] and [16]). Notably, the so-called Jiang Su-algebras of unital projectionless simple ASH-algebra Z whose Elliott invariant is the same as that of complex field. However, until very recently, they were classified as separate classes of simple C ∗ -algebras. In particular, it was not known that one C ∗ -algebra in one of these classes should be isomorphic to ones in a different class with the same Elliott invariant in general. Furthermore, methods used in classification of unital simple AH-algebras and unital simple dimension drop algebras dealt with maps from dimension drop algebras to dimension drop algebras and maps between some homogeneous C ∗ -algebras. These methods do not work for maps from dimension drop algebras into homogeneous C ∗ -algebras or maps from homogeneous C ∗ -algebras into dimension drop algebras in general. Therefore most classification theorems for simple inductive limits of basic building blocks do not deal with mixed building blocks. In this paper we consider a general class of inductive limits of subhomogeneous C ∗ -algebras with Hausdorff spectrum which includes both AH-algebras and all known unital simple inductive limits of dimension drop algebras as well as inductive limits with mixed basic building blocks. We will give a classification theorem for this class of unital simple C ∗ -algebras. Let X be a compact metric space. Let k 1 be an integer, let ξ1 , ξ2 , . . . , ξn ∈ X and let m1 , m2 , . . . , mn 1 be integers such that mj |k, j = 1, 2, . . . , n. One may write Mk = Mmj ⊗ Mk/m(j ) . Let Bj ∼ = Mmj ⊗ 1k/m(j ) be a unital C ∗ -subalgebra of Mk (1Bj = 1Mk ). Definition 1.1. Define Dk (X, ξ1 , ξ2 , . . . , ξn , m1 , m2 , . . . , mn ) = f ∈ C(X, Mk ): f (ξj ) ∈ Bj , 1 j n . In the above form, if X is an interval, it is known as (general) dimension drop interval algebra, and if X = T, it is known as (general) dimension drop circle algebra. Here we allow high dimensional dimension drop algebras. So when X is a connected finite CW complex, algebras Dk (X, ξ1 , ξ2 , . . . , ξn , m1 , m2 , . . . , mn ) and finite direct sums of their unital hereditary C ∗ -subalgebras may be called general dimension drop algebras. When mj = k, 1 j d(j ), j = 1, 2, . . . , N, it is a homogeneous C ∗ -algebra. Thus AH-algebras, inductive limits of dimension drop interval algebras and inductive limits of dimension drop circle algebras are inductive limit of general dimension drop algebras. A classification theorem for unital simple inductive limits of those general dimension drop algebras would not only generalize many previous known classification theorems but more importantly it would unify these classification results. However, one may consider more general C ∗ -algebras. Let X1 , X2 , . . . , Xn ⊂ X be disjoint compact subsets. Let k, m1 , m2 , . . . , mn and B1 , B2 , . . . , Bn be as above. Define Definition 1.2. D¯ X,k,{Xj },n = f ∈ C(X, Mk ): f (x) ∈ Bj for all x ∈ Xj , 1 j n .

(e1.1)


1911

By a generalized dimension drop algebra, we mean C ∗ -algebras of the form: N

Pj D¯ Xj ,r(j ),{Xj,i },d(j ) Pj

j =1

where Pj ∈ D¯ Xj ,r(j ),{Xi,j },d(j ) is a projection, Xj is a compact metric space with finite dimension and with property (A) (see 4.1 bellow) and {Xj,i } is a collection of finitely many disjoint compact subsets of Xj . Note that, in the following, A ⊗ Z and A have exactly the same Elliott invariant whenever K0 (A) is weakly unperforated. We prove the following theorem: Theorem 1.3. Let A and let B be two unital simple inductive limits of generalized dimension drop algebras. Then A ⊗ Z ∼ = B ⊗ Z if and only if K0 (A), K0 (A)+ , [1A ], K1 (A), T (A), rA ∼ = K0 (B), K0 (B)+ , [1B ], K1 (B), T (B), rB (see 2.10). Denote by D the class of inductive limits of generalized dimension drop algebras. By a recent result of W. Winter [19], unital simple ASH-algebras with no dimension growth are Z-stable. Therefore we also have the following: Theorem 1.4. Let A, B ∈ D be two unital simple C ∗ -algebras with no dimension growth. Then A∼ = B if and only if K0 (A), K0 (A)+ , [1A ], K1 (A), T (A), rA ∼ = K0 (B), K0 (B)+ , [1B ], K1 (B), T (B), rB . This unifies the previous known classification theorems for unital simple ASH-algebras with Hausdorff spectrum such as [3,2,4,15,13], etc. In fact, we show that, if A ∈ D is a unital simple C ∗ -algebra, then the tracial rank of A ⊗ Q is no more than one, where Q is the UHF-algebra with (K0 (Q), K0 (Q)+ , [1Q ]) = (Q, Q+ , 1). We then obtain the above mentioned classification theorems by applying a recent classification theorem in [10] (see also [18], [9] and [11]) for unital simple C ∗ -algebras A with the tracial rank of A ⊗ Q no more than one. Therefore one may view our result here as an application of the classification result in [10]. 2. Preliminaries 2.1. Let {An } be a sequence of C ∗ -algebras and let ϕn : An → An+1 be homomorphisms. Denote by A = limn→∞ (An , ϕn ) the C ∗ -algebra of the inductive limit. Put ϕn,m = ϕm ◦ ϕm−1 ◦ · · · ◦ ϕn : An → Am for m > n. Denote by ϕn,∞ : An → A the homomorphism induced by the inductive limit system. 2.2. Let X be a compact metric space, let ξ ∈ X and let d > 0. Set B(ξ, d) = x ∈ X: dist(x, ξ ) < d .

1912


Suppose that 0 < d2 < d1 . Set A(ξ, d1 , d2 ) = x ∈ X: d1 > dist(x, ξ ) > d2 . Definition 2.3. Let X be a compact metric space, let A be a C ∗ -algebra and let B ⊂ C(X, A) be a C ∗ -subalgebra. For each x ∈ X, denote by πx : C(X, A) → A the homomorphism defined by πx (f ) = f (x) for all f ∈ C(X, A). Suppose that Cx = πx (B) ⊂ A. Then πx : B → Cx is a homomorphism. 2.4. Let A be a C ∗ -algebra, let x, y ∈ A be two elements and let > 0. We write x ≈ y, if x − y < . Suppose that S ⊂ A is a subset. We write x ∈ S, if x ≈ y for some y ∈ S. Let B be another C ∗ -algebra, let ϕ, ψ : A → B be two maps, let S ⊂ A be a subset and let > 0. We write ϕ ≈ ψ

on S,

if ψ(x) ≈ ϕ(x) for all x ∈ S. 2.5. Let A be a unital C ∗ -algebra. Denote by U (A) the unitary group of A. Denote by U0 (A) the connected component of U (A) containing the identity. 2.6. Let A be a unital C ∗ -algebra. We say B is a unital C ∗ -subalgebra of A if B ⊂ A is a unital C ∗ -algebra and 1B = 1A . Definition 2.7. Let A be a unital C ∗ -algebra and let C be another C ∗ -algebra. Suppose that ϕ1 , ϕ2 : C → A are two homomorphisms. We say that ϕ1 and ϕ2 are asymptotically unitarily equivalent if there exists a continuous path of unitaries {u(t): t ∈ [0, ∞)} ⊂ A such that lim u(t)∗ ϕ2 (c)u(t) = ϕ1 (c)

t→∞

for all c ∈ C.

(e2.2)

We say that ϕ1 and ϕ2 are strongly asymptotically unitarily equivalent if in (e2.2) u(0) = 1A . Of course, ϕ1 and ϕ2 are strongly asymptotically unitarily equivalent if they are asymptotically unitarily equivalent and if U (A) = U0 (A). Definition 2.8. Denote by I the class of those C ∗ -algebras with the form ni=1 Mri (C(Xi )), where each Xi is a finite CW complex with (covering) dimension no more than one. A unital simple C ∗ -algebra A is said to have tracial rank no more than one (we write TR(A) 1) if for any finite subset F ⊂ A, > 0, any nonzero positive element a ∈ A, there


1913

is a C ∗ -subalgebra C ∈ I such that if denote by p the unit of C, then for any x ∈ F , one has (1) xp − px , (2) there is b ∈ C such that b − pxp and (3) 1 − p is Murray–von Neumann equivalent to a projection in aAa. Definition 2.9. For a supernatural number p, denote by Mp the UHF algebra associated with p (see [1]). A supernatural number p is said to be of infinite type, if q is a prime factor of p then q m is a factor of p for all integer m 1. We use Q throughout the paper for the UHFalgebra with (K0 (Q), K0 (Q)+ , [1Q ]) = (Q, Q+ , 1). We may identify Q with the inductive limit limn→∞ (Mn! , hn ), where hn : Mn! → M(n+1)! is defined by a → a ⊗ 1n+1 for a ∈ Mn! . Let A be the class of all unital separable simple amenable C ∗ -algebras A in N for which TR(A ⊗ Mp ) 1 for some supernatural number p of infinite type. Definition 2.10. Let A be a unital stably finite separable simple amenable C ∗ -algebra. Denote by T (A) the tracial state space of A. We also use τ for τ ⊗ Tr on A ⊗ Mk for any integer k 1, where Tr is the standard trace on Mk . By Ell(A) we mean the following: K0 (A), K0 (A)+ , [1A ], K1 (A), T (A), rA , where rA : T (A) → S[1A ] (K0 (A)) is a surjective continuous affine map such that rA (τ )([p]) = τ (p) for all projections p ∈ A ⊗ Mk , k = 1, 2, . . . , where S[1A ] (K0 (A)) is the state space of K0 (A). Suppose that B is another stably finite unital separable simple C ∗ -algebra. A map Λ : Ell(A) → Ell(B) is said to be a homomorphism if Λ gives an order homomorphism λ0 : K0 (A) → K0 (B) such that λ0 ([1A ]) = [1B ], a homomorphism λ1 : K1 (A) → K1 (B), a continuous affine map λ ρ : T (B) → T (A) such that λ ρ (τ )(p) = rB (τ ) λ0 [p] for all projection in A ⊗ Mk , k = 1, 2, . . . , and for all τ ∈ T (B). We say that such Λ is an isomorphism, if λ0 and λ1 are isomorphisms and λ ρ is an affine homeomorphism. In this case, there is an affine homeomorphism λρ : T (A) → T (B) such that

λ−1 ρ = λρ . Theorem 2.11. (See Corollary 11.9 of [10].) Let A, B ∈ A. Then A⊗Z ∼ =B ⊗Z if Ell(A ⊗ Z) = Ell(B ⊗ Z).

1914


3. Approximate unitary equivalence The following is known. Lemma 3.1. Let ϕ, ψ : Q → Q be two unital homomorphisms. Then they are strongly asymptotically unitarily equivalent. Proof. Since ϕ and ψ are unital, one computes that ϕ and ψ induce the same identity map on K0 (Q). Since K0 (Q) is divisible and K1 (Q) = {0}, one then checks that [ϕ] = [ψ] in KK(Q, Q). With K1 (Q) = {0}, it follows from Corollary 11.2 of [8] that ϕ and ψ are strongly asymptotically unitarily equivalent. 2 Lemma 3.2. For any > 0 and any finite subset F ⊂ Q, there exists δ > 0 and a finite subset G ⊂ Q satisfying the following: suppose that u ∈ U (Q) and ϕ : Q → Q is a unital homomorphism such that

u, ϕ(b) < δ

for all b ∈ G.

(e3.3)

Then there exists a continuous path of unitaries {u(t): t ∈ [0, 1]} in Q such that u(0) = u, u(1) = 1,

u(t), ϕ(a) <

for all a ∈ F .

(e3.4)

Moreover, Length u(t) 2π. Proof. Note that K1 (Q) = {0} and Ki (Q ⊗ C(T)) ∼ = Q is torsion free divisible group, i = 0, 1. One computes that Bott(ϕ, u) = 0. It follows from [7] that such {u(t)} exists.

2

Theorem 3.3. Let X be a compact subset of [0, 1] with a = inf{t: t ∈ X} and b = sup{t: t ∈ X}. Suppose that ϕ, ψ : Q → C(X, Q) are two unital monomorphisms. Then ϕ and ψ are approximately unitarily equivalent. Moreover, if πa ◦ ϕ = πa ◦ ψ (or πb ◦ ϕ = πb ◦ ψ, or both hold) then there exists a sequence of unitaries {un } ∈ C(X, Q) such that πa (un ) = 1 (or πb (un ) = 1, or πa (un ) = πb (un ) = 1) and lim ad un ◦ ϕ(x) = ψ(x)

n→∞

for all x ∈ Q.

Furthermore, if F ⊂ Q is a full matrix algebra as a unital C ∗ -subalgebra, πa ◦ϕ|F = πa ◦ψ|F and πb ◦ ϕ|F = πb ◦ ψ|F , one can find, for each > 0, a unitary u such that


1915

ad u ◦ ϕ(f ) − ψ(f ) < for all f ∈ F and πa (u) = πb (u) = 1. Proof. To simplify the notation, without loss of generality, we may assume that a = 0 and b = 1. Fix > 0 and a finite subset F . Without loss of generality, we may assume that f 1 for all f ∈ F . Let δ > 0 and G be the finite subset required by 3.2 corresponding to and F . Without loss of generality, we may assume that < δ and F ⊂ G. Let ϕ, ψ : Q → C(X, Q) be two unital homomorphisms. Denote by πt : C(X, Q) → Q the point-evaluation map at the point t (∈X). There is a partition: 0 = t0 < t1 < · · · < tn = 1 with ti ∈ X such that either (ti−1 , ti ) ∩ X = ∅, or πt ◦ ϕ(b) − πt ◦ ϕ(b) < δ/16 and πt ◦ ψ(b) − πt ◦ ψ(b) < δ/16 i i

(e3.5)

for all t ∈ [ti−1 , ti ] ∩ X, i = 1, 2, . . . , n, and all b ∈ G. Since K0 (Q) = Q, [1Q ] = 1 and K1 (Q) = {0}, one computes that [πt ◦ ϕ] = [πt ◦ ψ] in KK(Q, Q) for all t ∈ X. It follows from 3.1 that there is, for each i, a unitary vi ∈ Q such that ad vi ◦ πt ◦ ϕ(b) − πt ◦ ψ(b) < δ/16, i i

(e3.6)

i = 0, 1, . . . , n. Put wi = ui−1 u∗i if (ti−1 , ti ) ∩ X = ∅. Then, if (ti−1 , ti ) ∩ X = ∅, wi∗ πti−1 ◦ ϕ(b) wi ≈δ/16 ui πti−1 ◦ ψ(b) u∗i ≈δ/16 ui πti ◦ ψ(b) u∗i ≈δ/16 πti ◦ ϕ(b) ≈δ/16 πti−1 ◦ ϕ(b)

(e3.7) (e3.8) (e3.9) (e3.10)

for all b ∈ G. It follows from 3.2 that there exists a continuous path of unitaries {wi (t): t ∈ [ti−1 , ti ]} in Q such that wi (ti−1 ) = wi , wi (ti ) = 1 and wi (t), π ◦ πt

i−1

(a) < /16

(e3.11)

for all t ∈ [ti−1 , ti ] ∩ X and a ∈ F , i = 1, 2, . . . , n. Define u(ti ) = ui and (if (ti−1 , ti ) ∩ X = ∅) u(t) = wi (t)ui

for all t ∈ [ti−1 , ti ] ∩ X.

(e3.12)

1916


Note that u(t) is continuous. Moreover, when (ti−1 , ti ) ∩ X = ∅, for t ∈ [ti−1 , ti ] ∩ X, ad u(t) ◦ πt ◦ ϕ(a) ≈/16 ad u(t) ◦ πti ◦ ϕ(a)

(e3.13)

≈/16 ad ui ◦ πti ◦ ϕ(a)

(e3.14)

≈/16 πti ◦ ψ(a)

(e3.15)

≈/16 πt ◦ ψ(a)

(e3.16)

for all a ∈ F , where the first estimate follows from (e3.5), the second estimate follows from (e3.11), the third follows from (e3.6) and the last one follows from (e3.5). Note that u(t) ∈ C(X, Q). It follows that ϕ and ψ are approximately unitarily equivalent. To prove the second part of the statement, we use what we have just proved. Let > 0 and F ⊂ Q be a finite subset. Let δ and G be required by 3.2 for /4 and F . Without loss of generality, we may assume that F ⊂ G. Put 1 = min{δ/4, /4}. There exists η > 0 such that πt ◦ ϕ(a) − πt ◦ ϕ(a) < 1 /2 πt ◦ ψ(a) − πt ◦ ψ(a) < 1 /2

and

(e3.17) (e3.18)

for all a ∈ G, whenever |t − t | η and t, t ∈ X. From what we have shown, there exists a unitary w ∈ C(X, Q) such that ad w ◦ ψ(a) − ϕ(a) < 1 /2

for all a ∈ G.

If 0 is an isolated point in X, then we may assume that π0 (w) = 1. Otherwise, let η1 = sup t: t ∈ (0, η] ∩ X . Since π0 ◦ ϕ = π0 ◦ ψ and π1 ◦ ϕ = π1 ◦ ψ, we compute that πη (w)πη ◦ ψ(a) − πη ◦ ψ(a)πη (w) < 1 1 1 1 1 for all a ∈ G. It follows from 3.2 that there exists a continuous path of unitaries {W (t): t ∈ [0, η1 ]} ⊂ Q such that W (0) = 1, W (η1 ) = w(η1 ).

W (t), ψ(x) < /4

for all x ∈ F

and t ∈ [0, 1]. Define v ∈ C(X, Q) by πt (v) = W (t) if t ∈ [0, η1 ] ∩ X and πt (v) = πt ◦ w for t ∈ (η1 , 1] ∩ X. Then v ∈ C(X, Q) ad v ◦ ψ(a) − ϕ(a) < /2 for all a ∈ F and v(0) = 1. We can deploy the same argument for the end point 1. Thus we obtain a unitary u ∈ C(X, Q) such that u(0) = 1 = u(1) and ad u ◦ ψ(a) − ϕ(a) < for all a ∈ F .


1917

Let F ⊂ Q be a full matrix algebra as a unital C ∗ -subalgebra. Let {ei,j : 1 i, j k} be a system of matrix unit. From the above, there is a unitary W ∈ C(X, Q) such that ∗ W ψ(ei,j )W − ϕ(ei,j ) < , 8k

1 i, j k.

One has a unitary v ∈ C(X, Q) such that ϕ(e1,1 ) = v ∗ ψ(e1,1 )v. Define W1 =

k

ψ(ei,1 )vϕ(e1,i ).

i=1

Then W1 is a unitary such that W1 ψ(f )W1∗ = ϕ(f )

for all f ∈ F.

For any > 0, there exists 1 > δ > 0 such that W1 (t) − W1 (t ) < /4

(e3.19)

for all t, t ∈ X and |t − t | < δ. Since U (ϕ(ei,i )Qϕ(ei,i )) = U0 (ϕ(ei,i )Qϕ(ei,i )), it is easy to obtain a continuous path of unitaries {v(t): t ∈ [0, δ]} ⊂ U (Q) such that v(δ) = W1 (0)∗ ,

v(0) = 1

and u(t)πa ◦ ϕ(f ) = πa ◦ ϕ(f )u(t)

for all f ∈ F and all t ∈ [0, δ]. Define now u(t) ∈ C(X, Q) by u(t) = v(t) if t ∈ [0, δ) ∩ X and t−δ ∗ ) if t ∈ [δ, 1] ∩ X. By (e3.19), u(t) = W1 ( 1−δ ad u ◦ ψ(f ) − ϕ(f ) < One may arrange the other end point b similarly.

for all f ∈ F.

2

3.4. One can avoid to using Lemma 3.2 by modifying the last part of the above proof to obtain 3.3. 4. Approximate factorizations Definition 4.1. Let X be a connected compact metric space. We say that X has the property (A), if, for any ξ ∈ X and any δ > 0, there exist δ > δ1 > δ2 > δ3 > 0 satisfying the following: there exists a homeomorphism χ : B(ξ, δ1 ) \ {ξ } → A(ξ, δ1 , δ3 ) such that χ|A(ξ,δ1 ,δ2 ) = idA(ξ,δ1 ,δ2 ) . Every locally Euclidean connected compact metric space has the property (A). All connected simplicial complex has the property (A).

1918


Definition 4.2. Let X be a compact metric space and let k 1 be an integer. Fix ξ1 , ξ2 , . . . , ξn ∈ X and fix positive integers m1 , m2 , . . . , mn for which mj |k, j = 1, 2, . . . , n. Denote Dk,n = Dk (X, ξ1 , ξ2 , . . . , ξn , m1 , m2 , . . . , mn ). Note, if n 1, Dk,n ⊂ Dk,n−1 . If X1 , X2 , . . . , Xn ⊂ X are disjoint compact subsets, put D¯ k,{Xj },n = Dk (X, X1 , X2 , . . . , Xn , m1 , m2 , . . . , mn ), when {Xj : 1 j n} and {mj : 1 j n} are understood. Let P ∈ D¯ k,{Xj },n be a projection. Then f (x) · P (x) ∈ P D¯ k,n,{Xj } P for all f ∈ C(X). Similarly, if P ∈ Dk,n is a projection, then f (x) · P (x) ∈ P Dk,n P . Denote by Ck,{Xj },n the subalgebra f · 1D¯ k,n,{X } : f ∈ C(X) . j

Then Ck,n,{Xj } P is the center of P D¯ k,n,{Xj } P . Similarly, set Ck,n = f · 1Dk,n : f ∈ C(X) . Then Ck,n · P is the center of P Dk,n P . Lemma 4.3. Let X be a connected compact metric space with property (A), let ξ1 , ξ2 , . . . , ξn ∈ X, let k, m1 , m2 , . . . , mn 1 be integers with mj |k, j = 1, 2, . . . , n. Let P ∈ Dk,n be a projection. Then, for any > 0, any finite subset F ⊂ P Dk,n P ⊗ Q, there exists a unital monomorphism ψ : P Dk,n−1 P ⊗ Q → P Dk,n P ⊗ Q such that idP Dk,n P ⊗Q ≈ ψ ◦ ı

on F ,

where ı : P Dk,n P ⊗ Q → P Dk,n−1 P ⊗ Q is the embedding (note that Dk,n ⊂ Dk,n−1 ). Moreover, ψ maps (Ck,n−1 P ) ⊗ 1Q into (Ck,n P ) ⊗ 1Q . Proof. We may assume that f 1 for all f ∈ F . Fix > 0 and a finite subset F ⊂ P Dk,n P ⊗ Q. We may assume that P ∈ F . We may assume that f 1 for all f ∈ F and F ⊂ P Dk,n P ⊗ Mm! , where Mm! is a unital C ∗ -subalgebra of Q. There is δ > 0 such that f (x) − f (x ) < /16 for all f ∈ F , whenever dist(x, x ) δ.


1919

Without loss of generality, we may also assume that dist(ξj , ξn ) > δ, j = 1, 2, . . . , n − 1. There are δ > δ1 > δ2 > δ3 > 0 such that there exists a homeomorphism χ : B(ξn , δ1 ) \ {ξn } → A(ξn , δ1 , δ3 ) such that χ|A(ξn ,δ1 ,δ2 ) = id|A(ξn ,δ1 ,δ2 ) . For each f ∈ C(X, Mk ) ⊗ Q define f˜(x) = f (x) if dist(x, ξn ) > δ2 , f˜(x) = f (ξn ) if dist(x, ξn ) δ3 and f˜(x) = f (χ −1 (x)) if x ∈ A(ξn , δ2 , δ3 ). Note that, for each f ∈ F , f˜ − f < /16.

(e4.20)

Moreover P˜ is also a projection. Note that, for each x ∈ X, πx (P˜ Dk,m P˜ ⊗ Q) ∼ = Q for m 1. There are δ3 > δ4 > δ5 > 0 such that there is a homeomorphism χ1 : B(ξn , δ3 ) \ {ξn } → A(ξn , δ3 , δ5 ) such that (χ1 )|A(ξn ,δ3 ,δ4 ) = id|A(ξn ,δ3 ,δ4 ) . Let 0 < δ6 < δ5 . There is a unital isomorphism λ : πξn (P˜ Dk,n−1 P˜ ⊗Q) → πξn (P˜ Dk,n P˜ ⊗Q). Note that πξn (P˜ Dk,n P˜ ⊗ Q) is a unital C ∗ -subalgebra. In particular, (λ|πξ

˜

˜

n (P Dk,n P ⊗Q)

)∗0 = idK0 (πξ

˜

˜

n (P Dk,n P ⊗Q))

.

(e4.21)

Therefore, there is a unitary U0 ∈ πξn (P˜ Dk,n P˜ ⊗ Q) such that ad U0 ◦ λ(a) = a for all a ∈ πξn (P˜ Dk,n P˜ ⊗ Mm! ). To simplify the notation, we may assume that λ(a) = a

for all a ∈ πξn (P˜ Dk,n P˜ ⊗ Mm! ).

(e4.22)

On the other hand, by embedding πξn (P˜ Dk,n P˜ ⊗ Q) into πξn (P˜ Dk,n−1 P˜ ⊗ Q) unitally, we may view λ maps πξn (P˜ Dk,n−1 P˜ ⊗ Q) into πξn (P˜ Dk,n−1 P˜ ⊗ Q). Then (ı ◦ λ)∗0 = idK0 (πξ

˜

˜

n (P Dk,n−1 P ⊗Q))

.

It then follows from 3.1, there is a continuous path of unitaries

u(t): t ∈ [0, δ6 ) ⊂ πξn (P˜ Dk,n−1 P˜ ⊗ Q)

such that u(0) = 1 and lim ad u(t) ◦ λ(f ) = f

t→δ6

for all f ∈ πξn (P˜ Dk,n−1 P˜ ⊗ Q). Define a projection P1 ∈ C(X, Mk )⊗Q as follows: P1 (x) = P˜ (x)⊗1Q for all dist(x, ξn ) δ4 , P1 (x) = P˜ (χ1−1 (x)) ⊗ 1Q if x ∈ A(ξn , δ4 , δ5 ) and P1 (x) = u∗ (t) P˜ (ξn ) ⊗ 1Q u(t) = P (ξn ) ⊗ 1Q if dist(x, ξn ) = t for 0 t δ6 , and P1 (x) = P˜ (ξn ) ⊗ 1Q if δ5 > dist(x, ξn ) δ6 . Thus P1 = P˜ ⊗ 1Q .

1920


Define Φ : P˜ Dk,n−1 P˜ ⊗ Q = P1 (Dk,n−1 ⊗ Q)P1 → P1 (Dk,n ⊗ Q)P1 as follows: Φ(f )(x) = f (x) if dist(x, ξn ) δ4 , Φ(f )(x) = f (χ1−1 (x)) if x ∈ A(ξn , δ4 , δ5 ), Φ(f )(x) = f (ξn ) if δ6 dist(x, ξn ) < δ5 and Φ(f )(x) = u∗ (t)λ(f )u(t) if dist(x, ξn ) = t for 0 t < δ6 . It follows that Φ(f˜)(x) − f˜(x) < /16

(e4.23)

for all f ∈ F and for those x ∈ X with dist(x, ξn ) δ6 . Define a continuous map Γ : B(ξn , δ5 ) → [0, δ5 ] by Γ (x) = dist(x, ξn ) for x ∈ B(ξn , δ5 ). Put Y = Γ (B(ξn , δ6 )) and Y1 = Γ (B(ξn , δ5 )). Define a projection P0 ∈ C(Y, πξn (P1 (Dk,n−1 ⊗ Q)P1 )) by P0 (t) = u∗ (t) P˜ (ξn ) ⊗ 1Q u(t). We note that P0 (t) = P˜ (ξn ) ⊗ 1Q for all t ∈ Y. Define ψ0 : πξn (P˜ Dk,n P˜ ⊗ Q) → C(Y, P1 (ξn )Mk (Q)P1 (ξn )) (∼ = C(Y, Q)) by ψ0 (a)(t) = a for all a ∈ πξn (P˜ Dk,n P˜ ⊗ Q). Define ψ1 : πξn (P˜ Dk,n P˜ ⊗ Q) → C(Y, P1 (ξn )Mk (Q)P1 (ξn )) by ψ1 (a)(t) = u∗ (t)λ(a)u(t) for a ∈ πξn (P˜ Dk,n P˜ ⊗ Q), t ∈ Y \ {1} and ψ(a)(δ6 ) = a for all a ∈ πξn (P˜ Dk,n P˜ ⊗ Q). It follows from 3.3 that there exists a unitary U1 ∈ C Y, P1 (ξn )Mk (Q)P1 (ξn ) such that U1 (0) = U1 (δ6 ) = P1 (ξn ) and (U1 )∗ ψ1 U1 ≈/16 ψ0

on f (ξn ): f ∈ F .

(e4.24)

Put U2 = U1 + (1Mk (Q) − P1 (ξn )). Let U3 ∈ C(X, Mk (Q)) be defined by U3 (x) = 1 if dist(x, ξn ) δ6 and U3 (x) = U2 (t) if dist(x, ξn ) = t for t ∈ [0, δ6 ). We have, by (e4.23), ∗ U (x)Φ( ˜ f˜)U3 (t) − f˜(x) < /16 3

(e4.25)

for all f ∈ F and x ∈ X with dist(x, ξn ) δ6 , and, by (e4.24), ∗ U (x)Φ(f˜)U3 (x) − f˜(x) < /16 3

for all f ∈ F and x ∈ X with dist(x, ξn ) < δ6 .

(e4.26)


1921

Note that P − P˜ < /16. There is a unitary U4 ∈ C(X, Mk ) such that √ U − 1 < 2/16 4 and ∗ U4 P˜ U4 = P (see Lemma 6.2.1 of [12] for example). Then U4 (x) = P (x) if dist(x, ξn ) δ3 and U4 (ξn ) = P (ξn ). Define U4 = (1C(X,Mk )⊗Q − 1C(X,Mk ) ) + U4

∈ C(X, Mk ) ⊗ 1Q . Define ψ : P Dk,n−1 P ⊗ Q → P Dk,n P ⊗ Q by ψ(f ) = ad U4 U3 ◦ Φ(f )

for all f ∈ P Dk,n−1 P ⊗ Q.

Now we estimate (by applying (e4.25) and (e4.26)) that f − ψ(f ) f − f˜ + f˜ − ψ(f˜) √ < /16 + 2/16 + f˜ − U3∗ Φ(f˜)U3 √ (1 + 2) < + 2/16 < 16

(e4.27) (e4.28) (e4.29)

for all f ∈ F . Note that ψ is a unital monomorphism. This proves the first part of the lemma. The last part follows from the construction that ψ maps (Ck,n−1 P ) ⊗ 1Q into (Ck,n P ) ⊗ 1Q . 2 Lemma 4.4. Let X be a connected compact metric space with property (A), let ξ1 , ξ2 , . . . , ξn ∈ X, let k, m1 , m2 , . . . , mn 1 be integers with mj |k, j = 1, 2, . . . , n. Let P ∈ Dk (X, ξ1 , ξ2 , . . . , ξn , m1 , m2 , . . . , mn ) be a projection. Then, for any > 0, any finite subset F ⊂ P C(X, Mk )P ⊗ Q, there exists a unital monomorphism ψ : P C(X, Mk )P ⊗ Q → P Dk,n P ⊗ Q such that idP Dk,n P ⊗Q ≈ ψ ◦ ı

on F ,

where ı : P Dk,n P ⊗ Q → P C(X, Mk )P ⊗ Q is the embedding. Moreover, ψ maps (C(X) · P ) ⊗ 1Q into (Ck,n P ) ⊗ 1Q .

1922


Proof. We prove this by induction on n. If n = 1, then the lemma follows from 4.3 immediately. Suppose that the lemma holds for integers 1 n N. Denote by Dk,l = Dk (X, ξ1 , ξ2 , . . . , ξl , m1 , m2 , . . . , ml ), l = 1, 2, . . . . Let > 0, let F ⊂ P C(X, Mk )P ⊗ Q and let F ⊂ Q be a unital finite dimensional C ∗ subalgebra. By the inductive assumption, there exists a unital monomorphism ψ1 : P C(X, Mk )P ⊗ Q → P Dk,N P ⊗ Q such that ψ1 (f ) − f < /2

(e4.30)

for all f ∈ F . Moreover there exists a unital finite dimensional C ∗ -subalgebra F0 ⊂ Q such that ψ1 P C(X, Mk )P ⊗ F ⊂ CF . By Lemma 4.3, there exists a unital monomorphism ψ2 : P Dk,N P ⊗ Q → P Dk,N +1 P ⊗ Q such that ψ2 (f ) − f < /2

(e4.31)

for all f ∈ W1∗ ψ1 (F )W1 and a unital finite dimensional C ∗ -subalgebra F1 ∈ Q such that ψ2 (P Dk,N P ⊗ F0 ) ⊂ CF0 . Define ψ : P C(X, Mk )P ⊗ Q → P Dk,N +1 P ⊗ Q by ψ = ad W1∗ ◦ ψ2 ◦ ad W1 ◦ ψ1 . Then ψ(f ) − f = W1 ψ2 W ∗ ψ1 (f )W1 W ∗ − f 1 1 W1 ψ2 W1∗ ψ1 (f )W1 W1∗ − ψ1 (f ) + ψ1 (f ) − f ψ2 W1∗ ψ1 (f )W1 − W1∗ ψ1 (f )W1 + /2 < /2 + /2 =

(e4.32) (e4.33) (e4.34)

for all f ∈ F . Moreover, by 4.3, ψ maps (C(X) · P ) ⊗ 1Q into (Dk,n P ) ⊗ 1Q . This completes the induction. 2 Lemma 4.5. Let X be a connected compact metric space with property (A), let X1 , X2 , . . . , Xn ⊂ X be compact subsets, let k, m1 , m2 , . . . , mn 1 be integers with mj |k, j = 1, 2, . . . , n. Let P ∈ D¯ k,{Xj },n = D¯ k (X, X1 , X2 , . . . , Xn , m1 , m2 , . . . , mn ) be a projection. Then, for any > 0, any finite subset F ⊂ P C(X, Mk )P ⊗ Q, there exists a unital monomorphism ψ : P C(X, Mk )P ⊗ Q → P D¯ k,{Xj },n P ⊗ Q such that idP D¯ k,{X

j },n

where

P ⊗Q

≈ ψ ◦ ı

on F ,


1923

ı : P D¯ k,{Xj },n P ⊗ Q → P C(X, Mk )P ⊗ Q is the embedding. Moreover, ψ maps (C(X) · P ) ⊗ 1Q into (Ck,{Xj },n P ) ⊗ 1Q . Furthermore, if F ⊂ Q is a unital finite dimensional C ∗ -subalgebra, we may choose ψ so that there exists another unital finite dimensional C ∗ -subalgebra F1 ⊂ Q and a unital C ∗ -subalgebra CF ⊂ P Dk,n P ⊗ Q such that ψ P C(X, Mk )P ⊗ F ⊂ CF and there is a unitary W ∈ M2 (P D¯ k,{Xj },n P ⊗ Q) such that W ∗ CF W = P1 M2 (P D¯ k,{Xj },n P ⊗ F1 )P1 for some projection P1 ∈ M2 (P D¯ k,{Xj },n P ⊗ F1 ). Proof. Let {ξi : i = 1, 2, . . .} be a dense sequence of Xn . Put l

Dk,{Xj }n−1 ,{ξi },l = D¯ k (X, X1 , X2 , . . . , Xn−1 , ξ1 , ξ2 , . . . , ξl , m1 , m2 , . . . , ml−1 , mn , mn , . . . , mn ). j =1

Note that P Dk,{Xj }n−1 ,{ξi },l+1 P ⊂ P Dk,{Xj }n−1 ,{ξi },l P and j =1

j =1

∞

P Dk,{Xj }n−1 ,{ξi },l P = P D¯ k,{Xj },n P .

l=1

j =1

Let > 0 and let a finite subset F ⊂ P D¯ k,{Xj }n−1 ,n−1 P ⊗ Q be given. j =1

Let {F0,m } be an increasing sequence of finite subsets of P D¯ k,{Xj }n−1 ,n−1 P ⊗ Q whose j =1

union is dense in P D¯ k,{Xj }n−1 ,n−1 P ⊗ Q. Let {F∞,m } be an increasing sequence of finite j =1

subsets of P D¯ k,{Xj }nj=1 ,n P ⊗ Q whose union is dense in P D¯ k,{Xj }nj=1 ,n P ⊗ Q. Let {Fl,m } be an increasing sequence of finite subsets of P Dk,{Xj }n−1 ,{ξi },l P ⊗ Q whose union is dense in j =1 P Dk,{Xj }n−1 ,{ξi },l P ⊗ Q. By replacing Fj,m by sj Fs,m , we may assume that Fj +1,m ⊂ Fj,m , j =1

j = 0, 1, 2, . . . , andF∞,m ⊂ Fj,m , j = 1, 2, . . . . Let {n } be a sequence of decreasing positive numbers such that ∞ n=1 n < /2. We may also assume that F = F0,1 . By applying 4.4, one obtains, for each l, a unital monomorphism ψl : P Dk,{Xj }n−1 ,{ξi },l−1 P ⊗ j =1

Q → P Dk,{Xj }n−1 ,{ξi },l P ⊗ Q which maps Ck,{Xj }n−1 ,{ξi },l−1 · P into Ck,{Xj }n−1 ,{ξi },l P such that j =1

j =1

idP D

k,{Xj }n−1 j =1 ,{ξi },l

P ⊗Q

≈l /2 ψn ◦ ı

j =1

on Gl−1,l ,

where G0,1 = F0,1 = F and Gl−1,l = Fl−1,l ∪ ψl−1 (Gl−1,l ), l = 2, 3, . . . .

(e4.35)

1924


Define A = liml→∞ (P Dk,{Xj }n−1 ,{ξi },l−1 P ⊗ Q, ψl ). We claim that A ∼ = P D¯ k,{Xj },n P ⊗ Q. j =1

Put Bl = P Dk,{Xj }n−1 ,{ξi },l−1 P ⊗ Q, l = 1, 2, . . . , and put C = C(X, Mk ) ⊗ Q. Note that j =1

B1 = P D¯ k,{Xj }n−1 ,n−1 P ⊗ Q. j =1

Consider the following diagram:

B1

ψ1

B2

ı

B3

ı id

C

ψ2

···

A

id

···

C

ı id

C

ψ3

C

It follows from (e4.35) that the diagram is one-sided approximately intertwining. Therefore, by an argument of Elliott (see Theorem 1.10.4 of [5], for example) there exists a homomorphism Φ : A → C = C(X, Mk ) ⊗ Q such that ı Φ ◦ ψl,∞ ≈∞ j =l l

on Gl−1,l

and

(e4.36)

Φ ◦ ψl,∞ (b) = lim ı ◦ ψl,s (b)

(e4.37)

s→∞

for all b ∈ Bl−1 . Combining (e4.35) and (e4.37), we conclude that dist Φ ◦ ψl,∞ (b), Cl = 0

(e4.38)

for all b ∈ Bl . Since Cm ⊂ Cl , if m l, we conclude that dist Φ ◦ ψm,∞ (b), Cl = 0

(e4.39)

for all b ∈ Cm , m = l, l + 1, . . . . It follows that dist Φ(A), Cl = 0

for all l.

(e4.40)

Therefore Φ(A) ⊂

∞

Cl = P Dk,{Xj },n P ⊗ Q.

(e4.41)

l=1

Put D = P Dk,{Xj },n P ⊗ Q. Then ı : D → Bl for each l. Thus, as above, we obtain the following one-sided approximately intertwining: id

D ı

B1

id

D

B2

id

···

D

ψ3

···

A

ı

ı ψ1

D

ψ2

B3


1925

From this we obtain a homomorphism J : D → A such that J (d) = lim ψl,∞ ◦ ı(d) l→∞

(e4.42)

for all d ∈ D. We then compute (by (e4.37) and (e4.35)) that Φ ◦ J = idD .

(e4.43)

This, in particular, implies that Φ maps A onto D. We then conclude that Φ gives an isomorphism from A to D and Φ −1 = J. Put Ψ = Φ ◦ ψ1,∞ : P Dk,{Xj }n−1 ,n−1 P ⊗ Q → D. Then, by (e4.36), we have j =1

idD ≈/2 Ψ ◦ ı

on F = F0,1 .

(e4.44)

Moreover, Ψ is a unital monomorphism. Furthermore, we also note that Ψ maps Ck,{Xj }n−1 ,n−1 P ⊗ 1Q into Ck,{Xj },n P ⊗ 1Q . j =1

We now use the induction. The same argument used in the proof of 4.4 proves that there exists a unital monomorphism ψ : P C(X, Mk )P ⊗ Q → P Dk,{Xj },n P ⊗ Q such that idD ≈ ψ ◦ ı

on F

(e4.45)

and ψ maps (C(X, Mk ) · P ) ⊗ 1Q into (Ck,{Xj },n · P ) ⊗ 1Q . This proves the first two parts of the statement. To prove the last part of the statement, let F0 ⊂ Q be a unital finite dimensional C ∗ subalgebra. We may assume, without loss of generality, that F0 ⊂ Mm! . We may also assume that m > 2(dim(X) + 1) and 2(rank(P ) × m! − dim(X)) rank(P ) × m!. Put r = rank(P ) × m! − dim(X). There is a system of matrix units {ei,j }ri,j =1 ⊂ P C(X, Mk )P ⊗ Mm! such that ei,i is a rank one trivial projections and P ⊗ 1Mm! = q0 ⊕

r

ei,i ,

i=1

where q0 is a projection of rank dim(X). Let E be the C ∗ -subalgebra generated by (C(X) · P ⊗ 1Mm! ) · e1,1 and {ei,j : 1 i, j r}. There is a projection q00 ∈ E with rank dim(X) and a unitary w0 ∈ P C(X, Mk )P ⊗ Mm! such that w0∗ q00 w0 = q0 . It follows that there exists a projection P1 ∈ M2 (E) and a unitary W0 ∈ M2 (P C(X, Mk )P ⊗ Mm! ) such that W0∗ P1 M2 (E)P1 W0 = P C(X, Mk )P ⊗ Mm! . For any δ > 0, there is an integer N m such that dist ψ(ei,j ), D ⊗ MN ! < δ

(e4.46)

1926


for some integer N m, 1 i, j r. It follows from Lemma 2.5.10 of [5] that, by choosing small δ, there exists a unitary W1 ∈ D ⊗ MN ! such that W1 − 1 < /2

and W1∗ ψ(ei,j )W1 ⊂ D ⊗ MN ! ,

i, j = 1, 2, . . . , r.

Since ψ(C(X) · P ⊗ 1Q ) is in the center of D ⊗ Q, ψ f W1∗ ei,j W1 = ψ(f )ψ(W1 )∗ ψ(ei,j )ψ(W1 ) ∗

(e4.47) ∗

= ψ(W1 ) ψ(f )ψ(ei,j )ψ(W1 ) = ψ(W1 ) ψ(f ei,j )ψ(W1 )

(e4.48)

for all f ∈ C(X) · P ⊗ 1Q and 1 i, j r. It follows that ψ(W1 )∗ ψ(E)ψ(W1 ) ⊂ P Dk,{Xj },n P ⊗ MN ! . Put P2 = (ψ ⊗ idM2 )(P1 ). Then P2 ∈ M2 (P Dk,{Xj },n P ⊗ MN ! ). Define W3 = By (e4.46),

ψ(W1 ) 0

ψ P C(X, Mk )P ⊗ Mm! = (ψ ⊗ idM2 ) P C(X, Mk )P ⊗ Mm! = (ψ ⊗ idM2 ) W0∗ P1 M2 (E)P1 W0 = (ψ ⊗ idM2 )(W0 )∗ P2 M2 ψ(E) P2 (ψ ⊗ idM2 )(W0 ) = (ψ ⊗ idM2 )(W0 )∗ W3 W3∗ P2 M2 ψ(E) P2 W3 W3∗ (ψ ⊗ idM2 )(W0 ) ⊂ ψ ⊗ idM2 (W0 )∗ W3 P2 M2 (P Dk,{Xj },n P ⊗ MN ! )P2 W3∗ (ψ ⊗ idM2 )(W0 ). Put W = W3∗ (ψ ⊗ idM2 (W0 )).

0 ψ(W1 ) .

(e4.49) (e4.50) (e4.51) (e4.52) (e4.53)

2

5. The main results Theorem 5.1. Let A = limn→∞ (An , ϕn ) be a unital simple inductive limit of generalized dimension drop algebras. Then TR(A ⊗ Q) 1. ∗ Proof. Write Q = ∞ n=1 Fn , where each Fn is a unital simple finite dimensional C -subalgebras and Fn ⊂ Fn+1 , n = 1, 2, . . . . Let Φn = ϕn ⊗ idQ : An ⊗ Q → An+1 ⊗ Q, n = 1, 2, . . . . Write An =

m(n)

Pn,j D Xn,j , r(n, j ) Pn,j ,

j =1

where Xn,j is a connected compact metric space with finite dimension and with property (A), r(n, j ) 1 is an integer, D Xn,j , r(n, j ) = D¯ r(n,j ) Xn,j , r(n, j ), Yn,j,1 , , . . . , Yn,j,d(n,j ) , mn,j,1 , . . . , mn,j,d(n,j ) ,


1927

Yn,j,1 , Yn,j,2 , . . . , Yn,j,d(n,j ) are disjoint compact subsets of Xn,j and Pn,j ∈ D(Xn,j , r(n, j )) is a projection, n = 1, 2, . . . . Define Bn =

m(n)

Pn,j C(Xn,j , Mr(n,j ) )Pn,j ,

j =1

n = 1, 2, . . . . Denote by ın : An → Bn the embedding. Denote dn = max dim(Xn,j ): 1 j m(n) . sequence of finite subsets of A ⊗ Q for which Fn ⊂ Φn,∞ (An ⊗ Q), Let {Fn } be an increasing n = 1, 2, . . . , and ∞ n=1 Fn is dense in A ⊗ Q. Let Gn ⊂ An ⊗ Q be a finite subset such that Φn,∞ (Gn ) = Fn , n = 1, 2, . . . . Without loss of generality, we may assume that Gn ⊂ An ⊗ Fn , n = 1, 2, . . . . Let {n } be a sequence of decreasing positive numbers such that ∞ n=1 n < 1/2. It follows from 4.5 that there exists a unital monomorphism Ψ1 : B1 ⊗ Q → A1 ⊗ Q such that Ψ1 ◦ ı1 ≈1 idA1 ⊗Q

on G1 .

(e5.54)

Moreover, by 4.5, there exists a unital finite dimensional C ∗ -subalgebra E1,1 ⊂ Q, a unital C ∗ subalgebra CF1 ⊂ A1 ⊗ Q, a projection P1 ∈ M2 (A1 ⊗ E1,1 ) and a unitary W1 ∈ M2 (A1 ⊗ Q) such that Ψ1 (B1 ⊗ F1 ) ⊂ CF1

and W1∗ CF1 W1 = P1 M2 (A1 ⊗ E1,1 )P1 .

Put U1 = 1B1 and C1 = B1 ⊗ F1 . Define θ1 = Φ1 ◦ Ψ1 : B1 ⊗ Q→ A2 ⊗ Q. Suppose that {S1,k } is an increasing sequence of finite subsets of B1 ⊗ Q such that ∞ k=1 S1,k is dense in B1 ⊗ Q. We may assume that

S1,k ⊂

m(1)

P1,j C(X1,j , Mr(1,j ) )P1,j ⊗ Fk .

j =1

Define S1 = S1,1 ∪ ı1 (G1 ) and G2 = G2 ∪ θ1 (S1 ). To simplify the notation, we may assume, without loss of generality, that there exists a finite subset G2

⊂ A2 ⊗ F2 such that G2 ⊂2 /2 G2

. By applying 4.5, there exists a unital monomorphism Ψ2 : B2 ⊗ Q → A2 ⊗ Q such that Ψ2 ◦ ı2 ≈2 idB2

on G2 ∪ G2

.

(e5.55)

Moreover, by 4.5, there exists a unital finite dimensional C ∗ -subalgebra E2,1 ⊂ Q, a unital C ∗ subalgebra CF2 ⊂ A2 ⊗ Q, a projection P2 ∈ M2 (A2 ⊗ E2,1 ) and a unitary W2 ∈ M2 (P2 (A2 ⊗ Q)P2 ) such that

1928


Ψ2 (B2 ⊗ E1,1 ) ⊂ CF2

and

W2∗ CF2 W2 = P2 M2 (A2 ⊗ E2,1 )P2 .

(e5.56) (e5.57)

Define ψ1 = ı2 ◦ Φ1 ◦ Ψ1 : B1 ⊗ Q → B2 ⊗ Q. Define θ2 = Φ2 ◦ Ψ2 . We have ı2 ◦ θ1 = ψ1 .

(e5.58)

Also, by (e5.54) and (e5.55), we have θ1 ◦ ı1 ≈1 Φ1

on G1

and

(e5.59)

θ2 ◦ ı2 ≈2 Φ2

on G2 ∪ G2

.

(e5.60)

Let {S2,k } be an increasing sequence of finite subsets of B2 ⊗ Q such that its union is dense in B2 ⊗ Q. We may assume that ψ1 (S1,2 ), ı2 G2 ∪ G2

⊂3 /2 S2,2 .

(e5.61)

We may also assume that (Φ1 ⊗ idM2 )(W1∗ )S2,k (Φ1 ⊗ idM2 )(W1 ) ⊂ (Φ1 ⊗ idM2 )(P1 )M2 (B2 ⊗ E2,k )(Φ1 ⊗ idM2 )(P1 ), where {E2,k } is an increasing sequence of finite dimensional C ∗ subalgebras of Q whose union is dense in Q. We assume that E1,1 ⊂ E2,2 and E2,2 is simple and d23 < 1/22 . rank(E2,2 ) Define C2 = W1 (Φ1 ⊗ idM2 )(P1 )M2 (B2 ⊗ E2,2 )(Φ1 ⊗ idM2 )(P1 )W1∗ ⊂ B2 ⊗ Q. Define U2 = W1∗ . Then C2 = U2∗ (Φ1 ⊗ idM2 )(P1 )M2 (B2 ⊗ E2,2 )(Φ1 ⊗ idM2 )(P1 )U2 . Moreover, ψ1 (C1 ) ⊂ C2 . Define S2 = S2,2 ∪ ı2 (G2 ) ⊂ C2 and G3 = G3 ∪ θ2 (S2 ). We may assume that there is a finite subset G3

⊂ A3 ⊗ F3 such that G3 ⊂3 /2 G3

. We will use induction. Suppose that unital monomorphism Ψn : Bn ⊗ Q → An ⊗ Q, ψn = ın+1 ◦ Φn ◦ Ψn : Bn ⊗ Q → Bn+1 ⊗ Q, θn = Ψn ◦ Φn : Bn ⊗ Q → An+1 ⊗ Q (n = 1, 2, . . . , N ), a unital finite dimensional C ∗ -subalgebra En,1 ⊂ Q (n = 1, 2, . . . , N − 1), a projection Pn ∈ M2 (An ⊗ En,1 ) and a unitary Wn ∈ M2 (An ⊗ En,1 ) (n = 1, 2, . . . , N ) have been constructed such that θn ◦ ın ≈n Φn

on Gn ∪ Gn

,

(e5.62)

ψn = ın+1 ◦ θn ,

(e5.63)

Wn∗ Ψn (Bn ⊗ En−1,n−1 )Wn ⊂ Pn M2 (An ⊗ En,1 )Pn ,

(e5.64)

where {En,k } is an increasing sequence of unital finite dimensional C ∗ -subalgebras of Q whose union is dense in Q, with En−1,n−1 ⊂ En,1 , En,n is a full matrix algebra and dn3 < 1/2n , rank(En,n )

(e5.65)


1929

where {Sn,k } is an increasing sequence of finite subsets of Bn ⊗ Q whose union is dense in Bn ⊗ Q, n−1

ψj,n−1 (Sj,n ) ∪ ın Gn ∪ Gn

⊂n+1 /2 Sn,n ,

(e5.66)

j =1

Un Sn,k Un∗ ⊂ P (n−1) M2n−1 (Bn ⊗ En,k )P (n−1) ,

(e5.67)

∗ ⊗ 1 )ψ (n−1) ∈ where Un = (Φn−1 ⊗ idM2n−1 )(Wn−1 M2 n−1 (Un−1 ), n = 3, 4, . . . , N + 1, where P M2n−1 (Bn ⊗ En,n ) with (n−1)

Un∗ P (n−1) Un = 1Bn ⊗Q = 1Bn ⊗En,n ,

, G

= Gn+1 ∪ θn (Sn ), Gn+1 ⊂n+1 /2 Gn+1 where Gn+1 n+1 ⊂ An+1 ⊗ Fn+1 is a finite subset, n =

1, 2, . . . , N + 1 (with G1 = G1 ), where ψn = ψn ⊗ idM2i , i = 1, 2, . . . . Furthermore, Cn = Un∗ P (n−1) M2n−1 (Bn ⊗ En,n )P (n−1) Un and ψn−1 (Cn−1 ) ⊂ Cn , n = 2, 3, . . . , N + 1. By applying 4.5, we obtain a unital monomorphism ΨN +1 : BN +1 ⊗ Q → AN +1 ⊗ Q such that (i)

ΨN +1 ◦ ıN +1 ≈N+1 idBN+1

on GN +1 ∪ GN +1 .

(e5.68)

Moreover, there exists a unital finite dimensional C ∗ -subalgebra EN +1,1 ⊂ Q, a projection PN +1 ∈ M2 (AN +1 ⊗ EN +1,1 ) and a unitary WN +1 ∈ M2 (AN +1 ⊗ EN +1,1 ) such that WN∗ +1 ΨN +1 (BN +1 ⊗ EN +1,N +1 ) WN +1 ⊂ PN +1 M2 (AN +1 ⊗ EN +1,1 )PN +1 .

(e5.69)

Define θN +1 = ΦN +1 ◦ ΨN +1 : BN +1 ⊗ Q → AN +2 ⊗ Q and ψN +1 = ıN +2 ◦ θN +1 : BN +1 ⊗ Q → BN +2 ⊗ Q. Then, by (e5.62) θN +1 ◦ ıN +1 ≈N+1 ΦN +1

on GN +1 .

(e5.70)

Let {EN +1,k } be an increasing sequence of unital finite dimensional C ∗ -subalgebras of Q whose union is dense in Q and EN,N ⊂ EN +1,1 , EN +1,N +1 is a full matrix algebra with dN +1 < 1/2N +1 . rank(EN +1,N +1 )

(e5.71)

Let {SN +1,k } be an increasing sequence of finite subsets of BN +1 ⊗ Q whose union is dense in BN +1 ⊗ Q. We may assume that N

ψj,N (Sj,n ) ∪ ın Gn ∪ Gn

⊂N+1 /2 SN +1,N +1 .

j =1

Since UN∗ +1 P (N ) UN +1 = 1BN+1 ⊗1Q , we may further assume that UN +1 SN +1,k UN∗ +1 ⊂ (ΦN +1 ⊗ idM2 ) P (N ) M2 (BN +1 ⊗ EN +1,k )(ΦN +1 ⊗ idM2 ) P (N ) .

1930


(N +1) Define UN +2 = (ΦN +1 ⊗ idM2N )(WN∗ +1 ⊗ 1M2N )ψN +1 (UN +1 ). Put U¯ N +i = (ψN +i ⊗ (i) (N ) idM2 )(UN +i ), i = 1, 2, and Φm = Φm ⊗ idM2i , m, i = 1, 2, . . . , and P¯ (N ) = ψN +1 (P (N ) ). Then (using (e5.64))

ψN +1 (CN +1 ) (N ) = ψN +1 UN∗ +1 P (N ) M2N (BN +1 ⊗ EN +1,N +1 )P (N ) UN +1 (N ) = U¯ N∗ +1 P¯ (N ) ψN +1 M2N (BN +1 ⊗ EN +1,N +1 ) P¯ (N ) U¯ N +1 (2) (2) ⊂ U¯ N∗ +1 P¯ (N ) M2N ΦN +1 (WN +1 PN +1 )M2 (BN +2 ⊗ EN +2,N +2 )ΦN +1 PN +1 WN∗ +1 × P¯ (N ) U¯ N +1 (2) ∗ Note that ΦN +1 (WN +1 PN +1 WN +1 ) has the form

1BN+2 ⊗ 1EN+2,N+2 0

0 . 0

Therefore (2) P¯ (N ) ΦN +1 WN +1 PN +1 WN∗ +1 ⊗ 1M2N = P¯ (N ) .

(e5.72)

Put (N +1) W¯ N +1 = ΦN +1 (WN +1 ⊗ 1M2N )

and P (N +1) = W¯ N∗ +1 P¯ (N ) W¯ N +1 .

(e5.73)

One has (N +1) (N +1) UN∗ +2 P (N +1) UN +2 = ψN +1 (UN +1 )∗ W¯ N +1 P (N +1) W¯ N∗ +1 ψN +1 (UN +1 ) (N +1)

(N +1)

= ψN +1 (UN +1 )∗ P¯ (N ) ψN +1 (UN +1 )∗ (N +1) (N +1) = ψN +1 UN∗ +1 P (N ) UN +1 = ψN +1 (1BN+1 ⊗Q ) = 1BN+2 ⊗Q .

(e5.74) (e5.75) (e5.76)

It follows that (see (e5.73)) (e5.77)

ψN +1 (CN +1 ) ⊂ U¯ N∗ +1 P¯ (N ) W¯ N +1 M2N+1 (BN +2 ⊗ EN +2,N +2 )W¯ N∗ +1 P¯ (N ) U¯ N +1

(e5.78)

= U¯ N∗ +1 W¯ N +1 P (N +1) M2N+1 (BN +2

(e5.79)

⊗ EN +2,N +2 )P

(N +1)

W¯ N∗ +1 U¯ N +1

= UN∗ +2 P (N +1) M2N+2 (BN +2 ⊗ EN +2,N +2 )P (N +1) UN +2 . Define CN +1 = UN∗ +2 P (N +1) M2N+2 (BN +2 ⊗ EN +2,N +2 )P (N +1) UN +2 .

(e5.80)


1931

Therefore ψN +1 (CN +1 ) ⊂ CN +2 . Put En = An ⊗ Q and Bn = Bn ⊗ Q and define C = limn→∞ (Bn , ψn ). It follows from (e5.62) and (e5.63) that the following diagram Φ1

E1

Φ2

E2

ı1

E3

B1

θ2

ψ1

···

ı3

ı2 θ1

Φ3

B2

ψ2

B3 · · ·

is approximately intertwining in the sense of Elliott. It follows that A ⊗ Q ∼ = C. In particular, C is a unital simple C ∗ -algebra. By (e5.67), Sn,n ⊂ Cn ,

n = 1, 2, . . . .

It follows from this and (e5.66) that ∞

ψn,∞ (Cn ) = C.

n=1

Thus C = lim Cn , (ψn )|Cn . n→∞

It follows from (e5.65) that C is a unital simple AH-algebra with very slow dimension growth. It follows from a theorem of Guihua Gong (see Theorem 2.5 of [6]) that TR(C) 1. 2 Now we are ready to prove Theorem 1.3. 5.2. The proof of Theorem 1.3: Proof. It follows from 5.1 that A, B ∈ A. Thus the theorem follows from 2.11 (see [10]).

2

5.3. A unital simple inductive limit of generalized dimension drop algebra A is said to have no dimension growth, if A = limn→∞ (An , ϕn ), where An =

m(n)

Pn,j D¯ n,j Pn,j ,

j =1

D¯ n,j = Dk(n) (Xn,j , Yn,1 , Yn,2 , . . . , Yn,s(n,j ) , mn,1 , mn,2 , . . . , mn,s(n,j ) ),

1932


Xn,j is a finite dimensional connected compact metric space with property (A), {Yn,i : 1 i s(n, j )} is a collection of finitely many disjoint compact subsets of Xn,j , Pn,j ∈ D¯ n,j is a projection and {dim(Xn,j )} is bounded. 5.4. The proof of Theorem 1.4: Proof. By [17], ASH-algebras with no dimension growth are of finite decomposition rank. It follows from another recent result of W. Winter [19] that A and B are Z-stable. Therefore the theorem follows from 1.3 immediately. 2 Acknowledgments This work was mostly done when the author was in the East China Normal University during the summer of 2008. It was partially supported by an NSF grant (DMS074813), the Chang–Jiang Professorship from East China Normal University and Shanghai Priority Academic Disciplines. References [1] J. Dixmier, On some C ∗ -algebras considered by Glimm, J. Funct. Anal. 1 (1967) 182–203. [2] G.A. Elliott, G. Gong, Guihua, X. Jiang, H. Su, A classification of simple limits of dimension drop C ∗ -algebras, in: Operator Algebras and Their Applications, Waterloo, ON, 1994/1995, in: Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 125–143. [3] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C ∗ -algebras, II: The isomorphism theorem, Invent. Math. 168 (2) (2007) 249–320. [4] X. Jiang, H. Su, On a simple unital projectionless C ∗ -algebra, Amer. J. Math. 121 (2) (1999) 359–413. [5] H. Lin, An Introduction to the Classification of Amenable C ∗ -Algebras, World Scientific Publishing Co., River Edge, NJ, 2001. [6] H. Lin, Simple nuclear C ∗ -algebras of tracial topological rank one, J. Funct. Anal. 251 (2007) 601–679. [7] H. Lin, Approximate homotopy of homomorphisms from C(X) into a simple C ∗ -algebra, Mem. Amer. Math. Soc., in press, arxiv:math/0612125. [8] H. Lin, Asymptotic unitary equivalence and asymptotically inner automorphisms, Amer. J. Math., in press, arXiv:math/0703610. [9] H. Lin, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, II—An appendix, arXiv: 0709.1654v3, 2007. [10] H. Lin, Asymptotically unitary equivalence and classification of simple amenable C ∗ -algebras, arXiv:0806.0636, 2008. [11] H. Lin, Z. Niu, The range of a class of classifiable separable simple amenable C ∗ -algebras, Adv. Math. 219 (2008) 1729–1769. [12] G.J. Murphy, C ∗ -Algebras and Operator Theory, Academic Press, Inc., Boston, MA, ISBN 0-12-511360-9, 1990, x+286 pp. [13] J. Mygind, Classification of certain simple C ∗ -algebras with torsion K1 , Canad. J. Math. 53 (6) (2001) 1223–1308. [14] H. Su, On the classification of C ∗ -algebras of real rank zero: Inductive limits of matrix algebras over non-Hausdorff graphs, Mem. Amer. Math. Soc. 114 (547) (1995), viii+83 pp. [15] K. Thomsen, Limits of certain subhomogeneous C ∗ -algebras, Mém. Soc. Math. Fr. (N.S.) 71 (1997), vi+125 pp. [16] J. Villadsen, The range of the Elliott invariant of the simple AH-algebras with slow dimension growth, K-Theory 15 (1998) 1–12. [17] W. Winter, Decomposition rank of subhomogeneous C ∗ -algebras, Proc. Lond. Math. Soc. 89 (2004) 427–456. [18] W. Winter, Localizing the Elliott conjecture at strongly self-absorbing C ∗ -algebras, arXiv:0708.0283v3, 2007. [19] W. Winter, Decomposition rank and Z-stability, preprint, arXiv:0806.2948, 2008.


Small ball probability estimates, ψ2 -behavior and the hyperplane conjecture Nikos Dafnis a , Grigoris Paouris b,∗ a Department of Mathematics, University of Athens, Panepistimioupolis 157 84, Athens, Greece b Department of Mathematics, Texas A & M University, College Station, TX 77843, USA

Received 21 January 2009; accepted 10 June 2009 Available online 25 November 2009 Communicated by K. Ball

Abstract We introduce a method which leads to upper bounds for the isotropic constant. We prove that a positive answer to the hyperplane conjecture is equivalent to some very strong small probability estimates for the Euclidean norm on isotropic convex bodies. As a consequence of our method, we obtain an alternative proof of the result of J. Bourgain that every ψ2 -body has bounded isotropic constant, with a slightly better n n−1 and estimate: If K is a symmetric √ convex body in R such that ·, θq β·, θ2 for every θ ∈ S every q 2, then LK Cβ log β, where C > 0 is an absolute constant. © 2009 Published by Elsevier Inc. Keywords: Hyperplane conjecture; ψ2 -bodies; Small ball probability

1. Introduction A convex body K in Rn is called isotropic if it has volume |K| = 1, center of mass at the origin, and its inertia matrix is a multiple of the identity. Equivalently, if there is a constant LK > 0 such that x, θ 2 dx = L2K K


E-mail addresses: [email protected] (N. Dafnis), [email protected] (G. Paouris). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.06.038

(1.1)

1934

N. Dafnis, G. Paouris / Journal of Functional Analysis 258 (2010) 1933–1964

for every θ in the Euclidean unit sphere S n−1 . It is not hard to see that for every convex body K in Rn there exists an affine transformation T of Rn such that T (K) is isotropic. Moreover, this isotropic image is unique up to orthogonal transformations; consequently, one may define the isotropic constant LK as an invariant of the affine class of K. The isotropic constant is closely related to the hyperplane conjecture (also known as the slicing problem) which asks if there exists an absolute constant c > 0 such that maxθ∈S n−1 |K ∩ θ ⊥ | c for every convex body K of volume 1 in Rn with center of mass at the origin. This is because, by Brunn’s principle, for any convex body K in Rn and any θ ∈ S n−1 , 1 the function t → |K ∩ (θ ⊥ + tθ )| n−1 is concave on its support, and this implies that

−2 x, θ 2 dx K ∩ θ ⊥ .

(1.2)

K

Using this relation one can check that an affirmative answer to the slicing problem is equivalent to the following statement: There exists an absolute constant C > 0 such that LK C for every convex body K. We refer to the article [16] of Milman and Pajor for background information about isotropic convex bodies. The isotropic constant and the hyperplane conjecture can be studied in the more general set ting of log-concave measures. Let f : Rn → R+ be an integrable function with Rn f (x) dx = 1. We say that f is isotropic if f has center of mass at the origin and

x, θ 2 f (x) dx = 1

(1.3)

Rn

for every θ ∈ S n−1 . It is well known that the hyperplane conjecture for convex bodies is equivalent to the following statement: There exists an absolute constant C > 0 such that, for every isotropic log-concave function f on Rn , 1

f (0) n C.

(1.4)

It is known that LK LB2n c > 0 for every convex body K in Rn (we use the letters c, c1 , C, √ etc. to denote absolute constants). Bourgain proved in [3] that LK c 4 n log n and, a few years √ ago, Klartag [8] obtained the estimate LK c 4 n. The approach of Bourgain in [3] is to reduce √ the problem to the case of convex bodies that satisfy a ψ2 -estimate (with constant β = O( 4 n )). We say that K satisfies a ψ2 -estimate with constant β if ·, y

ψ2

β ·, y2

(1.5)

for all y ∈ Rn . Bourgain proved in [4] that, if (1.5) holds true, then LK Cβ log β.

(1.6)

The purpose of this paper is to introduce a different method which leads to upper bounds for LK . We prove that a positive answer to the hyperplane conjecture is equivalent to some


1935

very strong small probability estimates for the Euclidean norm on isotropic convex bodies; for −n < p ∞, p = 0, we define Ip (K) :=

1 p

x2 dx

p

(1.7)

K

and, for δ 1, we consider the parameter q−c (K, δ) := max p 1: I2 (K) δI−p (K) .

(1.8)

Then, the hyperplane conjecture is equivalent to the following statement: There exist absolute constants C, ξ > 0 such that, for every isotropic convex body K in Rn , q−c (K, ξ ) Cn. The main results of [22] and [23] show that there exists a parameter q∗ √ := q∗ (K) (related to the Lq -centroid bodies of K) with the following properties: (i) q∗ (K) c n, (ii) q−c (K, ξ ) q∗ (K) for some absolute constant ξ 1, and hence, I2 (K) ξ I−q∗ (K). The question that arises is to understand what happens with I−p (K) when p lies in the interval [q∗ , n], where there are no general estimates available up to now. In the case where K is a ψ2 -body, one has q∗ n and the problem is automatically resolved. The main idea in our approach is to start from an extremal isotropic convex body K in Rn with maximal isotropic constant LK Ln := sup{LK : K is a convex body in Rn }. Building on ideas from the work [5] of Bourgain, Klartag and Milman, we construct a second isotropic convex body K1 which is also extremal and, at the same time, is in α-regular M-position in the sense of Pisier (see [24]). Then, we use the fact that small ball probability estimates are closely related to estimates on covering numbers. This gives the estimate LK1 I−c

n (2−α)t α

√ (K1 ) Ct n,

(1.9)

for t C(α), where c, C > 0 are absolute constants. The construction of K1 from K can be done inside any subclass of isotropic log-concave measures which is stable under the operations of taking marginals or products. This leads us to the definition of a coherent class of probability measures (see Section 4): a subclass U of the class of probability measures P is called coherent if it satisfies two conditions: 1. If μ ∈ U is supported on Rn then, for all k n and F ∈ Gn,k , πF (μ) ∈ U . 2. If m ∈ N and μi ∈ U , i = 1, . . . , m, then μ1 ⊗ · · · ⊗ μm ∈ U . It should be noted that the class of isotropic convex bodies is not coherent. This is the reason for working with the more general class of log-concave measures. The basic tools that enable us to pass from one language to the other come from K. Ball’s bodies and are described in Section 2.

1936


Our main result is the following: Theorem 1.1. Let U be a coherent subclass of isotropic log-concave measures and let n 2 and δ 1. Then,

1 n

sup fμ (0) Cδ sup

μ∈U[n]

μ∈U[n]

en n log , q−c (μ, δ) q−c (μ, δ)

(1.10)

where C > 0 is an absolute constant and U[n] denotes the subclass of n-dimensional measures in U . √ Since one has that q−c (μ, c) n for any log-concave isotropic measure in Rn (where c > 0 is an absolute constant), then Theorem 1.1 has the following consequence: For every isotropic log-concave measure in Rn , √ 1 fμ (0) n C 4 n log n.

(1.11)

Moreover, in Section 4, for every α ∈ (1, 2] we introduce a coherent class Pα (β), of isotropic log-concave measures which is contained in the class of ψα -measures with constant β. Then, from Theorem 1.1 we get: Theorem 1.2. Let α ∈ (1, 2], let β > 0 and μ ∈ (Pα (β) ∩ IL)[n] . Then, 1 n

fμ (0) C n

2−α 2

βα

2−α log n 2 β α ,

(1.12)

where C > 0 is an absolute constant. For the special case α = 2, we prove that for symmetric measures the coherent class P2 (β) is essentially the same with the ψ2 -class. Then by Theorem 1.2 we have that: If μ is a symmetric log-concave ψ2 -measure with constant β > 0, then 1 fμ (0) n Cβ log β. From Theorems 1.1 and 1.2 we immediately deduce two facts: 1. If a symmetric convex body K satisfies a ψ2 -estimate with constant β, then LK Cβ log β. 2. For every convex body K in Rn , √ LK C 4 n log n.

(1.13)


1937

The first fact √ slightly improves Bourgain’s estimate from [4]. The second one is weaker than Klartag’s 4 n-bound in [8]; nevertheless, our method has the advantage that it can take into account any additional information on the ψα behavior of K. 2. Background material 2.1. Basic notation We work in Rn , which is equipped with an Euclidean structure ·,·. We denote by · 2 the corresponding Euclidean norm, and write B2n for the Euclidean unit ball, and S n−1 for the unit sphere. Volume is denoted by | · |. We write ωn for the volume of B2n and σ for the rotationally invariant probability measure on S n−1 . The Grassmann manifold Gn,k of k-dimensional subspaces of Rn is equipped with the Haar probability measure μn,k . Let k n and F ∈ Gn,k . We will denote by PF the orthogonal projection from Rn onto F . The letters c, c , c1 , c2 , etc. denote absolute positive constants which may change from line to line. In order to facilitate reading, we will denote by c, η, κ, ξ, τ , etc. some (absolute) positive constants that appear in more than one places. Whenever we write a b, we mean that there exist absolute constants c1 , c2 > 0 such that c1 a b c2 a. Also if K, L ⊆ Rn we will write K L if there exist absolute constants c1 , c2 > 0 such that c1 K ⊆ L ⊆ c2 K. 2.2. Probability measures We denote by P[n] the class of all probability measures in Rn which are absolutely continuous with respect to the Lebesgue measure. We write An for the Borel σ -algebra in Rn . The density of μ ∈ P[n] is denoted by fμ . The subclass SP [n] consists of all symmetric measures μ ∈ P[n] ; μ is called symmetric if fμ is an even function on Rn . The subclass CP [n] consists of all μ ∈ P[n] that have center of mass at the origin; so, μ ∈ CP [n] if x, θ dμ(x) = 0

(2.1)

Rn

for all θ ∈ S n−1 . Let μ ∈ P[n] . For every 1 k n − 1 and F ∈ Gn,k , we define the F -marginal πF (μ) of μ as follows: for every A ∈ AF ,

πF (μ)(A) := μ PF−1 (A) .

(2.2)

It is clear that πF (μ) ∈ P[dim F ] . Note that, by the definition, for every Borel measurable function f : Rn → [0, ∞) we have

f (x) dπF (μ)(x) = F

Rn

f PF (x) dμ(x).

(2.3)

1938


The density of πF (μ) is the function πF (fμ )(x) := fπF (μ) (x) =

fμ (y) dy.

(2.4)

x+F ⊥

Let μ1 ∈ P[n1 ] and μ2 ∈ P[n2 ] . We will write μ1 ⊗ μ2 for the measure in P[n1 +n2 ] which satisfies (μ1 ⊗ μ2 )(A1 × A2 ) = μ1 (A1 )μ2 (A2 )

(2.5)

for all A1 ∈ An1 and A2 ∈ An2 . It is easily checked that fμ1 ⊗μ2 = fμ1 fμ2 . 2.3. Log-concave measures Rn

We denote by L[n] the class of all log-concave probability measures on Rn . A measure μ on is called log-concave if for any A, B ∈ An and any λ ∈ (0, 1),

μ λA + (1 − λ)B μ(A)λ μ(B)1−λ .

(2.6)

A function f : Rn → [0, ∞) is called log-concave if log f is concave on its support {f > 0}. It is known that if μ ∈ L[n] and μ(H ) < 1 for every hyperplane H , then μ ∈ P[n] and its density fμ is log-concave (see [2]). As an application of the Prékopa–Leindler inequality [10,25,26] one can check that if f is log-concave then, for every k n − 1 and F ∈ Gn,k , πF (f ) is also log-concave. As before, we write CL[n] or SL[n] for the centered or symmetric non-degenerate μ ∈ L[n] respectively. 2.4. Convex bodies A convex body in Rn is a compact convex subset C of Rn with non-empty interior. We say that C is symmetric if x ∈ C implies that −x ∈ C. We say that C has center of mass at the origin if C x, θ dx = 0 for every θ ∈ S n−1 . The support function hC : Rn → R of C is defined by hC (x) = max{x, y: y ∈ C}. The mean width of C is defined by W (C) =

(2.7)

hC (θ ) σ (dθ ).

S n−1

For each −∞ < p < ∞, p = 0, we define the p-mean width of C by Wp (C) =

1 p hC (θ ) σ (dθ )

p

.

(2.8)

S n−1

The radius of C is the quantity R(C) = max{x2 : x ∈ C} and, if the origin is an interior point of C, the polar body C ◦ of C is C ◦ := y ∈ Rn : x, y 1 for all x ∈ C .

(2.9)


1939

Note that if K is a convex body in Rn then the Brunn–Minkowski inequality implies that 1K ∈ L[n] . [n] the subclass of bodies of We will denote by K[n] the class of convex bodies in Rn and by K volume 1. Also, CK[n] is the class of convex bodies with center of mass at the origin and SK[n] is the class of origin symmetric convex bodies in Rn . We refer to the books [28,18,24] for basic facts from the Brunn–Minkowski theory and the asymptotic theory of finite dimensional normed spaces. 2.5. Lq -centroid bodies Let μ ∈ P[n] . For every q 1 and θ ∈ S n−1 we define hZq (μ) (θ ) :=

x, θ q f (x) dx

1 q

,

(2.10)

Rn

where f is the density of μ. If μ ∈ L[n] then hZq (μ) (θ ) < ∞ for every q 1 and every θ ∈ S n−1 . We define the Lq -centroid body Zq (μ) of μ to be the centrally symmetric convex set with support function hZq (μ) . Lq -centroid bodies were introduced, with a different normalization, in [12] (see also [11] where an Lq affine isoperimetric inequality was proved). Here we follow the normalization (and notation) that appeared in [21]. The original definition concerned the class of measures 1K ∈ L[n] where K is a convex body of volume 1. In this case, we also write Zq (K) instead of Zq (1K ). If K is a compact set in Rn and |K| = 1, it is easy to check that Z1 (K) ⊆ Zp (K) ⊆ Zq (K) ⊆ Z∞ (K) for every 1 p q ∞, where Z∞ (K) = conv{K, −K}. Note that if T ∈ SLn then Zp (T (K)) = T (Zp (K)). Moreover, if K is a convex body, as a consequence of the Brunn– Minkowski inequality (see, for example, [21]), one can check that Zq (K) ⊆ c0 qZ2 (K)

(2.11)

q Zq (K) ⊆ c0 Zp (K) p

(2.12)

for every q 2 and, more generally,

for all 1 p < q, where c0 1 is an absolute constant. Also, if K has its center of mass at the origin, then Zq (K) ⊇ cK

(2.13)

for all q n, where c > 0 is an absolute constant. For a proof of this fact and additional information on Lq -centroid bodies, we refer to [20] and [22]. 2.6. Isotropic probability measures Let μ ∈ CP [n] . We say that μ is isotropic if Z2 (μ) = B2n . We write I[n] and IL[n] for the classes of isotropic probability measures and isotropic log-concave probability measures on Rn respectively.

1940


[n] is isotropic if Z2 (K) is a multiple of the Euclidean ball. We say that a convex body K ∈ CK We define the isotropic constant of K by LK :=

|Z2 (K)| |B2n |

1

n

(2.14)

.

So, K is isotropic if and only if Z2 (K) = LK B2n . We write IK[n] for the class of isotropic convex bodies in Rn . Note that K ∈ IK[n] if and only if LnK 1 K ∈ IL[n] . A convex body K is LK

called almost isotropic if K has volume one and K T (K) where T (K) is an isotropic linear transformation of K. We refer to [16,7,22] for additional information on isotropic convex bodies. 2.7. The bodies Kp (μ) A natural way to pass from log-concave measures to convex bodies was introduced by K. Ball in [1]. Here, we will give the definition in a somewhat more general setting: Let μ ∈ P[n] . For every p > 0 we define a set Kp (μ) as follows:

∞

Kp (μ) := x ∈ R : p n

fμ (rx)r

p−1

dr fμ (0) .

(2.15)

0

It is clear that Kp (μ) is a star shaped body with gauge function xKp (μ) :=

p fμ (0)

− 1

∞

p

fμ (rx)r

p−1

dr

.

(2.16)

0

Let 1 k < n and F ∈ Gn,k . For θ ∈ SF we define θ Bk+1 (μ,F ) := θ Kk+1 (πF (μ)) .

(2.17)

In the following proposition we give some basic properties of the star-shaped bodies Kp (μ). We refer to [1,16,22,23] for the proofs and additional references. Proposition 2.1. Let μ ∈ P[n] , p > 0, 1 k < n and F ∈ Gn,k . (i) If μ ∈ L[n] then Kp (μ) ∈ K[n] . Moreover, if μ ∈ SL[n] then Kp (μ) ∈ SK[n] . [n] . (ii) If μ ∈ CL[n] then Kn+1 (μ) ∈ CK[n] . If μ ∈ SIL[n] then K n+2 (μ) ∈ SK

(iii) If μ ∈ IL[n] then K n+1 (μ) is almost isotropic.

1 (iv) Let 1 p n and μ ∈ CL[n] . Then, fμ (0) n Zp (μ) Zp (K n+1 (μ)). (v) Let 1 p k < n, F ∈ Gn,k , μ ∈ CL[n] and K ∈ CL[n] . Then,

1 1 fπF (μ) (0) k PF Zp (μ) fμ (0) n Zp B k+1 (μ, F )

(2.18)


1941

and 1

K ∩ F ⊥ k PF Zp (K) Zp B k+1 (K, F ) .

(2.19)

(vi) Let 1 k < n, F ∈ Gn,k and K ∈ IK[n] . Then, 1 L k+1 (K,F ) K ∩ F ⊥ k B . LK

(2.20)

(vii) If μ ∈ IL[n] , then 1

LKn+1 (μ) fμ (0) n .

(2.21)

2.8. ψα -norm Let μ ∈ P[n] . Given α 1, the Orlicz norm f ψα of a measurable function f : Rn → R with respect to μ is defined by f ψα

|f (x)| α dμ(x) 2 . = inf t > 0: exp t

(2.22)

Rn

It is not hard to check that f ψα sup

f p : p α . p 1/α

(2.23)

Let θ ∈ S n−1 . We say that μ satisfies a ψα -estimate with constant βα,μ,θ in the direction of θ if ·, y

ψα

βα,μ,θ ·, y2 .

(2.24)

We say that μ is a ψα -measure with constant βα,μ where βα,μ := supθ∈S n−1 βα,μ,θ , provided that this last quantity is finite. [n] we define Similarly, if K ∈ K hZp (K) (θ ) . 1/α hZ2 (K) (θ ) θ∈S n−1 pα p

βα,K := sup sup

(2.25)

Note that βα,μ is an affine invariant, since βα,μ◦T −1 = βα,μ for all T ∈ SLn . Finally, we define P[n] (α, β) := {μ ∈ P[n] : βα,μ β}

(2.26)

[n] : βα,K β}. K[n] (α, β) := {K ∈ K

(2.27)

and

1942


2.9. The parameter k∗ (C) Let C be a symmetric convex body in Rn . Define k∗ (C) as the largest positive integer k n for which

n

n 1 n μn,k F ∈ Gn,k : W (C) B2 ∩ F ⊆ PF (C) ⊆ 2W (C) B2 ∩ F . (2.28) 2 n+k Thus, k∗ (C) is the maximal dimension k such that a “random” k-dimensional projection of C is 4-Euclidean. The parameter k∗ (C) is completely determined by the global parameters W (C) and R(C): There exist c1 , c2 > 0 such that c1 n

W (C)2 W (C)2 k∗ (C) c2 n 2 R(C) R(C)2

(2.29)

for every symmetric convex body C in Rn . The lower bound appears in Milman’s proof of Dvoretzky’s theorem (see [13]) and the upper bound was proved in [19]. 3. Negative moments of the Euclidean norm Let μ ∈ P[n] . If −n < p ∞, p = 0, we define Ip (μ) :=

1 p p x2 dμ(x) .

(3.1)

Rn

As usual, if K is a Borel subset of Rn with Lebesgue measure equal to 1, we write Ip (K) := Ip (1K ). Definition 3.1. Let μ ∈ P[n] and δ 1. We define

q∗ (μ) := max k n: k∗ Zk (μ) k , 1 q−c (μ, δ) := max p 1: I−p (μ) I2 (μ) , δ

k q∗ (μ, δ) := max k n: k∗ Zk (μ) 2 . δ One of the main results of [23] asserts that the moments of the Euclidean norm on log-concave measures satisfy a strong reverse Hölder inequality up to the value q∗ : Theorem 3.2. Let μ ∈ CL[n] . Then for every p q∗ (μ), Ip (μ) CI−p (μ), where C > 0 is an absolute constant.

(3.2)


1943

It is clear from the statement that in order to apply Theorem 3.2 in a meaningful way one should have some non-trivial estimate for the parameter q∗ . The next proposition (see [22, Proposition 3.10] or [23, Proposition 5.7]) gives a lower bound for q∗ , with a dependence on the ψα constant, in the isotropic case. Proposition 3.3. Let μ ∈ I[n] ∩ P[n] (α, β). Then α

q∗ (μ) c

n2 , βα

(3.3)

where c > 0 is an absolute constant. Definition 3.4. Let μ ∈ P[n] . We will say that μ is of small diameter (with constant A > 0) if for every p 2 one has Ip (μ) AI2 (μ).

(3.4)

The definition that we give here is a direct generalization of the one given in [21] for the case of convex bodies. Let μ ∈ P[n] and set B := 4I2 (μ)B2n . Note that 34 μ(B) 1. We define a new measure μ¯ on An in the following way: for every A ∈ An we set μ(A) ¯ :=

μ(A ∩ B) . μ(B)

Assume that, additionally, μ ∈ L[n] . Then, it is not hard to check that ¯ I2 (μ) I2 (μ),

Z2 (μ) Z2 (μ) ¯

1

1

and fμ¯ (0) n fμ (0) n .

(3.5)

Therefore, if μ ∈ L[n] , we can always find a measure μ¯ ∈ L[n] which is of small diameter (with 1 1 an absolute constant C > 0) and satisfies fμ¯ (0) n fμ (0) n . Moreover, if μ is isotropic, then μ¯ is almost isotropic. As a consequence of [23, Theorem 5.6] we have the following: Proposition 3.5. Let μ ∈ L. Then, q∗ (μ, ¯ ξ1 ) q−c (μ, ¯ ξ2 ), where ξ1 , ξ2 1 are absolute constants.

(3.6)

1944


4. Coherent classes of measures Our starting point is a simple but crucial observation from the paper [5] of Bourgain, Klartag and Milman. First of all, one may observe that Ln := sup{LK : K is a convex body in Rn } is, essentially, an increasing function of n: for every k n, Lk CLn , where C > 0 is an absolute constant. So, using (2.20) we see that if K0 is an isotropic convex body in Rn such that LK0 Ln , then, for all F ∈ Gn,k , L(K ,F ) Lk k+1 0 K0 ∩ F ⊥ 1/k B C1 C2 . LK0 Ln

(4.1)

Building on the ideas of [5] one can use this property of a body K0 with “extremal isotropic constant” to get upper bounds for the negative moments of the Euclidean norm on K0 . Since we want to apply this argument in different situations, we will first introduce some terminology. ∞ Definition 4.1. We define P := ∞ i=1 P[n] . Similarly, IP := i=1 IP[n] , etc. Let U be a subclass of P. Set U[n] = U ∩ P[n] . We say that U is coherent if it satisfies the following two conditions: 1. If μ ∈ U[n] then, for all k n and F ∈ Gn,k , πF (μ) ∈ U[dim F ] . 2. If m ∈ N and μi ∈ U[ni ] , i = 1, . . . , m, then μ1 ⊗ · · · ⊗ μm ∈ U[n1 +···+nm ] . We also agree that the null class is coherent. Note that if U1 and U2 are coherent then U1 ∩ U2 is also coherent. The following proposition is a translation of well-known results to this language. Proposition 4.2. The classes SP, CP, L, I are coherent. Note that the class K :=

∞

n=1 {μ ∈ P[n] :

μ = 1K; K ∈ K[n] } is not coherent.

Proposition 4.3. Let U be a coherent class of measures. If n is even then, for every μ ∈ U[n] , k = n2 and F ∈ Gn,k , 1

1

fπF (μ) (0) k sup fμ (0) n . μ∈U[n]

(4.2)

1

Moreover, if ρn (U) := supμ∈U[n] fμ (0) n , then 1 ρn−1 (U) n ρn−1 (U) ρn (U) . ρ1 (U)

(4.3)


1945

Proof. For the first assertion use the fact that πF (μ) ⊗ πF (μ) ∈ U[n] and 2 f(πF (μ)⊗πF (μ)) (0) = fπF (μ) (0) . For the second assertion use the fact that if μ1 ∈ U[n−1] and μ2 ∈ U[1] then we have μ1 ⊗ μ2 ∈ U[n] and fμ1 ⊗μ2 (0) = fμ1 (0)fμ2 (0). 2 In particular if a class satisfies e−n ρn (U) en , it is enough to bound ρn (U) for n even. Note that IL is such a class. In this section we introduce a coherent subclass of ψa measures, Pα (β). Let μ ∈ CP [n] . For every θ ∈ S n−1 and every λ > 0 we define λx,θ hμ,θ (λ) := h(λ) = log e dμ(x) .

(4.4)

Rn

Next, if α ∈ (1, 2], we define 1 α∗ 1 1 1 λx,θ α log e ψα,μ (θ ) := sup h(λ) ∗ = sup dμ(x) , λ>0 λ λ>0 λ

(4.5)

Rn

where α∗ is the conjugate exponent of α, i.e.

1 α

+

1 α∗

= 1.

Definition 4.4. Let μ be a probability measure on Rn . For α ∈ (1, 2] we define μ,α := sup β θ∈S n−1

α,μ (θ ) ψ . hZ2 (μ) (θ )

(4.6)

We also define Pα (β) :=

∞

μ,α β}. {μ ∈ P[n] : β

n=1

Proposition 4.5. 1. If μ ∈ CP [n] , then for every α ∈ (1, 2] and every θ ∈ S n−1 we have that ·, θ

ψα

α,μ (θ ), ψ α,μ (−θ ) C max ψ

where C > 0 is an absolute constant.

(4.7)

1946


2. Let μ ∈ SP [n] , then for every θ ∈ S n−1 we have that 2,μ (θ ) ·, θ C2 ψ 2,μ (θ ), C1 ψ ψ 2

(4.8)

where C1 , C2 > 0 are absolute constants. Proof. Let α ∈ (1, 2] and let α∗ ∈ [2, ∞) be the conjugate exponent of α. We set ψ−1 := α,μ (−θ ), ψ1 := ψ α,μ (θ ), ψ0 := max{ψ α,μ (θ ), ψ α,μ (−θ )} and ψ2 := ·, θ ψα . ψ For every λ > 0,

eλx,θ dμ(x) exp λα∗ ψ1α∗ .

(4.9)

Rn

So, by Markov’s inequality we get that, for every t > 0, α α∗ α∗ α μ x: eλx,θ et eλ ψ1 e−t .

(4.10)

tα α α∗ −1 α∗ +λ μ x: x, θ ψ1 e−t . λ

(4.11)

Equivalently,

Choosing λ :=

t α−1 ψ1 ,

we get α μ x: x, θ 2tψ1 e−t .

(4.12)

Similarly, for every t > 0 we have α μ x: x, −θ 2tψ−1 e−t .

(4.13)

Therefore, μ x: x, θ 2tψ0 = μ x: x, θ 2tψ0 + μ x: x, −θ 2tψ0 μ x: x, θ 2tψ1 + μ x: x, −θ 2tψ−1 2e−t . α

The last inequality implies that ψ2 C1 ψ0 and we are finished with the first part of the proposition. For the second part we assume that μ is symmetric and α = 2. We only have to prove the right-hand inequality in (4.8). Using the fact that μ is symmetric we have that for every odd k∈N x, θ k dμ(x) = 0. Rn


1947

So, eλx,θ dμ(x) =

∞ ∞ λk x, θ k λ2k x, θ 2k dμ(x) dμ(x) = k! (2k)! k=0 Rn

Rn

∞ k=0

∞ k=0

k=0

(λ)2k (2k)k ψ22k (2k)!

∞ k=0

Rn

(λ)2k (2e)k k!ψ22k (2k)!

(2eλ2 ψ22 )k = exp 2eλ2 ψ22 . k!

It follows that 1 √ 2 1 log eλx,θ dμ(x) 2eψ2 . λ>0 λ

ψ1 := sup

(4.14)

Rn

This completes the proof.

2

Corollary 4.6. For every α ∈ (1, 2], CP α (β) ⊆ CP(α, cβ)

(4.15)

SP(2, c2 β) ⊆ SP 2 (β) ⊆ SP(2, c1 β),

(4.16)

and

where c, c1 , c2 > 0 are universal constants. Proof. Indeed, if μ ∈ CP α (β) then Proposition 4.5 implies that sup θ∈S n−1

α,μ (θ ) hψα (μ) (θ ) ψ c sup cβ hZ2 (μ) (θ ) θ∈S n−1 hZ2 (μ) (θ )

(4.17)

which means that μ ∈ CP(α, cβ) (recall (2.26)). The second part is proved in a similar way.

2

Next, we prove that the class Pα (β) is coherent. α,μ for products of measures is described by the following: The behavior of ψ Proposition 4.7. Let k be a positive integer and let μi ∈ CP [ni ] and θi ∈ S ni −1 , i = 1, . . . , k. If α,μi (θi ) < ∞ for all i k and some α ∈ (1, 2], then ψ α,μ ((θ1 , . . . , θk )) ψ

k i=1

where μ = μ1 ⊗ · · · ⊗ μk .

α∗ α,μ ψ (θi ) i

1 α∗

,

(4.18)

1948


Proof. For every λ > 0 we can write 1 λ ki=1 xi ,θi log ··· e dμk (xk ) · · · dμ1 (x1 ) λα∗ Rn1

Rnk

as follows: k k 1 1 λxi ,θi log e dμi (xi ) = α log eλxi ,θi dμi (xi ) λα∗ λ ∗ i=1 Rni

i=1

Rni

k 1 α∗ α∗ α,μ (θi ) λ ψ i λα∗ i=1

k

α∗ α,μ ψ (θi ). i

i=1

Taking the supremum with respect to λ > 0 we get the result.

2

The behavior of marginals is described by the following: Proposition 4.8. Let μ ∈ CP [n] . Let F ∈ Gn,k and θ ∈ SF . If α ∈ (1, 2], then α,πF (μ) (θ ) ψ α,μ (θ ). ψ

(4.19)

Proof. Note that, for every λ > 0, λx,θ e dμ(x) = eλx,θ dπF (μ)(x). Rn

(4.20)

F

It follows that 1 log λα∗

eλx,θ dπF (μ)(x) =

1 log λα∗

α∗ α,μ eλx,θ dμ(x) ψ (θ ).

(4.21)

Rn

F

Taking the supremum with respect to λ > 0 we get the result.

2

Proposition 4.9. Let α ∈ (1, 2] and let β > 0. Then the class Pα (β) is coherent. Proof. Let μ ∈ (Pα (β))[n] . Fix 1 k < n and F ∈ Gn,k . Then, using (4.18) and the fact that hZ2 (πF (μ)) (θ ) = hZ2 (μ) (θ ) for θ ∈ SF , we see that πF (μ),α = sup β

θ∈SF

So, πF (μ) ∈ Pα (β).

α,πF (μ) (θ ) α,μ (θ ) ψ ψ μ,α . sup β hZ2 (πF (μ)) (θ ) θ∈SF hZ2 (μ) (θ )

(4.22)


1949

Next, let μi ∈ (Pα (β))[ni ] , i := 1, . . . , k and set N := n1 + · · · + nk . Since hZ2 (μ1 ⊗···⊗μk ) (θ1 , 1 . . . , θk ) = ( ki=1 h2Z2 (μi ) (θi )) 2 , we have μ1 ⊗···⊗μk ,α = β

sup (θ1 ,...,θk )∈S N−1

sup (θ1 ,...,θk )∈S N−1

β

α,μ1 ⊗···⊗μk (θ1 , . . . , θk ) ψ hZ2 (μ1 ⊗···⊗μk ) (θ1 , . . . , θk ) 1 α∗ α∗ α,μ ( ki=1 ψ i (θi )) 1 ( ki=1 h2Z2 (μi ) (θi )) 2

sup (θ1 ,...,θk )∈S N−1

1 ( ki=1 hαZ∗2 (μi ) (θi )) α∗ 1 ( ki=1 h2Z2 (μi ) (θi )) 2

β since α∗ ∈ [2, ∞), and xkα xk . So, μ1 ⊗ · · · ⊗ μk ∈ Pα (β). ∗

2

2

5. M-positions and extremal bodies All the results in this section are stated for the case where the dimension is even. Proposition 4.3 shows that this is sufficient for our purposes. However, with minor changes in the proofs, all the results remain valid in the case where the dimension is odd. Our main goal in this section is to prove the following: Proposition 5.1. Let U ⊆ IL be a coherent class of probability measures, let n 2 even, α ∈ C0 α1 (1, 2) and t ( 2−α ) . Then, there exists μ1 ∈ U[n] such that 1

1

fμ1 (0) n C1 sup fν (0) n

(5.1)

ν∈U[n]

and I−c2

n (2−α)t α

√ 1 (μ1 ) C3 t nfμ1 (0)− n ,

(5.2)

where C0 , C1 , C3 > 0 and c2 2 are absolute constants. Moreover, if U = IL, μ1 can be chosen to be of small diameter (with an absolute constant C4 > 0). Recall that if K and C are convex bodies in Rn , then the covering number of K with respect to C is the minimum number of translates of C whose union covers K: N(K, C) := min k ∈ N: ∃z1 , . . . , zk ∈ R : K ⊂ n

k i=1

(zi + C) .

(5.3)

1950


Let K be a convex body of volume 1 in Rn . Milman (see [14,15,17] for the not necessarily symmetric case) proved that there exists an ellipsoid E with |E| = 1, such that log N (K, E) κn,

(5.4)

where κ > 0 is an absolute constant. We will use the existence of α-regular M-ellipsoids for convex bodies. More precisely, we need the following theorem of Pisier (see [24]; the result is stated and proved in the case of symmetric convex bodies but it can be easily extended to the non-symmetric case): Theorem 5.2. Let K be a convex body of volume 1 in Rn with center of mass at the origin. For every α ∈ (0, 2) there exists an ellipsoid E with |E| = 1 such that, for every t 1, log N (K, tE)

κ(α) n, tα

(5.5)

where κ(α) > 0 is a constant depending only on α. One can take κ(α) absolute constant.

κ 2−α ,

where κ > 0 is an

We will also need the following facts about ellipsoids: Lemma 5.3. Let E be an ellipsoid in Rn . Assume that there exists a diagonal matrix T with entries λ1 · · · λn > 0 such that E = T (B2n ). Then, k λi max |E ∩ F | = max PF (E) = ωk

F ∈Gn,k

F ∈Gn,k

(5.6)

i=1

and min |E ∩ F | = min PF (E) = ωk

F ∈Gn,k

F ∈Gn,k

n

λi

(5.7)

i=n−k+1

for all 1 k n − 1. Proof. A proof of the equality minF ∈Gn,k |E ∩ F | = ωk ni=n−k+1 λi is outlined in [9, Lemma 4.1]. Let Fs (k) = span{en−k+1 , . . . , en }. Then, for every F ∈ Gn,k we have PF

s (k)

(E) = E ∩ Fs (k) |E ∩ F | PF (E).

(5.8)

This shows that min PF (E) = PFs (k) (E) = ωk

F ∈Gn,k

and completes the proof of (5.7).

n i=n−k+1

λi

(5.9)


1951

Observe that E ◦ = T −1 (B2n ) is also an ellipsoid; since the diagonal entries of T −1 are λ−1 n −1 · · · λ1 > 0, the same reasoning shows that k −1 ◦ ◦ λi . min E ∩ F = min PF E = ωk

F ∈Gn,k

F ∈Gn,k

(5.10)

i=1

Since PF (E) is an ellipsoid in F and E ◦ ∩ F is its polar in F , by the affine invariance of the product of volumes of a body and its polar, we get |PF (E)| · |E ◦ ∩ F | = |B2n ∩ F |2 = ωk2 for every F ∈ Gn,k . This observation and (5.10) prove (5.6). 2 Lemma 5.4. Let n be even and let E be an ellipsoid in Rn . Assume that there exists a diagonal matrix T with entries λ1 · · · λn > 0 such that E = T (B2n ). Then, there exists F ∈ Gn,n/2 such that PF (E) = λn/2 (B2n ∩ F ). Proof. The proof can be found in [30, pp. 125–126], but we sketch it for the reader’s convenience. We may assume that λ1 > · · · > λn > 0. Write n = 2s. Then, E ◦ ∩ en⊥ = {x ∈ R2s−1 : 2s−1 2 2 i=1 λi xi 1} (the reason for this step is that the argument in [30, pp. 125–126] works in odd dimensions). Since λi > λs > λ2s−i for every i s − 1, we can define b1 , . . . , bs−1 > 0 by the equations

λ2i bi2 + λ22s−i = λ2s bi2 + 1 .

(5.11)

Consider the subspace F = span{v1 , . . . , vs } ∈ G2s,s , where vs = es and vi =

bi ei + e2s−i , bi2 + 1

i = 1, . . . , s − 1.

(5.12)

It is easy to check that {v1 , . . . , vs } is an orthonormal basis for F and, using (5.11) and (5.12), we see that, for every x ∈ F , λ2s x22 = λs

s 2s−1 x, vi 2 = λ2i x, ei 2 = x2E . i=1

(5.13)

i=1

n n n This proves E ◦ ∩ F = λ−1 s (B2 ∩ F ) and, by duality, PF (E) = λs (B2 ∩ F ) = λn/2 (B2 ∩ F ).

2

[n] . Let 1 k n − 1 and set Proposition 5.5. Let K ∈ IK 1 γ := max K ∩ F ⊥ k . F ∈Gn,k

(5.14)

Then, 1 min K ∩ H ⊥ n−k γ

H ∈Gn,n−k

where 0 < η < 1 is an absolute constant.

n η n−k , γ

(5.15)

1952


Proof. Fix α = 1 and consider an α-regular M-ellipsoid E for K given by Theorem 5.2. By the invariance of the isotropic position under orthogonal transformations, we may assume that there exists a diagonal matrix T with entries λ1 · · · λn > 0 such that E = T (B2n ). Recall that |E| = 1. Let F ∈ Gn,k , 1 k n − 1. Since projecting a covering creates a covering of the projection, we have |PF (K)| N (K, E) eκn . |PF (E)|

(5.16)

We will use the Rogers–Shephard inequality (see [27]) for K and E: since |K| = 1, we know that 1 1

n k en ⊥ k c1 K ∩ F PF (K) , k k

(5.17)

where c1 > 0 is a universal constant (see [29] or [17] for the left-hand side inequality). From (5.17) and the definition of γ in (5.14), we see that 1 PF (K) k c1 . γ

(5.18)

1 κn c1 e k PF (E) k . γ

(5.19)

1 c1 κn min PF (E) k e− k . γ

(5.20)

Using (5.16) we get

In other words,

F ∈Gn,k

We can now apply the upper bound from (5.17) to get 1 1 κ1 n κn κn en c1 E ∩ F ⊥ k e k PF (E)E ∩ F ⊥ k e k e k . γ k

(5.21)

It follows that max

|E ∩ H |

e κ1 n γ k

.

(5.22)

e κ1 n γ k , max PH (E) H ∈Gn,n−k c1k

(5.23)

H ∈Gn,n−k

c1k

Lemma 5.3 implies that


1953

and hence, κ2 n k PH (K) eκn PH (E) e γ c1k

(5.24)

for every H ∈ Gn,n−k , where we have used again (5.16). Applying (5.17) once again, we have 1 1 κ2 n

c1 K ∩ H ⊥ PH (K) n−k K ∩ H ⊥ n−k e n−k

γ c1

k n−k

.

(5.25)

This proves that 1 min K ∩ H ⊥ n−k γ

H ∈Gn,n−k

with η = c1 e−κ2 , as claimed.

n η n−k γ

(5.26)

2

[n] . Assume that, for some s > 0, Lemma 5.6. Let K ∈ CK

rs := log N K, sB2n < n.

(5.27)

I−rs (K) 3es.

(5.28)

Then,

Proof. Let z0 ∈ Rn such that |K ∩ (−z0 + sB2n )| |K ∩ (z + sB2n )| for every z ∈ Rn . It follows that

(K + z0 ) ∩ sB n · N K, sB n |K| = 1. 2

(5.29)

2

Let q := rs < n. Then, using Markov’s inequality, the definition of I−q (K + z0 ) and (5.27), we get (K + z0 ) ∩ 3−1 I−q (K + z0 )B n 3−q < e−q = e−rs 2

1 . N (K, sB2n )

(5.30)

From (5.29) we obtain (K + z0 ) ∩ 3−1 I−q (K + z0 )B n < (K + z0 ) ∩ sB n ,

(5.31)

3−1 I−q (K + z0 ) s.

(5.32)

2

2

and this implies

Since K has center of mass at the origin, as an application of Fradelizi’s theorem (see [6]), we have that I−k (K + z) 1e I−k (K) for any 1 k < n and z ∈ Rn (a proof appears in [23, Proposition 4.6]). This proves the lemma. 2

1954


[n] . Set Theorem 5.7. Let n be even and let K ∈ IK 2 γ := max K ∩ F ⊥ n .

(5.33)

F ∈Gn, n 2

[n] such that: Then, there exists K1 ∈ IK (i) ηγ1 LK LK1 η2 γ LK , where η1 , η2 > 0 are absolute constants. (ii) If α ∈ (1, 2) one has that for every t C1 γ 2

√ κ(α)n log N K1 , t nB2n C2 γ 2 α , t κ and C1 , C2 > 0 are absolute constants. where κ(α) 2−α (iii) If K is a body of small diameter (with some constant A > 1) then K1 is also a body of small diameter (with constant C3 γ 2 A > 1, where C3 is an absolute constant).

Proof. Let E be an α-regular M-ellipsoid for K given by Theorem 5.2. As in the proof of Proposition 5.5, we assume that E = T (B2n ) for some diagonal matrix T with entries λ1 · · · λn > 0. From (5.20) and Lemma 5.3 we have n

n

ω n2 (λ n2 ) 2 ω n2

1/k

and hence (recall that ωk

λi = min |PF E| e−κn

F ∈Gn, n

i= n2 +1

2

c1 γ

n 2

,

(5.34)

√ 1/ k), λ n2

√ c2 n . γ

(5.35)

Similarly, (5.22) and Lemma 5.3 imply that

n

n 2

ω n2 (λ n2 ) ω n2

2

i=1

λi = max |E ∩ H | e H ∈Gn, n 2

κ1 n

γ c1

n 2

,

(5.36)

and hence, λ n2 c3 γ

√ n.

(5.37)

Then, by Lemma 5.4 we can find F0 ∈ Gn, n2 such that √ √ c2 n n B2 ∩ F0 ⊆ PF0 (E) ⊆ c3 γ n B2n ∩ F0 . γ

(5.38)


1955

n Let K0 := B (K, F0 ) and K1 := T (K0 × K0 ) ∈ Rn , where T ∈ SLn is such that K1 is 2 +1 isotropic. Note that K0 × K0 has volume 1, center of mass at the origin and is almost isotropic. In other words T is almost an isometry. We will show that K1 satisfies (i), (ii) and (iii). (i) From Proposition 2.1(vi) we know that

2 2 c2 LK K ∩ F0⊥ n LK0 c1 LK K ∩ F0⊥ n ,

(5.39)

where c1 , c2 > 0 are absolute constants. Then, Proposition 5.5 shows that η1 LK LK0 η2 γ LK , γ

(5.40)

where η1 = η2 c2 , η2 = c1 . Note that LK1 = LK0 . This completes the proof of (i). (ii) From Proposition 2.1(v) and from the fact that c conv{C, −C} ⊆ Z n2 (C) ⊆ conv{C, −C} [ n ] , we get for all C in CK 2 1 n conv{K0 , −K0 } ⊆ Z n2 B +2 (K, F0 ) 2 c 2

1 K ∩ F0⊥ n PF0 Z n2 (K) ⊆ cc3

1 γ PF0 conv{K, −K} ⊆ cc3 and, similarly,

n conv{K0 , −K0 } ⊇ Z n2 B (K, F0 ) 2 +2 ⊇

2

1 K ∩ F0⊥ n PF0 Z n2 (K) c4

⊇

η2 c 1 PF conv{K, −K} , c4 2c0 γ 0

where we have used the fact that Z n2 (K) ⊇

1 c 2c0 Zn (K) ⊇ 2c0

conv{K, −K}. In other words,

c5 PF0 conv{K, −K} ⊆ conv{K0 , −K0 } ⊆ c6 γ PF0 conv{K, −K} , γ where c5 , c6 > 0 are absolute constants. For s > 0 we have

√ √ N K1 , s nB2n = N T (K0 × K0 ), s nB2n

√ N K0 × K0 , cs nB2n √ √

N K0 × K0 , 2cs n B2n ∩ F0 × B2n ∩ F0 2

√ N K0 , c s nB2n ∩ F0 ,

(5.41)

1956


where we have used the fact that T is almost an isometry, and hence, T (K0 × K0 ) ⊆ 1c (K0 × K0 ). Moreover, we have used the fact that if K, C are convex bodies, then N (K × K, C × C) N (K, C)2

(5.42)

and B2k × B2k ⊇ √1 B22k . 2 Recall that c2 and c3 are the constants in (5.38). For every r > 0,

√ √ N K0 , c3 rγ n B2n ∩ F0 N conv{K0 , −K0 }, c3 rγ n B2n ∩ F0

N conv{K0 , −K0 }, rPF0 (E)

N c6 γ PF0 conv{K, −K} , rPF0 (E)

N c6 γ conv{K, −K}, r E r N K − K, E c6 γ 2 r E . N K, 2c6 γ So, we can write

√ N K1 , t nB2n N K, for every t > 0, where c7 = we have

t E c7 γ 2

4 (5.43)

√ 2c2 c6 . Since E is an α-regular ellipsoid for K, for every t c7 γ 2

√ n log N K1 , t nB2 4 log N K,

t 4c7 κ(α)γ 2 n E . tα c7 γ 2

(5.44)

This completes the proof of (ii). √ (iii) We have that R(K0 ) cγ A nLK . Indeed, by Proposition 2.1,

n R(K0 ) = R B (K, F0 ) 2 +1

n cR Z n2 +1 B (K, F0 ) 2 +1

2 c K ∩ F0⊥ n R PF0 Z n2 +1 (K)

c γ R conv{K, −K} √ 2c γ R(K) cγ A nLK .

Also, R(K1 ) = R(K0 × K0 ) =

√ 2R(K0 ).

(5.45)


1957

To see this, write R 2 (K0 × K0 ) =

max

(x,y)∈K0 ×K0

x22 + y22 = 2R 2 (K0 ).

(5.46)

So, using (i) we get that R(K1 ) This completes the proof.

√ √ √ √ 2R(K0 ) c 2γ A nLK C3 γ 2 A nLK1 .

(5.47)

2

Lemma 5.8. Let μ ∈ IL[n] . Fix 1 k < n − 1 and F ∈ Gn,k . Then, 1

1 fπF (μ) (0) k ⊥ k K , n+1 (μ) ∩ F 1 fμ (0) n

1

LBk+1 (μ,F ) fπF (μ) (0) k LB

k+1 (K n+1 (μ),F )

(5.48)

(5.49)

,

and

1 fμ (0) n B k+1 (μ, F ) B k+1 K n+1 (μ), F .

(5.50)

Proof. (i) We will make use of the following facts (see Proposition 4.2 and Theorem 4.4 in [23]): If μ ∈ IL[n] , then 1 1 fπF (μ) (0) k PF Zk (μ) k 1,

(5.51)

1 1 K ∩ F ⊥ k PF Zk (K) k 1.

(5.52)

[n] then and if K ∈ CK

Then, taking into account Proposition 2.1(iv), we get 1

1 1 k

− 1 ⊥ k k fμ (0)− n1 PF Zk (μ)− k fπF (μ) (0) . K PF Zk K n+1 (μ) ∩ F n+1 (μ) 1 fμ (0) n This proves (5.48). (ii) Using Proposition 2.1(v) and (iv), we have that 1

⊥ k K PF Z2 K Z2 B k+1 K n+1 (μ), F n+1 (μ) ∩ F n+1 (μ) 1

fπF (μ) (0) k fμ (0)

1 n

1 fμ (0) n PF Z2 (μ)

(5.53)

1958


1 = fπF (μ) (0) k PF Z2 (μ) 1

= fπF (μ) (0) k BF , because Z2 (μ) = B2n . Taking volumes we see that 1

LB

k+1 (K n+1 (μ),F )

fπF (μ) (0) k

(5.54)

and we conclude by Proposition 2.1(vii) and (2.17). (iii) By Proposition 2.1(v), 1

πF (μ)(0) k PF Zk (μ) B k+1 (μ, F ) Zk B k+1 (μ, F ) 1 fμ (0) n

(5.55)

and, by Proposition 2.1(v) and then (iv),

B k+1 K n+1 (μ), F Zk B k+1 K n+1 (μ), F 1 ⊥ k PF Zk K K n+1 (μ) ∩ F n+1 (μ) 1

πF (μ)(0) k fμ (0)

1 n

1 fμ (0) n PF Zk (μ)

1 = πF (μ)(0) k PF Zk (μ) . We have thus shown that

1 B k+1 K n+1 (μ), F πF (μ)(0) k PF Zk (μ) .

(5.56)

Combining (5.55) and (5.56) we see that

1 fμ (0) n B k+1 (μ, F ) B k+1 K n+1 (μ), F . This completes the proof.

(5.57)

2 1

1

Proof of Proposition 5.1. (i) Let ν ∈ U[n] such that supμ∈U[n] fμ (0) n = fν (0) n . Let

K := T K n+1 (ν) ,

(5.58)

[n] . Note that, from Proposition 2.1, T is almost an isometry where T ∈ SLn is such that K ∈ IK 1 and LK fν (0) n . ν )). By Proposition 2.1 and (2.14) we have that LK If U = IL we take K := T (K n+1 (¯ 1 fν (0) n . The proof of the first two assertions is identical in both cases. We write μ for either ν or ν¯ .


1959

[ n ] and K1 ∈ IK [n] as in the proof of Theorem 5.7. Let μ1 := Let F0 ∈ Gn, n2 , K0 ∈ IK 2 πF0 (μ) ⊗ πF0 (μ). Assume that the two copies of πF0 (μ) live on F and F ⊥ respectively, where F ∈ G2n,n . Since μ ∈ U and U is coherent, we have μ1 ∈ U . Moreover, using again Proposition 2.1, we have that 1

2

fμ1 (0) n = fπF0 (μ) (0) n LK n (π

F0 (μ))

2 +1

= LB LB n n (μ,F ) 2 +1

2 +1

0

(K n+1 (μ),F0 )

LB n +1 (K,F0 ) = LK0 = LK1 2

1

fμ (0) n . This settles the first assertion of the proposition. (ii) Since U is coherent, for every F ∈ Gn, n2 we have 2

1

fπF (μ) (0) n fμ (0) n .

(5.59)

2

Set γ := maxF ∈Gn, n |K ∩ F ⊥ | n . Then, 2

γ

LB n

2 +1

(K n+1 (μ),F )

LK

2

fπF (μ) (0) n 1

fμ (0) n

C,

(5.60)

where we have used again Lemma 5.8. So, by Theorem 5.7 we have that

√ log N K, t nB2n

Cn . t α (2 − α)

(5.61)

Note that, for every p > 0 and every pair of probability measures ν1 , ν2 living in F, F ⊥ respectively, we have PF Zp (ν1 ⊗ ν2 ) = Zp (ν1 ) and PF ⊥ Zp (ν1 ⊗ ν2 ) = Zp (ν2 ). Indeed, if θ ∈ SF , we have that

p

hZp (ν1 ⊗ν2 ) (θ ) =

x + y, θ p dν2 (y) dν1 (x)

F F⊥

=

x, θ p dν1 (x) = hp

Zp (ν1 ) (θ ).

F

Note that for every convex body K and F ∈ Gn,k one has K ⊆ PF (K) ⊗ PF ⊥ (K).

(5.62)

1960


So, we have that

K n+1 (μ1 ) ⊆ PF K n+1 (μ1 ) × PF ⊥ K n+1 (μ1 )

PF Z n2 K n+1 (μ1 ) × PF ⊥ Z n2 K n+1 (μ1 )

1 fμ (0) n PF Z n2 πF0 (μ) ⊗ πF0 (μ)

1 × fμ (0) n PF ⊥ Z n2 πF0 (μ) ⊗ πF0 (μ)

1 1 fμ (0) n Z n2 πF0 (μ) × fμ (0) n Z n2 πF0 (μ) n n fμ (0) n B (μ, F0 ) × fμ (0) n B (μ, F0 ) 2 +1 2 +1

n B +1 Kn+1 (μ), F0 × B n +1 Kn+1 (μ), F0 1

1

2

2

n n (K, F0 ) × B (K, F0 ) B 2 +1 2 +1

= K0 × K0 = K1 . Therefore,

R K n+1 (μ1 ) cR(K1 )

(5.63)

and

√ n √ n log N K n+1 (μ1 ), t nB2 log N K, ct nB2

Cn t α (2 − α)

.

(5.64)

We have assumed that t α (2 − a) C, and hence, by Lemma 5.6 we have

√ I−p K n+1 (μ) 3et n,

(5.65)

Cn < n. Note that if μ ∈ CL then for every 1 p n − 1 one has (see Proposiwhere p = t α (2−a) tion 3.4 in [23])

1 I−p (μ)fμ (0) n I−p K n+1 (μ) .

(5.66)

√ 1 (μ1 ) C t nfμ1 (0)− n ,

(5.67)

It follows that I−

Cn t α (2−a)

and the proof of the second assertion is complete.


1961

For the rest of the proof we set μ = ν¯ . In this case, K is a body of small diameter. Indeed, for p 2, by Proposition 2.1(iv) we have

√ 1 1 ν ) Ip (μ)fν¯ (0) n nfν¯ (0) n I2 K ν ) I2 (K). Ip (K) Ip K n+1 (¯ n+1 (¯ From Theorem 5.7 we have that K1 is a body of small diameter, and this implies that Also, by the first assertion we have that 1

(5.68)

R(K1 ) I2 (K1 )

1

LK1 fν¯ (0) n fμ1 (0) n LK .

1.

(5.69)

Then, from (5.63) we see that for p 2, Ip (μ1 ) Ip (K R(K R(K1 ) n+1 (μ1 )) n+1 (μ1 )) √ 1. c √ 1 I2 (μ1 ) I2 (K1 ) nLK1 nfμ1 (0) n So, μ1 is a measure of small diameter. The proof is complete.

(5.70)

2

6. Proof of the main result We are now ready to state and prove the main result of the paper: Theorem 6.1. Let U be a coherent subclass of IL and let n 2 and δ 1. Then,

1 n

sup fμ (0) Cδ sup

μ∈U[n]

μ∈U[n]

en n log , q−c (μ, δ) q−c (μ, δ)

(6.1)

where C > 0 is an absolute constant. Moreover if U = IL then the supremum on right-hand side can be taken over all ν¯ ∈ IL. Proof. By Proposition 4.3 we can assume that n is even. Let q := infμ∈U[n] q−c (μ, δ). Let α := 1 n en 2 − log(e n and t = C1 q log q , where the absolute constant C1 > 0 can be chosen large enough ) q

to ensure that t α (2 − α) C0 , where C0 > 0 is the constant that appears in Proposition 5.1. We have t α (2 − α)

en 1 n n log = , q q log en q q

(6.2)

and hence, n q. t α (2 − α)

(6.3) 1

1

By Proposition 5.1 there exists a measure μ1 ∈ U[n] such that fμ1 (0) n supμ∈U fμ (0) n and I−q (μ1 ) = I−

cn t α (2−α)

√ 1 (μ1 ) C t nfμ1 (0)− n C

1 en √ n log nfμ1 (0)− n . q q

(6.4)

1962


On the other hand, by the definition of q, we have √ n 1 = I2 (μ1 ) I−q−c (μ1 ,δ) (μ1 ) I−q (μ1 ). δ δ Combining the above we get the result.

(6.5)

2 1

Remark. Observe that for the choice δ = supμ∈U[n] fμ (0) n we have inf q−c (μ, δ) n

μ∈U[n]

(see Proposition 4.8 in [23]). This shows that the preceding result is sharp (up to a universal constant). Theorem 3.2 shows that there exists an absolute constant ξ > 0 such that q−c (μ, ξ ) q∗ (μ) for every μ ∈ IL. So we get the following: Corollary 6.2. Let U be a coherent subclass of IL. Then for any n 1,

1 n

sup fμ (0) C sup

μ∈U[n]

μ∈ U[n]

en n log , q∗ (μ) q∗ (μ)

(6.6)

where C > 0 is an absolute constant. Corollary 6.3. Let α ∈ (1, 2], let β > 0 and μ ∈ (Pα (β) ∩ IL)[n] . Then,

1 n

fμ (0) C n

2−α 2

2−α β α log n 2 β α ,

(6.7)

where C > 0 is an absolute constant. Proof. Since μ ∈ CP α (β), by Corollary 4.6 we have that μ ∈ CP(α, cβ). Then, Proposition 3.3 α

shows that q∗ (μ) c nβ 2α . Therefore, the result follows from Corollary 6.2.

2

Theorem 6.4. For every isotropic log-concave measure μ, 1 1 fμ (0) n Cn 4 log n.

(6.8)

Moreover, if μ is symmetric and ψ2 with constant β > 0, then 1 fμ (0) n Cβ log β.

(6.9)

Proof. (6.8) is a direct consequence of Corollary 6.2, Proposition 3.3 and the fact that every log-concave measure is ψ1 with an absolute constant. Recall that, from Corollary 4.6, if μ ∈ SP(2, β) then μ ∈ SP 2 (c1 β). Then (6.9) follows from Corollary 6.3. 2


1963

Remark. In the proof of Corollary 6.2 we have used the fact that q∗ (μ) q−c (μ). One may check that in general this is not sharp (for example one may check that for fμ := 1Bn one has 1 q∗ (μ) q−c (μ, ξ ) for ξ 1). As Proposition 3.5 shows, this is not the case for measures of small diameter. We conclude with the following: Theorem 6.5. The following statements are equivalent: (a) There exists C1 > 0 such that 1

sup sup fμ (0) n C1 . n μ∈IL[n]

(b) There exist C2 , ξ1 > 0 such that sup sup

n μ∈IL[n]

n C2 . q−c (μ, ξ1 )

(c) There exist C3 , ξ2 > 0 such that sup sup

n μ∈IL[n]

n C3 . q∗ (μ, ¯ ξ2 )

Proof. The claim that (a) implies (b) is an immediate consequence of the remark after Theorem 6.1. The fact that (b) implies (c) follows from Proposition 3.5. Finally from Proposition 3.5 and Theorem 6.1 we get that (c) implies (a). 2 We close by noting that there is a strong connection between the existence of supergaussian directions and small ball probability estimates, and hence, in view of Theorem 6.5, with the hyperplane conjecture as well. This connection will appear elsewhere. Acknowledgments We would like to thank Apostolos Giannopoulos for many interesting discussions. Also, the second named author wants to thank Assaf Naor for several valuable comments on an earlier version of this paper. References [1] K.M. Ball, Logarithmically concave functions and sections of convex sets in Rn , Studia Math. 88 (1988) 69–84. [2] C. Borell, Convex set functions in d-space, Period. Math. Hungar. 6 (2) (1975) 111–136. [3] J. Bourgain, On the distribution of polynomials on high dimensional convex sets, in: J. Lindenstrauss, V.D. Milman (Eds.), Geom. Aspects of Funct. Analysis, in: Lecture Notes in Math., vol. 1469, 1991, pp. 127–137. [4] J. Bourgain, On the isotropy constant problem for ψ2 -bodies, in: V.D. Milman, G. Schechtman (Eds.), Geom. Aspects of Funct. Analysis, in: Lecture Notes in Math., vol. 1807, 2003, pp. 114–121. [5] J. Bourgain, B. Klartag, V.D. Milman, Symmetrization and isotropic constants of convex bodies, in: Geometric Aspects of Functional Analysis, in: Lecture Notes in Math., vol. 1850, 2004, pp. 101–116. [6] M. Fradelizi, Sections of convex bodies through their centroid, Arch. Math. 69 (1997) 515–522.

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[7] A. Giannopoulos, Notes on isotropic convex bodies, Warsaw University Notes, 2003. [8] B. Klartag, On convex perturbations with a bounded isotropic constant, Geom. Funct. Anal. 16 (2006) 1274–1290. [9] B. Klartag, V.D. Milman, Rapid Steiner symmetrization of most of a convex body and the slicing problem, Combin. Probab. Comput. 14 (5–6) (2005) 829–843. [10] L. Leindler, On a certain converse of Hölder’s inequality, in: Linear Operators and Approximation, Proc. Conf., Oberwolfach, 1971, in: Internat. Ser. Numer. Math., vol. 20, Birkhäuser, Basel, 1972, pp. 182–184. [11] E. Lutwak, D. Yang, G. Zhang, Lp affine isoperimetric inequalities, J. Differential Geom. 56 (2000) 111–132. [12] E. Lutwak, G. Zhang, Blaschke–Santaló inequalities, J. Differential Geom. 47 (1997) 1–16. [13] V.D. Milman, A new proof of A. Dvoretzky’s theorem in cross-sections of convex bodies, Funktsional. Anal. i Prilozhen. 5 (4) (1971) 28–37 (in Russian). [14] V.D. Milman, Inegalité de Brunn–Minkowski inverse et applications à la théorie locale des espaces normés, C. R. Acad. Sci. Paris 302 (1986) 25–28. [15] V.D. Milman, Isomorphic symmetrization and geometric inequalities, in: J. Lindenstrauss, V.D. Milman (Eds.), Geom. Aspects of Funct. Analysis, in: Lecture Notes in Math., vol. 1317, 1988, pp. 107–131. [16] V.D. Milman, A. Pajor, Isotropic positions and inertia ellipsoids and zonoids of the unit ball of a normed n-dimensional space, in: GAFA Seminar 87–89, in: Lecture Notes in Math., vol. 1376, Springer, 1989, pp. 64– 104. [17] V.D. Milman, A. Pajor, Entropy and asymptotic geometry of non-symmetric convex bodies, Adv. Math. 152 (2000) 314–335. [18] V.D. Milman, G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Lecture Notes in Math., vol. 1200, Springer, Berlin, 1986. [19] V.D. Milman, G. Schechtman, Global versus local asymptotic theories of finite-dimensional normed spaces, Duke Math. J. 90 (1997) 73–93. [20] G. Paouris, Ψ2 -estimates for linear functionals on zonoids, in: Geom. Aspects of Funct. Analysis, in: Lecture Notes in Math., vol. 1807, 2003, pp. 211–222. [21] G. Paouris, On the Ψ2 -behavior of linear functionals on isotropic convex bodies, Studia Math. 168 (3) (2005) 285– 299. [22] G. Paouris, Concentration of mass on convex bodies, Geom. Funct. Anal. 16 (2006) 1021–1049. [23] G. Paouris, Small ball probability estimates for log-concave measures, preprint. [24] G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Math., vol. 94, 1989. [25] A. Prékopa, Logarithmic concave measures with application to stochastic programming, Acta Sci. Math. (Szeged) 32 (1971) 301–316. [26] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973) 335–343. [27] C.A. Rogers, G.C. Shephard, Convex bodies associated with a given convex body, J. London Soc. 33 (1958) 270– 281. [28] R. Schneider, Convex Bodies: The Brunn–Minkowski Theory, Encyclopedia Math. Appl., vol. 44, Cambridge University Press, Cambridge, 1993. [29] J. Spingarn, An inequality for sections and projections of a convex set, Proc. Amer. Math. Soc. 118 (1993) 1219– 1224. [30] C. Zong, Strange Phenomena in Convex and Discrete Geometry, Universitext, Springer, 1996.


Classification of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III Toshihiko Masuda a,∗,1 , Reiji Tomatsu b,1 a Graduate School of Mathematics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka 819-0395, Japan b Faculty of Science and Technology, Department of Mathematics, Tokyo University of Science, Noda,

Chiba 278-8510, Japan Received 2 February 2009; accepted 16 November 2009 Available online 28 November 2009 Communicated by S. Vaes

Abstract We classify a certain class of minimal actions of a compact Kac algebra with amenable dual on injective factors of type III. The structural analysis of type III factors and the canonical extension of endomorphisms introduced by Izumi are our main technical tools. © 2009 Elsevier Inc. All rights reserved. Keywords: Operator algebra; Factor of type III; Compact Kac algebra; Minimal action

1. Introduction In this paper, we extend the main result of our previous paper [18] to type III factors. Namely, we show uniqueness of certain minimal actions of a compact Kac algebra with amenable dual on injective type III factors. Among compact group actions on type III factors, there are some preceding works relevant with our works. The complete classification for compact abelian groups was obtained by * Corresponding author.

E-mail addresses: [email protected] (T. Masuda), [email protected] (R. Tomatsu). 1 Supported by JSPS.


1966

T. Masuda, R. Tomatsu / Journal of Functional Analysis 258 (2010) 1965–2025

Y. Kawahigashi and M. Takesaki in [15]. Recently, M. Izumi showed conjugacy for certain minimal actions of compact groups in [12]. More precisely, if minimal actions of a compact group on a type III0 factor have common faithful Connes–Takesaki modules introduced in [5], then they are conjugate. He also showed that those minimal actions are dual actions of free and centrally trivial actions. In this paper, we classify minimal actions whose dual actions are approximately inner and centrally free, which generalizes classification for invariant-less case in [15]. Our approach is on the whole same as [18], that is, we mainly handle actions of an amenable instead of a compact Kac algebra G, and obtain our main result through discrete Kac algebra G on type III factors to a duality argument as [6]. More precisely, we extend given actions of G larger von Neumann algebras, which are the crossed products by abelian group actions. Then we classify the composed actions of them and dual actions. Splitting the dual actions and taking the partial crossed products, we show that all approximately inner and centrally free actions come from a free action on the injective type II1 factor. In these processes, what play crucial roles are the structural analysis of type III factors developed in [5,26], Izumi’s theory on canonical extension of endomorphisms introduced in [12] and the characterization of approximate innerness and central triviality of endomorphisms shown in [19]. This paper is organized as follows. In Section 2, our main results and their applications are stated. In Section 3, we prove some results for the study in latter sections. In particular, the relative Rohlin theorem proved in Section 3.3 plays an important role for our argument for splitting a model action. In Section 4, type IIIλ case (0 < λ < 1) is studied. Taking the discrete decomposition, we can and the integer group Z reduce our problem to classifying actions of the direct product of G on the injective type II∞ factor. The Z-action has non-trivial Connes–Takesaki module, and the main theorem of [18] is not immediately applicable. However, we can show the model action splitting as in [2] that enables us to cancel the Connes–Takesaki module and to use the main theorem of [18]. In Section 5, type III0 case is studied. We make use of the continuous decomposition, and represent a flow of weights as a flow built under a ceiling function. Then we can regard this problem as reduced to type II case as in [23,24]. We classify actions of the direct product of G and an AF ergodic groupoid on the injective type II∞ factor by using the main result of [18] and Krieger’s cohomology lemma in [13]. In Section 6, type III1 case is studied. Following the lines of the theory of Connes and Haagerup on classification of injective factors of type III1 in [4,9], we take the discrete decompo sition of the type IIIλ factor by the type III1 . Then we classify actions of the direct product of G and the torus coming from the dual action by showing the model action splitting in Sections 6.3 and 6.4. The key point to show this result is approximate innerness of modular automorphisms. In Appendix A, we prove some basic results of the canonical extension in order that readers can smoothly shift from theory of endomorphisms to that of actions of discrete Kac algebras. Most of them can be directly shown from [19] by using a Hilbert space in a von Neumann algebra introduced in [21]. 2. Notations and main theorem Throughout this paper, we treat only separable von Neumann algebras, except for ultraproduct , ϕ) = (L∞ (G), von Neumann algebras. We freely use the notations in [18]. For example, G denotes a discrete Kac algebra. Although some of our results are applicable to a general discrete


1967

except Appendix A. See [18] and the Kac algebra, we always assume the amenability of G references therein for the definition of a discrete (or compact) Kac algebra and its amenability. For a von Neumann algebra M, we denote by U (M) the set of unitary elements in M. By W (M), we denote the set of faithful normal semifinite weights on M. By [3,16,4,9], injective type III factors are determined by their flow of weights. We denote by R0 , R0,1 , Rλ and R∞ the injective factor of type II1 , type II∞ , type IIIλ (0 < λ < 1) and type III1 , respectively. Let M be a factor. For a finite dimensional Hilbert space K, let Mor0 (M, M ⊗ B(K)) be a set of all homomorphisms with finite index. When M is properly infinite, we can identify Mor0 (M, M ⊗ B(K)) with End0 (M), the set of endomorphisms of M with finite index. (See Appendix A.) By TrK and trK , we denote the non-normalized trace and the normalized trace on B(K), respectively. We define an isometric intertwiner Tπ,π ∈ (1, π · π) by Tπ,π =

i∈Iπ

1 √ επ i ⊗ επi . dπ

2.1. Actions and cocycle actions We recall some definitions and notations used in [18] for readers’ convenience. Let M be a and u ∈ U (M ⊗ L∞ (G) ⊗ L∞ (G)). The pair von Neumann algebra, α ∈ Mor(M, M ⊗ L∞ (G)) on M if the following holds: (α, u) (or simply α) is called a cocycle action of G (1) (α ⊗ id) ◦ α = Ad u ◦ (id ⊗ ) ◦ α; (2) (u ⊗ 1)(id ⊗ ⊗ id)(u) = α(u)(id ⊗ id ⊗ )(u); (3) u·,1 = u1,· = 1. Here, α(u) := (α ⊗ id ⊗ id)(u), which is one of our conventions frequently used in our paper, that is, we will omit id when the place where α acts is apparent. If u = 1, α is called an action. We introduce a left inverse Φπα : M ⊗ B(Hπ ) → M of απ for each π ∈ Irr(G) as follows: ∗ ∗ Φπα (x) = 1 ⊗ Tπ,π uπ,π απ (x)uπ ,π (1 ⊗ Tπ,π ) for x ∈ M ⊗ B(Hπ ). Then Φπα is a faithful normal unital completely positive map with Φπα ◦ απ = idM [18, Lemma 2.4]. In general, a left inverse of απ is not uniquely determined, but we always use the left inverse above. If M is a factor, then Φπα is standard, that is, the conditional expectation απ ◦ Φπα : M ⊗ B(Hπ ) → απ (M) is minimal (see Proposition A.10). The another easy but useful remark is the fact that u is evaluated in (M α ) ∩ M, where M α := {x ∈ M | α(x) = x ⊗ 1} is the fixed point algebra. This means that (α|(M α ) ∩M , u) is a cocycle action on (M α ) ∩ M. 2.2. Approximate innerness and central freeness We collect basic notions of homomorphisms and actions from [18]. Definition 2.1. Let M be a von Neumann algebra and α ∈ Mor0 (M, M ⊗B(K)) with the standard left inverse Φ α . We say that

1968


(1) α is properly outer if there exists no non-zero element a ∈ M ⊗ B(K) such that a(x ⊗ 1) = α(x)a for any x ∈ M; (2) α is approximately inner if there exists a sequence {U ν }ν ⊂ U (M ⊗ B(K)) such that ∗ lim (ϕ ⊗ trK ) ◦ Ad U ν − ϕ ◦ Φ α = 0 for all ϕ ∈ M∗ ;

ν→∞

(3) α is centrally trivial if α ω (x) = x ⊗ 1 for all x ∈ Mω ; (4) α is centrally non-trivial if α is not centrally trivial; (5) α is properly centrally non-trivial if there exists no non-zero element a ∈ M ⊗ B(K) such that α ω (x)a = (x ⊗ 1)a for all x ∈ Mω . be a cocycle action of G. We say that Definition 2.2. Let α ∈ Mor(M, M ⊗ L∞ (G)) (1) α is free if απ is properly outer for all π ∈ Irr(G) \ {1}; (2) α is approximately inner if απ is approximately inner for all π ∈ Irr(G); (3) α is centrally free if απ is properly centrally non-trivial for all π ∈ Irr(G) \ {1}. on a factor [18, Definition 2.7], then απ is Note the following fact. If α is a free action of G irreducible for each π ∈ Irr(G) [18, Lemma 2.8]. If απ is irreducible, then central non-triviality is equivalent to properly central non-triviality [18, Lemma 8.3]. Hence a free action α on a factor is centrally free if and only if απ is centrally non-trivial for each π ∈ Irr(G) \ {1}. 2.3. Main theorem We recall the notion of the cocycle conjugacy for two (cocycle) actions. and Definition 2.3. Let M and N be von Neumann algebras. Let α ∈ Mor(M, M ⊗ L∞ (G)) be cocycle actions of G with 2-cocycles u and v, respectively. β ∈ Mor(N, N ⊗ L∞ (G)) (1) α and β are said to be conjugate if there exists an isomorphism θ : M → N such that • α = (θ −1 ⊗ id) ◦ β ◦ θ ; • u = (θ −1 ⊗ id ⊗ id)(v). We write α ≈ β if α and β are conjugate. (2) α and β are said to be cocycle conjugate if there exist an isomorphism θ : M → N and such that w ∈ U (M ⊗ L∞ (G)) • Ad w ◦ α = (θ −1 ⊗ id) ◦ β ◦ θ ; • wα(w)u(id ⊗ )(w ∗ ) = (θ −1 ⊗ id ⊗ id)(v). We write α ∼ β if α and β are cocycle conjugate. is called an α-cocycle if (v ⊗ 1)α(v) = (id ⊗ )(v). When α is an action, v ∈ U (M ⊗ L∞ (G)) The following is the main theorem of this paper which asserts the uniqueness of approximately inner and centrally free actions. Theorem 2.4. Let G be a compact Kac algebra with amenable dual, M an injective factor, α an on R0 . on M, and α (0) a free action of G approximately inner and centrally free action of G (0) Then α is cocycle conjugate to idM ⊗ α .


1969

This implies the following in view of the duality theorem [6, Theorem IV.3]. Theorem 2.5. Let G be a compact Kac algebra with amenable dual, M an injective factor, α a minimal action of G on M, and α (0) a minimal action of G on R0 . If the dual action of α is approximately inner and centrally free, then α is cocycle conjugate to idM ⊗ α (0) . If α is a dual action, then α and idM ⊗ α (0) are conjugate. As a corollary, we obtain the following classification of minimal actions of compact Lie groups on R∞ . Corollary 2.6. Let G be a semisimple connected compact Lie group. Then any two minimal actions of G on R∞ are conjugate. Proof. This follows from Theorem 2.5, [19, Theorem 3.15, 4.12] and [12, Corollary 5.14].

2

Our main purpose is to prove Theorem 2.4. In [18, Theorem 7.1], we have proved the main theorem in type II1 case. The remaining cases are type II∞ , IIIλ (0 < λ < 1), III0 and III1 . Type II∞ case is easily shown as follows. Proof of Theorem 2.4 for R0,1 . Let α be an approximately inner and centrally free action of on R0,1 . Let τ be a normal trace on R0,1 . Since α is approximately inner, we have τ ◦ Φπα = G τ ⊗ trπ for π ∈ Irr(G) by Corollary A.7. Hence τ is invariant under α. Let {ei,j }∞ i,j =1 ⊂ R0,1 be a system of matrix units with a finite projection e11 . Since (τ ⊗ such that trπ )(e11 ⊗ 1) = (τ ⊗ trπ )(απ (e11 )) for each π ∈ Irr(G), we can take v ∈ R0,1 ⊗ L∞ (G) ∞ ∗ ∗ vv = e11 ⊗ 1 and v v = α(e11 ). Set a unitary V = i=1 (ei1 ⊗ 1)vα(e1i ). Then the perturbed cocycle action Ad V ◦ α fixes the type I factor B := {ei,j } i,j . Therefore Ad V ◦ α|B ∩R0,1 is an approximately inner and centrally free cocycle action on an injective type II1 factor B ∩ R0,1 . By [18, Theorem 6.2], we can perturb Ad V ◦ α|B ∩R0,1 to be an action. Then this action is cocycle conjugate to the model action α (0) . Therefore we have α ∼ idB(2 ) ⊗ α (0) . Using α (0) ∼ idR0 ⊗ α (0) , we obtain α ∼ idR0,1 ⊗ α (0) . 2 By Theorem 2.4, any two approximately inner and centrally free actions α and β on an injective factor M are cocycle conjugate. This can be more precisely stated as [18, Theorem 7.1]. an amenable discrete Kac algebra. Let α and β Theorem 2.7. Let M be an injective factor and G on M. Then there exist θ ∈ Int(M) and be approximately inner and centrally free actions of G ∞ an α-cocycle v ∈ M ⊗ L (G) such that Ad v ◦ α = θ −1 ⊗ id ◦ β ◦ θ. Proof. Since M is injective, M is isomorphic to R0 ⊗ M. Fix an isomorphism Ψ : M → M ⊗ R0 . on R0 . Set γ := (Ψ −1 ⊗ id) ◦ (idM ⊗ α (0) ) ◦ Ψ , which is an Let α (0) be a free action of G approximately inner and centrally free action on M. By Theorem 2.4, we can take θ0 ∈ Aut(M) and an α-cocycle v such that Ad v ◦ α = (θ0−1 ⊗ id) ◦ γ ◦ θ0 . To prove the theorem, it suffices to show the statement for β = γ . Set θ1 := Ψ ◦ θ0 ◦ Ψ −1 ∈ Aut(M ⊗ R0 ). Note that the core of M ⊗ R0 canonically coincides is surjective by [25], there ⊗ R0 . Since the module map mod : Aut(M) → Autθ (Z(M)) with M

1970


exists θ2 ∈ Aut(M) such that mod(θ1 ) = mod(θ2 ⊗ idR0 ). Set θ3 := Ψ −1 ◦ (θ2 ⊗ idR0 ) ◦ Ψ ∈ Aut(M). Then θ3−1 θ0 = Ψ −1 ◦ (θ2−1 ⊗ idR0 )θ1 ◦ Ψ implies mod(θ3−1 θ0 ) = id. Putting θ := θ3−1 θ0 , we have Ad v ◦ α = (θ −1 ⊗ id) ◦ γ ◦ θ because θ3 commutes with γ . Moreover θ ∈ Int(M) by [14, Theorem 1 (1)]. 2 3. Preliminaries The results in this section are frequently used in the later sections. One of the most important results is the relative Rohlin theorem (Theorem 3.13). 3.1. Basic results on cocycle conjugacy on a properly infinite von Neumann algebra M. Lemma 3.1. Let (α, u) be a cocycle action of G Let H be a Hilbert space. Then (α, u) and (idB(H ) ⊗ α, 1 ⊗ u) are cocycle conjugate. Proof. Take a Hilbert space H ⊂ M with support 1 and the same dimension d ∞ as H [21]. Let {ξi }di=1 be an orthonormal basis of H. Then we have an isomorphism Ψ : B(H ) ⊗ M → M such that Ψ (eij ⊗ x) = ξi xξj∗ for all x ∈ M and i, j ∈ N, where {eij }ij is a canonical system of matrix units of B(H ). Define the unitary v := di=1 (ξi ⊗ 1)α(ξi∗ ). We check that Ψ and v satisfy the statement. For x ∈ M and i, j ∈ N, we have Ad v ◦ α ◦ Ψ (eij ⊗ x) = vα ξi xξj∗ v ∗ = (ξi ⊗ 1)α(x) ξj∗ ⊗ 1 = (Ψ ⊗ id) ◦ (id ⊗ α)(eij ⊗ x). Hence (1) holds. On (2), we have (v ⊗ 1)α(v)u(id ⊗ ) v ∗ =

d

(ξi ⊗ 1 ⊗ 1) α ξi∗ ⊗ 1 · α(ξj ) ⊗ 1 α α ξj∗ · u(id ⊗ ) v ∗

i,j =1

=

d (ξi ⊗ 1 ⊗ 1)u(id ⊗ ) α ξi∗ (id ⊗ ) v ∗ i=1

d = (ξi ⊗ 1 ⊗ 1)u ξi∗ ⊗ 1 ⊗ 1 = (Ψ ⊗ id ⊗ id)(1 ⊗ u).

2

i=1

Lemma 3.2. Let (α, u) be a cocycle action on a properly infinite von Neumann algebra M. Then u is a coboundary. Proof. By the previous lemma, it suffices to prove that (idB(L2 (G)) ⊗ α, 1 ⊗ u) can be perturbed to an action. Write α = idB(L2 (G)) ⊗ α and u ¯ = 1 ⊗ u. Then we set a unitary v := W31 u∗231 ∈ ⊗ M ⊗ L∞ (G), where W ∈ L∞ (G) ⊗ L∞ (G) is the multiplicative unitary defined B(L2 (G))


1971

in [18, Section 2]. Using the 2-cocycle relation of u and (x) = W ∗ (1 ⊗ x)W for x ∈ L∞ (G), we have v123 α(v)u(id ¯ ⊗ id ⊗ ) v ∗ = W31 u∗231 · W41 α u∗ 2341 · u234 · (id ⊗ ⊗ id)(u)2341 ( ⊗ id) W ∗ 341 = W31 u∗231 · W41 α u∗ · (u ⊗ 1) · (id ⊗ ⊗ id)(u) 2341 ( ⊗ id) W ∗ 341 = W31 u∗231 · W41 (id ⊗ id ⊗ )(u) 2341 ( ⊗ id) W ∗ 341 ∗ = W31 u∗231 · W41 W34 u124 W34 2341 ( ⊗ id) W ∗ 341 ∗ = W31 u∗231 · W41 · W41 u231 W41 · ( ⊗ id) W ∗ 341 = W31 W41 · ( ⊗ id) W ∗ 341 = 1. 2 Next we discuss the cocycle conjugacy of extended actions. For definition of the canonical extension of a cocycle action, readers are referred to [12] and Appendix A. on a properly infinite von Neumann algebra M. Then the Lemma 3.3. Let α be an action of G θ R is cocycle conjugate to α. second canonical extension α on M Proof. This is immediately obtained from Lemma 3.1 and Corollary A.15.

2

We close this subsection with the following lemma. 2 be discrete Kac algebras. Let α i and β i be actions of G i on von Neu1 and G Lemma 3.4. Let G mann algebras M and N , respectively. Assume the following: • α 1 and α 2 commute; • β 1 and β 2 commute; 1 × G 2 -actions α := (α 1 ⊗ id) ◦ α 2 and β := (β 1 ⊗ id) ◦ β 2 are cocycle conjugate. • The G 2 (resp. β 2 on M β 2 G 2 ). Then the action α 1 (resp. β 1 ) extends to the action α 1 on M α 2 G Moreover, α 1 and β 1 are cocycle conjugate. Proof. Let v be an α-cocycle and Ψ : M → N be an isomorphism such that Ad v ◦ α = (Ψ −1 ⊗ 1 ) and v r := v1⊗· ∈ M ⊗ L∞ (G 2 ), which id) ◦ β ◦ Ψ . Set unitaries v := v·⊗1 ∈ M ⊗ L∞ (G 1 2 1 → are α -cocycle and α -cocycle, respectively. Then we define an isomorphism Θ : M α 2 G 2 r r ∗ 2 N β 2 G by Θ(x) = (Ψ ⊗ id)(v x(v ) ). We set a unitary u := (α ⊗ id)(v ) ∈ (M α 2 G2 ) ⊗ 1 ). Then u is an α 1 -cocycle. By direct calculation, we have Ad u ◦ α 1 = (Θ −1 ⊗ id) ◦ L∞ (G β 1 ◦ Θ. 2 3.2. Rohlin property See [18, Section 3] for notions of ultraproduct algebras and actions on them. First we recall the following definition [18, Definition 3.4, 3.13].

1972


be an action of G. We say that Definition 3.5. Let γ ∈ Mor(M ω , M ω ⊗ L∞ (G)) (1) γ is strongly free when for any π ∈ Irr(G) \ {1} and any countably generated von Neumann subalgebra S ⊂ M ω , there exists no non-zero a ∈ M ω ⊗ B(Hπ ) such that γπ (x)a = a(x ⊗ 1) for all x ∈ S ∩ Mω ; (2) γ is semi-liftable when for any π ∈ Irr(G), there exist elements β ν , β ∈ Mor0 (M, M ⊗ B(Hπ )), ν ∈ N, such that β ν converges to β and γπ ((x ν )ν ) = (β ν (x ν ))ν for all (x ν )ν ∈ M ω . is centrally free if and only if α ω is Note that a cocycle action α ∈ Mor(M, M ⊗ L∞ (G)) ν ω strongly free [18, Lemma 8.2]. For (x )ν ∈ M , we set τ ω (x) := limν→ω x ν . Then τ ω : M ω → M is a faithful normal conditional expectation. be a set of all projections in L∞ (G) with finite support. For F ∈ Let Projf(L∞ (G)) ∞ ∞ Projf(L (G)) and ε > 0, a projection S ∈ Projf(L (G)) is said to be (F, ε)-invariant if we have (F ⊗ 1)(S) − F ⊗ S < ε|F |ϕ |S|ϕ . ϕ⊗ϕ an be an amenable discrete Kac algebra and γ ∈ Mor(Mω , Mω ⊗ L∞ (G)) Definition 3.6. Let G ∞ action. We say that γ has the Rohlin property when for any central F ∈ Projf(L (G)), δ > 0, with K e1 , any countable subset S ⊂ M ω and any (F, δ)-invariant central K ∈ Projf(L∞ (G)) such that faithful state φ ∈ M∗ , there exists a projection E ∈ Mω ⊗ L∞ (G) (R1) E is supported on K, that is, E = E(1 ⊗ K); (R2) E almost intertwines γ and in the following sense: γF (E) − (id ⊗ F )(E)

φ◦τ ω ⊗ϕ⊗ϕ

5δ 1/2 |F |ϕ ;

that is, if we decompose E as (R3) E gives a copy of L∞ (G)K, E=

d(π)−1 Eπi,j ⊗ eπi,j ,

π∈supp(K) i,j ∈Iπ

then, for all i, j ∈ Iπ , k, ∈ Iρ and π, ρ ∈ supp(K), we have Eπi,j Eρk, = δπ,ρ δj,k Eπi, ; (R4) (id ⊗ ϕπ )(E) ∈ S ∩ Mω for any π ∈ supp(K); (R5) E gives a partition of unity of S ∩ Mω , that is, (id ⊗ ϕ)(E) = 1. The above projection E is called a Rohlin projection. be an action. Assume that Mω is globally Definition 3.7. Let γ ∈ Mor(M ω , M ω ⊗ L∞ (G)) invariant under γ and γ |Mω has the Rohlin property. We say that γ has the joint Rohlin prop δ > 0, (F, δ)-invariant central K ∈ Projf(L∞ (G)) with erty when for any F ∈ Projf(L∞ (G)), ω K e1 , any countable set S ⊂ M and any countable family of γ -cocycles C which are evalu such that ated in M ω , there exists a projection E ∈ Mω ⊗ L∞ (G)


1973

(S1) E satisfies (R1)–(R5); (S2) For any v ∈ C, a projection vEv ∗ also satisfies (R3); (S3) For any v ∈ C and π ∈ supp(K), we have (id ⊗ ϕπ )(vEv ∗ ) = (id ⊗ ϕπ )(E). Lemma 3.8. If γ has the joint Rohlin property and E is a projection as above, then the element (id ⊗ ϕ)(vE) is a unitary for all v ∈ C. Proof. Set μ := (id ⊗ ϕ)(vE) and E := vEv ∗ . Then,

μ=

vπi,j Eπj,i =

π∈Irr(G) i,j ∈Iπ

π∈Irr(G) i,j ∈Iπ

Eπ i,j vπj,i .

Using (R3) for E and E , we can check μμ∗ = 1 = μ∗ μ as follows,

μμ∗ =

vπi,j Eπj,i Eπ∗k, vπ∗k, =

vπi,j Eπj,k vπ∗k,i

π∈Irr(G) i,j,k∈Iπ

π∈Irr(G) i,j,k,∈Iπ

= (id ⊗ ϕ) E = 1, and μ∗ μ =

∗ vπ∗j,i E π E π vπ,k = i,j

k,

π∈Irr(G) i,j,k,∈Iπ

vπ∗j,k E π vπ,k j,

π∈Irr(G) j,k,∈Iπ

= (id ⊗ ϕ) v ∗ E v = (id ⊗ ϕ)(E) = 1.

2

Such an element (id ⊗ ϕ)(vE) is called a Shapiro unitary. 1 := G = (L∞ (G), ), G 2 = (L∞ (G 2 ), 2 ) be amenable discrete Kac algebras with Let G 1 2 ×G 2 is naturally the invariant weights ϕ := ϕ and ϕ , respectively. The product Kac algebra G ˜ = 2 , constructed. The invariant weight and the coproduct are denoted by ϕ˜ = ϕG× G 2 and G×G respectively. i ) for i = 1, 2, respectively. Lemma 3.9. Take (Fi , δi )-invariant central projections Ki ∈ L∞ (G Then K1 ⊗ K2 is (F1 ⊗ F2 , δ1 + δ2 )-invariant. Proof. (F1 ⊗ F2 ⊗ 1 ⊗ 1)(K ˜ 1 ⊗ K2 ) − F1 ⊗ F2 ⊗ K1 ⊗ K2 ϕ⊗ ˜ ϕ˜ ˜ 1 ⊗ K2 ) − (F1 ⊗ F2 ⊗ K1 ⊗ 1)2 (K2 )24 (F1 ⊗ F2 ⊗ 1 ⊗ 1)(K ϕ⊗ ˜ ϕ˜ 2 + (F1 ⊗ F2 ⊗ K1 ⊗ 1) (K2 )24 − F1 ⊗ F2 ⊗ K1 ⊗ K2 ϕ⊗ ˜ ϕ˜ = (F1 ⊗ 1)(K1 ) − (F1 ⊗ K1 ) ϕ 1 ⊗ϕ 1 (F2 ⊗ 1)2 (K2 ) ϕ 2 ⊗ϕ 2 + (F2 ⊗ 1)2 (K2 ) − (F2 ⊗ K2 ) ϕ 2 ⊗ϕ 2 |F1 ⊗ K1 |ϕ 1 ⊗ϕ 1 < (δ1 + δ2 )|F1 ⊗ F2 |ϕ˜ |K1 ⊗ K2 |ϕ˜ .

2

1974


The following elementary lemma is useful. Lemma 3.10. Let P , Q be von Neumann algebras. Let φ ∈ P∗ and ψ ∈ Q∗ be faithful positive functionals, respectively. Let X, Y ∈ P ⊗ Q be given. If X ∈ (P ⊗ Q)φ⊗ψ , then one has (id ⊗ ψ)(Y X) Y |X|φ⊗ψ . φ Proof. Let (id ⊗ ψ)(Y X) = w|(id ⊗ ψ)(Y X)| be the polar decomposition. Since X commutes with φ ⊗ ψ, we have (id ⊗ ψ)(Y X) = φ w ∗ (id ⊗ ψ)(Y X) = (φ ⊗ ψ) w ∗ ⊗ 1 Y X φ w ∗ ⊗ 1 Y |X|φ⊗ψ Y |X|φ⊗ψ . 2 3.3. Relative Rohlin theorem Throughout this subsection, we are given the following: (A1) (A2) (A3) (A4) (A5)

A von Neumann algebra M; 2 -action γ 2 on M ω , and they are commuting; A G-action γ 1 on M ω and a G 2 1 ×G -action γ := (γ ⊗ id) ◦ γ 2 is strongly free and semi-liftable; The G Mω is globally invariant under γ ; γ τ ω ◦ Φ(π,ρ) = τ ω ⊗ trπ ⊗ trρ on Mω ⊗ B(H(π,ρ) ) for all (π, ρ) ∈ Irr(G) × Irr(G2 ).

The assumption (A3) restricts not only γ but also Mω . For example, Mω = C case is excluded. When M is a factor, (A5) automatically holds. Indeed by semi-liftability, we can take (β ν )ν , a sequence of homomorphisms on M converging to some homomorphism β and defining γ(π,ρ) , that is, γ(π,ρ) (x) = (β ν (x))ν for x = (x ν )ν ∈ M ω . Then by [18, Lemma 3.3], we obtain τ ω ◦ γ Φ(π,ρ) = Φ β ◦ (τ ω ⊗ id) on M ω ⊗ B(H(π,ρ) ). Since M is a factor, τ ω |Mω is a trace. Hence for x ∈ Mω and y ∈ B(H(π,ρ) ), we have γ τ ω ◦ Φ(π,ρ) (x ⊗ y) = Φ β ◦ τ ω ⊗ id (x ⊗ y) = τ ω (x)Φ β (1 ⊗ y) γ = τ ω (x)τ ω Φ(π,ρ) (1 ⊗ y) = τ ω (x)(trπ ⊗ trρ )(y). γ

This shows τ ω ◦ Φ(π,ρ) = τ ω ⊗ trπ ⊗ trρ as desired. Our aim is to prove the relative Rohlin theorem which assures that a Rohlin projection for γ 1 γ2

can be evaluated in Mω . Let us take Fi , Ki and δi for i = 1, 2 as in Lemma 3.9. We may assume that Ki e1 for each i (see [18, §2.3]). Set F = F1 ⊗ F2 , δ = δ1 + δ2 and K = K1 ⊗ K2 . Set Ki = supp(Ki ) for each i and K = supp(K) = K1 × K2 . We fix a faithful state φ ∈ M∗ and set ψ := φ ◦ τ ω . Let C be a countable family of γ 1 -cocycles. Let S ⊂ M ω and T ⊂ (M ω )γ a countably generated ×G 2 ), we denote by Eˆ the sliced von Neumann subalgebras. For a projection E ∈ M ω ⊗ L∞ (G 2 element (id ⊗ id ⊗ ϕ )(E). ×G 2 )K such that E ∈ J if Define the set J consisting of projections in (T ∩ Mω ) ⊗ L∞ (G ˆ and only if E satisfies (R1), (R3) and (R4) and, in addition, E satisfies (S2) and (S3) for C. Since 0 ∈ J, J is non-empty. Define the following functions a, b, c and d from J to R+ :


1975

˜ aE = |F |−1 ; ϕ γF (E) − (id ⊗ F )(E) ψ⊗ϕ⊗ ˜ ϕ˜ bE = |E|ψ⊗ϕ˜ ; γ 2 (E) ˆ 132 − Eˆ ⊗ F2 ; cE = |F2 |−1 ψ⊗ϕ˜ ϕ 2 F2 γ 1 (E) − (id ⊗ F ⊗ id)(E) . dE = |F1 |−1 1 G ψ⊗ϕ 1 ⊗ϕ˜ ϕ 1 F1 Lemma 3.11. Let E ∈ J. Assume that bE < 1 − δ 1/2 . Then there exists E ∈ J such that (1) aE − aE 3δ 1/2 (bE − bE ); (2) 0 < (δ 1/2 /2)|E − E|ψ⊗ϕ bE − bE ; 1/2 (3) cE − cE 4δ2 (bE − bE ); 1/2

(4) dE − dE 3δ1 (bE − bE ). Proof. Our proof is similar to the one presented in [20]. We may assume that S contains T and the matrix entries of all v ∈ C, and that S is globally γ -invariant. We add the matrix entries of E ˜ Again we may and do to S and denote the new countably generated von Neumann algebra by S. assume that S˜ is globally γ -invariant. Take δ3 > 0 such that bE < (1 − δ3 )(1 − δ 1/2 ). Recall our assumptions (A3) and (A5). Then by [18, Lemma 5.3], there is a partition of unity q {ei }i=0 ⊂ S˜ ∩ Mω such that (1) |e0 |ψ < δ3 ; (2) (ei ⊗ 1π ⊗ 1ρ )γ(π,ρ) (ei ) = 0 for all 1 i q and (π, ρ) ∈ K · K \ {1}. ˜ K (ei )) ∈ S˜ ∩ Mω . We claim that at least one i with 1 i q Set a projection fi := (id ⊗ ϕ)(γ satisfies |E(fi ⊗ 1 ⊗ 1)|ψ⊗ϕ˜ < (1 − δ 1/2 )|fi |ψ . Since E(fi ⊗ 1 ⊗ 1)

ψ⊗ϕ˜

= (ψ ⊗ ϕ) ˜ E(fi ⊗ 1) = ψ (id ⊗ ϕ)(E)(id ˜ ⊗ ϕ) ˜ γK (ei ) γ d(π)2 d(ρ)2 ψ (id ⊗ ϕ)(E)Φ ˜ = (π ,ρ) (ei ⊗ 1π ⊗ 1ρ ) (π,ρ)∈K

=

(π,ρ)∈K

=

γ ˜ (ei ⊗ 1π ⊗ 1ρ ) d(π)2 d(ρ)2 ψ Φ(π,ρ) γ(π,ρ) (id ⊗ ϕ)(E) ˜ (ei ⊗ 1π ⊗ 1ρ ) d(π)2 d(ρ)2 (ψ ⊗ trπ ⊗ trρ ) γ(π,ρ) (id ⊗ ϕ)(E)

(π,ρ)∈K

˜ (ei ⊗ 1 ⊗ 1) , = (ψ ⊗ ϕ) ˜ γK (id ⊗ ϕ)(E) we have the following: q E(fi ⊗ 1 ⊗ 1) i=1

ψ⊗ϕ˜

⊥ ˜ e0 ⊗ 1 ⊗ 1 = (ψ ⊗ ϕ) ˜ γK (id ⊗ ϕ)(E) (ψ ⊗ ϕ) ˜ γK (id ⊗ ϕ)(E) ˜ = |K|ϕ˜ (ψ ⊗ ϕ)(E) ˜

1976


= bE |K|ϕ˜

< (1 − δ3 ) 1 − δ 1/2 |K|ϕ˜ . If |E(fi ⊗ 1 ⊗ 1)|ψ⊗ϕ˜ (1 − δ 1/2 )|fi |ψ (= (1 − δ 1/2 )|ei |ψ |K|ϕ˜ ) for all 1 i q, then we have 1 − δ 1/2 e0⊥ ψ |K|ϕ˜ < (1 − δ3 ) 1 − δ 1/2 |K|ϕ˜ . This is a contradiction with |e0 |ψ < δ3 . Hence there exists fi such that |E(fi ⊗ 1 ⊗ 1)|ψ⊗ϕ˜ < (1 − δ 1/2 )|fi |ψ . Set e := ei and f := (id ⊗ ϕ)(γ ˜ K (e)) ∈ S˜ ∩ Mω . ×G 2 ) by Define the projection E ∈ Mω ⊗ L∞ (G E = E f ⊥ ⊗ 1 ⊗ 1 + γK (e). ×G 2 )K. Then E satisfies (R1), Since T ⊂ (M ω )γ and e ∈ S˜ ∩ Mω , E ∈ (T ∩ Mω ) ⊗ L∞ (G ˆ (R3) and (R4) by [18, Lemma 5.7]. We have to check E satisfies (S2) and (S3). Set a projection e = (id ⊗ ϕ 2 )(γK2 2 (e)) ∈ S˜ ∩ Mω . If we show (e ⊗ 1)γπ1 (e ) = 0 for each π ∈ K1 · K1 \ {1}, then we are immediately done in view of [18, Lemma 5.7]. This is verified as follows. First we compute the following: for π ∈ K1 · K1 \ {1} and ρ ∈ K2 , (e ⊗ 1 ⊗ 1)γρ2 γπ1 e = (e ⊗ 1 ⊗ 1)γρ2 γπ1 id ⊗ ϕ 2 γK2 2 (e) id ⊗ id ⊗ id ⊗ ϕσ2 (e ⊗ 1 ⊗ 1 ⊗ 1)γρ2 γπ1 γσ2 (e) = σ ∈K2

= id ⊗ id ⊗ id ⊗ ϕρ2 (e ⊗ 1 ⊗ 1 ⊗ 1)γρ2 γπ1 γρ2 (e) = 0, where we have used the starting condition for e. Using this, we get e ⊗ 1 γπ1 e = id ⊗ ϕ 2 γK2 2 (e) ⊗ 1 γπ1 e γ2 = d(ρ)2 Φρ (e ⊗ 1) ⊗ 1 γπ1 e ρ∈K2

=

γ2 d(ρ)2 Φρ ⊗ id (e ⊗ 1 ⊗ 1)γρ2 γπ1 e

ρ∈K2

= 0. Therefore Eˆ satisfies (S2) and (S3), which means E ∈ J. Next we estimate bE as follows: bE − bE = (ψ ⊗ ϕ) ˜ E − E = (ψ ⊗ ϕ) ˜ −E(f ⊗ 1 ⊗ 1) + γK (e) > − 1 − δ 1/2 |f |ψ + |f |ψ = δ 1/2 |f |ψ .


1977

δ 1/2 |f |ψ < bE − bE .

(3.1)

Hence

We check the inequalities in the statements. The first, the second and the fourth ones are derived in a similar way as in the proof of [18, Lemma 5.11]. Thus we only present a proof for the third one. Since ⊥ ˆ 132 γF2 (f )⊥ γF22 Eˆ 132 − Eˆ ⊗ F2 = γF22 (E) 13 − f ⊗ 1 ⊗ F2 2 ˆ 132 − (Eˆ ⊗ F2 ) f ⊥ ⊗ 1 ⊗ F2 + γF22 (E) + γK1 1 id ⊗ id ⊗ ϕ 2 id ⊗ 2 γ 2 (e) (1 ⊗ F2 ⊗ K2 ) − γK1 1 id ⊗ id ⊗ ϕ 2 id ⊗ 2 γK2 2 (e) (1 ⊗ F2 ⊗ 1) , we have 2 γ Eˆ − Eˆ ⊗ F2 ψ⊗ϕ˜ F2 132 ⊥ ˆ 132 γF2 (f )⊥ γF22 (E) 13 − f ⊗ 1 ⊗ F2 ψ⊗ϕ˜ 2 ˆ 132 − (Eˆ ⊗ F2 ) f ⊥ ⊗ 1 ⊗ F2 + γF22 (E) ψ⊗ϕ˜ 1 2 2 2 id ⊗ γK ⊥ (e) (1 ⊗ F2 ⊗ K2 ) ψ⊗ϕ˜ + γK1 id ⊗ id ⊗ ϕ 2 + γ 1 id ⊗ id ⊗ ϕ 2 id ⊗ 2 γ 2 (e) 1 ⊗ F2 ⊗ K ⊥ . K1

K2

2

ψ⊗ϕ˜

Then we have the following estimates: (3.3) cE ,

and (3.4), (3.5) < δ2 |F2 |ϕ 2 |f |ψ .

On (3.2), we have ˆ 132 γF2 (f )13 − f ⊗ 1 ⊗ F2 (3.2) = (ψ ⊗ ϕ) ˜ γF22 (E) 2 2 γ (f ) − f ⊗ F2 = ψ ⊗ ϕ 2 γF22 (id ⊗ ϕ)(E) ˜ F2 ψ ⊗ ϕ 2 γF22 (f ) − f ⊗ F2 ψ ⊗ ϕ 2 id ⊗ ϕ 1 ⊗ id ⊗ ϕ 2 ◦ γK1 1 id ⊗ F2 2K2 γK2 ⊥ (e) 2 2 2 1 2 1 + ψ ⊗ϕ id ⊗ ϕ ⊗ id ⊗ ϕ ◦ γK1 (id ⊗ F2 K ⊥ ) γK2 (e) 2

δ2 |K1 |ϕ 1 |F2 |ϕ 2 |K2 |ϕ 2 |e|ψ + δ2 |K1 |ϕ 1 |F2 |ϕ 2 |K2 |ϕ 2 |e|ψ = 2δ2 |F2 |ϕ 2 |f |ψ . By using (3.1), we have 1/2

cE cE + 4δ2 |f |ψ < cE + 4δ2 (bE − bE ).

2

(3.2) (3.3) (3.4) (3.5)

1978


Thus we obtain the following as [18, Theorem 5.9]. Lemma 3.12. Let γ = (γ 1 ⊗ id) ◦ γ 2 , F , K, S, T and C be as before. Then the following statements hold: (1) γ has the Rohlin property; ×G 2 , we can take a Rohlin projection E from (T ∩ (2) In the setting of Definition 3.7 for G ∞ 2 ˆ Mω ) ⊗ L (G × G ) such that E satisfies (S1)–(S3) for C and 2 1/2 γ (E) ˆ 132 − Eˆ ⊗ F2 < 5δ2 |F2 |ϕ 2 ; F2 ψ⊗ϕ˜ 1 1/2 γ (E) − (id ⊗ F ⊗ id)(E) < 5δ1 |F1 |ϕ 1 . 1 F1 ψ⊗ϕ 1 ⊗ϕ˜ Our main theorem in this subsection is the following. Theorem 3.13 (Relative Rohlin theorem). Let M be a von Neumann algebra and γ = (γ 1 ⊗ id) ◦ ×G 2 on M ω such that: γ 2 an action of G • • • •

2 -action γ 2 on M ω ; The G-action γ 1 on M ω commutes with the G 2 1 2 ×G -action γ := (γ ⊗ id) ◦ γ is strongly free and semi-liftable; The G Mω is globally invariant under γ ; γ τ ω ◦ Φ(π,ρ) = τ ω ⊗ trπ ⊗ trρ on Mω ⊗ B(H(π,ρ) ) for all (π, ρ) ∈ Irr(G) × Irr(G2 ).

Then γ 1 has the joint Rohlin property. Moreover, for any countably generated von Neumann subγ2 satisfying (S1)–(S3). algebra T ⊂ (M ω )γ , γ 1 has a Rohlin projection E ∈ (T ∩ Mω ) ⊗ L∞ (G) Proof. Let Fi , Ki and δi (i = 1, 2) be given as before. Take a Rohlin projection E ∈ (T ∩ Mω ) ⊗ ×G 2 ) supported on K1 ⊗ K2 as in the previous lemma. Then we have L∞ (G 2 1/2 γ (E) ˆ 132 − Eˆ ⊗ F2 5δ2 |F2 |ϕ 2 ; F2 ψ⊗ϕ˜ 1 1/2 γ (E) − (id ⊗ F ⊗ id)(E) 5δ1 |F1 |ϕ 1 . 1 F1 ψ⊗ϕ 1 ⊗ϕ˜

(3.6) (3.7)

We set Eˆ = (id ⊗ id ⊗ ϕ 2 )(E). By (R3), Eˆ gives a partition of unity by matrix elements along with K1 . We estimate the equivariance of Eˆ with respect to γ 1 1 γ (E) ˆ − (id ⊗ F1 )(E) ˆ F1 ψ⊗ϕ 1 ⊗ϕ 1 1 2 = id ⊗ id ⊗ id ⊗ ϕ γF1 (E) − (id ⊗ F1 ⊗ id)(E) ψ⊗ϕ 1 ⊗ϕ 1 γF11 (E) − (id ⊗ F1 ⊗ id)(E) ψ⊗ϕ 1 ⊗ϕ˜ (by Lemma 3.10) 1/2 5δ1 |F1 |ϕ 1 by (3.7) . ∞ 2 Take an increasing sequence {F2 (n)}∞ n=1 ⊂ Projf(Z(L (G ))) with F2 (n) → 1 strongly as 1/2 n → ∞. Next we take δ2 (n) > 0 such that δ2 (n) |F2 (n)|ϕ 2 → 0 as n → ∞. Take a sequence


1979

of Rohlin projections {E(n)}n satisfying the above inequalities (3.6) and (3.7) for F2 (n) and ∞ ˆ δ2 (n). By using the index selection trick [18, Lemma 3.11] for (E(n)) n ∈ (N, Mω ), we obtain 2 γ supported on K1 such that a Rohlin projection E1 ∈ (T ∩ Mω ) ⊗ L∞ (G) 1 1/2 γ (E1 ) − (id ⊗ F )(E1 ) 5δ1 |F |ϕ 1 . 1 F1 ψ⊗ϕ 1 ⊗ϕ 1

2

on M ω . Corollary 3.14 (Rohlin theorem). Let M be a von Neumann algebra and γ an action of G Assume the following: • γ is strongly free and semi-liftable; • Mω is globally invariant under γ ; γ • τ ω ◦ Φπ = τ ω ⊗ trπ on Mω ⊗ B(Hπ ) for all π ∈ Irr(G). Then: (1) γ has the joint Rohlin property; (2) For any countably generated von Neumann subalgebra T ⊂ (M ω )γ , γ has a Rohlin projec satisfying (S1)–(S3); tion E ∈ (T ∩ Mω ) ⊗ L∞ (G) there exists a unitary (3) γ is stable on M ω , that is, for any γ -cocycle v ∈ M ω ⊗ L∞ (G), then μ can be taken from Mω . μ ∈ M ω such that v = (μ ⊗ 1)γ (μ∗ ). If v ∈ Mω ⊗ L∞ (G), 2 = {1}. Then (1) and (2) hold. Proof. In the previous theorem, we put G ∞ (3) Let {Fν }ν∈N ⊂ L (G) be an increasing family of finitely supported central projections such that Fν → 1 strongly as ν → ∞. For each ν, take an (Fν , 1/ν)-invariant finite projection with e1 Kν . Let E ν ∈ Mω ⊗ L∞ (G) be a Rohlin projection satisfying (S1)–(S3) Kν ∈ L∞ (G) for a faithful state ψ = φ ◦ τ ω ∈ (M ω )∗ , Fν , Kν and 1/ν 2 . Then we get the Shapiro unitary μν = (id ⊗ ϕ)(vE ν ), and we have vγFν μν − μν ⊗ Fν = (id ⊗ id ⊗ ϕ) (v ⊗ 1)γFν vE ν − (id ⊗ id ⊗ ϕ) (id ⊗ Fν ) vE ν = (id ⊗ id ⊗ ϕ) (id ⊗ Fν )(v)γFν E ν − (id ⊗ id ⊗ ϕ) (id ⊗ Fν ) vE ν = (id ⊗ id ⊗ ϕ) (id ⊗ Fν )(v) γFν E ν − (id ⊗ Fν ) E ν . Since the element γFν (E ν ) − (id ⊗ Fν )(E ν ) is in the centralizer of ψ ⊗ ϕ ⊗ ϕ, we can use Lemma 3.10, and we get vγF μν − μν ⊗ Fν γFν E ν − (id ⊗ Fν ) E ν ψ⊗ϕ⊗ϕ ν ψ⊗ϕ 5/ν. By using the index selection map for (μν )ν ∈ ∞ (N, M ω ), we get μ ∈ M ω such that vγ (μ) = μ ⊗ 1. When v is evaluated in Mω , each μν is in Mω , and so is μ by the property of the index selection map. 2 on M ω and θ ∈ Aut(M ω ). Corollary 3.15. Let M be a von Neumann algebra, γ an action of G ω Regard θ as an action of Z on M . Assume the following:

1980

• • • •


θ commutes with γ ; × Z-action γ ◦ θ is strongly free and semi-liftable; The G Mω is globally invariant under γ ◦ θ ; γ τ ω ◦ θ = τ ω on Mω and τ ω ◦ Φπ = τ ω ⊗ trπ on Mω ⊗ B(Hπ ) for all π ∈ Irr(G).

Then for any n > 0 and any countably generated von Neumann subalgebra T ⊂ (M ω )γ θ , there γ exists a partition of unity {Ei }n−1 i=0 ⊂ T ∩ Mω such that θ (Ei ) = Ei+1 , where En = E0 . Proof. For m > 0, set δm = 2/nm and Km := {0, 1, 2, nm − 1}. Then Km is a ({1}, δm )γ invariant subset of Z. By Theorem 3.13, we have a partition of unity {Fim }i∈Km in Mω such 1 m−1 m m m 2 that nm−2 i=0 |θ (Fi ) − Fi+1 |ψ 5δm . For 0 i n − 1, set Ei := k=0 Fkn+i . Then for 0 i n − 2, we have m−1 1 m θ (Fkn+i ) − Fkn+i+1 5δm2 . θ E − E m i i+1 ψ ψ k=0

Applying the index selection trick to {Eim }∞ m=1 , 0 i n − 1, we get θ (Ei ) = Ei+1 for 0 i n − 2. Then θ (En−1 ) = E0 follows automatically. 2 Recall the following result [18, Lemma 4.3]. The statement is slightly strengthened here, but the same proof is applicable if we replace Mω with A ∩ Mω . Note that A ∩ Mω is of type II1 for any countably generated von Neumann subalgebra A ⊂ M ω when Mω is of type II1 . Theorem 3.16 (2-cohomology vanishing). Let M be a von Neumann algebra such that Mω is of type II1 . Let A ⊂ M ω be a countably generated von Neumann subalgebra. Let (γ , w) be a on M ω . Assume the following: cocycle action of G • A ∩ Mω is globally invariant under γ ; ⊗ L∞ (G); • w ∈ (A ∩ Mω ) ⊗ L∞ (G) and β ∈ Mor(M ω , M ω ⊗ • γ is of the form γ = Ad U ◦ β, where U ∈ U (M ω ⊗ L∞ (G)) is semi-liftable. L∞ (G)) Then the 2-cocycle w is a coboundary in A ∩ Mω . on Mω . Let S ⊂ (M ω )γ Corollary 3.17. Let γ be a strongly free and semi-liftable action of G be a countably generated von Neumann subalgebra. If Mω is of type II1 , then the von Neumann γ algebra S ∩ Mω is also of type II1 . Proof. Let I be a finite index set. Since S ∩ Mω is of type II1 , we can take a system of matrix units {ei,j }i,j ∈I in S ∩ Mω . Let Q be a finite dimensional subfactor generated by {ei,j }i,j ∈I . Let π ∈ Irr(G). Take an index i0 ∈ I . Since {γπ (ei,j )}i,j ∈I and {ei,j ⊗ 1π }i,j ∈I are systems of matrix units in (S ∩ Mω ) ⊗ B(Hπ ), ei0 ,i0 ⊗ 1π and γπ (ei0 ,i0 ) are equivalent. Hence there exists vπ ∈ (S ∩ Mω ) ⊗ B(Hπ ) such that ei0 ,i0 ⊗ 1π = vπ vπ∗ and γπ (ei0 ,i0 ) = vπ∗ vπ . Set the unitary v˜π =

(ei,i0 ⊗ 1π )vπ γπ (ei0 ,i ). i∈I


1981

we have Then v˜π is satisfying v˜π γπ (ei,j )v˜π∗ = ei,j ⊗ 1. Setting v˜ = (v˜π )π ∈ Mω ⊗ L∞ (G), vγ ˜ (x)v˜ ∗ = x ⊗ 1 for all x ∈ Q. Hence the map Ad v˜ ◦ γ is a cocycle action on Q ∩ (S ∩ Mω ). Using the previous 2-cohomology such vanishing result for Q ∩ (S ∩ Mω ), we obtain a unitary w ∈ (Q ∩ (S ∩ Mω )) ⊗ L∞ (G) that w v˜ is a γ -cocycle. Now we have w vγ ˜ (x)v˜ ∗ w ∗ = x ⊗ 1 for all x ∈ Q. Since γ has the joint Rohlin property, the action γ |Mω is stable by Corollary 3.14. Hence the Mω -valued γ -cocycle w v˜ is of the form w v˜ = (ν ∗ ⊗ 1)γ (ν) for some ν ∈ U (S ∩ Mω ). This γ implies that a subfactor νQν ∗ is fixed by γ . Hence S ∩ Mω contains a subfactor with arbitrary finite dimension, and it is of type II1 . 2 3.4. Approximately inner actions an amenable discrete Kac algebra and Γ a discrete Let M be a von Neumann algebra, G amenable group with the neutral element e. In this subsection, we study the following situation: • We are given two actions α ∈ Mor(M, M ⊗ L∞ (G)), θ : Γ → Aut(M) and unitaries ∞ (vg )g∈Γ ∈ U (M ⊗ L (G)) such that (θg ⊗ id) ◦ α ◦ θg−1 = Ad vg∗ ◦ α; • • • • •

Mω is of type II1 and Z(M) ⊂ M θ ; (vg )g∈Γ is a (θ ⊗ id)-cocycle; vg∗ is an α-cocycle for each g ∈ Γ ; α is approximately inner; απ θg is properly centrally non-trivial for each (π, g) ∈ Irr(G) × Γ \ (1, e).

Take Uπν ∈ U (M ⊗ B(Hπ )), ν ∈ N, such that Ad Uπν converges to απ for each π ∈ Irr(G). Set where Uπ := (Uπν )ν ∈ M ω ⊗ B(Hπ ). Then α = Ad U on M. Our U := (Uπ )π ∈ M ω ⊗ L∞ (G) first task is to replace U with a new one which well behaves to the action θ ω . and g ∈ Γ , the sequence (vg (θg ⊗ id)(Uπν ))ν approxiLemma 3.18. For each π ∈ Irr(G) mates απ . In particular, U ∗ vg (θgω ⊗ id)(U ) ∈ Mω ⊗ L∞ (G). Proof. Take φ ∈ M∗ . We verify that (φ ⊗ trπ ) ◦ Ad(θg ⊗ id)((Uπν )∗ )vg∗ converges to φ ◦ Φπα as follows: ∗ lim (φ ⊗ trπ ) ◦ Ad(θg ⊗ id) Uπν vg∗ ν→∞ ∗ = lim θg−1 ⊗ id (vg )Uπν (φ ◦ θg ⊗ trπ ) Uπν θg−1 ⊗ id vg∗ ◦ θg−1 ⊗ id ν→∞ = θg−1 ⊗ id (vg ) φ ◦ θg ◦ Φπα θg−1 ⊗ id vg∗ ◦ θg−1 ⊗ id

1982


Ad v ∗ ◦α = vg φ ◦ θg ◦ Φπα ◦ θg−1 ⊗ id vg∗ = vg φ ◦ Φπ g vg∗ = vg φ ◦ Φπα ◦ Ad vg vg∗ = φ ◦ Φπα . The latter statement follows from [18, Lemma 3.6].

2

such that vg (θgω ⊗ id)(U u) = U u. Lemma 3.19. There exists u ∈ U (Mω ⊗ L∞ (G)) Proof. Since the Γ -action θ ω is strongly free, it has the joint Rohlin property. Let S ⊂ M ω be a von Neumann subalgebra generated by all matrix entries of (θgω ⊗id)(U ) and vg for all g ∈ Γ . Let F ⊂ Γ be a finite subset and δ > 0. Since Γ is amenable, there exists a finite subset K ⊂ Γ such that g∈F |gKK| < δ|F ||K|. Fix a faithful state φ ∈ M∗ and set ψ := φ ◦ τ ω . Take a Rohlin / K and ∈Γ |θgω (E ) − Eg |ψ projection (Eg )g∈Γ ⊂ (S ∩ Mω ) such that Eg = 0 for g ∈ ω ∗ by u = 5δ 1/2 . Define u ∈ M ω ⊗ L∞ (G) k∈K U vk (θk ⊗ id)(U )(Ek ⊗ 1). By the previous Then it is easy to see that u is a unitary element, and for g ∈ F we lemma, u is in Mω ⊗ L∞ (G). have U ∗ vg θgω ⊗ id (U ) · θgω ⊗ id (u) ω θgω ⊗ id U ∗ (θg ⊗ id)(vk ) θgk = U ∗ vg θgω ⊗ id (U ) ⊗ id (U ) θgω (Ek ) ⊗ 1 =

k∈K

=

k∈K

ω U vgk θgk ⊗ id (U ) θgω (Ek ) ⊗ 1 ∗

ω ∗ ω U ∗ vgk θgk ⊗ id (U ) θgω (Ek ) − Egk ⊗ 1 + U v θ ⊗ id (U )(E ⊗ 1).

k∈K

∈gK

such that Take a partial isometry wg ∈ M ω ⊗ L∞ (G) ∗ ω U vg θ ⊗ id (U ) · θ ω ⊗ id (u) − u = w ∗ U ∗ vg θ ω ⊗ id (U ) θ ω ⊗ id (u) − u . g

g

g

g

g

∗ be a faithful state. Then we have Let χ ∈ L∞ (G) ∗ ω U vg θ ⊗ id (U ) · θ ω ⊗ id (u) − u g

=

k∈K

−

g

ψ⊗χ

ω (ψ ⊗ χ) wg∗ U ∗ vgk θgk ⊗ id (U ) θgω (Ek ) − Egk ⊗ 1

(ψ ⊗ χ) wg∗ U ∗ v θω ⊗ id (U )(E ⊗ 1) .

∈K\gK

Since Ek ∈ (Mω )ψ , we can use Lemma 3.10, and we have ω θ (Ek ) ⊗ 1 − Egk ⊗ 1 (3.8) g

k∈K

By the assumption of K, we have

ψ⊗χ

5δ 1/2 .

(3.8) (3.9)


(3.9) |E |ψ

∈K\gK

∈KgK

|E |ψ

= E − Ek = Egk − 1 ∈gK

k∈K

ψ

1983

k∈K

ψ

Egk − θ ω (Ek ) 5δ 1/2 . Egk − θgω (Ek ) = g ψ ψ

k∈K

k∈Γ

Hence we have ∗ ω U vg θ ⊗ id (U ) · θ ω ⊗ id (u) − u g

g

ψ⊗χ

10δ 1/2 .

(3.10)

satisfying (3.10) for δ = 1/ν. Take an increasing For each ν ∈ N, take uν ∈ Mω ⊗ L∞ (G) F = Γ . Applying the index selection trick to (uν )ν , we get sequence Fν Γ with ∞ ν ν=1 ∞ ∗ ω with U vg (θg ⊗ id)(U u) = u for all g ∈ Γ . 2 u ∈ Mω ⊗ L (G) Replacing U with U u, we may assume that U = (U ν )ν also satisfies vg θgω ⊗ id (U ) = U. As in [18], we consider two cocycle actions γ −1 = Ad U ∗ ◦ α ω and γ 0 = Ad U ∗ on Mω . Their 2-cocycles w −1 and w 0 are given by ∗ w −1 = U ∗ ⊗ 1 α ω U ∗ (id ⊗ )(U ), w 0 = U ∗ ⊗ 1 U13 id ⊗ opp (U ). and G opp , respectively. Here note that γ −1 and γ 0 are cocycle actions of G Lemma 3.20. In the above setting, γ −1 and γ 0 are cocycle actions on Mωθω . Proof. At first, we show that γ −1 and γ 0 commute with θω . Using vg (θgω ⊗ id)(U ) = U , we have (θgω ⊗ id) ◦ γ −1 = γ −1 ◦ θgω on M ω . In particular, γ −1 commutes with θω on Mω . Let x ∈ Mω . Since vg commutes with θgω (x) ⊗ 1, we have ω θg ⊗ id γ 0 (x) = U ∗ vg θgω (x) ⊗ 1 vg∗ U = U ∗ θgω (x) ⊗ 1 U = γ 0 θgω (x) . Hence γ 0 also commutes with θ ω . Secondary, we check that the 2-cocycles w −1 and w 0 are evaluated in Mωθω . Since vg∗ is an α-cocycle, we have ω θg ⊗ id ⊗ id w −1 = θgω ⊗ id U ∗ ⊗ 1 · θgω ⊗ id ⊗ id α ω U ∗ · θgω ⊗ (U ) = U ∗ ⊗ 1 (vg ⊗ 1) · vg∗ ⊗ 1 α ω (θg ⊗ id) U ∗ (vg ⊗ 1) · (id ⊗ ) vg∗ U = U ∗ ⊗ 1 α ω U ∗ vg (vg ⊗ 1)(id ⊗ ) vg∗ (id ⊗ )(U ) = w −1 ,

1984


and ω θg ⊗ id ⊗ id w 0 = θgω ⊗ id U ∗ ⊗ 1 · θgω ⊗ id U ∗ 13 · θgω ⊗ opp (U ) ∗ (vg )13 · id ⊗ opp vg∗ U = U ∗ ⊗ 1 (vg ⊗ 1) · U13 ∗ = U ∗ ⊗ 1 U13 α(vg )132 (vg )13 (id ⊗ ) vg∗ 132 id ⊗ opp (U ) = w0 .

2

×G opp on Mω by γ := (γ −1 ⊗ id) ◦ γ 0 . Its 2-cocycle w is Define the cocycle action γ of G given by ∗ ω ∗ ω ∗ ω w := U12 α U ∗ 123 α ω U12 α U 1245 (id ⊗ G× G opp ) α (U )U12 . By direct computation, we have 0 −1 ∗ −1 w = γ −1 w123 w132 1234 w124 (id ⊗ ⊗ id ⊗ id) γ −1 w 0 12435 . Hence w is evaluated in Mωθω , that is, γ is a cocycle action on Mωθω . ×G opp ) such that Then we apply Theorem 3.16 to γ and get c ∈ Mωθω ⊗ L∞ (G ∗ c123 γ (c)w(id ⊗ G× G opp ) c = 1. Here we note that the proof of [18, Lemma 4.3] works in our case by replacing Mω with Mωθω . Also note that Mωθω is of type II1 . Then the proof similar to that Set the unitaries c := c·⊗1 and cr := c1⊗· in Mωθω ⊗ L∞ (G). of [18, Lemma 4.6] shows that • c U ∗ is an α ω -cocycle; • U (cr )∗ is a unitary representation of G; • U (cr )∗ is fixed by the perturbed action Ad(c U ∗ ) ◦ α ω . Replacing U with U (cr )∗ , we obtain the following. and Lemma 3.21. Let α, θ and (vg )g∈Γ be as before. Then there exist U ∈ U (M ω ⊗ L∞ (G)) ∞ such that c ∈ U (Mω ⊗ L (G)) (1) (2) (3) (4) (5)

(Ad Uπν )ν approximates απ for all π ∈ Irr(G); that is, we have (id ⊗ )(U ) = U12 U13 ; U is a unitary representation of G, ∗ ω cU is an α -cocycle; U is fixed by the perturbed action Ad cU ∗ ◦ α ω ; vg (θgω ⊗ id)(U ) = U and (θgω ⊗ id)(c) = c for all g ∈ Γ . Now we set the following maps on M ω as before: γ 1 := Ad cU ∗ ◦ α ω ,

γ 2 := Ad U ∗ (· ⊗ 1),

and G opp , respectively. They preserve Mω and Mωθω . which are actions of G


1985

Lemma 3.22. In the above settings, one has the following: (1) vg∗ U is a unitary representation of G; γ2

(2) For all π ∈ Irr(G) and X ∈ M ω ⊗ B(Hπ ), Φπ (X) = (id ⊗ trπ )(U XU ∗ ). Proof. (1) Since vg∗ is an α-cocycle, we have ∗ ∗ vg U 12 vg U 13 = vg∗ 12 α vg∗ U12 U13 = (id ⊗ ) vg∗ U . opp . For X ∈ M ω ⊗ B(Hπ ), (2) Let Sπ,π be an isometric intertwiner from 1 into π ⊗ π for G we have ∗ 2 ∗ U 12 X13 (U )12 (1 ⊗ Sπ,π ) Φπγ (X) = 1 ⊗ Sπ,π ∗ ∗ id ⊗ opp (U )(1 ⊗ Sπ,π ) = 1 ⊗ Sπ,π id ⊗ opp U ∗ U13 X13 U13 ∗ ∗ U13 X13 U13 (1 ⊗ Sπ,π ) = 1 ⊗ Sπ,π ∗ 2 = (id ⊗ trπ ) U XU . Our next aim is to replace U with a new one such that we can retake c = 1. Lemma 3.23. There exists z ∈ U (Mωθω ) such that U cU ∗ = (z ⊗ 1)α ω (z∗ ). γ 1 ◦θ ω

Proof. By definition of γ 1 , we have Φ(π,g) = θgω−1 ◦ Φπα ◦ Ad U c∗ . Since θg ◦ τ ω = τ ω on Mω , ω

γ 1 ◦θ ω

we get τ ω ◦ Φ(π,g) = τ ω ⊗ trπ on Mω ⊗ B(Hπ ) for all (π, g) ∈ Irr(G) × Γ . By Lemma 3.20, × Γ -action. It is easy to see that γ 1 ◦ θ ω is strongly free. Since Ad(cν U ν∗ ) ◦ α γ 1 ◦ θ ω is a G converges to the trivial action, γ 1 is semi-liftable. Hence γ 1 ◦ θ ω has the joint Rohlin property. be an (F, δ)-invariant Now we have two γ 1 -cocycles U c∗ and U . Let K ∈ Projf(Z(L∞ (G))) projection with K e1 . By Theorem 3.13, we can take a Rohlin projection E ∈ Mωθω ⊗ L∞ (G)K for C = {U, U c∗ }. Set the Shapiro unitaries μδ := (id⊗ϕ)(U E) and ν δ := (id⊗ϕ)(U c∗ E). Then we claim the following: Claim 1. μδ ν δ∗ = (id ⊗ ϕ) U EcU ∗ ,

μδ ν δ∗ ∈ Mωθω .

Indeed, the first equality is shown by using (R3). Next we show that μδ ν δ∗ ∈ Mωθω . By Lemma 3.22, we have d(π)2 (id ⊗ trπ ) U EcU ∗ μδ ν δ∗ = (id ⊗ ϕ) U EcU ∗ = =

π∈Irr(G) γ2

d(π)2 Φπ (Ec).

π∈Irr(G)

Since Ec ∈ (Mω )θω ⊗ B(Hπ ), μδ ν δ∗ is in Mω . Using the commutativity of γ 2 |Mω and θω , we have

1986


2 2 θgω μδ ν δ∗ = d(π)2 θgω Φπγ (Ec) = d(π)2 Φπγ θgω ⊗ id (Ec) π∈Irr(G)

=

π∈Irr(G) γ2

d(π)2 Φπ (Ec) = μδ ν δ∗ .

π∈Irr(G)

Next we claim the following: Claim 2. 1 δ U γ μ − μ δ ⊗ F F

ψ⊗ϕ

∗ 1 δ U c γ ν − ν δ ⊗ F F

5δ 1/2 ;

ψ⊗ϕ

5δ 1/2 .

(3.11) (3.12)

Let U γF1 (μδ ) − μδ ⊗ F = v|U γF1 (μδ ) − μδ ⊗ F | be the polar decomposition. Then we have 1 δ U γ μ − μ δ ⊗ F F = v ∗ U γF1 μδ − μδ ⊗ F = v ∗ (id ⊗ id ⊗ ϕ) U12 U13 γ 1 (E) − v ∗ (id ⊗ id ⊗ ϕ) U12 U13 (id ⊗ F )(E) = v ∗ (id ⊗ id ⊗ ϕ) U12 U13 γ 1 (E) − (id ⊗ F )(E) . Using Lemma 3.10, we have 1 δ U γ μ − μ δ ⊗ F F

ψ⊗ϕ

∗ = (id ⊗ id ⊗ ϕ) v12 U12 U13 γ 1 (E) − (id ⊗ F )(E) ψ⊗ϕ γ 1 (E) − (id ⊗ F )(E) ψ⊗ϕ⊗ϕ 5δ 1/2 .

Similarly we can prove (3.12). Now we use the index selection trick. For decreasing δn = 1/n → 0 and increasing finite as n → ∞, we take μ(n) := μ1/n and ν(n) := ν 1/n rank central projections Fn → 1 in L∞ (G) ω in U (M ) for n ∈ N. Set μ˜ = (μ(n))n and ν˜ = (ν(n))n . From them, we construct μ and ν in U (M ω ) by index selection. Since μ(n)ν(n)∗ ∈ Mωθω , μν ∗ ∈ Mωθω . By definition of an index selection map (i.e. it commutes with γ 1 ), we have U γ 1 (μ) = μ ⊗ 1 and U c∗ γ 1 (ν) = ν ⊗ 1. These imply α ω νμ∗ = U c∗ γ 1 νμ∗ cU ∗ = νμ∗ ⊗ 1 U cU ∗ . Therefore, z := μν ∗ is a desired solution.

2

By the previous lemma, we get z ∈ U (Mωθω ) such that U cU ∗ = (z ⊗ 1)α ω (z∗ ). Then we in M ω . By the previous lemma, consider V = (z∗ ⊗ 1)U (z ⊗ 1), which is a representation of G we have V ∗ = z∗ ⊗ 1 cU ∗ · U c∗ U ∗ (z ⊗ 1) = z∗ ⊗ 1 cU ∗ α ω (z).


1987

Since cU ∗ is an α ω -cocycle, so is V ∗ . Moreover we have vg (θg ⊗ id)(V ) = vg z∗ ⊗ 1 vg∗ U (z ⊗ 1) = z∗ ⊗ 1 U (z ⊗ 1) = V . Finally we again replace U with V = (z∗ ⊗ 1)U (z ⊗ 1), and we get the following. Theorem 3.24. Let M be a von Neumann algebra. Assume the following: θ : Γ → Aut(M) and unitaries • We are given two actions α ∈ Mor(M, M ⊗ L∞ (G)), ∞ (vg )g∈Γ ∈ U (M ⊗ L (G)) such that (θg ⊗ id) ◦ α ◦ θg−1 = Ad vg∗ ◦ α; • • • • •

Mω is of type II1 and Z(M) ⊂ M θ ; (vg )g∈Γ is a (θ ⊗ id)-cocycle; vg∗ is an α-cocycle for each g ∈ Γ ; α is approximately inner; απ θg is properly centrally non-trivial for each (π, g) ∈ Irr(G) × Γ \ (1, e).

such that Then there exists U = (U ν )ν ∈ U (M ω ⊗ L∞ (G)) (1) (2) (3) (4)

(Ad Uπν )ν converges to απ for all π ∈ Irr(G); that is, (id ⊗ )(U ) = U12 U13 ; U is a representation of G, ∗ α ω (U ∗ ) = (id ⊗ )(U ∗ ); ∗ ω U is an α -cocycle, that is, U12 ω vg (θg ⊗ id)(U ) = U for all g ∈ Γ .

be an α ω -cocycle. Take U ∈ U (M ω ⊗ L∞ (G)) as in Corollary 3.25. Let w ∈ M ω ⊗ L∞ (G) then there exists z ∈ U (Mωθω ) such that the previous theorem. If U ∗ wU is in Mωθω ⊗ L∞ (G), w = (z ⊗ 1)α ω (z∗ ). Proof. The proof is similar to that of Lemma 3.23. Let γ 1 = Ad U ∗ ◦ α ω , γ 2 = Ad U ∗ (· ⊗ 1) and γ = (γ 1 ⊗ id) ◦ γ 2 as before. be an (F, δ)-invariant Now we have two γ 1 -cocycles U and wU . Let K ∈ Projf(L∞ (G)) as central projection. By Theorem 3.13, we can take a Rohlin projection E ∈ Mωθω ⊗ L∞ (G)K δ in Definition 3.7 for C = {U, wU }. Set the Shapiro unitaries μ := (id ⊗ ϕ)(U E) and ν δ := (id ⊗ ϕ)(wU E). Then we have μδ ν δ∗ = (id ⊗ ϕ) U EU ∗ w ∗ ,

μδ ν δ∗ ∈ Mωθω .

Next we show that μδ ν δ∗ ∈ Mωθω . By Lemma 3.22, we have μδ ν δ∗ = (id ⊗ ϕ) U EU ∗ w ∗ = d(π)2 (id ⊗ trπ ) U EU ∗ w ∗ =

π∈Irr(G)

π∈Irr(G)

d(π)2 Φπ EU ∗ w ∗ U . γ2

1988


Since EU ∗ w ∗ U ∈ (Mω )θω ⊗ B(Hπ ) by our assumption on w, μδ ν δ∗ is in Mω . Using the commutativity of γ 2 |Mω and θω , we have θgω (μδ ν δ∗ ) = μδ ν δ∗ . Now we get μ and ν in U (M ω ) by the index selection as before. Then μν ∗ ∈ Mωθω . By definition of an index selection map (i.e. it commutes with γ 1 ), we have U γ 1 (μ) = μ ⊗ 1 and wU γ 1 (ν) = ν ⊗ 1. These imply wα ω (νμ∗ ) = wU γ 1 (νμ∗ )U ∗ = (νμ∗ ⊗ 1). Therefore, z := νμ∗ is a desired solution. 2 The previous result yields the following, which can be also proved by using [17, Theorem 7.2]. Corollary 3.26. Let M be an injective factor and α an approximately inner and centrally free on M. Let ϕ ∈ W (M) and T > 0. Then there exists a sequence {wn }n ⊂ cocycle action of G U (M) such that ϕ

• σT = limn→∞ Ad wn in Aut(M); • [Dϕ ◦ Φπα : Dϕ ⊗ trπ ]T = limn→∞ απ (wn )(wn∗ ⊗ 1) for all π ∈ Irr(G), where the latter limit is taken in the strong* topology. Proof. By [18, Theorem 6.2] and Lemma 3.2, we can perturb α to be an action. By the chain rule of Connes’ cocycles, we may and do assume that α is an action. Applying the previous such that Ad U ν theorem to α and Γ = {e}, we can take a unitary U = (U ν )ν in M ω ⊗ L∞ (G) ∗ ω approximates α and U is an α -cocycle. ϕ Take a sequence of unitaries {v ν }ν ⊂ M such that σT = limν→∞ Ad v ν . This is possible beϕ cause σT is approximately inner [4,9,14]. We set v := (v ν )ν ∈ M ω . For π ∈ Irr(G), we set a unitary wπν := ((v ν )∗ ⊗ 1)[Dϕ ◦ Φπα : Dϕ ⊗ trπ ]∗T απ (v ν ) in M ⊗ and w = (w ν )ν ∈ M ω ⊗ L∞ (G). Then by B(Hπ ), and also set w ν := (wπν )π ∈ M ⊗ L∞ (G) Lemma A.12, we see that w is an α ω -cocycle. We will check that U ∗ wU ∈ Mω ⊗ L∞ (G). Take any π ∈ Irr(G) and ψ ∈ M∗ . Recall the notation φ ν ∼ ψ ν for sequences (φ ν )ν , (ψ ν )ν ⊂ ϕ◦Φ α ϕ (M ⊗ B(Hπ ))∗ with limν→ω φ ν − ψ ν = 0. Using Φπα ◦ σT π = σT ◦ Φπα (see [19, §3.2]), we have ν ∗ ν ν ∗ ∗ Uπ wπ Uπ · (ψ ⊗ trπ ) · Uπν wπν Uπν ∗ ∗ ∼ Uπν wπν · ψ ◦ Φπα · wπν Uπν ∗ ∗ ν ∗ v = Uν ⊗ 1 Dϕ ◦ Φπα : Dϕ ⊗ trπ T · απ v ν · ψ ◦ Φπα ∗ Dϕ ◦ Φπα : Dϕ ⊗ trπ T v ν ⊗ 1 U ν · απ v ν ∗ ∗ ν ∗ ∗ v ◦ Φπα = Uν ⊗ 1 Dϕ ◦ Φπα : Dϕ ⊗ trπ T · v ν · ψ · v ν · Dϕ ◦ Φπα : Dϕ ⊗ trπ T v ν ⊗ 1 U ν ∗ ∗ ν ∗ ϕ v ∼ Uν ⊗ 1 Dϕ ◦ Φπα : Dϕ ⊗ trπ T · ψ ◦ σ−T ◦ Φπα · Dϕ ◦ Φπα : Dϕ ⊗ trπ T v ν ⊗ 1 U ν ∗ ∗ ϕ ∼ U ν · Dϕ ◦ Φπα : Dϕ ⊗ trπ T · ψ ◦ σ−T ◦ Φπα · Dϕ ◦ Φπα : Dϕ ⊗ trπ T ϕ⊗tr ◦ σT π · U ν


1989

∗ ϕ◦Φ α ϕ = U ν · ψ ◦ σ−T ◦ Φπα ◦ σT π · U ν ∗ = U ν · ψ ◦ Φπα · U ν ∼ ψ ⊗ trπ . Using Corollary 3.25, we can By [18, Lemma 3.6], we see that U ∗ wU is in Mω ⊗ L∞ (G). ω ∗ take a unitary z ∈ Mω such that w = (z ⊗ 1)α (z ), that is, [Dϕ ◦ Φπα : Dϕ ⊗ trπ ]∗T = (vz ⊗ 1)απω (z∗ v ∗ ). Then a representing sequence of vz satisfies the desired properties. 2 4. Classification for type IIIλ case 4.1. Canonical extension to discrete cores and the main result As explained in Introduction, our idea in type IIIλ case is that we reduce the classification problem to type II∞ case by using the discrete decomposition. For this purpose, we have to consider the canonical extension of endomorphisms of a type IIIλ factor to its discrete core. This is possible for endomorphisms with trivial Connes–Takesaki modules as follows [12, Proposition 4.5]. Readers are referred to Appendix A for relations between the results of [12] and [19]. Let R be a type IIIλ factor, 0 < λ < 1, and φ a generalized trace, that is, φ(1) = ∞ and φ σT = id, T = −2π/ log λ, hold. Then R σ φ T is called the discrete core. We denote by λφ (t) φ the unitary implementing σt for t ∈ T. Definition 4.1. Let R be a type IIIλ factor and K a finite dimensional Hilbert space. For β ∈ Mor0 (R, R ⊗ B(K)) with the standard left inverse Φ and mod(β) = id, we define the canonical ∈ Mor(R σ φ T, (R σ φ T) ⊗ B(K)) by extension β (1) (2)

(x) = β(x) for all x ∈ R; β β (λφ (t)) = [Dφ ◦ Φ : Dφ ⊗ trK ]t (λφ (t) ⊗ 1) for all t ∈ R/T Z.

we can prove that For a cocycle action α ∈ Mor(R, R ⊗ L∞ (G)), α := ( απ )π is a cocycle action in a similar way as in the proof of Theorem A.13. is an approximately inner and centrally free cocycle Lemma 4.2. If β ∈ Mor(Rλ , Rλ ⊗ L∞ (G)) is also approximately inner and centrally free. action, then β π ) = id for each π ∈ Irr(G). Let φˆ be the dual weight on M. Then Proof. We check mod(β φˆ σt = Ad λφ (t) for t ∈ T. Take a positive operator h such that λφ (t) = h−it for t ∈ T. Then φˆ h is a β

trace on M := Rλ σ φ T. Note that Φπ commutes with the dual action θ . Let Tθ : M → Rλ be the β β operator valued weight obtained by averaging the Z-action θ . Using φˆ ◦ Φπ = φ ◦ Φπ ◦ (Tθ ⊗ id), we can compute as follows: D φˆ h ◦ Φπβ : D φˆ h ⊗ trπ t = D φˆ h ◦ Φπβ : D φˆ ◦ Φπβ t D φˆ ◦ Φπβ : D φˆ ⊗ trπ t · [D φˆ ⊗ trπ : D φˆ h ⊗ trπ ]t

1990


π hit Dφ ◦ Φπβ ◦ (Tθ ⊗ id) : Dφ ◦ Tθ ⊗ trπ h−it ⊗ 1 =β t π λφ (t)∗ Dφ ◦ Φπβ : Dφ ⊗ trπ λφ (t) ⊗ 1 = 1. =β t is approximately inner. By Corollary A.7, β . If β π is not properly outer for some π = 1, then β π is Next we check the freeness of β actually implemented by a unitary. This fact is proved as in the proof of [12, Proposition 3.4] because of the irreducibility of βπ [18, Lemma 2.8]. Also note Lemma A.4. Using (Rλ )ω ⊂ Mω (see the proof of [19, Lemma 4.11]), we see that βπ is centrally trivial, and this is a contradiction. is cocycle is a centrally free action. The second canonical extension β We show that β is centrally free on M Z, and (β )ω is nonconjugate to β by Lemma 3.3. Hence β θ π trivial on (M θ Z)ω for any π = 1. Since (M θ Z)ω is naturally isomorphic to Mωθω and )ω | = (β )ω |Mω , (β π )ω is non-trivial on Mωθω for any π = 1. In particular, β is a centrally (β Mω free action because β is free. 2 is unique up to cocycle conjugacy, we need to consider the Z-action θ to Though the action β × Z-action β θ on R0,1 . The obtain the uniqueness of the original β. Our aim is to classify the G following is our main theorem in this section. on M, and β an Theorem 4.3. Let M ∼ = R0,1 with a trace τ , θ ∈ Aut(M), α be an action of G on R0 . Assume the following: action of G • • • •

θ satisfies τ ◦ θ = λτ , 0 < λ < 1; α is approximately inner and centrally free; α and θ commute; β is free.

Then αθ is cocycle conjugate to θ ⊗ β. Once proving the above theorem, we can show Theorem 2.4 for Rλ as follows. Proof of Theorem 2.4 for Rλ , 0 < λ < 1. Let ϕ be a generalized trace on Rλ , and M := Rλ σ ϕ T. Then M is isomorphic to R0,1 . Let θ be a dual action of Z on M, and α the canonical extension of α. Then α is approximately inner and centrally free by Lemma 4.2. Applying the previous theorem to α θ , we get α θ ∼ θ ⊗ β. By Lemma 3.4, the second extension α on M θ Z is cocycle conjugate to id ⊗ β on (M θ Z) ⊗ R0 . By Lemma 3.3, α is cocycle conjugate to α. Hence α is cocycle conjugate to idRλ ⊗ β. 2 4.2. Model action splitting The rest of this section is devoted to prove Theorem 4.3. Let M, τ , α and θ be as in that theorem. We also take a faithful normal state φ on M. We fix these notations from here. Since θ × Z-action αθ is not approximately inner. scales the trace, the G × Z-action αθ is centrally free. Lemma 4.4. The G


1991

Proof. Since τ ◦ θ = λτ with λ = 1, θ is centrally free. Assume that απ θ n is centrally trivial for some π ∈ Irr(G) and n ∈ Z. Then the map απ θ n is implemented by a unitary by Corollary A.7, but we have mod(απ θ n ) = mod(θ n ) because mod(απ ) = id. Hence n = 0, and π = 1 by central freeness of α. 2 as in Theorem 3.24 with Γ = Z and vg = 1. Define the G ×G opp Take U ∈ U (M ω ⊗ L∞ (G)) action γ = (γ 1 ⊗ id) ◦ γ 2 as before, where γ 1 (x) = U ∗ α ω (x)U,

γ 2 (x) = U ∗ (x ⊗ 1)U

for x ∈ M ω .

×G opp × Z-action Since U is fixed by θ ω , γ commutes with θ ω on M ω . Hence γ θ ω is a G ω on M . Applying Corollary 3.15 to the strongly free and semi-liftable action γ 1 ◦ θ and the set 1 T = {Uπi,j } i,j,π , we have the following result. Note that T ∩ (M ω )γ = (M ω )γ . γ

Lemma 4.5. For any n ∈ N, there exists a partition of unity {Ei }n−1 i=0 in Mω such that θω (Ei ) = Ei+1 for 0 i n − 1 (En := E0 ). As in [2], we obtain the following stability result by using the above lemma. γ

γ

Lemma 4.6. The Z-action θω on Mω is stable, that is, for any u ∈ U (Mω ), there exists w ∈ γ U (Mω ) such that u = wθω (w ∗ ). γ

Lemma 4.7. For any n ∈ N, there exists a system of matrix units {fij }n−1 i,j =0 ⊂ Mω with θω (fij ) = μi−j fij , where μ = e2π

√

−1/n .

γ θω

Proof. By Corollary 3.17 for γ 1 θ ω , we see that (T ∩ Mω )γ θ = Mω is of type II1 . Hence we n−1 i γ θω can take a system of matrix units {eij }n−1 i=0 μ eii , and by Lemma 4.6, i,j =0 ⊂ Mω . Set u := γ ∗ ∗ we have w ∈ U (Mω ) such that u = wθω (w ). Set fij := w eij w ∈ M γ . Then we have 1 ω

θω (fij ) = θω w ∗ eij θω (w) = w ∗ ueij u∗ w = μi−j fij .

2

Recall the following result [20, Proposition 7.1]. Lemma 4.8. Let e, f be projections in M ω with v ∗ v = e, vv ∗ = f for an element v ∈ M ω . Let e = (e(ν))ν and f = (f (ν))ν be representing sequences such that e(ν) and f (ν) are equivalent for each ν ∈ N. Then we can choose a representing sequence of v, v = (v(ν))ν , such that v ∗ (ν)v(ν) = e(ν) and v(ν)v(ν)∗ = f (ν) for each ν ∈ N. √

Lemma 4.9. Let n ∈ N and μ = e2π −1/n . Then for any F Irr(G), Ψ (M∗ )+ , and > 0, a unitary w ∈ M and a system of matrix units {fij }n−1 there exist a unitary u ∈ M ⊗ L∞ (G), i,j =0 in M such that (i) uπ − 1#φ⊗trπ < for all π ∈ F ; (ii) w − 1#φ < ; (iii) [fij , ψ] < for all ψ ∈ Ψ and 0 i, j n − 1;

1992


(iv) Ad u ◦ α(fij ) = fij ⊗ 1 for all 0 i, j n − 1; (v) Ad w ◦ θ (fij ) = μi−j fij for all 0 i, j n − 1. γ

Proof. Let {fij }n−1 i,j =0 be a system of matrix units in Mω as in Lemma 4.7. Then γ (fij ) = fij ⊗ 1

implies α ω (fij ) = fij ⊗ 1. Take a representing sequence of fij , (fij (ν))ν such that {fij (ν)}n−1 i,j =0 is a system of matrix units in M for all ν. By Lemma 4.8, for each π ∈ Irr(G), there exists vπ (ν) ∈ M ⊗B(Hπ ) such that vπ (ν)vπ (ν)∗ = f00 (ν) ⊗ 1, vπ (ν)∗ vπ (ν) = απ (f00 (ν)) and (vπ (ν))ν = f00 ⊗ 1. Set a unitary uπ (ν) := n−1 i=0 (fi0 (ν) ⊗ 1)vπ (ν)απ (f0i (ν)). Then Ad uπ (ν) ◦ απ (fij (ν)) = fij (ν) ⊗ 1 holds. We have (uπ (ν))ν = 1 in M ω ⊗ B(Hπ ). Indeed, n−1 n−1 uπ (ν) ν = fi0 (ν) ⊗ 1 ν vπ (ν) ν απω f0i (ν) ν = (fi0 ⊗ 1)(f00 ⊗ 1)(f0i ⊗ 1) = 1. i=0

i=0

Set a unitary u(ν) = (uπ (ν))π in M ⊗ L∞ (G). Next we construct w. Applying Lemma 4.8 to θω (f00 ) = f00 , we obtain v(ν) ∈ M such that v(ν)v(ν)∗ = f00 , v(ν)∗ v(ν) = θ (f00 (ν)) and (v(ν))ν = f00 . Set a unitary w(ν) := n−1 i i−j f (ν) holds for all 0 i, j ij i=0 μ fi0 (ν)v(ν)θ (f0i (ν)). Then Ad w(ν) ◦ θ (fij (ν)) = μ n − 1 and ν ∈ N. We can show w(ν) → 1 strongly* as ν → ω as above. Hence we can choose ν ∈ N such that u = u(ν), w = w(ν) and fij (ν) satisfy the desired conditions. 2 Let Ψn M∗ be an increasing subset such that Ψ = following result due to Connes [2, Lemma 2.3.6].

∞

n=1 Ψn

is total in M∗ . We recall the

. . . , Mn ⊂ M be mutually commuting finite dimensional subfactors. Lemma 4.10. Let M1 , M2 , ∞ M := N . If Denote ∞ k k=1 k=1 ψ ◦ EMk ∩M − ψ < ∞ for all ψ ∈ Ψ , then N is a hyperfinite subfactor of type II1 and we have M = N ∨ N ∩ M ∼ = N ⊗ N ∩ M. Here EMk ∩M = trMk ⊗ idMk ∩M . Let {nk }∞ k=1 ⊂ N be a sequence such that any n ∈ N appears infinitely many times. Set μk := √ nk −1 j 2π −1/n k . For a system of n × n -matrix units {e }nk −1 , set u e k k ij i,j =0 nk := j =0 μk ejj , and σ := ∞ ∞ ∼ k=1 Ad unk . Then σ is an aperiodic automorphism on k=1 Mnk (C) = R0 . We will prove the following model action splitting result. Lemma 4.11. The action αθ is cocycle conjugate to the action σ ⊗ αθ . Proof. Step 1. Let {k }∞ with ∞ k=1 k < ∞. k=1 be a decreasing sequence of positive numbers

∞ be a family of increasing finite subsets of Irr(G) such that F = Irr(G). Let {Fm }∞ m=1 m m=1 Recall that we have fixed a faithful normal state φ ∈ M∗ . We will construct the following families: n −1

(1) Matrix units, {fijk }i,jk =0 ⊂ M for k ∈ N, such that they are mutually commuting for k and satisfy [ψ, fijk ] k /nk for all 0 i, j nk , ψ ∈ Ψk and k ∈ N.

m nk −1 We set Mnk := ({fijk }i,j k=1 Mnk ; =0 ) and Em :=


1993

and w m ∈ E (2) Unitaries um ∈ (Em−1 ∩ M) ⊗ L∞ (G) m−1 ∩ M satisfying the following for each m ∈ N: # # m • um π − 1φ⊗trπ < m and w − 1φ < m for all π ∈ Fm ; • We set u¯ m := um um−1 · · · u1 and w¯ m := w m w m−1 · · · w 1 . Then we have, for all 0 i, j nk − 1 and 1 k m, Ad u¯ m ◦ α(fijk ) = fijk ⊗ 1; i−j

Ad w¯ m ◦ θ (fijk ) = μk fijk . Assume we have constructed up to k = m. Set α m := Ad u¯ m ◦ α, and θ(m) := Ad w¯ m ◦ θ . Since ∩ M. fixes Em , α m is a cocycle action on Em dim(E ) i ∗ Let {eˆ } a basis for Em . Let us decompose ψ ∈ Ψm+1 as ψ = i=1 m eî ⊗ ψi , ψi ∈ ∩ M) , and denote by Ψ ˆ m+1 the set of all such ψi . Fix δm+1 > 0 so that δm+1 (Em ∗ −1 m+1 (nm+1 dim Em ) . αm

Claim. There exist the following elements: n

−1

(1) A system of matrix units {fijm+1 }i,jm+1 =0 ψ ∈ Ψˆ m+1 .

∩ M such that [ψ, f m+1 ] δ ⊂ Em m+1 for ij

−1

Set Mm+1 := ({fijm+1 }i,jm+1 =0 ) and Em+1 = Em ∨ Mm+1 . ∩ M) ⊗ L∞ (G) ∩ M satisfying the following: and w m+1 ∈ Em (2) Unitaries um+1 ∈ (Em # # m+1 m+1 • uπ − 1φ⊗trπ < m+1 and w − 1φ < m+1 for all π ∈ Fm+1 ; n

• Ad um+1 ◦ α m (fijm+1 ) = fijm+1 for all 0 i, j nm+1 − 1; (i−j )

Ad w m+1 ◦ θ(m) (fijm+1 ) = μm+1 fijm+1 for all 0 i, j nm+1 − 1. ∩ M)ω = E ∩ M ω , Indeed, we can prove this as follows. Via the natural isomorphism (Em m we have ω Em ∩ M ω ⊂ Em = Mω ⊂ E m (4.1) ∩M ∩ Mω . ∩ M, we have a G-cocycle On Em action α m and a Z-action θ(m) . Using Lemma 4.7, we take a nm+1 −1 γ i−j system of matrix units {fij }i,j =0 ⊂ Mω such that θω (fij ) = μm+1 fij for 0 i, j nm+1 − 1.

ω (f ) = w Then we get θ(m) ¯ m θω (fij )(w¯ m )∗ = μm+1 fij . Since fij is fixed by γ , α ω (fij ) = fij ⊗ 1 ij as before. Hence we have (α m )ω (fij ) = u¯ m (fij ⊗ 1)(u¯ m )∗ = fij ⊗ 1. By using (4.1), we can n −1 n −1 ∩ M. Then we can take desired as sequences {(fij (ν))ν }i,jm+1 in Em represent {fij }i,jm+1 =0 =0 elements in the Claim as in Lemma 4.9. i−j

Now the condition (1) in the Claim implies [ψ, fijm+1 ] m /nm for ψ ∈ Ψm+1 . Thus we complete induction, and have constructed families um , w m and E m for m ∈ N. Since for ψ ∈ Ψk we have n −1 n −1 k k 1 1 ψ ◦ EMn ∩M − ψ = fijk ψfjki − ψ = fijk ψ, fjki k nk nk ij =0

1 nk

n k −1 ij =0

ij =0

ψ, f k 1 · n2 · k = k , ji nk k nk

1994


we can check ∞ k=1 ψ ◦ EMn k ∩M − ψ < ∞ for ψ ∈ Ψ . Then Lemma 4.10 implies that E :=

∞ ∼ k=1 Ek is isomorphic to R0 and yields a tensor product splitting M = E ∨ (E ∩ M) = E ⊗ (E ∩ M). Step 2. From the condition (2), the strong* limits u¯ ∞ = limm→∞ u¯ m and w¯ ∞ = limm→∞ w¯ m exist, and together with (1), we have Ad u¯ ∞ ◦ α(x) = x ⊗ 1 and Ad w¯ ∞ ◦ θ (x) = σ (x) for x ∈ E. Extend w¯ ∞ to a θ -cocycle naturally and denote it also by w¯ ∞ ∈ M ⊗ ∞ (Z). Then we get the × Z-action αθ to the G × Z-cocycle action (Ad u¯ ∞ α(w¯ ∞ ) ◦ αθ, v). Set perturbation from the G ∞ ∞ β := Ad u¯ α(w¯ ) ◦ αθ . Then β is of the form σ ⊗ β on E ⊗ (E ∩ M), where β = β|E ∩M . We claim that v is evaluated in E ∩ M, and (β , v) is a cocycle action. ) ◦ β. Let k ∈ N and 0 i, j nk − 1. By definition of v, (β ⊗ id) ◦ β = Ad v ◦ (id ⊗ G×Z Then we have the following: m(i−j ) (+m)(i−j ) k (β(π,) ⊗ id) β(ρ,m) fijk = μk fij ⊗ 1π ⊗ 1ρ β(π,) fijk ⊗ 1ρ = μk and (id ⊗ G×Z ) β fijk (π,),(ρ,m) = (id ⊗ ) β fijk (·,+m) π,ρ (+m)(i−j ) = μk (id ⊗ ) fijk ⊗ 1 π,ρ (+m)(i−j ) k fij ⊗ 1π ⊗ 1ρ . = μk Hence v is evaluated in Mk ∩ M for any k ∈ N, and hence in E ∩ M. We have shown that αθ is cocycle conjugate to the cocycle action σ ⊗ β . Since E ∩ M is of × Z-action β by Lemma 3.2. Hence αθ ∼ σ ⊗ β . Since type III, we can perturb (β , v) to a G σ ⊗ σ ≈ σ , we get αθ ∼ σ ⊗ σ ⊗ β ∼ σ ⊗ αθ . 2 Remark 4.12. We can use the Jones–Ocneanu cocycle argument in [20, Lemma 2.4] to obtain cocycle conjugacy αθ ∼ σ ⊗ αθ in Step 2 above. We set ν := u¯ ∞ α(w¯ ∞ ). Then we have Ad ν ◦ αθ = β = σ ⊗ β . Since σ is conjugate to σ ⊗ σ , there exists an isomorphism γ from E ⊗ E onto E with γ −1 ◦ σ ◦ γ = σ ⊗ σ . So we have −1 ◦ Ad ν ◦ αθ ◦ (γ ⊗ id) = γ −1 ◦ σ ◦ γ ⊗ β = σ ⊗ σ ⊗ β γ ⊗ id ⊗ idL∞ (G×Z) = σ ⊗ Ad ν ◦ αθ. Then the following holds: Ad(γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ν ◦ αθ = (γ ⊗ id ⊗ id) ◦ (σ ⊗ αθ ) ◦ γ −1 ⊗ id . We will verify that (γ ⊗ id ⊗ id)(1 ⊗ ν ∗ )ν is an αθ -cocycle. Here note that (γ ⊗ id ⊗ id)(1 ⊗ v) = v holds because the 2-cocycle v is evaluated in E ∩ M. Then the following holds: (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ν ⊗ 1 · αθ (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ν = (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ⊗ 1 · σ ⊗ β (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ (ν ⊗ 1)αθ (ν) = (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ⊗ 1 σ ⊗ σ ⊗ β 1 ⊗ ν ∗ v(id ⊗ )(ν)


1995

= (γ ⊗ id ⊗ id) 1 ⊗ αθ ν ∗ ν ∗ v(id ⊗ )(ν) = (γ ⊗ id ⊗ id) 1 ⊗ (id ⊗ ) ν ∗ v ∗ v(id ⊗ )(w) = (id ⊗ ) (γ ⊗ id ⊗ id) 1 ⊗ ν ∗ ν . Hence αθ and αθ ⊗ σ are cocycle conjugate. Proof of Theorem 4.3. Note that θ ⊗ θ −1 is cocycle conjugate to idB(2 ) ⊗ σ by Connes [2]. Then the following holds: αθ ∼ idB(2 ) ⊗ αθ

(by Lemma 3.1)

∼ idB(2 ) ⊗ σ ⊗ αθ ∼θ ⊗θ

−1

(by Lemma 4.11)

⊗ αθ.

Since the action θ −1 ⊗ αθ preserves the trace of R0,1 , it is approximately inner. The central freeness is clear. Then θ −1 ⊗ αθ is cocycle conjugate to idB(2 ) ⊗ σ ⊗ β by Theorem 2.4 for type II∞ case, and the following holds: θ ⊗ θ −1 ⊗ αθ ∼ θ ⊗ idB(2 ) ⊗ σ ⊗ β ∼θ ⊗σ ⊗β ∼ θ ⊗ β. Therefore we get αθ ∼ θ ⊗ β.

2

We close this section with the following lemma which is used in Section 6. Lemma 4.13. Let N be a type IIIλ factor with 0 < λ < 1 and α an approximately inner action on N . Let ψ be a generalized trace on N . Then there exists a G-action of G β on N such that • β ∼ α; β • ψ ◦ Φπ = ψ ⊗ trπ for all π ∈ Irr(G). Proof. Since α is approximately inner, we see that ψ ◦ Φπα is a generalized trace for all π ∈ Irr(G). Hence there exists a unitary vπ ∈ N ⊗ B(Hπ ) such that ψ ◦ Φπα = (ψ ⊗ trπ ) ◦ Ad vπ . and consider the cocycle action δ := Ad v ◦ α, whose 2-cocycle is Set v = (vπ )π ∈ N ⊗ L∞ (G), given by u := (v ⊗ 1)α(v)(id ⊗ )(v ∗ ). Then we have ψ ◦ Φπδ = ψ ⊗ trπ , and σ ψ and δ commute in particular. We check that u is evaluated in Nψ as follows: for π, ρ ∈ Irr(G), u∗ (ψ ⊗ trπ ⊗ trρ )u = u∗ · ψ ◦ Φρδ ◦ Φπδ ⊗ id · u = (id ⊗ )(v)απ vρ∗ vπ∗ ⊗ 1 · ψ ◦ Φρδ ◦ Φπδ ⊗ id · (vπ ⊗ 1)απ (vρ )(id ⊗ ) v ∗ = (id ⊗ )(v)απ vρ∗ · ψ ◦ Φρδ ◦ Φπα ⊗ id · απ (vρ )(id ⊗ ) v ∗

1996


= (id ⊗ )(v) · vρ∗ · ψ ◦ Φρδ · vρ ◦ Φπα ⊗ id · (id ⊗ ) v ∗ = (id ⊗ )(v) · ψ ◦ Φρα ◦ Φπα ⊗ id · (id ⊗ ) v ∗ =

σ ≺π⊗ρ S∈ONB(σ,π⊗ρ)

=


=


=


d(σ ) (id ⊗ )(v)(1 ⊗ S) · ψ ◦ Φσα · 1 ⊗ S ∗ (id ⊗ ) v ∗ d(π)d(ρ) d(σ ) (1 ⊗ S)vσ · ψ ◦ Φσα · vσ∗ 1 ⊗ S ∗ d(π)d(ρ) d(σ ) (1 ⊗ S) · ψ ◦ Φσδ · 1 ⊗ S ∗ d(π)d(ρ) d(σ ) (1 ⊗ S) · (ψ ⊗ trσ ) · 1 ⊗ S ∗ d(π)d(ρ)

= ψ ⊗ trπ ⊗ trρ . ⊗ L∞ (G), and (δ|Nψ , u) is a cocycle action on the type II∞ factor Nψ . Hence u ∈ Nψ ⊗ L∞ (G) perturbing (δ, u) to the action (Ad w ◦ By Lemma 3.2, there exists a unitary w ∈ Nψ ⊗ L∞ (G) δ, 1). Then wv is an α-cocycle and we set β := Ad wv ◦ α. We check that β satisfies the second β we have condition. Since Φπ = Φπδ ◦ Ad wπ∗ and w ∈ Nψ ⊗ L∞ (G), ψ ◦ Φπβ = ψ ◦ Φπδ ◦ Ad wπ∗ = (ψ ⊗ trπ ) ◦ Ad wπ∗ = ψ ⊗ trπ .

2

5. Groupoid actions and type III0 case θ, τM Let M be an injective factor of type III0 and {M, } the canonical core of M. Let ∞ (X, ν, Ft ) be the flow of weights for M, that is, Z(M) = L (X, ν), θt (f )(x) = f (F−t x) and ν is a measure on X. We represent (X, ν, Ft ) as a flow built under the ceiling function, that is, there exist a measure space (Y, μ), f ∈ L∞ (Y, μ) with f (x) R for some R > 0, and a nonsingular transformation T on (Y, μ) such that X is identified with {(y, t) | y ∈ Y, 0 t < f (y)}, ν = μ × dt, and Ft (y, s) = (y, t + s) where we identify (y, f (y)) and (T y, 0). Then we have two kinds of measured groupoids, G := R F X and G := Z T Y . In fact, G is characterized as G = {γ ∈ G | s(γ ), r(γ ) ∈ Y }. Here for a Γ -space Z, the groupoid Γ Z is defined as (g, hx)(h, x) = (gh, x) for g, h ∈ Γ and x ∈ Z. The source map s and the range map r are defined by s(g, x) = x and r(g, x) = gx, respectively. on M. Then mod(α) = id by Theorem A.6, Let α be an approximately inner action of G ∞ that is, the canonical extension α fixes L (X, ν). We first discuss the reduction of the study of × R-action G α θ to the groupoid actions. = ⊕ M(x) dx be the central decomposition. Since M(x) Let M are injective for almost every X ∼ x ∈ X, M(x) = R0,1 holds for almost every x ∈ X. As in [24], we obtain a family of actions on M(x) determined by { αx }x∈X of G ⊕

⊕ α (a)(x) dμ(x) =

α (a) = X

X

αx a(x) dμ(x),


1997

and an action {θγ }γ ∈ G of G by ⊕ θt (a) =

⊕ θt (a)(x) dμ(x) =

X

θγ a(F−t x) dμ(x),

X

where γ = (t, F−t x). Of course θγ is an isomorphism from M(s(γ )) onto M(r(γ )). Then θγ and αx commute in the following sense: αr(γ ) ◦ θγ = (θγ ⊗ id) ◦ αs(γ ) . Since α preserves τM by We denote the π -component of αx by απ,x . Lemma A.14, each αx preserves τx , a trace on M(x). We introduce the notion of a G-G-action. a discrete Kac algebra, and G a groupoid. Definition 5.1. Let R be a von Neumann algebra, G and {αγ }γ ∈G an action of G on R. We say that α (1) Let {αx }x∈G(0) be a family of actions of G is a G-G-action if αr(γ ) ◦ αγ = (αγ ⊗ id) ◦ αs(γ ) for all γ . We denote αr(γ ) αγ and απ,r(γ ) αγ by α·,γ and απ,γ for simplicity. We say {αx }x∈G(0) and {αγ }γ ∈G the G-part and the G-part of α, respectively. (2) For two G-G-actions α and β on R, we say that α and β are cocycle conjugate if there exist a Borel map σ : X → Aut(R), a βx -cocycle ux for x ∈ X and a βγ -cocycle uγ for γ ∈ G satisfying r(γ ) −1 βr(γ ) (uγ ) ◦ β·,γ (σr(γ ) ⊗ id) ◦ α·,γ ◦ σs(γ ) = Ad u and ur(γ ) βr(γ ) (uγ ) = (uγ ⊗ 1)(βγ ⊗ id) us(γ ) for all x ∈ X and γ ∈ G. In this case, we simply say that ur(γ ) βr(γ ) (uγ ) is a β-cocycle. The following can be shown as [24, p. 430]. on a type III0 injective factor M. Suppose that mod(α) = Lemma 5.2. Let α, β be actions of G id = mod(β). × R-actions are cocycle conjugate if and only if the Gθ on M (1) The G α θ and β G-actions r(γ ) θγ on R0,1 are cocycle conjugate. αr(γ ) θγ and β r(γ ) θγ on R0,1 are cocycle conjugate, then they are also (2) If the G-G-actions αr(γ ) θγ and β G-actions. cocycle conjugate as G r(γ ) θγ on R0,1 . Here the GHence we only have to classify two G-G-actions αr(γ ) θγ and β parts preserve the trace, and the G-parts come from θ , which are independent from α and β. Now we consider the following situation: • • • •

We are given two G-G-actions α and β on R0,1 ; The G-parts of α and β are free actions; The G-parts of α and β preserve the trace on R0,1 ; mod(αγ ) = mod(βγ ) for γ ∈ G.

1998


Note that G is an ergodic, approximately finite (AF), orbitally discrete principal groupoid, and the following Krieger’s cohomology lemma provides a powerful tool for study of actions of such groupoids [16]. (Also see [13, Appendix].) Theorem 5.3. Let G be a Polish group, and N a normal subgroup. Let G be an ergodic AF orbitally discrete principal groupoid. Let θ 1 and θ 2 be homomorphisms from G to G with θγ1 ≡ −1 θγ2 mod N . Then there exist Borel maps σ : G(0) → N and u : G → N such that σr(γ ) θγ1 σs(γ )= 2 uγ θγ . on R0,1 . We need some preparations as in [13,23]. Let σ be a trace preserving free action of G (1) Let Cσ be the set of pairs (θ, v), where θ ∈ Int(R0,1 ) and v is a σ -cocycle such that Ad v ◦ (1) σ = (θ ⊗ id) ◦ σ ◦ θ −1 . We define the multiplication on Cσ by (θ1 , v1 )(θ2 , v2 ) := (θ1 θ2 , (θ1 ⊗ id)(v2 )v1 ). Let Autσˆ (R0,1 σ G) be the set of all automorphisms which commute with the dual (1) in a canonical way, and Cσ(1) is a Polish action of G. Then we have Cσ ⊂ Autσˆ (R0,1 σ G) (1) ∩ Ker(mod) holds. Let Cσ(0) := {(Ad v, (v ⊗ 1)σ (v ∗ )) | group. In fact, Cσ = Autσˆ (R0,1 σ G) (0) (1) v ∈ U (R0,1 )}. Then Cσ is a normal subgroup of Cσ . Lemma 5.4. Cσ(0) is dense in Cσ(1) . Proof. Since σ is trace preserving and free, σ is approximately inner and centrally free by Corol as in Theorem 3.24 with lary A.7. Then we can take a unitary U = (U ν )ν ∈ Rω0,1 ⊗ L∞ (G) Γ = {e}. (1) Take (θ, v) ∈ Cσ and choose {v ν }ν ⊂ U (R0,1 ) with θ = limν→∞ Ad v ν . Then ∗ Ad v ◦ σ = (θ ⊗ id) ◦ σ ◦ θ −1 = lim Ad v ν ⊗ 1 ◦ σ ◦ Ad v ν ν→∞ ∗ ◦ σ. = lim Ad v ν ⊗ 1 σ v ν ν→∞

Set V := (v ν )ν ∈ Rω0,1 . Then w := v ∗ (V ⊗ 1)σ ω (V ∗ ) is a σ ω -cocycle, and U ∗ wU ∈ (R0,1 )ω ⊗ By Corollary 3.25, there exists z ∈ (R0,1 )ω such that w = (z ⊗ 1)σ ω (z∗ ). This implies L∞ (G). ∗ (z V ⊗ 1)σ ω (V ∗ z) = v. Let (μν )ν be a representing sequence of z∗ V . Then θ = limν→ω Ad μν and v = limν→ω (μν ⊗ 1)σ ((μν )∗ ). 2 Lemma 5.5. Suppose that βx is constant, that is, βx = βx0 for some x0 ∈ G(0) . Then there exist Borel families of automorphisms {σx }x∈G(0) ⊂ Int(R0,1 ) and βx -cocycles {w x }x∈G(0) ⊂ U (R0,1 ⊗ such that (σx ⊗ id) ◦ αx ◦ σx−1 = Ad w x ◦ βx . L∞ (G)) and N (x) := R0,1 αx G for each x ∈ X. Note that N and N (x) Proof. Set N := R0,1 βx0 G 2 2 act on the common Hilbert space L (R0,1 ) ⊗ L (G). Let Bx be the set of pairs (σ, v), where σ ∈ Aut(R0,1 ) and v is a 1-cocycle for αx such that (σ −1 ⊗ id) ◦ Ad v ◦ αx ◦ σ = βx0 . Then Bx is non-empty because of Theorem 2.4 for R0,1 and it is identified with the set of isomorphisms from N onto N (x) preserving R0,1 . Moreover, Bx is a Polish space because it is identified with a closed subset of unitary maps L2 (N ) onto L2 (N (x)) which intertwine N and N (x), preserve positive cones and L2 (R0,1 ) and commute with modular conjugation [8]. Then thanks to the measurable cross section theorem [27, Theorem A.16,


1999

vol. I], we can choose a Borel family (σx , v x ) ∈ Bx as in the proof of [27, Theorem IV.8.28, Proposition IV.8.29]. 2 Theorem 5.6. Let α and β be G-G-actions on R0,1 as before. Assume that βx is constant. Then α and β are cocycle conjugate as G-G-actions. Proof. By the previous lemma, we can take Borel families of automorphisms {σx }x∈G(0) ⊂ such that (σx ⊗ id) ◦ αx ◦ σx−1 = Int(R0,1 ) and, βx -cocycles {w x }x∈G(0) ⊂ U (R0,1 ⊗ L∞ (G)) −1 Ad w x ◦ βx . By replacing αr(γ ) αγ with (σr(γ ) ⊗ id) ◦ αr(γ ) αγ ◦ σs(γ ) , we may assume αx = Ad w x ◦ βx and mod(αγ ) = mod(βγ ). Since (βγ ⊗ id)βs(γ ) = βr(γ ) βγ , we can regard βγ as a by γ → (βγ , 1). We also have homomorphism from G to Autσˆ (R0,1 βx0 G) (αγ ⊗ id)βs(γ ) = (αγ ⊗ id) ◦ Ad w s(γ )∗ ◦ αs(γ ) = Ad(αγ ⊗ id) w s(γ )∗ ◦ (αγ ⊗ id)αs(γ ) = Ad(αγ ⊗ id) w s(γ )∗ ◦ αr(γ ) αγ = Ad(αγ ⊗ id) w s(γ )∗ w r(γ ) ◦ βr(γ ) αγ , where (αγ ⊗ id)(w s(γ )∗ )w r(γ ) is a βr(γ ) -cocycle. So we can regard α as a homomorphism from G by γ → (αγ , (αγ ⊗ id)(w s(γ )∗ )w r(γ ) ). Here note that C (1) = C (1) because to Autσˆ (R0,1 βx0 G) βx βx0 βx is constant. (1) We next show that αγ ≡ βγ mod (Cβr(γ ) ). Since mod(αγ ) = mod(βγ ), it is clear that αγ βγ−1 ∈ Int(R0,1 ). By the above computation, we also have the following: αγ βγ−1 ⊗ id ◦ βr(γ ) = (αγ ⊗ id) ◦ βs(γ ) βγ−1 = Ad(αγ ⊗ id) w s(γ )∗ w r(γ ) ◦ βr(γ ) αγ βγ−1 . Hence αγ βγ−1 ∈ Cβr(γ ) . Applying Theorem 5.3 and Lemma 5.4 to the two maps α, β : G → Cβx (1)

(0)

(1)

(1)

0

and Cβx , we get Borel maps G(0) x → (σx , v x ) ∈ Cβx and u : G γ → uγ ∈ U (R0,1 ) such 0 that Ad uγ , uγ βr(γ ) u∗γ · (βγ , 1) −1 −1 s(γ )∗ = σr(γ ) , v r(γ ) · αγ , (αγ ⊗ id) w s(γ )∗ w r(γ ) · σs(γ . ) , σs(γ ) v The left-hand side is equal to (Ad uγ ◦ βγ , uγ βr(γ ) (u∗γ )). We compute the right-hand side. For simplicity we write αγ for αγ ⊗ id and so on. −1 −1 s(γ )∗ σr(γ ) , v r(γ ) · αγ , αγ w s(γ )∗ w r(γ ) · σs(γ ) , σs(γ ) v s(γ )∗ r(γ ) r(γ ) −1 −1 s(γ )∗ w v · σs(γ ) , σs(γ ) v = σr(γ ) αγ , σr(γ ) αγ w −1 −1 s(γ )∗ σr(γ ) αγ w s(γ )∗ w r(γ ) v r(γ ) . = σr(γ ) αγ σs(γ ) , σr(γ ) αγ σs(γ ) v

2000


−1 By comparing the first component, we have Ad uγ ◦ βγ = σr(γ ) ◦ αγ ◦ σs(γ ) . Since G is generated by a single transformation, we may assume that uγ is a β-cocycle. The second component is computed as follows:

−1 s(γ )∗ σr(γ ) αγ σs(γ σr(γ ) αγ w s(γ )∗ w r(γ ) v r(γ ) ) v = Ad uγ βγ v s(γ )∗ · σr(γ ) αγ w s(γ )∗ σr(γ ) w r(γ ) v r(γ ) = Ad uγ βγ v s(γ )∗ σs(γ ) w s(γ )∗ σr(γ ) w r(γ ) v r(γ ) . Set ux := σx (w x )v x . By σx ◦ βx ◦ σx−1 = Ad v x ◦ βx , it follows that ux is a βx -cocycle and σx ◦ αx ◦ σx−1 = Ad ux ◦ βx . By comparing the second component, we have βγ (u∗γ ) = βγ (us(γ )∗ )u∗γ ur(γ ) , and equivalently ur(γ ) βγ (uγ ) = uγ βγ (us(γ ) ). This shows that u·,γ is a βcocycle, and σr(γ ) ◦ α·,γ ◦ σs(γ ) = Ad u·,γ ◦ β·,γ . Thus α and β are cocycle conjugate. 2 Proof of Theorem 2.4 for type III0 factors. Let M, α and α (0) be as in Theorem 2.4. Then α (0) (0) trivially and free on M by Theorem A.6. By using an and idM ⊗ α = idM act on Z(M) ⊗α isomorphism R0,1 ∼ αx )x∈X and (idM(x) ⊗ α (0) )x∈X are satisfying the = R0,1 ⊗ R0 , we see that ( α θ and θ ⊗ α (0) are condition of Theorem 5.6. Then the two G-G-actions on R0,1 arising from cocycle conjugate. This implies the cocycle conjugacy of the G × R-actions α θ and (θ ⊗ α (0) ) (0) as in the by Lemma 5.2. Considering the partial crossed product by θ , we get α ∼ idM ⊗α (0) proof of Lemma 3.4. Thus α and idM ⊗ α are cocycle conjugate by Lemma 3.3. 2 Remark 5.7. In general, there may appear some obstructions in combining the G-part and the Gpart. In [13,23,15], model actions absorbing obstructions are constructed. In our case, however, we are treating only free actions, and no obstructions appear. Hence we do not need such model actions. 6. Classification for type III1 case 6.1. Basic results on canonical extensions In Section 4, we obtained the classification of approximately inner and centrally free actions of an amenable discrete Kac algebra on the injective factor of type IIIλ . Using this result together with ideas of [4,9] (also see [17]), we classify actions on the injective factor of type III1 . Let M ∼ = R∞ and ϕ be a faithful normal state on M. Fix T > 0. Set N := M σ ϕ Z, which T

2π

σT . is an injective factor of type IIIλ , λ := e− T , and let U ∈ N be the unitary implementing √ − −1t The dual action of the torus T = R/2πZ is denoted by θ , which acts on U by θt (U ) = e U for t ∈ T. Using the averaging expectation Eθ : N → M by θ , we extend ϕ to ϕˆ := ϕ ◦ Eθ . Throughout this section, we keep these notations. Now we introduce the extension ˆ: End0 (M) → End0 (N ) defined by ρ(x) = ρ(x)

for all x ∈ M;

ρ(U ˆ ) = d(ρ)iT [Dϕ ◦ φρ : Dϕ]T U.

ϕ


2001

by U = Note that ρˆ is one of the variants of the canonical extension. Indeed, regarding N ⊂ M λϕ (T ), we see that ρˆ = ρ |N . Lemma 6.1. For any ρ ∈ End0 (M), mod(ρ) ˆ = id. ϕˆ

Proof. Since σT = Ad U . We can take a positive operator h affiliated with Nϕˆ such that U = hiT . We set ψ := ϕˆ h−1 , whose modular automorphism has the period T . Note that Eθ ◦φρˆ = φρˆ ◦Eθ = φρ ◦ Eθ because φρˆ |M = φρ (see Theorem 6.3 (2)). Then we can compute [Dψ ◦ φρˆ : Dψ]T as follows: ˆ T [D ϕˆ : Dψ]T [Dψ ◦ φρˆ : Dψ]T = [Dψ ◦ φρˆ : D ϕˆ ◦ φρˆ ]T [D ϕˆ ◦ φρˆ : D ϕ] = ρˆ [Dψ : D ϕ] ˆ T [D ϕˆ ◦ φρˆ : D ϕ] ˆ T [D ϕˆ : Dψ]T ∗ = ρˆ U [Dϕ ◦ φρ ◦ Eθ : Dϕ ◦ Eθ ]T U = ρˆ U ∗ [Dϕ ◦ φρ : Dϕ]T U = d(ρ)−iT . By [11, Theorem 2.8] d(ρ) = d(ρ), ˆ so the above equality means mod(ρ) ˆ = id.

2

We denote by Endθ0 (N ) the set of endomorphisms with finite indices on N which commute with θ , and by Ker(mod) the set of endomorphisms with finite indices in End(N )CT with trivial Connes–Takesaki modules. Note that ρˆ ∈ Endθ0 (N ) for all ρ ∈ End0 (N ). We will analyze the ∩ N , which admits the torus action θ . Define the following linear relative commutant ρ(N) ˆ space for each n ∈ Z: √ In := a ∈ ρ(N) ˆ ∩ N θt (a) = e −1nt a for all t ∈ T .

Lemma 6.2. For each n ∈ Z, one has In = U −n (ρ, σnT ρ). ϕ

Proof. Take a ∈ In . Then θt (U n a) = U n a for t ∈ T, and b := U n a ∈ M. We check b ∈ ϕ (ρ, σnT ρ) as follows: for x ∈ M, ϕ bρ(x) = U n ρ(x)a = U n ρ(x)U n∗ U n a = σnT ρ(x) b. Hence In ⊂ U −n (ρ, σnT ρ). Next we show the converse inclusion. Set a unitary u := d(ρ)iT [Dϕ ◦ φρ : dϕ]T . Take b ∈ ϕ ϕ ϕ (ρ, σnT ρ). By direct computation, we see that U −n b ∈ In if and only if b = σnT (u)σT (b)u∗ ϕ ϕ ϕ ϕ ∗ holds. Consider the map μ : (ρ, σnT ρ) b → σnT (u)σT (b)u ∈ (ρ, σnT ρ). Then μ is a wellϕ defined unitary, here the inner product is given by a, b = φρ (b∗ a) for a, b ∈ (ρ, σT ρ). Hence ϕ it suffices to prove that μ is actually an identity map. Since (ρ, σnT ρ) is finite dimensional, it is √ spanned by eigenvectors of μ. Let b be an eigenvector μ(b) = e −1s b for some s ∈ [0, 2π). We claim that U −n b ∈ (θ−s ρ, ˆ ρ). ˆ For x ∈ M, we have the following ϕ

ϕ −n ˆ = U −n σnT ρ(x) b = ρ(x)U −n b = ρ(x)U ˆ b. U −n bθ−s ρ(x) We also have the following

2002

T. Masuda, R. Tomatsu / Journal of Functional Analysis 258 (2010) 1965–2025 √ ˆ ) = U −n bθˆ−s (uU ) = U −n b · e −1s uU = U −n μ(b)uU U −n bθˆ−s ρ(U

= uU −n σT (b)U = uU 1−n b = ρ(U ˆ )U −n b. ϕ

Thus we have verified the claim. By the Frobenius reciprocity, dim(θ−s ρ, ˆ ρ) ˆ = dim(θ−s , ρˆ ρ), ˆ¯ ¯ and hence ρˆ ρˆ contains θ−s as an irreducible component. However by the previous lemma, ρˆ has a trivial Connes–Takesaki module, and mod(θ−s ) = mod(ρˆ ρ) ˆ¯ = id. This is possible only if s = 0. Therefore μ = id. 2 Theorem 6.3. Let ρ ∈ End0 (M). Then one has the following: (1) ρˆ is irreducible if and only if ρ is irreducible. In this case, the inclusion ρ(M) ⊂ N is irreducible; (2) The standard left inverse φρˆ is given by φρˆ xU n = d(ρ)−inT φρ x[Dϕ ◦ φρ : Dϕ]∗nT U n

for all x ∈ M, n ∈ Z;

(3) The extension ˆ· is a bijection from End0 (M) onto Endθ0 (N ) ∩ Ker(mod); (4) ρˆ ∈ Cnd(N ) if and only if ρ ∈ Cnd(M). Proof. (1) If ρˆ is irreducible, then I0 = C, and (ρ, ρ) = C follows from the previous lemma. Conversely if ρ is irreducible, then ρρ contains no non-trivial modular automorphisms because ϕ the T -set T (M) is trivial. This means (ρ, ρ) = C, and (σnT ρ, ρ) = 0 for n = 0. Hence I0 = C, and In = 0 for n = 0. Since ρ(N) ˆ ∩ N is densely spanned by {In }n∈Z , ρˆ is irreducible. We prove the latter statement in (1). Take x ∈ ρ(M) ∩ N and let x = n∈Z xn∗ U n be the ϕ formal decomposition. Then for each n ∈ Z, xn ∈ (ρ, σnT ρ). From the above argument, x0 ∈ C and xn = 0 for n = 0. Hence ρ(M) ∩ N = C. (2) By [19, Lemma 3.5], the map φρˆ is well defined. By [11, Theorem 2.8], ρφ ˆ ρˆ is the minimal conditional expectation, and it follows that φρˆ is standard. (3) Let ψ be a periodic weight constructed as in the proof of Lemma 6.1. By Lemma 6.1, we see that ρˆ ∈ Endθ0 (N ) ∩ Ker(mod). So, the given map is well defined. We show that the map is a bijection. Clearly it is injective, and it suffices to show the surjectivity. Let σ ∈ Endθ0 (N ) ∩ Ker(mod). Since mod(σ ) = id, we have d(σ )iT [Dψ ◦ φσ : Dψ]T = 1. This is equivalent to σ (U ) = d(σ )iT [Dϕ ◦ φσ |M : Dϕ]T U.

(6.1)

Set ρ = σ |M . The action θ of T on σ (N ) is dominant, and d(σ ) = d(ρ) follows from [11, Theorem 2.8 (2)]. In the proof of [11, Theorem 2.8 (2)], it is also shown that σ ◦ φσ |M is the minimal expectation from M onto ρ(M). Hence φρ = φσ |M . Then the equality (6.1) yields σ = σ |M . (4) Let ρ ∈ Cnd(M). We may and do assume that ρ is irreducible. Then by [19, Theoϕ ϕ ϕˆ rem 4.12], there exists t ∈ R such that [ρ] = [σt ]. Then [ρ] ˆ = [σ t ] = [σt ], and ρˆ ∈ Cnd(N ). Conversely we assume that ρˆ ∈ Cnd(N ). Thanks to (1), we may and do assume that ρˆ is ϕˆ irreducible. By [19, Theorem 4.12], there exist t ∈ R and u ∈ U (N ) such that ρˆ = Ad u ◦ σt . ϕ Considering the formal decomposition of u, we see that (σnT +t , ρ) = 0 for some n ∈ Z. Since ρ ϕ is irreducible by (1), this means [ρ] = [σnT +t ], and ρ ∈ Cnd(M). 2


2003

Let K be a finite dimensional Hilbert space. Following the procedure introduced in Appendix A, we define the canonical extension β ∈ Mor(N, N ⊗ B(K)) for β ∈ Mor0 (M, M ⊗ B(K)) by β(x) = β(x) for all x ∈ M; β(U ) = d(β)iT Dϕ ◦ Φ β : Dϕ ⊗ TrK T (U ⊗ 1). By Morθ0 (N, N ⊗ B(K)), we denote the set of homomorphisms in Mor0 (N, N ⊗ B(K)) commuting with θ . The following is a direct consequence of the previous theorem. The fourth statement follows from the third one and Theorem A.6. Lemma 6.4. Let K be a finite dimensional Hilbert space. Then one has the following: (1) Let β ∈ Mor0 (M, M ⊗ B(K)). Then β is irreducible if and only if β is irreducible. In this case, the inclusion β(M) ⊂ N ⊗ B(K) is irreducible; (2) Let β ∈ Mor0 (M, M ⊗ B(K)). Then d(β) = d(β) and the standard left inverse Φ β is given by the following equality: for x ∈ M ⊗ B(K) and n ∈ Z, Φ β x U n ⊗ 1 = d(β)−inT Φ β x Dϕ ⊗ TrK : Dϕ ◦ Φ β nT U n ; (3) The extension · is a bijection from Mor0 (M, M ⊗ B(K)) onto Morθ0 (N, N ⊗ B(K)) ∩ Ker(mod); (4) Let β ∈ Mor0 (M, M ⊗ B(K)). If d(β) = dim(K), then β ∈ Int(N, N ⊗ B(K)); (5) Let β ∈ Mor0 (M, M ⊗ B(K)). Then β ∈ Cnt(M, M ⊗ B(K)) if and only if β ∈ Cnt(N, N ⊗ B(K)). 6.2. Reduction to the classification of actions on Rλ ∼ R∞ . Then α is automatically approx on M = Let α a centrally free cocycle action of G imately inner by Corollary A.7. For each π ∈ Irr(G), we consider the canonical extension α π ∈ Mor0 (N, N ⊗ B(K)) as before. Then α is a cocycle action on N with the same 2-cocycle. on M. Then α is an approximately Proposition 6.5. Let α be a centrally free cocycle action of G inner and centrally free cocycle action of G on N . Proof. For each π ∈ Irr(G) \ {1}, α π is approximately inner and centrally non-trivial by Lemma 6.4 (4) and (5). Since απ is properly outer, απ is irreducible [18, Lemma 2.8]. Hence so is α π by Lemma 6.4 (1). Then by [18, Lemma 8.3], α π is properly centrally non-trivial. Thus the cocycle action α is centrally free. 2 Our main theorem of this section is the following: on M. Then the G × T-action αθ on N is Theorem 6.6. Let α be a centrally free action of G (0) (0) cocycle conjugate to θ ⊗ α , where α is a free action of G on R0 .

2004


Proof of Theorem 2.4 for R∞ . Since the natural extension of α to N θ T is cocycle conjugate to α by the Takesaki duality, we see that Theorem 6.6 implies Theorem 2.4 considering the partial crossed product by θ as before. 2 The rest of this section is devoted to show Theorem 6.6. The essential part of our proof is the model action splitting result in Proposition 6.10. The following lemma shows that the canonical extension well behaves to cocycle perturbations. Lemma 6.7. For i = 1, 2, let M i be a type III1 factor, ϕ i ∈ W (M i ) and (α i , ui ) be a cocycle ϕi 1 1 on M i . We set N i := M i ϕ i Z and the dual action θ i := σ action of G T . If (α , u ) is cocycle σT

conjugate to (α 2 , u2 ), then there exist an isomorphism Ψ : N 1 → N 2 and a unitary v ∈ M 2 ⊗ such that L∞ (G) • Ψ ◦ θt1 = θt2 ◦ Ψ for all t ∈ T; • (Ψ ⊗ id) ◦ α 1 ◦ Ψ −1 = Ad v ◦ α 2 ; • (Ψ ⊗ id ⊗ id)(u1 ) = (v ⊗ 1)α 2 (v)u2 (id ⊗ )(v ∗ ). × T-cocycle action α 1 θ 1 is cocycle conjugate to α 2 θ 2 . In particular, the G Proof. Since (α 1 , u1 ) ∼ (α 2 , u2 ), there exist an isomorphism Ψ0 : M 1 → M 2 and v ∈ M 2 ⊗ such that L∞ (G) • (Ψ0 ⊗ id) ◦ α 1 ◦ Ψ0−1 = Ad v ◦ α 2 ; • (Ψ0 ⊗ id ⊗ id)(u1 ) = (v ⊗ 1)α 2 (v)u2 (id ⊗ )(v ∗ ). We set ψ 2 := ϕ 1 ◦ Ψ0−1 ∈ W (M 2 ). Then there exists an isomorphism Ψ : N 1 → M 2

ψ2

σT

1

2

1

2

Z ϕ1

such that Ψ (xU ϕ ) = Ψ0 (x)U ψ , where U ϕ and U ψ are the implementing unitaries for σT ψ2

and σT , respectively. Then Ψ intertwines the dual actions. Regard M 2

ψ2

σT

Z = N 2 in the

2 . It suffices to show the second equality holds on the implementing unitary U ψ . This is core M checked as follows: for π ∈ Irr(G), we have 2

2 1 (Ψ ⊗ id) ◦ α 1π ◦ Ψ −1 U ψ = (Ψ ⊗ id) α 1π U ϕ 1 1 = (Ψ ⊗ id) Dϕ 1 ◦ Φπα : Dϕ 1 ⊗ trπ T U ϕ ⊗ 1 2 1 = Dϕ 1 ◦ Φπα ◦ Ψ0−1 ⊗ id : Dϕ 1 ◦ Ψ0−1 ⊗ trπ T U ψ ⊗ 1 2 1 = Dψ 2 ◦ Ψ0 ◦ Φπα ◦ Ψ0−1 ⊗ id : Dψ 2 ⊗ trπ T U ψ ⊗ 1 2 (Ψ0 ⊗id)◦α 1 ◦Ψ0−1 = Dψ 2 ◦ Φπ : Dψ 2 ⊗ trπ T U ψ ⊗ 1 2 2 = Dψ 2 ◦ ΦπAd v◦α : Dψ 2 ⊗ trπ T U ψ ⊗ 1 2 2 = Dψ 2 ◦ Φπα ◦ Ad vπ∗ : Dψ 2 ⊗ trπ T U ψ ⊗ 1


2005

2 2 2 ψ◦Φ α = vπ σT π vπ∗ Dψ 2 ◦ Φπα : Dψ 2 ⊗ trπ T U ψ ⊗ 1 ψ 2 ⊗trπ ∗ ψ 2 2 = vπ Dψ 2 ◦ Φπα : Dψ 2 ⊗ trπ T σT vπ U ⊗ 1 2 2 = vπ Dψ 2 ◦ Φπα : Dψ 2 ⊗ trπ T U ψ ⊗ 1 vπ∗ 2 2 = Ad vπ ◦ α 2 U ψ .

The following lemma is an equivariant version of [4, Lemma I.2]. Recall that α (0) is a free on R0 . action of G Lemma 6.8. One has the following: on Rλ and γ ∈ Aut(Rλ ) such that (1) Let δ be an action of G • δ commutes with γ ; • Rλ γ Z ∼ = Rλ ; • The natural extension δ of δ to Rλ γ Z is approximately inner and centrally free; × T-action δ γˆ is centrally free on Rλ γ Z. • The G × Z-action δγ on Rλ is cocycle conjugate to idR ⊗ γ (0) ⊗ α (0) , where γ (0) is an Then G λ aperiodic automorphism on R0 . on Rλ , and β an action of T on Rλ such that (2) Let δ be an action of G • δ is approximately inner and centrally free; • δ commutes with β; × T-action δβ is centrally free on Rλ ; • The G • Rλ β T ∼ = Rλ . (0) ⊗ α (0) . Then the G × T-action δβ is cocycle conjugate to idRλ ⊗ γ × Z-action δγ . Let W ∈ R γ Z be the unitary Proof. (1) Set R := Rλ which admits the G implementing γ . Step 1. We show that γ is approximately inner and centrally free. This follows from [4, Lemma I.2]. Also see [27, Lemma XVIII.4.18]. × Z-action δγ is approximately inner. Step 2. We show that the G It is known that R and R γ Z have the common flow of weights [15,22]. Since δ is approximately inner on R γ Z, mod(δ) = mod(δ) = id. Hence δ is approximately inner on R by Theorem A.6, and so is δγ . × Z-action δγ is centrally free. Step 3. We show that the G Fix a generalized trace ψ on R. Note that our assumption of (1) is satisfied for any perturbed actions of δγ . By Lemma 4.13, we may and do assume that ψ is invariant by δγ . For each π ∈ Irr(G), we set Qπ := δπ (R) ∩ (R γ Z ⊗ B(Hπ )). We can show that Qπ is finite dimensional in a similar way to the proof of Theorem 6.3 (1), where the freeness of γ is crucial. Also we can show that Ad(W ⊗ 1) ergodically acts on Qπ , and the torus action γˆ preserves Qπ . Therefore, there exist atoms {pi }m i=1 ⊂ Qπ such that pi ∈ R ⊗ B(Hπ ), γ (pi ) = pi+1 for 1 i m − 1 and Qπ = δπ (R) ∩ R ⊗ B(Hπ ) = Cp1 + · · · + Cpm .

(6.2)

2006


Take an isometry V1 ∈ N ⊗ B(Hπ ) such that V1 V1∗ = p1 . Set Vi := (γ i−1 ⊗ id)(V1 ) for 1 i m. Then we have Vi Vi∗ = pi . Now assume that δπ γ n is not properly centrally non-trivial on R for some π ∈ Irr(G) and ∗ n n ∈ Z. Set βi := Vi δπ (γ∗ (·))Vi for each i. Then βi ∈ Mor0 (R, R ⊗ B(Hπ )) is irreducible and δπ γ n = m V β (·)V . i=1 i i i Then βi is not properly centrally non-trivial for some i. We may and do assume i = 1. Since β1 is irreducible, β1 is centrally trivial [18, Lemma 8.3]. Then by Corolψ lary A.7, we see that β1 = Ad u ◦ σt0 for some u ∈ U (R) and t0 ∈ R. ψ So we have δπ (γ n (x))V1 u = V1 uσt0 (x) for x ∈ R. Applying γ i−1 to the both sides, we ψ have δπ (γ n (γ i−1 (x)))Vi γ i−1 (u) = Vi γ i−1 (u)σt0 (γ i−1 (x)) for x ∈ R, where we have used the fact that γ commutes with σ ψ . By definition of βi , we obtain βi (γ i−1 (x))γ i−1 (u) = ψ ψ γ i−1 (u)σt0 (γ i−1 (x)), that is, βi = Ad γ i−1 (u) ◦ σt0 . Hence {βi }m i=1 define the equivalent sectors. By (6.2), this is possible when m = 1, that is, δπ γ n is irreducible. Hence we may assume ψ that δπ γ n = Ad u ◦ σt0 . Since ψ is invariant under δγ , u ∈ Rψ , and γ (u) = e

ψˆ that δ π ◦ Ad W n = Ad u ◦ σt0

√

−1s0 u

for some s0 ∈ R. We can check

◦ γˆ−s0 holds on R γ Z by direct computation. So δ π γˆs0 is centrally ψ trivial, and the assumption (1) yields π = 1, s0 = 0, and γ n = Ad u ◦ σt0 . Then we get n = 0 by central freeness of γ .

Step 4. We use the classification result for actions on Rλ . × Z-action δγ on Rλ is approximately inner and centrally free. So δγ is cocycle The G conjugate to idN ⊗ γ (0) ⊗ α (0) by Theorem 2.4 for Rλ . . Extend the action δ to N , which is also denoted by δ. (2) Let N = Rλ β T and γ = β Using the Takesaki duality [26], we see that all the assumptions of (1) are fulfilled. Then we ∼ idRλ ⊗ γ (0) ⊗ α (0) . Comparing the crossed products by β and γ (0) , we obtain δβ ∼ get δ β (0) (0) idRλ ⊗ γ ⊗ α . 2 ∼ R∞ , N = M ϕ Z as before, and α a centrally free action of G on M. Lemma 6.9. Let M = σT (0) ⊗ α (0) . × T-action θ−t ⊗ αθt on N ⊗ N is cocycle conjugate to idN ⊗ γ Then the G t Proof. We can identify (N ⊗ N ) θt ⊗θ−t T with (M ⊗ M) σ ϕ ⊗σ ϕ Z [4, Lemma 1(b)]. Hence T T (N ⊗ N) θt ⊗θ−t T is a factor of type IIIλ . By Proposition 6.5, α is approximately inner and centrally free, hence so is id ⊗ α. It is obvious that θ−t ⊗ αθt is a centrally free action. Then the previous lemma can be applied. 2 Proposition 6.10. Let M, N , α, θ be as above. Let βt be a product type action of T on R0 ∼ = 1 0 ∞ ∞ √ for t ∈ R. Then M (C) given by β = Ad αθ is cocycle conjugate to id ⊗ 2 t R −1t λ i=1 i=1 0e β ⊗ αθ . The proof of Proposition 6.10 will be presented in the sequel subsections. Here we prove Theorem 6.6 assuming Proposition 6.10. Proof of Theorem 6.6. Note that β is a minimal action of T, hence is dual, and conjugate to (0) . Since θ γ −t ⊗ θt (resp. θt ) is cocycle conjugate to idRλ ⊗ βt (resp. θt ⊗ βt ⊗ idRλ ) by the theory of Connes [4, Lemma 5] and Haagerup [9], we have


αθt ∼ idRλ ⊗ βt ⊗ αθt

2007

(by Proposition 6.10)

∼ θt ⊗ θ−t ⊗ αθt ∼ θt ⊗ βt ⊗ idRλ ⊗ α (0)

(by Lemma 6.9)

∼ θt ⊗ α (0) . Hence αθt is cocycle conjugate to θt ⊗ α (0) .

2

Therefore the proof of Theorem 2.4 has been reduced to that of Proposition 6.10. We will show that αθ ∼ idRλ ⊗ αθ in Corollary 6.15, and αθ ∼ β ⊗ αθ in Theorem 6.17, and complete the proof of Proposition 6.10. 6.3. λ-stability As an analogue of the property L a in [1], we introduce the following notion. be a discrete Kac algebra, P a factor, and α a cocycle action of G on P . Definition 6.11. Let G λ For 0 < λ < 1, set a = 1+λ . We say that (P , α) satisfies the property L a if we have the following: For any ε > 0, any finite sets F Irr(G) and Ψπ (P ⊗ B(Hπ ))∗ for π ∈ F , there exists a partial isometry u ∈ P such that for ψ ∈ Ψπ , π ∈ F , uu∗ + u∗ u = 1,

u2 = 0; (u ⊗ 1) · ψ − λψ · (u ⊗ 1) < ε; u ⊗ 1 − απ (u) · ψ < ε; ψ · u ⊗ 1 − απ (u) < ε. Note that the property L a is stable under perturbations of a cocycle action. on R∞ . Then (R∞ , α) has the propLemma 6.12. Let α be a centrally free cocycle action of G λ erty L a , a = 1+λ , for any 0 < λ < 1. Proof. Since M := R∞ is properly infinite, we may and do assume that α is an action. Take π ∈ Irr(G), and set π := d(π)1 ⊕ π , a direct sum representation of G. Consider an inclusion απ (M) ⊂ M ⊗ B(Hπ ). We can identify M ⊗ B(Hπ ) with M2 (M ⊗ B(Hπ )) and α (M) = π

x ⊗ 1π 0π

0π απ (x)

x∈M .

Then απ (M) ⊂ M ⊗ B(Hπ ) is an inclusion of injective factors of type III1 with minimal index 4d(π)2 . The minimal expectation E π is given by Eπ

a c

b d

1 = απ (id ⊗ trπ )(a) + Φπ (d) . 2

2008


T = −2π/ log λ For a fixed 0 < λ < 1, we construct the type IIIλ factor N := M σ ϕ Z ⊂ M, T

ϕ◦Φπα

as before. The implementing unitary is denoted by U ϕ = λϕ (T ). Set γ := σT απ−1 ◦ E π . Then γ globally preserves the inclusion απ (M) ⊂ M ⊗ B(Hπ ).

, where Φπα =

Claim 1. We show that the inclusion απ (M) γ Z ⊂ (M ⊗ B(Hπ )) γ Z is isomorphic to α π (N ) ⊂ N ⊗ B(Hπ ). We identify (M ⊗ B(Hπ )) γ Z with (M ⊗ B(Hπ )) ϕ⊗trπ Z in the core algebra Q of M ⊗ B(Hπ ). Then

σT

α απ (M) γ Z = απ (M) ∨ λϕ◦Φπ (T ) = απ (M) ∨ Dϕ ◦ Φπα : Dϕ ⊗ trπ T λϕ⊗trπ (T ) . ⊗ B(Hπ ) satisfies Ψ |M⊗B(H ) = id and The canonical isomorphism Ψ : Q → M π ϕ⊗tr ϕ π (T )) = λ (T ) ⊗ 1. Hence Ψ (λ α Ψ λϕ◦Φπ (T ) = Dϕ ◦ Φπα : Dϕ ⊗ trπ T λϕ (T ) ⊗ 1 = α π λϕ (T ) . Then we have Ψ (απ (M) γ Z) = α π (N ) and Ψ ((M ⊗ B(Hπ )) γ Z) = N ⊗ B(Hπ ). Claim 2. We show that the inclusion α π (N ) ⊂ N ⊗ B(Hπ ) is relatively λ-stable. Since α is approximately inner and centrally free on N by Proposition 6.5, α is cocycle conjugate to idRλ ⊗ α by Theorem 2.4 for type IIIλ case. Hence the inclusion α π (N ) ⊂ N ⊗ B(Hπ ) is relatively λ-stable in the sense that α π (N ) ⊂ N ⊗ B(Hπ ) ∼ = Rλ ⊗ α π (N ) ⊂ (Rλ ⊗ N ) ⊗ B(Hπ ). Claim 3. We show that γ is an approximately inner automorphism on the subfactor απ (M) ⊂ M ⊗ B(Hπ ). ϕ By Corollary 3.26, we can choose {wn }n ⊂ U (M) such that σT = limn→∞ Ad wn and [Dϕ ◦ Φπ : Dϕ ⊗ trπ ]T = limn→∞ απ (wn )(wn∗ ⊗ 1) for all π ∈ Irr(G). Since 2ϕ ◦ απ−1 ◦ E π is nothing but a balanced functional ϕ ⊗ trπ ⊕ ϕ ◦ Φπ , ϕ 0π 1 ◦ σT ⊗ idπ . γ = Ad π 0π [Dϕ ◦ Φπ : Dϕ ⊗ trπ ]T Thus γ = limn→∞ Ad απ (wn ), and γ is approximately inner in a subfactor sense. By the previous three claims, we can show that the inclusion απ (M) ⊂ M ⊗ B(Hπ ) is relatively λ-stable. Indeed, the proof is similar to that of [4, Corollary II.3]. (Also see [17, Theorem 3.6].) Hence for any ε > 0 and any {ψi }ni=1 ⊂ (M ⊗ B(Hπ ))∗ , there exists u ∈ M such that u2 = 0, uu∗ + u∗ u = 1 and απ (u) · ψi − λψi · απ (u) < ε, for all 1 i n. For ψ ∈ (M ⊗ B(Hπ ))∗ , define ψij ∈ (M ⊗ B(Hπ ))∗ by ψij (a) = ψ(aij ) via identification of M ⊗ B(Hπ ) with M2 (M ⊗ B(Hπ )) and απ (x) = diag(x ⊗ 1π , απ (x)) for x ∈ M. Assume we have chosen u so that απ (u) · ψij − λψij · απ (u) < ε for all i, j = 1, 2.


2009

Then we obtain the following four inequalities (u ⊗ 1π ) · ψ − λψ · (u ⊗ 1) < ε, απ (u) · ψ − λψ · (u ⊗ 1) < ε,

(u ⊗ 1) · ψ − λψ · απ (u) < ε; απ (u) · ψ − λψ · απ (u) < ε.

It is easy to deduce that u satisfies the condition in Definition 6.11 for ψ. So far, we have considered a single element π ∈ Irr(G). For a finite subset F Irr(G), define Π := π∈F π , and consider the similarly defined inclusion αΠ (M) ⊂ M ⊗ B(HΠ ). Then the same argument is applicable. 2 Lemma 6.13. Let P be a properly infinite factor, H a finite dimensional Hilbert space, α ∈ Mor0 (P , P ⊗ B(H )) and Φ (P ⊗ B(H ))∗ a finite set of faithful states. Let 0 < ε < 1 and 0 < λ 1. Assume that there exists u ∈ P such that uu∗ + u∗ u = 1, u2 = 0 and for all ϕ ∈ Φ, (u ⊗ 1) · ϕ − λϕ · (u ⊗ 1) λε, u ⊗ 1 − α(u) · ϕ λε,

ϕ · (u ⊗ 1) − λ−1 (u ⊗ 1) · ϕ λε; ϕ · u ⊗ 1 − α(u) λε.

Then there exists a √ unitary v ∈ P ⊗ B(H ) such that Ad v ◦ α = id on the type I2 subfactor {u} # and v − 1ϕ < 12 4 ε for all ϕ ∈ Φ. Proof. In the following, we frequently use the inequalities x2ϕ xx · ϕ, x · ϕ √ ϕxϕ . First we show uu∗ ⊗ 1 and α(uu∗ ) are close as follows: ∗ uu ⊗ 1 − α uu∗ · ϕ uu∗ ⊗ 1 · ϕ − λ−1 α(u) · ϕ · (u ⊗ 1) + λ−1 α(u) · ϕ · u∗ ⊗ 1 − α uu∗ · ϕ = uu∗ ⊗ 1 ϕ − λ−1 (u ⊗ 1) · ϕ · u∗ ⊗ 1 + λ−1 (u ⊗ 1) · ϕ · u∗ ⊗ 1 − λ−1 α(u) · ϕ · u∗ ⊗ 1 + λ−1 ϕ · u∗ ⊗ 1 − u∗ ⊗ 1 · ϕ + u∗ ⊗ 1 ϕ − α u∗ · ϕ 4ε. Since x2ϕ xϕx, we have uu∗ ⊗ 1 − α(uu∗ )2ϕ 8ε. In the same way, we have u∗ u ⊗ √ 1 − α(u∗ u)2ϕ 8ε. Hence we have uu∗ ⊗ 1 − α(uu∗ )#ϕ 2 2ε and u∗ u ⊗ 1 − α(u∗ u)#ϕ √ 2 2ε. By [2, Lemma 1.1.4] and [20, Lemma 8.1.1], there exists a partial isometry w ∈ P ⊗ B(H ) with ww ∗ = uu∗ ⊗ 1, w ∗ w = α(uu∗ ), w − uu∗ ⊗ 1#ϕ 7uu∗ ⊗ 1 − α(uu∗ )ϕ for ϕ ∈ Φ. √ Hence we have w − uu∗ ⊗ 1#ϕ 14 2ε. Set v := (uu∗ ⊗ 1)wα(uu∗ ) + (u∗ ⊗ 1)wα(u). It is standard to see Ad √v ◦ α(x) = x ⊗ 1 for x ∈ {u} . We estimate (v − 1) · ϕ and ϕ · (v − 1). Since xϕ 2x#ϕ , and xϕ √ ϕxϕ , we have √ √ w − uu∗ ⊗ 1 · ϕ w − uu∗ ⊗ 1 2w − uu∗ ⊗ 1# 28 ε. ϕ ϕ Since [uu∗ ⊗ 1, ϕ] 2ε, we get

2010


∗ uu ⊗ 1 wα uu∗ − uu∗ ⊗ 1 · ϕ wα uu∗ − uu∗ ⊗ 1 · ϕ wα uu∗ − w uu∗ ⊗ 1 · ϕ + w uu∗ ⊗ 1 − uu∗ ⊗ 1 · ϕ 4ε + w − uu∗ ⊗ 1 ϕ, uu∗ ⊗ 1 + w − uu∗ ⊗ 1 · ϕ · uu∗ ⊗ 1 √ √ 4ε + 4ε + 28 ε 36 ε and ∗ u ⊗ 1 wα(u) − u∗ u ⊗ 1 · ϕ wα(u) − u ⊗ 1 · ϕ wα(u) − w(u ⊗ 1) · ϕ + w(u ⊗ 1) − u ⊗ 1 · ϕ ε + w − uu∗ ⊗ 1 (uϕ − λϕu) + w − uu∗ ⊗ 1 · λϕ · (u ⊗ 1) √ √ ε + 2ε + 28 ε 31 ε. √ √ √ √ Hence (v − 1) · ϕ 36 ε + 31 ε = 67 ε, and v − 12ϕ 134 ε holds. Next we estimate ϕ · (v − 1) as follows: ∗ ϕ · uu ⊗ 1 wα uu∗ − uu∗ ⊗ 1 ϕ · uu∗ ⊗ 1 wα uu∗ − uu∗ ⊗ 1 ϕ, uu∗ ⊗ 1 wα uu∗ − uu∗ ⊗ 1 + uu∗ ⊗ 1 · ϕ · wα uu∗ − uu∗ ⊗ 1 4ε + uu∗ ⊗ 1 · ϕ · wα uu∗ − uu∗ ⊗ 1 4ε + ϕ · w − α uu∗ α uu∗ + ϕ · uu∗ ⊗ 1 α uu∗ − uu∗ ⊗ 1 √ 4ε + 28 ε + ϕ · uu∗ ⊗ 1 α uu∗ − uu∗ ⊗ 1 √ 32 ε + 4ε + ϕ · α uu∗ − uu∗ ⊗ 1 √ √ 32 ε + 4ε + 4ε 40 ε and ∗ ϕ · u ⊗ 1 wα(u) − u∗ u ⊗ 1 ϕ · u∗ ⊗ 1 − λ u∗ ⊗ 1 · ϕ · wα(u) − u ⊗ 1 + λu∗ ϕ · wα(u) − u ⊗ 1 2ε + ϕ · wα(u) − u ⊗ 1 2ε + ϕ · w − uu∗ ⊗ 1 α(u) + ϕ · uu∗ ⊗ 1 α(u) − u ⊗ 1 √ 2ε + 28 ε + ϕ · uu∗ ⊗ 1 · α(u) − u ⊗ 1 √ √ 30 ε + 4ε + ϕ · α(u) − u ⊗ 1 35 ε. √ √ Hence ϕ · (v − 1) 75 ε, and v ∗ − 12ϕ 150 ε holds. This implies that v − 1#2 ϕ = √ √ 1 2 ∗ 2 # 4 2 (v − 1ϕ + v − 1ϕ ) 142 ε, and v − 1ϕ 12 ε. 2


2011

on R∞ . Then α is cocycle conjugate to Theorem 6.14. Let α be a centrally free action of G idRλ ⊗ α for all 0 < λ < 1. Proof. Set M := R∞ , εn := 16−n . Let {Fn }∞ n=1 be an increasing sequence of finite sets of Irr(G) ∞ with ∞ n=1 Fn = Irr(G). Let {ψn }n=1 ⊂ (M∗ )+ be a countable dense subset such that ψ1 is a faithful state. For each k ∈ N, we will construct a sequence of mutually commuting 2 × 2-matrix with the following five conditions: units {eij (k)}2i,j =1 , and unitaries v k , v¯ k ∈ M ⊗ L∞ (G) v¯ n = v n v n−1 · · · v 1 ; Ad v¯πn ◦ απ eij (k) = eij (k) ⊗ 1π , i, j = 1, 2, 1 k n, π ∈ Fn ; n √ v − 1# < 12 4 εn , π ∈ Fn ; ψ1 ⊗trπ

π

n v − 1 # π (ψ

√ < 12 4 εn , ψk · eij (n) − λi−j eij (n) · ψk < 2εn , n−1∗ 1 ⊗trπ )◦Ad v¯ π

π ∈ Fn ; 1 k n.

Since (M, α) has the property L a for any 0 < a < 1/2 by Lemma 6.12, we can choose u ∈ M such that uu∗ + u∗ u = 1, u2 = 0; u ⊗ 1 − απ (u) · (ψ1 ⊗ trπ ) < λε1 , (ψ1 ⊗ trπ ) · u ⊗ 1 − απ (u) < λε1 ,

π ∈ F1 ; π ∈ F1 ;

u · ψ1 − λψ1 · u < λ2 ε1 . √ Then by Lemma 6.13, there exists a unitary vπ1 such that vπ1 − 1#ψ1 ⊗trπ < 12 4 ε1 , π ∈ F1 , and Ad vπ1 ◦ απ (u) = u ⊗ 1. We define vρ1 , ρ ∈ / F1 , in a similar way as in the proof of Lemma 6.13. Set {e11 (1), e12 (1), e21 (1), e22 (1)} := {uu∗ , u, u∗ , u∗ u}. Note that [eii (1), ψ1 ] < 2ε1 , so the first step is complete.

Suppose we have done up to the n-th step. Set En := nk=1 ({eij (k)}2i,j =1 ) , α n+1 := Ad v¯ n ◦α, and Mn+1 := En ∩ M. Then α n+1 is a centrally free cocycle action on Mn+1 ∼ = R∞ . Hence n (Mn+1 , α n+1 ) has the property L a by Lemma 6.12. Let {w }4=1 be a basis for En∗ with w 1, n and decompose ψk = 4=1 w ⊗ ψk . Take u ∈ Mn+1 satisfying uu∗ + u∗ u = 1, u2 = 0 and the following conditions: for any π ∈ Fn+1 , u ⊗ 1 − α n+1 (u) · (ψ1 ⊗ trπ ) < λεn+1 ; π (ψ1 ⊗ trπ ) · u ⊗ 1 − α n+1 (u) < λεn+1 ; π u ⊗ 1 − α n+1 (u) · (ψ1 ⊗ trπ ) ◦ Ad v¯ n∗ < λεn+1 ; π π (ψ1 ⊗ trπ ) ◦ Ad v¯ n∗ · u ⊗ 1 − α n+1 (u) < λεn+1 ; π π (u ⊗ 1) · (ψ1 ⊗ trπ ) ◦ Ad v¯ n∗ − λ (ψ1 ⊗ trπ ) ◦ Ad v¯ n∗ · (u ⊗ 1) < λ2 εn+1 ; π

u · ψk − λψk · u < 4

π

−n 2

λ εn+1 ,

1 k n + 1, 1 4n .

2012


Here we have regarded ψ1 and (ψ1 ⊗ trπ ) ◦ Ad v¯πn∗ as states on Mn+1 and Mn+1 ⊗ B(Hπ ), respectively. The last inequality yields u · ψk − λψk · u λ2 εn+1 , and in particular, (u ⊗ 1) · (ψ1 ⊗ tr) − λ(ψ1 ⊗ tr) · (u ⊗ 1) λ2 εn+1 . By Lemma 6.13, there exists a unitary vπn+1 ∈ Mn+1 ⊗ B(Hπ ) for π ∈ Fn+1 such that Ad vπn+1 ◦ απn (u) = u ⊗ 1, π ∈ Fn+1 ; n+1 # √ v − 1ψ ⊗tr < 12 4 εn+1 , π ∈ Fn+1 ; π π 1 # n+1 √ v − 1(ψ ⊗tr )◦Ad v¯ n∗ < 12 4 εn+1 , π ∈ Fn+1 . π π

1

π

Set {e11 (n + 1), e12 (n + 1), e21 (n + 1), e22 (n + 1)} := {uu∗ , u, u∗ , u∗ u}. Then ψk · eij (n + 1) − λi−j eij (n + 1) · ψk < 2εn+1 holds for 1 k n + 1. Define v n+1 by extending vπn+1 , π ∈ Fn+1 , as before. Thus we have finished the (n + 1)-st and this completes our induction. step, ∞ i−j e (k) · ψ < ∞ for all {e (k)} . Since ψ Define E∞ := ∞ n · eij (k) − λ ij n k=1 ij k=1 i,j =1,2 ∩M ∼ n ∈ N, E∞ is an injective factor of type IIIλ , and we have the factorization M = E∞ ∨ E∞ = E∞ ⊗ E∞ ∩ M by [1, Theorem 1.3]. (Also see [27, Lemma XVIII.4.5].) Next we show the convergence of {v¯πn }∞ n=1 . If π ∈ Fn , we have n+1 v¯ − v¯ n π

π ψ1 ⊗trπ

= vπn+1 − 1 v¯πn ψ⊗tr = vπn+1 − 1(ψ⊗tr )◦Ad v¯ n∗ π π π √ √ < 12 2 4 εn+1

and n+1 ∗ v¯ − v¯ n π

π

ψ1 ⊗trπ

∗ = vπn+1 − 1 ψ⊗tr π √ √ 4 < 12 2 εn+1 .

Hence for each π ∈ Irr(G), {v¯πn }∞ n=1 is a Cauchy sequence in the strong* topology, and set v¯ π := n By the choice of vπn , α := Ad v¯ ◦ α acts trivially limn→∞ v¯π . Set v¯ = (v¯π )π ∈ M ⊗ L∞ (G). ∩ M with a 2-cocycle u = v¯ (12) (α ⊗ id)(v)(id ¯ ⊗ on E∞ . Hence α is a cocycle action on E∞ ∗ )(v¯ ). Since E∞ ∩ M is of type III, u is a coboundary by Lemma 3.2. Hence α is cocycle ∩ M. Since E ⊗ E ∼ E ∼ R , on E∞ conjugate to idE∞ ⊗ β for some action β of G ∞ ∞ = ∞ = λ α ∼ idE∞ ⊗ β ≈ idE∞ ⊗ idE∞ ⊗ β ∼ idRλ ⊗ α. 2 Corollary 6.15. Let M ∼ = R∞ , N = M σ ϕ Z, T = −2π/ log λ and θ be as before. Let α be a T on R∞ . Then the G × T-action αθ is cocycle conjugate to idR ⊗ αθ . centrally free action of G λ Proof. This is immediate from Lemma 6.7 and Theorem 6.14 when we consider the state of the form ϕλ ⊗ ϕ on Rλ ⊗ M, where ϕλ is a periodic state on Rλ . 2 6.4. Model action splitting ∼ R∞ . Then there exists a on M = Lemma 6.16. Let α be a centrally free cocycle action of G ∗ + u∗ u = 1, u2 = 0, θ (u ) = } ⊂ N with u u centralizing sequence of partial isometries {u n n n n t n n n n √ e −1t un for all t ∈ R and limn→∞ α(un ) − un ⊗ 1 = 0 in the σ -strong* topology.


2013

Proof. Since M is properly infinite, α is cocycle conjugate to an action α . Then α ∼ idRλ ⊗ α by Theorem 6.14 and Rλ ∼ = R0 ⊗ Rλ , and hence α is cocycle conjugate to idR0 ⊗ α via an M. By Lemma 6.7, it suffices to show the statements for idR0 ⊗ α isomorphism R0 ⊗ M ∼ = and N = (R0 ⊗ M) σ tr⊗ϕ Z assuming that α is an action. We denote by U the implementing T unitary. ∗ Let {vn }∞ n=1 ⊂ R0 ⊗C1 ⊂ R0 ⊗M be a centralizing sequence of partial isometries with vn vn + ∞ ∗ 2 vn vn = 1, vn = 0, and (idR0 ⊗ α)(vn ) = vn ⊗ 1. Let {wn }n=1 ⊂ C1 ⊗ M be as in Corollary 3.26. tr⊗ϕ Set un := U ∗ wn vn∗ for each n ∈ N. Since [wn , vn ] = 0 and U vn U ∗ = σT (vn ) = vn , we have un u∗n = vn vn∗ , u∗n un = vn∗ vn ∈ M, un u∗n + u∗n un = 1 and u2n√= 0. Since (U ∗ wn )n is centralizing, −1t u for all t ∈ T. Take a faithful {un }∞ n n=1 is a centralizing sequence in N , and θt (un ) = e normal state ψ on N ⊗ B(Hπ ). Then we have ψ · (id ⊗ α π )(un ) − un ⊗ 1 = ψ · U ∗ ⊗ 1 [Dϕ ◦ Φπ : Dφ ⊗ trπ ]∗T (id ⊗ απ )(wn ) vn∗ ⊗ 1 − wn vn∗ ⊗ 1 = ψ · U ∗ ⊗ 1 [Dϕ ◦ Φπ : Dφ ⊗ trπ ]∗T − (wn ⊗ 1)(id ⊗ απ ) wn∗ →0 as n → ∞. In a similar way, we get limn→∞ ((id ⊗ α π )(un ) − un ⊗ 1) · ψ = 0. These implies that α π (un ) − un ⊗ 1 converges to 0 σ -strongly*. 2 on M. Let β be the Theorem 6.17. Let M, N , θ be as before. Let α be a centrally free action of G × T-action infinite tensor product type action of T on R0 given in Proposition 6.10. Then the G αθ is cocycle conjugate to β ⊗ αθ . Proof. The proof is similar to that of Theorem 6.14. Set εn := 16−n . Let {ψn }∞ n=1 ⊂ (M∗ )+ be a countable dense subset such that ψ1 is a faithful state. We will construct a sequence of mutually commuting 2 × 2-matrix units {eij (k)}2i,j =1 ⊂ N , and unitaries v k , v¯ k ∈ M ⊗ B(Hπ ), k ∈ N, with the following: v¯ k = v k v k−1 · · · v 1 ; Ad v¯πn ◦ α π eij (k) = eij (k) ⊗ 1 for all i, j = 1, 2, 1 k n, π ∈ Fn ; n √ v − 1 < 12 4 εn for all π ∈ Fn ; π ψ1 ⊗trπ n √ v − 1 < 12 4 εn for all π ∈ Fn ; π (ψ1 ⊗trπ )◦Ad v¯ n−1∗ 1 √0 eij (k) for all t ∈ R, i, j = 1, 2, k ∈ N; θt eij (k) = Ad −1t 0 e ψk · eij (n) − eij (n) · ψk < 2εn for all i, j = 1, 2, 1 k n. By Lemma 6.16, there exists a partial isometry u ∈ N such that u2 = 0, uu∗ + u∗ u = 1, √ −1t u, uu∗ , u∗ u ∈ M and θt (u) = e u ⊗ 1 − α π (u) · (ψ1 ⊗ trπ ) < ε1

for all π ∈ F1 ;

2014


(ψ1 ⊗ trπ ) · u ⊗ 1 − α π (u) < ε1

for all π ∈ F1 ;

u · ψ1 − ψ1 · u < ε1 . Since uu∗ ⊗ 1π , α π (uu∗ ) = απ (uu∗ ) ∈ M ⊗ B(Hπ ), we can take w from M in the proof of Lemma 6.13. Then vπ1 constructed in Lemma 6.13 is in M ⊗ B(Hπ ), and we have 1 4 v − 1# < 12√ ε1 π ψ 1

for all π ∈ F1 ;

Ad vπ1 ◦ α π (u) = u ⊗ 1 for all π ∈ F1 . by extendSet {e11 (1), e12 (1), e21 (1), e22 (1)} := {u∗ u, u∗ , u, uu∗ }. Define v 1 ∈ M ⊗ L∞ (G) 1 ing vπ , π ∈ F1 , as before. Note that [eij (1), ψ1 ] < 2ε1 for i, j = 1, 2. So the first step is complete. Set α 2 := Ad v 1 ◦ α, and N2 := {u} ∩ N . Take w ∈ M an isometry with ww ∗ = e11 (1). Set s1 := e11 (1)w and s2 := e21 (1)w. Then si sj∗ = eij (1), θt (s1 ) = s1 and θt (s2 ) = e−it s2 hold. Let ρ(x) := 2i=1 si xsi∗ . Then ρ is an isomorphism between N and N2 which intertwines θ . Then ϕ

M2 := N2θ = ρ(M) is the injective factor of type III1 , and θ is the dual action for σT where ϕ := ϕ ◦ ρ −1 ∈ (M2 )∗ . Since θ commutes with α 2 because of vπ1 ∈ M ⊗ B(Hπ ), α 2 preserves M2 . Note that v 1 α(v 1 )(id ⊗ )((v 1 )∗ ), a 2-cocycle of α 2 is in N2 and fixed by θ , and it is indeed ) = Z(N 2 ). Hence in M2 . This means that α|M2 is a cocycle action. Obviously we have Z(N 2 2 α has trivial Connes–Takesaki module, and α is approximately inner. By Lemma 6.4, α 2 is the canonical extension of α 2 := α 2 |M2 . Since α is centrally free, α 2 is centrally free, and α 2 is centrally free on M2 by Lemma 6.4. Then we can apply Lemma 6.16 to M2 , α 2 , and θ . The rest of the proof is same as that of Theorem 6.14. 2 Acknowledgments

A part of this work was done while the first named author stayed at Fields Institute and the second named author stayed at Katholieke Universiteit Leuven. They would like to thank their warm hospitality. Appendix A We discuss relations between the canonical extension of endomorphisms and homomor phisms. In this section, we do not assume the amenability of G. A.1. Canonical extension of homomorphisms Let M be a properly infinite factor and H a finite dimensional Hilbert space with dim H = n. be the canonical core of M [7, Definition 2.5]. We denote by TrH and trH the nonLet M normalized and the normalized traces on B(H ), respectively. Then we can introduce an isomor and M ⊗ B(H ) ⊂ M ⊗ B(H ) as follows. Fix isometries phism between the inclusions M ⊂ M n n ∗ ⊗ B(H ), M) by {vi }i=1 ⊂ M with orthogonal ranges and i=1 vi vi = 1. Define σ ∈ Mor(M σ (x) =

n i,j =1

vi xij vj∗ .


It is easy to see that σ is an isomorphism with σ −1 (x) = the following bijection:

n

∗ i,j =1 vi xvj

σ∗ : Mor0 M, M ⊗ B(K) → End0 (M),

2015

⊗ eij . The map σ derives

α → σ ◦ α.

We can check that d(α) = d(σ ◦ α) and the standard left inverse of ρ := σ ◦ α is given by φρ = Φα ◦ σ −1 . Hence Φ α (x) = φρ ◦ σ (x) = ni,j =1 φρ (vi xij vj∗ ) holds. Recall the topology on End0 (M) introduced in [19, Definition 2.1]. We also introduce a topology on Mor0 (M, M ⊗ B(H )) similarly. Lemma A.1. The map σ∗ is a homeomorphism. Proof. Take arbitrary ϕ ∈ M∗ . Assume that α ν → α in Mor0 (M, M ⊗ B(H )) as ν → ∞, that ν is, we have the norm convergence ϕ ◦ Φ α → ϕ ◦ Φ α in (M ⊗ B(H ))∗ . Write ρ ν = σ∗ (α ν ) ν and ρ = σ∗ (α). Using φρ ν = Φ α ◦ σ −1 and φρ = Φ α ◦ σ −1 , we have the norm convergence ϕ ◦ φρ ν → ϕ ◦ φρ , that is, ρ ν → ρ as ν → ∞. Hence σ∗ is continuous. Similarly we can prove that σ∗−1 is continuous. 2 Lemma A.2. Let ϕ be a faithful normal state on M. Then one has n ϕ α Dϕ ◦ Φ : Dϕ ⊗ Tr t = vi∗ [Dϕ ◦ φσ∗ (α) : Dϕ]t σt (vj ) ⊗ eij

for all t ∈ R.

i,j =1

ϕ Proof. Set ρ := σ∗ (α) and a unitary ut := ni,j =1 vi∗ [Dϕ ◦ φρ : Dϕ]t σt (vj ) ⊗ eij . Then ut is a ϕ⊗Tr σ -cocycle. We verify that ut satisfies the relative modular condition. Let D := {z ∈ C | 0 < Im(z) < 1}, and A(D) := f (z) f (z) is analytic on D, bounded, continuous on D . Take x, y ∈ M ⊗ B(H ). By the relative modular condition for [Dϕ ◦ Φ α : Dϕ]t , n ∗ j =1 yj vj , we can choose F (z) ∈ A(D) such that F (t) =

n

ϕ ϕ ◦ φρ [Dϕ ◦ φρ : Dϕ]t σt (vk xk )yj vj∗

n

k=1 vk xk

for all t ∈ R

j,k=1

and F (t +

√

−1 ) =

n

ϕ ϕ yj vj∗ [Dϕ ◦ φρ : Dϕ]t σt (vk xk )

j,k=1

Set F (z) :=

n

=1 F (z) ∈ A(D).

Then we have

n ϕ⊗Tr ϕ⊗Tr (x)y = ϕ ◦ φρ vi ut σt (x)y ij vj∗ ϕ ◦ Φ α ut σt i,j =1

for all t ∈ R.

and

2016


=

n

ϕ ϕ ◦ φρ vi ut,ik σt (xk )yj vj∗

i,j,k,=1

=

n

ϕ ϕ ϕ ◦ φρ vi vi∗ [Dϕ ◦ φρ : Dϕ]t σt (vk )σt (xk )yj vj∗

i,j,k,=1

=

n

ϕ ϕ ◦ φρ [Dϕ ◦ φρ : Dϕ]t σt (vk xk )yj vj∗

j,k,=1

= F (t), and n n ϕ⊗Tr ϕ⊗Tr ϕ (ϕ ⊗ Tr) yut σt (x) = ϕ yut σt (x) = ϕ yj ut,j k σt (xk ) =1

=

j,k,=1

n

ϕ ϕ ϕ yj vj∗ [Dϕ ◦ φρ : Dϕ]t σt (vk )σt (xk )

j,k,=1

= F (t +

√ −1 ).

This shows that ut satisfies the relative modular condition.

2

be the canonical extension [12, Theorem 2.4]. We define the Let ∼ : End(M)0 → End(M) M ⊗ B(K)) by map ∼ : Mor0 (M, M ⊗ B(K)) → Mor(M, α = σ −1 ◦ σ ◦α

for all α ∈ Mor0 M, M ⊗ B(K) .

In fact, α does not depend on σ as follows. Theorem A.3. One has the following: (1) α (x) = α(x) for all x ∈ M; (2) α (λϕ (t)) = d(α)it [Dϕ ◦ Φ α : Dϕ ⊗ TrK ]t (λϕ (t) ⊗ 1) for all t ∈ R. Proof. Set ρ := σ∗ (α). Then by definition, we have ρ (x) = ρ(x) for all x ∈ M; ϕ ρ λ (t) = d(ρ)it [Dϕ ◦ φρ : Dϕ]t λϕ (t) for all t ∈ R. Since σ −1 ◦ ρ = α, (1) follows. On (2), we have n ϕ α λϕ (t) = σ −1 ρ λ (t) = vk∗ ρ λϕ (t) v ⊗ ek k,=1


=

n

2017

d(ρ)it vk∗ [Dϕ ◦ φρ : Dϕ]t λϕ (t)v ⊗ ek

k,=1

=

n

ϕ d(ρ)it vk∗ [Dϕ ◦ φρ : Dϕ]t σt (v ) ⊗ ek λϕ (t) ⊗ 1

k,=1

= d(α)it Dϕ ◦ Φ α : Dϕ ⊗ TrK t λϕ (t) ⊗ 1 (by Lemma A.2).

2

We say that α ∈ Mor0 (M, M ⊗ B(K)) is inner if there exists a unitary U ∈ M ⊗ B(K) such that α = U (· ⊗ 1)U ∗ . Denote by Int(M, M ⊗ B(K)), Int(M, M ⊗ B(K)) and Cnt(M, M ⊗ B(K)) the set of inner homomorphisms, approximately inner homomorphisms and centrally trivial homomorphisms in Mor0 (M, M ⊗ B(K)), respectively. (See Definition 2.1.) Then we have the following bijective correspondence. See [19] for the notations used here. Lemma A.4. The bijection σ∗ : Mor0 (M, M ⊗ B(K)) → End0 (M) yields the following bijective maps: (1) σ∗ : Int(M, M ⊗ B(K)) → Intdim(K) (M); (2) σ∗ : Int(M, M ⊗ B(K)) → Intdim(K) (M); (3) σ∗ : Cnt(M, M ⊗ B(K)) → Cnd(M). Proof. (1) Assume that α = Ad U (· ⊗ 1) for some unitary U ∈ M ⊗ B(K). Set ρ := σ∗ (α) and a Hilbert space H ⊂ M which is spanned by wk := ni=1 vi Uik , k = 1, . . . , n. Then for x ∈ M, we have n ρ(x) = σ α(x) = σ U (x ⊗ 1)U ∗ = σ Uik xUj∗k ⊗ eij i,j,k=1

=

n

vi Uik xUj∗k vj∗ =

i,j,k=1

n

wk xwk∗ = ρH (x).

k=1

Hence ρ = ρH ∈ Intdim(K) (M). Conversely if we have ρ = ρH with dim H = n, then setting Uik := vi∗ wk for some orthonormal basis {wk }nk=1 ⊂ H, we have σ −1 ◦ ρ = Ad U (· ⊗ 1). (2) This follows from (1) and Lemma A.1. (3) Assume that α ∈ Cnt(M, M ⊗ B(K)). Set ρ := σ∗ (α). Take an ω-centralizing sequence ν ν (x ν )ν in M. Then as ν → ω. Hence ρ(x ν ) − σ (x ν ⊗ 1) → 0. Since n α(x ) −ν x∗ ⊗ 1 → 0 strongly* ν ν σ (x ⊗ 1) = i,j =1 vi x vi , we see that ρ(x ) − x ν → 0, that is, ρ ∈ Cnd(M). The converse can be proved similarly. 2 We define the following set: MorCT M, M ⊗ B(K) = α ∈ Mor0 M, M ⊗ B(K) σ∗ (α) ∈ End(M)CT . The following lemma shows that this set does not depend on σ .

2018


Lemma A.5. Let α ∈ Mor0 (M, M ⊗ B(K)). Then the following are equivalent: (1) α ∈ MorCT (M, M ⊗ B(K)); such that α (z) = γ (z) ⊗ 1 for all z ∈ Z(M). (2) There exists γ ∈ Autθ (Z(M)) Proof. Assume that α ∈ MorCT (M, M ⊗ B(K)). Set ρ := σ∗ (α). Then ρ has a Connes–Takesaki For z ∈ Z(M), we have module mod(ρ). By definition, σ −1 (z) = z ⊗ 1 for z ∈ Z(M). (z) = σ −1 mod(ρ)(z) = mod(ρ)(z) ⊗ 1. α (z) = σ −1 ρ Conversely, assume that such γ exists. Then we have ρ (z) = σ α (z) = σ γ (z) ⊗ 1 = γ (z). Hence ρ has the Connes–Takesaki module γ , that is, α ∈ MorCT (M, M ⊗ B(K)).

2

In this situation, we say that α has the Connes–Takesaki module mod(α) := γ . Theorem A.6. Let M be a properly infinite injective factor. Then one has the following: (1) α ∈ Int(M, M ⊗ B(K)) ⇔ α ∈ MorCT (M, M ⊗ B(K)) with mod(α) = θlog(dim(K)/d(α)) ; (2) α ∈ Cnt(M, M ⊗ B(K)) such that α = U (· ⊗ 1)U ∗ . ⇔ There exists a unitary U ∈ Md(α),dim(K) (M) Proof. This follows from [18, Theorem 3.15, 4.12]. Note that if α ∈ Cnt(M, M ⊗ B(K)), then d(α) is an integer [12, Theorem 3.3 (5)]. 2 We obtain the following corollary. Corollary A.7. The following statements hold: (1) If M = R0,1 , then • α ∈ Int(M, M ⊗ B(K)) ⇔ τ ◦ Φ α = τ ⊗ trK , where τ is a trace on M; • α ∈ Cnt(M, M ⊗ B(K)) ⇔ there exist n ∈ N and a unitary U ∈ M ⊗ Mdim(K),n (C) such that α(x) = U (x ⊗ 1)U ∗

for all x ∈ M.

(2) If M = Rλ with 0 < λ < 1, then • α ∈ Int(M, M ⊗ B(K)) ⇔ [Dϕ ◦ Φ α : Dϕ ⊗ trK ]T = 1, where ϕ is a generalized trace on M and T = −2π/ log λ; • α ∈ Cnt(M, M ⊗ B(K)) ⇔ there exist n ∈ N, a unitary U ∈ M ⊗ Mdim(K),n (C) and {si }ni=1 ⊂ R such that α(x) = U diag σsϕ1 (x), . . . , σsϕn (x) U ∗

for all x ∈ M.


2019

(3) If M = R∞ , then • Int(M, M ⊗ B(K)) = Mor0 (M, M ⊗ B(K)); • α ∈ Cnt(M, M ⊗ B(K)) ⇔ there exist n ∈ N, a unitary U ∈ M ⊗ Mdim(K),n (C) and {si }ni=1 ⊂ R such that α(x) = U diag σsϕ1 (x), . . . , σsϕn (x) U ∗

for all x ∈ M.

A.2. Canonical extension of cocycle actions on a We discuss canonical extension of cocycle actions. Let (α, u) be a cocycle action of G factor M. For π ∈ Irr(G), we define the left inverse Φπα for απ by ∗ ∗ uπ,π (απ ⊗ id)(x)uπ ,π (1 ⊗ Tπ ,π ) Φπα (x) = 1 ⊗ Tπ,π

for all x ∈ M ⊗ B(Hπ ),

where Tπ,π is an isometry intertwining 1 and π ⊗ π [18, p. 491]. Then απ ◦ Φπα is a faithful normal conditional expectation from M ⊗ B(Hπ ) onto απ (M). Set d(π) := dim(Hπ ). Recall the of u [18, Definition 5.5]: diagonal operator a ∈ M ⊗ L∞ (G) (a ⊗ 1) 1 ⊗ (e1 ) = u 1 ⊗ (e1 ) . Lemma A.8. One has Ind(απ ◦ Φπα ) = d(π)2 for all π ∈ Irr(G). d(π)

Proof. Set Eπ := απ ◦ Φπα and d(π) = dim Hπ . Let {eπij }i,j =1 be a system of matrix units of B(Hπ ). We show that {d(π)1/2 (1 ⊗ eπij )aπ∗ }i,j =1 is a quasi basis for Eπ [28, Definition 1.2.2]. For any y ∈ M and 1 k, d(π), we have d(π)

Φπα aπ (1 ⊗ eπj i )(y ⊗ eπk ) = δik 1 ⊗ Tπ∗,π u∗π ,π απ (aπ ) απ (y) ⊗ eπj uπ ,π (1 ⊗ Tπ ,π ) = δik 1 ⊗ Tπ∗,π aπ∗ ⊗ 1 απ (aπ ) απ (y) ⊗ eπj (aπ ⊗ 1)(1 ⊗ Tπ,π ) = δik 1 ⊗ Tπ∗,π απ (y)aπ ⊗ eπj (1 ⊗ Tπ,π ) by [18, Lemma 5.6 (1)] = d(π)−1 δik απ (y)aπ π . j

Using this equality, we have d(π)

d(π)(1 ⊗ eπij )aπ∗ Eπ aπ (1 ⊗ eπj i )(y ⊗ ek )

i,j =1

=

d(π) i,j =1

=

d(π) j =1

d(π) δik (1 ⊗ eπij )aπ∗ απ απ (y)aπ π = (1 ⊗ eπkj )aπ∗ απ απ (y)aπ π j

j =1

(1 ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j aπ∗ ⊗ 1 απ απ (y)aπ (1 ⊗ 1 ⊗ επ )

j

2020


=

d(π) j =1

=

d(π) j =1

=

d(π) j =1

=

d(π) j =1

=

d(π) j =1

(1 ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j u∗π,π απ απ (y)aπ (1 ⊗ 1 ⊗ επ ) (1 ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j (id ⊗ ) α(y) u∗π,π απ (aπ )(1 ⊗ 1 ⊗ επ ) (y ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j u∗π,π απ (aπ )(1 ⊗ 1 ⊗ επ ) (y ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j aπ∗ ⊗ 1 απ (aπ )(1 ⊗ 1 ⊗ επ ) (y ⊗ επk ) 1 ⊗ επ∗j ⊗ επ∗ j (1 ⊗ 1 ⊗ επ ) = y ⊗ eπk .

Hence {d(π)1/2 (1 ⊗ eπij )aπ∗ }i,j =1 is a quasi basis for Eπ , and we have d(π)

Ind(Eπ ) =

d(π)

d(π)1/2 (1 ⊗ eπij )aπ∗ · d(π)1/2 aπ 1 ⊗ eπ∗ ij

i,j =1

= d(π)2 (id ⊗ trπ ) aπ∗ aπ = d(π)2

by [18, Lemma 5.6 (2)] .

2

is standard when the Definition A.9. We say that a cocycle action α ∈ Mor(M, M ⊗ L∞ (G)) α left inverse Φπ is standard for each π ∈ Irr(G). be a cocycle action. Then the following hold: Proposition A.10. Let α ∈ Mor(M, M ⊗ L∞ (G)) then α (1) If α is cocycle conjugate to a standard cocycle action β ∈ Mor(M 2 , M 2 ⊗ L∞ (G)), is standard; (2) If α is free, then α is standard; is amenable, then α is standard. (3) If G Proof. (1) Let π ∈ Irr(G). Since the inclusion βπ (M 2 ) ⊂ M 2 ⊗ B(Hπ ) is isomorphic to απ (M) ⊂ M ⊗ B(Hπ ), [M ⊗ B(Hπ ) : απ (M)]0 = [βπ (M 2 ) : M 2 ⊗ B(Hπ )]0 = d(π)2 . Hence α is standard. (2) For any π ∈ Irr(G), the expectation απ ◦ Φπα is minimal because of the irreducibility of the inclusion απ (M) ⊂ M ⊗ B(Hπ ) [18, Lemma 2.8]. Hence α is standard. (3) Since [B(2 ) ⊗ M ⊗ B(Hπ ) : B(2 ) ⊗ απ (M)]0 = [M ⊗ B(Hπ ) : απ (M)]0 , we may and do assume that M is properly infinite by considering id ⊗ α. Then α is cocycle conjugate to an action β on M by Lemma 3.2. By (1), it suffices to show that β is standard. We check Eπ−1 =


2021

d(π)2 Eπ on Qπ := βπ (M) ∩ (M ⊗ B(Hπ )) to use [10, Theorem 1 (2)]. Take x ∈ Qπ . Then by [28, p. 62, Remark], we have Eπ−1 (x) =

d(π)

d(π)1/2 (1 ⊗ eπij )xd(π)1/2 1 ⊗ eπ∗ ij = d(π)2 (id ⊗ trπ )(x) ⊗ 1 .

i,j =1

So, Eπ is minimal if and only if the following holds: (id ⊗ trπ )(x) = Φπβ (x) ∈ C.

(A.1)

If we can find a β-invariant state ψ ∈ M ∗ , the proof is finished. Indeed, applying ψ to Φπ , we have β

∗ (ψ ⊗ id ⊗ id) βπ (x) Tπ ,π = Tπ∗,π 1π ⊗ (ψ ⊗ id)(x) Tπ,π ψ Φπβ (x) = Tπ,π = (ψ ⊗ trπ )(x). ∗ . Take Hence (A.1) holds. Such a state ψ is constructed by using an invariant mean m ∈ L∞ (G) α a state ϕ on M and set ψ := m((ϕ ⊗ id)(β(x))). Then we have (ψ ⊗ id)(β(x)) = ψ(x)1 for all x ∈ M, that is, ψ is invariant under β. 2 on a factor standard? Problem A.11. Is any cocycle action of G be a standard cocycle action with a 2-cocycle u. Now for Let α ∈ Mor(M, M ⊗ L∞ (G)) M ⊗ B(Hπ )). Collecting ( απ )π , π ∈ Irr(G), we consider the canonical extension απ ∈ Mor(M, which is called the canonical extension of the M ⊗ L∞ (G)), we obtain a map α ∈ Mor(M, action α. We have the following equalities: απ (x) = απ (x) for all x ∈ M; ϕ απ λ (t) = Dϕ ◦ Φπα : Dϕ ⊗ trπ t λϕ (t) ⊗ 1 for all t ∈ R, ϕ ∈ W (M). The following two results even for actions of general Kac algebras are obtained in [29], where operator valued weight theory is fully used, but we can directly prove them for a discrete Kac We present their proofs for readers’ convenience. algebra G. by Take ϕ ∈ W (M). For t ∈ R, we define wt = (wt,π )π ∈ U (M ⊗ L∞ (G)) ∗ wt,π = Dϕ ◦ Φπα : Dϕ ⊗ trπ t . Lemma A.12. The unitary wt satisfies the following: ϕ (wt ⊗ 1)α(wt )u(id ⊗ ) wt∗ = σt ⊗ id (u). Proof. By the chain rule of Connes’ cocycles, we may and do assume that ϕ is a state. Let π, ρ ∈ Irr(G). Using the isomorphism απ−1 : απ (M) → M, we have ∗ απ (wt,ρ ) = Dϕ ◦ Φρα ◦ απ−1 ⊗ id : Dϕ ◦ απ−1 ⊗ trρ t .

2022


Since Eπ := απ ◦ Φπα : M ⊗ B(Hπ ) → απ (M) is a conditional expectation, we have ∗ απ (wt,ρ ) = Dϕ ◦ Φρα ◦ απ−1 ⊗ id ◦ (Eπ ⊗ id) : Dϕ ◦ απ−1 ◦ Eπ ⊗ trρ t ∗ = Dϕ ◦ Φρα ◦ Φπα ⊗ id : Dϕ ◦ Φπα ⊗ trρ t = Dϕ ◦ Φπα ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id t . Then we have (wt,π ⊗ 1ρ )απ (wt,ρ ) = Dϕ ⊗ trπ : Dϕ ◦ Φπα t ⊗ 1ρ Dϕ ◦ Φπα ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id t = Dϕ ⊗ trπ ⊗ trρ : Dϕ ◦ Φπα ⊗ trρ t Dϕ ◦ Φπα ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id t = Dϕ ⊗ trπ ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id t . Multiplying (σt ⊗ id ⊗ id)(u∗π,ρ ) and uπ,ρ to the both sides, we have ϕ

ϕ σt ⊗ id ⊗ id u∗π,ρ (wt,π ⊗ 1ρ )απ (wt,ρ )uπ,ρ ϕ = σt ⊗ id ⊗ id u∗π,ρ uπ,ρ · Dϕ ⊗ trπ ⊗ trρ ◦ Ad uπ,ρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id ◦ Ad uπ,ρ t = Dϕ ⊗ trπ ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id ◦ Ad uπ,ρ t . (A.2) Recall the following formula [18, Lemma 2.5]: for X ∈ M ⊗ B(Hπ ) ⊗ B(Hρ ), Φρα ◦ Φπα ⊗ id uπ,ρ Xu∗π,ρ =

σ ≺π·ρ S∈ONB(σ,π·ρ)

d(σ ) Φσα 1 ⊗ S ∗ X(1 ⊗ S) . d(π)d(ρ)

Hence for S ∈ ONB(σ, π · ρ), we have Φρα Φπα ⊗ id uπ,ρ 1 ⊗ SS ∗ Xu∗π,ρ =

d(σ ) Φσα 1 ⊗ S ∗ X(1 ⊗ S) d(π)d(ρ) = Φρα Φπα ⊗ id uπ,ρ X 1 ⊗ SS ∗ u∗π,ρ .

In particular, 1 ⊗ SS ∗ is in the centralizer of ϕ ◦ Φρα ◦ (Φπα ⊗ id) ◦ Ad uπ,ρ . Trivially, it is also in the centralizer of ϕ ⊗ trπ ⊗ trρ . Hence we see that the both sides of (A.2) commutes with 1 ⊗ SS ∗ , and we have (A.2) =


=

Dϕ ⊗ trπ ⊗ trρ : Dϕ ◦ Φρα ◦ Φπα ⊗ id t 1 ⊗ SS ∗ D(ϕ ⊗ trπ ⊗ trρ )1⊗SS ∗


: D ϕ ◦ Φρα ◦ Φπα ⊗ id ◦ Ad uπ,ρ 1⊗SS ∗ t ,

where the last cocycles are evaluated in (M ⊗ B(Hπ ) ⊗ B(Hρ ))1⊗SS ∗ .

(A.3)


2023

Let ΘS : B(Hσ ) → (B(Hπ ) ⊗ B(Hρ ))SS ∗ be the isomorphism defined by ΘS (x) = SxS ∗ for x ∈ B(Hσ ). Using (trπ ⊗ trρ )SS ∗ =

d(σ ) trσ ◦ ΘS−1 , d(π)d(ρ)

ϕ ◦ Φρα ◦ Φπα ⊗ id ◦ Ad uπ,ρ 1⊗SS ∗ =

d(σ ) ϕ ◦ Φσα ◦ id ⊗ ΘS−1 , d(π)d(ρ)

we have (A.3) =


=


=

Dϕ ⊗ trσ ◦ ΘS−1 : Dϕ ◦ Φσα ◦ id ⊗ ΘS−1 t (id ⊗ ΘS ) Dϕ ⊗ trσ : Dϕ ◦ Φσα t (id ⊗ )(wt ) 1 ⊗ SS ∗ = (id ⊗ π ρ )(wt ).


Thus we get ϕ σt ⊗ id ⊗ id u∗π,ρ (wt,π ⊗ 1ρ )απ (wt,ρ )uπ,ρ = (id ⊗ π ρ )(wt ).

2

on a factor M. Then the canonical Theorem A.13. Let (α, u) be a standard cocycle action of G extension ( α , u) is a cocycle action on M. Proof. We will check ( α ⊗ id) ◦ α = Ad u ◦ (id ⊗ ) ◦ α . This holds on M, since α = α on M. For t ∈ R, α(λϕ (t)) = wt∗ (λϕ (t) ⊗ 1). The previous lemma yields ϕ ( α ⊗ id) α λ (t) = ( α ⊗ id) wt∗ λϕ (t) ⊗ 1 = (α ⊗ id) wt∗ wt∗ ⊗ 1 λϕ (t) ⊗ 1 ⊗ 1 ϕ = u(id ⊗ ) wt∗ σt ⊗ id ⊗ id u∗ λϕ (t) ⊗ 1 ⊗ 1 ϕ ∗ 2 = u(id ⊗ ) α λ (t) u . on M. The canonical trace τ on M Lemma A.14. Let (α, u) be a standard cocycle action of G α is invariant under α , that is, τ ◦ Φπ = τ ⊗ trπ for all π ∈ Irr(G). ϕˆ such that hit = λϕ (t). Then the Proof. Let ϕ ∈ W (M). Take a positive operator h affiliated in M canonical trace is given by τ := ϕˆh−1 , which does not depend on the choice of the weight ϕ. Let → M be the averaging operator valued weight for θ . Then ϕˆ = ϕ ◦ Tθ . Since θ commutes Tθ : M with α , we have α : D ϕˆ ⊗ trπ t = Dϕ ◦ Φπα ◦ (Tθ ⊗ id) : Dϕ ◦ Tθ ⊗ trπ t = Dϕ ◦ Φπα : Dϕ ⊗ trπ t . D ϕˆ ◦ Φπ This implies

2024


α Dτ ◦ Φπ : Dτ ⊗ trπ

t

α α α D ϕˆ ◦ Φπ = Dτ ◦ Φπ : D ϕˆ ◦ Φπ : D ϕˆ ⊗ trπ t [D ϕˆ ⊗ trπ : Dτ ⊗ trπ ]t t = απ h−it Dϕ ◦ Φπα : Dϕ ⊗ trπ t hit ⊗ 1 = 1. 2

θ R. We call it the second canonical Since α commutes with θ , α extends to an action on M extension and denote that by α. be a standard action. The second canonical Corollary A.15. Let α ∈ Mor(M, M ⊗ L∞ (G)) extension α is cocycle conjugate to idB(2 ) ⊗ α. = M σ ϕ R. Define Proof. Let ϕ be a faithful normal semifinite weight on M. We regard M by w(t) = w−t for t ∈ R. Then w is an idB(L2 (R)) ⊗ α-cocycle. w(·) ∈ U (L∞ (R) ⊗ M ⊗ L∞ (G)) θ R ∼ By the Takesaki duality, there exists a canonical isomorphism M = B(L2 (R)) ⊗ M intertwining the actions α and Ad w ◦ (id ⊗ α). 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]

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[24] C.E. Sutherland, M. Takesaki, Actions of discrete amenable groups on injective factors of type IIIλ , λ = 1, Pacific J. Math. 137 (1989) 405–444. [25] C.E. Sutherland, M. Takesaki, Right inverse of the module of approximately finite-dimensional factors of type III and approximately finite ergodic principal measured groupoids, in: Operator Algebras and Their Applications, II, Waterloo, ON, 1994/1995, in: Fields Inst. Commun., vol. 20, Amer. Math. Soc., Providence, RI, 1998, pp. 149–159. [26] M. Takesaki, Duality for crossed products and the structure of von Neumann algebras of type III, Acta Math. 131 (1973) 249–310. [27] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, Berlin, Heidelberg, New York, 2002; Theory of Operator Algebras II, III, Springer-Verlag, Berlin, Heidelberg, New York, 2003. [28] Y. Watatani, Index for C ∗ -subalgebras, Mem. Amer. Math. Soc. 83 (424) (1990). [29] T. Yamanouchi, Canonical extension of actions of locally compact quantum groups, J. Funct. Anal. 201 (2003) 522–560.


Anti-periodic solutions to nonlinear evolution equations ✩ Liu Zhenhai Faculty of Mathematics and Computer Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, PR China Received 13 March 2009; accepted 20 November 2009 Available online 2 December 2009 Communicated by C. Kenig

Abstract We deal with anti-periodic problems for nonlinear evolution equations with nonmonotone perturbations. The main tools in our study are the maximal monotone property of the derivative operator with anti-periodic conditions and the theory of pseudomonotone perturbations of maximal monotone mappings. © 2009 Elsevier Inc. All rights reserved. Keywords: Nonlinear evolution equalities; Anti-periodic solutions; Nonmonotone perturbations

1. Introduction In this paper, we study the following anti-periodic problem of nonlinear evolution equations with nonmonotone perturbations u (t) + Au(t) + Gu(t) = f, a.e. t ∈ (0, T ), (1) u(0) = −u(T ), in a real reflexive Banach space V , A is monotone and G is not. If we use the periodic condition u(0) = u(T ) or the zero-initial valued condition in problem (1), these kinds of problems have considered by many authors, for example, if G = 0, the ✩

Project supported partially by: NNSF of China Grant No. 10971019. E-mail address: [email protected].


L. Zhenhai / Journal of Functional Analysis 258 (2010) 2026–2033

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existence of solutions for the Cauchy problem (1) was obtained by Barbu [3] using the theory of monotone operators. While (1) with zero-initial valued condition was considered by Liu in [7,8] by use of the theory of pseudomonotone operators. H. Okochi [9] initiated the study for anti-periodic solutions of evolution equation in Hilbert spaces (see also [10,11]). Following Okochi’s work, Haraux [6] proved some existence and uniqueness theorems for anti-periodic solutions by using Brouwer’s or Schauder’s fixed point theorem. Later, Aizicovici and Pavel [2] studied the anti-periodic solutions of second order evolution equations in Hilbert and Banach spaces by using monotone and accretive operator theory, see Aizicovici, McKibben and Reich [1] for nonmonotone cases. In [4,5], Chen et al. studied by fixed point theorem the anti-periodic solution for the following first order semilinear evolution equation:

u (t) + Au(t) + ∂Gu(t) + F (t, u) = 0, a.e. t ∈ (0, T ), u(0) = −u(T ),

in a real separable Hilbert space H , where A : D(A) ⊆ H → H is a linear dense self-adjoint operator such that D(A) is compactly embedded into H , and ∂G is a continuous bounded mapping in H , and F : [0, T ] × H → H is a continuous mapping which is bounded above by some L2 functions. Our first purpose in this paper is to show the maximal monotone property of the derivative operator with anti-periodic conditions. Then we prove the existence of solutions for the antiperiodic problem (1) by using the theory of pseudomonotone perturbations of maximal monotone mappings. 2. Preliminaries Let V be a real reflexive Banach space densely and continuously embedded in a real Hilbert space H . Identifying H with its dual, we have V ⊆ H ⊆ V , where V stands for the dual of V . The norm of any Banach space B is denoted by · B . The duality pairing between B and its dual B is denoted by ·,·B . Let p, q and T be constants such that T > 0, p 2 and p1 + q1 = 1. Let X = Lp (0, T ; V ), X = Lq (0, T ; V ), I = [0, T ]. Let W = {u ∈ X; u ∈ X }. Then W with the norm uW = uX + u X is a Banach space (see Proposition 23.23 of [12]). The norm convergence is denoted by → and the weak convergence by . We give the basic assumptions. (H1 ) A : V → V is monotone and demicontinuous. (H2 ) G : V → V is both continuous and weakly continuous. Furthermore, for any sequence {un } in V with un u in V , we have lim supGun , un − uV 0. (H3 ) There exist positive constants c1 , c2 , c3 and c4 such that p−1 AuV c1 uV + 1 ,

p−1 GuV c2 uV + 1 , p

Au + Gu, uV c3 uV − c4

∀u ∈ V .

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3. Main results We define D(L) = u ∈ W : u(0) = −u(T ) .

Lu = u ,

(2)

Here v stands for the generalized derivative of v, i.e., T

T

v (t)φ(t) dt = − 0

v(t)φ (t) dt

∀φ ∈ C0∞ (I ).

0

Proposition 1. Let V ⊆ H ⊆ V be an evolution triple and let X = Lp (0, T ; V ) where 1 < p < ∞ and 0 < T < ∞. Then the linear operator L : D(L) ⊆ X → X defined by (2) is maximal monotone. Proof. We make essential use of the integration by parts formula T Lu, uX =

u (t), u(t) V dt

0

2

2

= 2−1 u(T ) H − u(0) H

(3)

for all u ∈ W , which shows the operator L is monotone by u(0) = −u(T ) for all u ∈ D(L). In the sequel we shall show that L is maximal monotone. To prove this, suppose that (v, v ∗ ) ∈ X × X and 0 v ∗ − Lu, v − u X

∀u ∈ D(L).

(4)

We have to show that v ∈ D(L) and v ∗ = Lv, i.e., v ∗ = v . To this end, choose where ϕ ∈ C0∞ (0, T ) and z ∈ V .

u = ϕz,

Then u = ϕ z and u ∈ D(L). By (3), Lu, uX = 0. From (4) it follows that

∗

0 v ,v

X

T −

ϕ (t)v(t) + ϕ(t)v ∗ (t), z V dt

∀z ∈ V .

(5)

0

In this connection, note that x, yV = x, yH = y, xV ∀x, y ∈ V by (23.17) of [12]. From (5) we get T 0

ϕ (t)v(t) + ϕ(t)v ∗ (t) dt = 0 ∀ϕ ∈ C0∞ (0, T ).


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Hence v = v∗ and v ∈ W , since v ∗ ∈ X . It remains to show that v ∈ D(L). Using integration by parts, we obtain from (3) that 0 v − u , v − u X

2

2

∀u ∈ D(L). = 2−1 v(T ) − u(T ) H − v(0) − u(0) H Therefore, we obtain

2

2 0 v(T ) − u(T ) H − v(0) − u(0) H

2

2

= v(T ) − v(0) + 2 u(0), v(0) + v(T ) H

H

H

∀u ∈ D(L).

(6)

Note that u(0) = −u(T ). In particular, we can choose u(t) = (1 − T2 t)y for arbitrary y ∈ V . Hence v(0) = −v(T ), i.e., v ∈ D(L). Note that V is dense in H . The proof is complete. 2 Then we have Proposition 2. Suppose that the assumptions (H1 )–(H3 ) hold. Then the mapping A + G : X → X is coercive, bounded, demicontinuous, and pseudomonotone. Proof. By (H1 )–(H3 ) we easily obtain that the sum operator A + G is coercive, bounded and demicontinuous. We only need to show that A + G is pseudomonotone. Suppose un u in X and lim supAun + Gun , un − uX 0.

(7)

We shall prove the pseudomonotonicity of A + G by showing Aun + Gun , wX → Au + Gu, wX

∀w ∈ X,

(8)

and lim infAun + Gun , un X = Au + Gu, uX .

(9)

Define hn (t) = Aun (t) + Gun (t), un (t) − u(t) V

∀t ∈ I.

First we show that for almost all t ∈ I , lim inf hn (t) 0.

(10)

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Suppose that the assertion is false, that is, there exists a subset Iε with its measure |Iε | > 0 such that for ∀t0 ∈ Iε , lim inf hn (t0 ) < 0.

(11)

hn (t0 ) = Aun (t0 ) + Gun (t0 ), un (t0 ) − u(t0 ) X

p

p−1

c3 un (t0 ) V − c4 − (c1 + c2 ) un (t0 ) V + 1 u(t0 ) V .

(12)

It follows from (H3 ) that

By (11) and (12), we get that {un (t0 )} is bounded in V . Therefore, passing to a subsequence if necessary we can assume that un (t0 ) u(t0 ) in V . Using assumption (H2 ), we obtain lim inf G(un )(t0 ), un (t0 ) − u(t0 ) V 0. It follows from the monotonicity of A and the weak convergence of un (t0 ) that lim inf A(un )(t0 ), un (t0 ) − u(t0 ) V 0. Hence, we have from the two inequalities above lim inf hn (t0 ) lim inf A(un )(t0 ), un (t0 ) − u(t0 ) V + lim inf G(un )(t0 ), un (t0 ) − u(t0 ) V 0 which contracts (11) and, thereby, proves (10). Then by Fatou’s Lemma, we obtain from (7) and (10) T 0

lim inf hn (t) dt 0

T lim inf

hn (t) dt 0

T lim sup

hn (t) dt 0

= lim supAun + Gun , un − uX 0. The inequalities above imply that lim inf hn (t) = 0 for almost all t ∈ I . Passing to a subsequence if necessary, we have lim hn (t) = 0 a.e. t ∈ I.

n→∞

(13)


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By use of (12) and (13), it follows that {un (t)} is bounded in V for almost all t ∈ I . Therefore, there exists a positive constant C, such that

un (t) C V

a.e. t ∈ I,

(14)

and hence, we may assume un (t) u(t) in V for almost all t ∈ I . Since G is weakly continuous, we have that for any w ∈ X, G(un )(t) − G(u)(t), w(t) V → 0 a.e. t ∈ I. Using (14) and (H3 ), we obtain

G(un )(t) − G(u)(t), w(t) G(un )(t) − G(u)(t) w(t)

V V V

p−1

c2 u(t) V + k1 w(t) V , where k1 is also a positive constant. By the Dominated Convergence Theorem, it follows from the above facts that T Gun − Gu, wX =

G(un )(t) − G(u)(t), w(t) V dt → 0 ∀w ∈ X,

0

i.e., Gun Gu

in X .

(15)

Similarly, using (H2 ), (H3 ) and (14), we have

G(un )(t), un (t) − u(t) G(un )(t) un (t) − u(t)

V V V

c2 u(t) + k2 , V

and lim inf G(un )(t), un (t) − u(t) V 0. Using Fatou’s Lemma again, we obtain T 0

lim inf G(un )(t), un (t) − u(t) V dt,

0

T lim inf

G(un )(t), un (t) − u(t) V dt,

0

i.e., lim infGun , un X Gu, uX .

(16)

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So from (7) and (16), we have lim supAun , un − uX 0. By the definition of A, we readily get that A is a monotone and demicontinuous operator on the real reflexive Banach space X. It follows from Proposition 32.7 in [12] that A is maximal monotone. Therefore, using Lemma 1.3 of Chapter II in [3], we have Aun Au

(17)

lim Aun , un X = Au, uX .

(18)

lim supGun , un − uX 0.

(19)

lim Gun , un − uX = 0.

(20)

and

n→∞

From (7) and (18), we obtain

So by (16), we get

n→∞

Hence from (15)–(20) we obtain (8) and (9), which proves our Proposition 2.

2

Now we are in a position to obtain our main results. Theorem. Let f ∈ X be given. Under assumptions (H1 )–(H3 ), problem (1) has at least one solution u ∈ X such that u ∈ Lp (0, T ; V ), u ∈ Lq (0, T ; V ). Proof. By Proposition 1, we know that L is a maximal monotone mapping. By means of the operator L, we can rewrite (1) as Lv + Av + Gv = f,

v ∈ D(L).

(21)

In virtue of Theorem 32.A of [12] and our Proposition 2, we obtain that Eq. (21) has a solution u ∈ D(L), which implies problem (1) has a solution u ∈ X and u ∈ X . The proof of the main theorem is complete. 2 In order to illustrate our theoretical result, we have the following example. 1,p A nonlinear anti-periodic parabolic problem. We denote H := L2 (Ω), V := W0 (Ω), where n Ω is a bounded domain in R with a smooth boundary ∂Ω and let p 2 and 1/p + 1/q = 1. It is well known (see, e.g., Zeidler [12]) that uV = ( Ω |∇u|p dx)1/p is an equivalent norm on V . Let X = Lp (0, T ; V ), X = Lq (0, T ; V ).


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We consider the following problem: ⎧ p−2 ∇u = f (x, u), ⎪ ⎨ ut − div |∇u| u(x, t) = 0, ⎪ ⎩ u(x, 0) = −u(x, T ),

on Ω × (0, T ), on ∂Ω × (0, T ), on Ω.

(22)

The p-Laplacian div(|∇u|p−2 ∇u) arises in many applications such as non-Newtonian fluids, quasi-regular and quasi-conformal mapping theory and Finsler geometry, etc. Let us define the following two operators on V , Au, vV :=

|∇u|p−2 ∇u · ∇v dx

∀u, v ∈ V ,

Ω

F u, vV :=

f (x, u)v dx

∀u, v ∈ V .

Ω

If f : Ω × R → R satisfies suitable conditions, we may get that the assumptions (H1 )–(H3 ) hold for the operators A and G. Then we can show the existence of anti-periodic solutions to the problem (22) by using our abstract results. References [1] S. Aizicovici, M. McKibben, S. Reich, Anti-periodic solutions to nonmonotone evolution equations with discontinuous nonlinearities, Nonlinear Anal. 43 (2001) 233–251. [2] S. Aizicovici, N.H. Pavel, Anti-periodic solutions to a class of nonlinear differential equations in Hilbert space, J. Funct. Anal. 99 (1991) 387–408. [3] V. Barbu, Nonlinear Semigroup and Differential Equations in Banach Spaces, Noordhoff, Leyden, 1976. [4] Yuqing Chen, Anti-periodic solutions for semilinear evolution equations, J. Math. Anal. Appl. 315 (2006) 337–348. [5] Yuqing Chen, Juan J. Nieto, Donal O’Regan, Anti-periodic solutions for full nonlinear first-order differential equations, Math. Comput. Modelling 46 (2007) 1183–1190. [6] A. Haraux, Anti-periodic solutions of some nonlinear evolution equations, Manuscripta Math. 63 (1989) 479–505. [7] Zhenhai Liu, Nonlinear evolution variational inequalities with nonmonotone perturbations, Nonlinear Anal. 29 (1997) 1231–1236. [8] Zhenhai Liu, Existence for implicit differential equations with nonmonotone perturbations, Israel J. Math. 129 (2002) 363–372. [9] H. Okochi, On the existence of periodic solutions to nonlinear abstract parabolic equations, J. Math. Soc. Japan 40 (1988) 541–553. [10] H. Okochi, On the existence of anti-periodic solutions to a nonlinear evolution equation associated with odd subdifferential operators, J. Funct. Anal. 91 (1990) 246–258. [11] H. Okochi, On the existence of anti-periodic solutions to nonlinear parabolic equations in noncylindrical domains, Nonlinear Anal. 14 (1990) 771–783. [12] E. Zeidler, Nonlinear Functional Analysis and Its Applications, vols. IIA and IIB, Springer, New York, 1990.


Shift-invariant spaces on LCA groups Carlos Cabrelli a,b,∗ , Victoria Paternostro a,b,1 a Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires,

Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina b CONICET, Argentina Received 7 May 2009; accepted 13 November 2009 Available online 26 November 2009 Communicated by N. Kalton

Abstract In this article we extend the theory of shift-invariant spaces to the context of LCA groups. We introduce the notion of H -invariant space for a countable discrete subgroup H of an LCA group G, and show that the concept of range function and the techniques of fiberization are valid in this context. As a consequence of this generalization we prove characterizations of frames and Riesz bases of these spaces extending previous results, that were known for Rd and the lattice Zd . © 2009 Elsevier Inc. All rights reserved. Keywords: Shift-invariant spaces; Translation invariant spaces; LCA groups; Range functions; Fibers

1. Introduction A shift-invariant space (SIS) is a closed subspace of L2 (R) that is invariant under translations by integers. The Fourier transform of a shift-invariant space is a closed subspace that is invariant under integer modulations (multiplications by complex exponentials of integer frequency). Spaces that are invariant under integer modulations are called doubly invariant spaces. Every re* Corresponding author at: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Buenos Aires, Argentina. E-mail addresses: [email protected] (C. Cabrelli), [email protected] (V. Paternostro). 1 The research of C. Cabrelli and V. Paternostro is partially supported by Grants: ANPCyT, PICT 2006–177, CONICET, PIP 5650, UBACyT X058 and X108.


C. Cabrelli, V. Paternostro / Journal of Functional Analysis 258 (2010) 2034–2059

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sult on doubly invariant spaces can be translated to an equivalent result in shift-invariant spaces via the Fourier transform. Doubly invariant spaces have been studied in the sixties by Helson [7] and also by Srinivasan [16,10], in the context of operators related to harmonic analysis. Shift-invariant spaces are very important in applications and the theory had a great development in the last twenty years, mainly in approximation theory, sampling, wavelets, and frames. In particular they serve as models in many problems in signal and image processing. In order to understand the structure of doubly invariant spaces, Helson introduced the notion of range function. This became an essential tool in the modern development of the theory. See [3,4,14,1]. Range functions characterize completely shift-invariant spaces and provide a series of techniques known in the literature as fiberization that allow to have a different view and a deeper insight of these spaces. Fiberization techniques are very important in the class of finitely generated shift-invariant spaces. A key feature of these spaces is that they can be generated by the integer translations of a finite number of functions. Using range functions allows us to translate problems on finitely generated shift-invariant spaces, into problems of linear algebra (i.e. finite dimensional problems). Shift-invariant spaces generalize very well to several variables where the invariance is understood to be under the group Zd . When looking carefully at the theory it becomes apparent that it is strongly based on the additive group operation of Rd and the action of the subgroup Zd . It is therefore interesting to see if the theory can be set in a context of general locally compact abelian groups (LCA groups). The locally compact abelian group framework has several advantages. First because it is important to have a valid theory for the classical groups such as Zd , Td and Zn . This will be crucial particularly in applications, as in the case of the generalization of the Fourier transform to LCA groups and also Kluvanek’s theorem, where the Classical Sampling theorem is extended to this general context (see [13,5]). On the other side, the LCA groups setting, unifies a number of different results into a general framework with a concise and elegant notation. This fact enables us to visualize hidden relationships between the different components of the theory, what, as a consequence, will translate in a deeper and better understanding of shift-invariant spaces, even in the case of the real line. In this paper we develop the theory of shift invariant spaces in LCA groups. Our emphasis will be on range functions and fiberization techniques. The order of the subjects follows mainly the treatment of Bownik in Rd , [1]. In [12] the authors study, in the context of LCA groups, principal shift-invariant spaces, that is, shift-invariant spaces generated by one single function. However they don’t develop the general theory. This article is organized in the following way. In Section 2 we give the necessary background on LCA groups and set the basic notation. In Section 3 we state our standing assumptions and prove the characterizations of H -invariant spaces using range functions. We apply these results in Section 4 to obtain a characterization of frames and Riesz bases of H -translations. 2. Background on LCA Groups In this section we review some basic known results from the theory of LCA groups, that we need for the remainder of the article. In this way we set the notation that we will use in the following sections. Most proofs of the results are omitted unless it is considered necessary. For details and proofs see [15,8,9].

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2.1. LCA Groups Throughout this article, G will denote a locally compact abelian, Hausdorff group (LCA) and its dual group. That is, Γ (or G) Γ = {γ : G → C: γ is a continuous character of G}, where a character is a function such that: (a) |γ (x)| = 1, ∀x ∈ G. (b) γ (x + y) = γ (x)γ (y), ∀x, y ∈ G. Thus, characters generalize the exponential functions γt (y) = e2πity , from the case G = (R, +). Since in this context, both the algebraic and topological structures coexist, we will say that two groups G and G are topologically isomorphic and we will write G ≈ G , if there exists a topological isomorphism from G onto G . That is, an algebraic isomorphism which is a homeomorphism as well. The following theorem states some important facts about LCA groups. Its proof can be found in [15]. Theorem 2.1. Let G be an LCA group and Γ its dual. Then, (a) The dual group Γ , with the operation (γ + γ )(x) = γ (x)γ (x), is an LCA group. (b) The dual group of Γ is topologically isomorphic to G, with the identification x ∈ G ↔ φx ∈ Γ, where φx (γ ) := γ (x). (c) G is discrete (compact) if and only if Γ is compact (discrete). As a consequence of item (b) of Theorem 2.1, it is convenient to use the notation (x, γ ) for the complex number γ (x), representing the character γ applied to x or the character x applied to γ . Next we list the most basic examples that are relevant to Fourier analysis. As usual, we identify the interval [0, 1) with the torus T = {z ∈ C: |z| = 1}. Example 2.2. (I) In case that G = (Rd , +), the dual group Γ is also (Rd , +), with the identification x ∈ Rd ↔ γx ∈ Γ , where γx (y) = e2πi x,y . (II) In case that G = T, its dual group is topologically isomorphic to Z, identifying each k ∈ Z with γk ∈ Γ , being γk (ω) = e2πikω . (III) Let G = Z. If γ ∈ Γ , then (1, γ ) = e2πiα for same α ∈ R. Therefore, (k, γ ) = e2πiαk . Thus, the complex number e2πiα identifies the character γ . This proves that Γ is T. (IV) Finally, in case that G = Zn , the dual group is also Zn . Let us now consider H ⊆ G, a closed subgroup of an LCA group G. Then, the quotient G/H is a regular (T3) topological group. Moreover, with the quotient topology, G/H is an LCA group and if G is second countable, the quotient G/H is also second countable.


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For an LCA group G and H ⊆ G a subgroup of G, we define the subgroup of Γ as follows: = γ ∈ Γ : (h, γ ) = 1, ∀h ∈ H . This subgroup is called the annihilator of H . Since each character in Γ is a continuous function on G, is a closed subgroup of Γ . Moreover, if H ⊆ G is a closed subgroup and is the annihilator of H , then H is the annihilator of (see [15, Lemma 2.1.3.]). The next result establishes duality relationships among the groups H , , G/H and Γ /. Theorem 2.3. If G is an LCA group and H ⊆ G is a closed subgroup of G, then: ). (i) is topologically isomorphic to the dual group of G/H , i.e.: ≈ (G/H . (ii) Γ / is topologically isomorphic to the dual group of H , i.e.: Γ / ≈ H Remark 2.4. According to Theorem 2.1, each element of G induces one character in Γ. In particular, if H is a closed subgroup of G, each h ∈ H induces a character that has the additional property of being -periodic. That is, for every δ ∈ , (h, γ + δ) = (h, γ ) for all γ ∈ Γ . The following definition will be useful throughout this paper. It agrees with the one given in [11]. Definition 2.5. Given G an LCA group, a uniform lattice H in G is a discrete subgroup of G such that the quotient group G/H is compact. The next theorem points out a number of relationships which occur among G, H , Γ , and their respective quotients. Theorem 2.6. Let G be a second countable LCA group. If H ⊆ G is a countable ( finite or countably infinite) uniform lattice, the following properties hold. (1) (2) (3) (4) (5) (6)

G is separable. H ⊆ G is closed. G/H is second countable and metrizable. ⊆ Γ , the annihilator of H , is closed, discrete and countable. ) ≈ . ≈ Γ / and (G/H H Γ / is a compact group.

Note that in particular, this theorem states that is a countable uniform lattice in Γ . 2.2. Haar Measure on LCA groups On every LCA group G, there exists a Haar measure. That is, a non-negative, regular borel measure mG , which is not identically zero and trans-lation-invariant. This last property means that, mG (E + x) = mG (E)

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for every element x ∈ G and every Borel set E ⊆ G. This measure is unique up to constants, in the following sense: if mG and mG are two Haar measures on G, then there exists a positive constant λ such that mG = λmG . Given a Haar measure mG on an LCA group G, the integral over G is translation-invariant in the sense that, f (x + y) dmG (x) = f (x) dmG (x) G

G

for each element y ∈ G and for each Borel-measurable function f on G. As in the case of the Lebesgue measure, we can define the spaces Lp (G, mG ), that we will denote as Lp (G), in the following way p p f (x) dmG (x) < ∞ . L (G) = f : G → C: f is measurable and G

If G is a second countable LCA group, Lp (G) is separable, for all 1 p < ∞. We will focus here on the cases p = 1 and p = 2 The next theorem is a generalization of the periodization argument usually applied in case G = R and H = Z (for details see [9, Theorem 28.54]). Theorem 2.7. Let G be an LCA group, H ⊆ G a closed subgroup and f ∈ L1 (G). Then, the Haar measures mG , mH and mG/H can be chosen such that

f (x) dmG (x) = f (x + h) dmH (h) dmG/H [x] , G

G/H H

where [x] denotes the coset of x in the quotient G/H . If G is a countable discrete group, the integral of f ∈ L1 (G) over G, is determined by the formula f (x) dmG (x) = mG {0} f (x), G

x∈G

since, due to the translations invariance, mG ({x}) = mG ({0}), for each element x ∈ G. Definition 2.8. A section of G/H is a set of representatives of this quotient. That is, a subset C of G containing exactly one element of each coset. Thus, each element x ∈ G has a unique expression of the form x = c + h with c ∈ C and h ∈ H . We will need later in the paper to work with Borel sections. The existence of Borel sections is provided by the following lemma (see [11] and [6]). Lemma 2.9. Let G be an LCA group and H a uniform lattice in G. Then, there exists a section of the quotient G/H , which is Borel measurable. Moreover, there exists a section of G/H which is relatively compact.


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A section C ⊆ G of G/H is in one to one correspondence with G/H by the cross-section map τ : G/H → C, [x] → [x] ∩ C. Therefore, we can carry over the topological and algebraic structure of G/H to C. Moreover, if C is a Borel section, τ : G/H → C is measurable with respect to the Borel σ -algebra in G/H and the Borel σ -algebra in G (see [6, Theorem 1]). Therefore, the set value function defined by m(E) = mG/H (τ −1 (E)) is well defined on Borel subsets of C. In the next lemma, we will prove that this measure m is equal to mG up to a constant. Lemma 2.10. Let G be an LCA group, H a countable uniform lattice in G and C a Borel section of G/H . Then, for every Borel set E ⊆ C

mG (E) = mH {0} mG/H τ −1 (E) , where τ is the cross-section map. In particular, mG (C) = mH ({0})mG/H (G/H ). Proof. According to Lemma 2.9, there exists a relatively compact section of G/H . Let us call it C . Therefore, if C is any other Borel section of G/H , it must satisfy mG (C) = mG (C ). Since C has finite mG measure, C must have finite measure as well. Now, take E ⊆ C a Borel set. Using Theorem 2.7,

mG (E) = χE (x) dmG (x) = χE (x + h) dmH (h) dmG/H [x] G

G/H H

= mH {0}

χE (x + h) dmG/H [x]

G/H h∈H

= mH {0}

χτ −1 (E) [x] dmG/H [x]

G/H

= mH {0} mG/H τ −1 (E) .

2

Remark 2.11. Notice that C, together with the LCA group structure inherited by G/H through τ , has the Haar measure m. We proved that mG |C , the restriction of mG to C, is a multiple of m. It follows that mG |C is also a Haar measure on C. In this paper we will consider C as an LCA group with the structure inherited by G/H and with the Haar measure mG . nA trigonometric polynomial in an LCA group G is a function of the form P (x) = j =1 aj (x, γj ), where γj ∈ Γ and aj ∈ C for all 1 j n. As a consequence of Stone–Weierstrass Theorem, the following result holds, (see [15, p. 24]). Lemma 2.12. If G is a compact LCA group, then the trigonometric polynomials are dense in C(G), where C(G) is the set of all continuous complex-valued functions on G. Another important property of characters in compact groups is the following. For its proof see proof of [15, Theorem 1.2.5].

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Lemma 2.13. Let G be a compact LCA group and Γ its dual. Then, the characters of G verify the following orthogonality relationship:

(x, γ ) x, γ dmG (x) = mG (G)δγ γ ,

G

for all γ , γ ∈ Γ , where δγ γ = 1 if γ = γ and δγ γ = 0 if γ = γ . Let us now suppose that H is a uniform lattice in G. If Γ is the dual group of G and is the annihilator of H, the following characterization of the characters of the group Γ / will be useful to understand what follows. For each h ∈ H , the function γ → (h, γ ) is constant on the cosets [γ ] = γ + . Therefore, it defines a character on Γ /. Moreover, each character on Γ / is of this form. Thus, this correspondence between H and the characters of Γ /, which is actually a topological isomorphism, shows the dual relationship established in Theorem 2.3. Furthermore, since Γ / is compact, we can apply Lemma 2.13 to Γ /. Then, for h ∈ H , we have

mΓ / (Γ /) if h = 0, h, [γ ] dmΓ / [γ ] = (1) 0 if h = 0. Γ /

2.3. The Fourier transform on LCA groups Given a function f ∈ L1 (G) we define the Fourier transform of f , as f(γ ) =

f (x)(x, −γ ) dmG (x),

γ ∈ Γ.

(2)

G

Theorem 2.14. The Fourier transform is a linear operator from L1 (G) into C0 (Γ ), where C0 (Γ ) is the subspace of C(Γ ) of functions vanishing at infinite, that is, f ∈ C0 (Γ ) if f ∈ C(Γ ) and for all ε > 0 there exists a compact set K ⊆ G with |f (x)| < ε if x ∈ K c . Furthermore, ∧ : L1 (G) → C0 (Γ ) satisfies f(γ ) = 0 ∀γ ∈ Γ

⇒

f (x) = 0 a.e. x ∈ G.

(3)

The Haar measure of the dual group Γ of G, can be normalized so that, for a specific class of functions, the following inversion formula holds (see [15, Section 1.5]), f (x) =

f(γ )(x, γ ) dmΓ (γ ).

Γ

In the case that the Haar measures mG and mΓ are normalized such that the inversion formula holds, the Fourier transform on L1 (G)∩L2 (G) can be extended to a unitary operator from L2 (G) onto L2 (Γ ), the so-called Plancherel transformation. We also denote this transformation by “∧”.


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Thus, the Parseval formula holds

f (x)g(x) dmG (x) =

f, g =

g (γ ) dmΓ (γ ) = f, g , f(γ )

Γ

G

where f, g ∈ L2 (G). Let us now suppose that G is compact. Then Γ is discrete. Fix mG and mΓ in order that the inversion formula holds. Then, using the Fourier transform, we obtain that

1 = mΓ {0} mG (G).

(4)

The following lemma is a straightforward consequence of Lemma 2.13, Eq. (1) and statement (3). Lemma 2.15. If G is a compact LCA group and its dual Γ is countable, then the characters {γ : γ ∈ Γ } form an orthogonal basis for L2 (G). For an LCA group G and a countable uniform lattice H in G, we will denote by Ω a Borel section of Γ /. In the remainder of this paper we will identify L2 (Ω) with the set {ϕ ∈ L2 (Γ ): ϕ = 0 a.e. Γ \ Ω} and L1 (Ω) with the set {φ ∈ L1 (Γ ): φ = 0 a.e. Γ \ Ω}. Let us now define the functions ηh : Γ → C, as ηh (γ ) = (h, −γ )χΩ (γ ). Using Lemma 2.15 we have: Proposition 2.16. Let G be an LCA group and H a countable uniform lattice in G. Then, {ηh }h∈H is an orthogonal basis for L2 (Ω). Remark 2.17. We can associate to each ϕ ∈ L2 (Ω), a function ϕ defined on Γ / as ϕ ([γ ]) = δ∈ ϕ(γ + δ). The correspondence ϕ → ϕ , is an isometric isomorphism up to a constant 2 2 between L (Ω) and L (Γ /), since

2 ϕ2L2 (Ω) = m {0} ϕ L2 (Γ /) . Combining the above remark, Proposition 2.16, and the relationships established in Theorem 2.3, we obtain the following proposition, which will be very important on the remainder of the paper. and Proposition 2.18. Let G be an LCA group, H countable uniform lattice on G, Γ = G the annihilator of H . Fix Ω a Borel section of Γ / and choose mH and mΓ / such that the inversion formula holds. Then

mH ({0})1/2

a2 (H ) = a h ηh

2 , mΓ (Ω)1/2 L (Ω) h∈H

for each a = {ah }h∈H ∈ 2 (H ).

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Proof. Let a ∈ 2 (H ). So, a L2 (Γ /) , a2 (H ) =

(5)

≈ Γ / and therefore ∧ : H → Γ /. since H Take ϕ(γ ) = h∈H ah (h, −γ )χΩ (γ ). Then, by Proposition 2.16, ϕ ∈ L2 (Ω). Furthermore, ϕ ([γ ]) = ϕ(γ ), a.e. γ ∈ Ω. So, as a consequence of Remark 2.17, we have

2

ϕ 2

L (Γ /)

Now, a ([γ ]) = mH ({0})

h∈H

=

1 ϕ2L2 (Ω) . m ({0})

(6)

ah (h, −[γ ]). Therefore, substituting in Eqs. (5) and (6),

a2 (H ) =

mH ({0}) ϕL2 (Ω) . m ({0})1/2

Finally, since mΓ (Ω) = m ({0})mΓ / (Γ /), using (4) we have that mH ({0}) mH ({0})1/2 = , m ({0})1/2 mΓ (Ω)1/2 which completes the proof.

2

We finish this section with a result which is a consequence of statement (3) and Theorem 2.7. (h) = 0 for all Proposition 2.19. Let G, H and Ω as in Proposition 2.18. If φ ∈ L1 (Ω) and φ h ∈ H , then φ(ω) = 0 a.e. ω ∈ Ω. 3. H -invariant spaces In this section we extend the theory of shift-invariant spaces in Rd to general LCA groups. We will develop the concept of range function and the techniques of fiberization in this general context. The treatment will be for shift-invariant spaces following the lines of Bownik [1]. The conclusions for doubly invariant spaces will follow via the Plancherel theorem for the Fourier transform on LCA groups. First we will fix some notation and set our standing assumptions that will be in force for the remainder of the manuscript. Standing Assumptions 3.1. We will assume throughout the next sections that. • G is a second countable LCA group. • H is a countable uniform lattice on G. We denote, as before, by Γ the dual group of G, by the annihilator of H , and by Ω a fixed Borel section of Γ /. The choice of particular Haar measure in each of the groups considered in this paper does not affect the validity of the results. However, different constants will appear in the formulas.


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Since we have the freedom to choose the Haar measures, we will fix the following normalization in order to avoid carrying over constants through the paper and to simplify the statements of the results. We choose m and mH such that m ({0}) = mH ({0}) = 1. We fix mΓ / such that mΓ / (Γ /) = 1 and therefore the inversion formula holds between H and Γ /. Then, we set mΓ such that Theorem 2.7 holds for mΓ , mΓ / and m . Finally, we normalize mG such that the inversion formula holds for mΓ and mG . As a consequence of the normalization given above and Lemma 2.10, it holds that mΓ (Ω) = 1. Note that under our Standing Assumptions 3.1, Theorem 2.6 applies. So we will use the properties of G, H , Γ and stated in that theorem. 3.1. Preliminaries The space L2 (Ω, 2 ()) is the space of all measurable functions Φ : Ω → 2 () such that

Φ(ω) 22 dmΓ (ω) < ∞, () Ω

where a function Φ : Ω → 2 () is measurable, if and only if for each a ∈ 2 () the function ω → Φ(ω), a is a measurable function from Ω into C. Remark 3.2. This is the usual notion of weak measurability for vector functions. If the values of the functions are in a separable space, as a consequence of Petti’s Theorem, the notions of weak and strong measurability agree. As we have seen in Section 2 and according to our hypotheses, is a countable uniform lattice on Γ . Therefore, 2 () is a separable Hilbert space. Then, in L2 (Ω, 2 ()) we have only one measurability notion. The space L2 (Ω, 2 ()), with the inner product Φ, Ψ := Φ(ω), Ψ (ω) 2 () dmΓ (ω) Ω

is a complex Hilbert space. Note that for Φ ∈ L2 (Ω, 2 ()) and ω ∈ Ω

1/2 Φ(ω) 2

Φ(ω) 2 = , () δ δ∈

where (Φ(ω))δ denotes the value of the sequence Φ(ω) in δ. If Φ ∈ L2 (Ω, 2 ()), the sequence Φ(ω) is the fiber of Φ at ω. The following proposition shows that the space L2 (Ω, 2 ()) is isometric to L2 (G). Proposition 3.3. The mapping T : L2 (G) −→ L2 (Ω, 2 ()) defined as T f (ω) = f(ω + δ) δ∈ , is an isomorphism that satisfies T f 2 = f L2 (G) .

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The next periodization lemma will be necessary for the proof of Proposition 3.3. Lemma 3.4. Let g ∈ L2 (Γ ). Define the function G(ω) = and moreover

δ∈ |g(ω

+ δ)|2 . Then, G ∈ L1 (Ω)

gL2 (Γ ) = GL1 (Ω) . =

Proof. Since Ω is a section of Γ /, we have that Γ Therefore, g(γ )2 dmΓ (γ ) =

δ∈ Ω

− δ, where the union is disjoint.

g(ω)2 dmΓ (ω)

δ∈Ω−δ

Γ

=

g(ω + δ)2 dmΓ (ω)

δ∈ Ω

=

g(ω + δ)2 dmΓ (ω).

Ω δ∈

This proves that G ∈ L1 (Ω) and gL2 (Γ ) = GL1 (Ω) .

2

Proof of Proposition 3.3. First we prove that T is well defined. For this we must show that, ∀f ∈ L2 (G), the vector function T f is measurable and T f 2 < ∞. According to Lemma 3.4, the sequence {f(ω + δ)}δ∈ ∈ 2 (), a.e. ω ∈ Ω, for all f ∈ 2 L (G). Then, given a = {aδ }δ∈ ∈ 2 (), the product T f (ω), a = δ∈ f(ω + δ)aδ is finite a.e. ω ∈ Ω. From here the measurability of f implies that ω → T f (ω), a is a measurable function in the usual sense. This proves the measurability of T f . If f ∈ L2 (G), as a consequence of Lemma 3.4, we have T

f 22

=

T f (ω) 22

()

dmΓ (ω)

Ω

f(ω + δ)2 dmΓ (ω) = Ω δ∈

=

f(γ )2 dmΓ (γ )

Γ

=

f (x)2 dmG (x).

G

Thus, T f 2 < ∞ and this also proves that T f 2 = f L2 (G) . What is left is to show that T is onto. So, given Φ ∈ L2 (Ω, 2 ()) let us see that there exists a function f ∈ L2 (G) such that T f = Φ. Using that the Fourier transform is an isometric isomorphism between L2 (G) and L2 (Γ ), it will be sufficient to find g ∈ L2 (Γ ) such that {g(ω + δ)}δ∈ = Φ(ω) a.e. ω ∈ Ω and then take f ∈ L2 (G) such that f= g.


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Given γ ∈ Γ , there exist unique ω ∈ Ω and δ ∈ such that γ = ω + δ. So, we define g(γ ) as

g(γ ) = Φ(ω) δ . The measurability of g is straightforward. Once again, according to Lemma 3.4,

g(γ )2 dmΓ (γ ) =

g(ω + δ)2 dmΓ (ω) Ω δ∈

Γ

Φ(ω)δ 2 dmΓ (ω) = Ω δ∈

=

Φ(ω) 22

()

dmΓ (ω)

Ω

= Φ22 < +∞. Thus, g ∈ L2 (Γ ) and this completes the proof.

2

The mapping T will be important to study the properties of functions of L2 (G) in terms of their fibers, (i.e. in terms of the fibers T f (ω)). 3.2. H -invariant spaces and range functions Definition 3.5. We say that a closed subspace V ⊆ L2 (G) is H -invariant if f ∈ V ⇒ th f ∈ V

∀h ∈ H,

where ty f (x) = f (x − y) denotes the translation of f by an element y of G. For a subset A ⊆ L2 (G), we define EH (A) = {th ϕ: ϕ ∈ A, h ∈ H } and S(A) = span EH (A). We call S(A) the H -invariant space generated by A. If A = {ϕ} , we simply write EH (ϕ) and S(ϕ), and we call S(ϕ) a principal H -invariant space. Our main goal is to give a characterization of H -invariant spaces. We first need to introduce the concept of range function. Definition 3.6. A range function is a mapping J : Ω −→ closed subspaces of 2 () . The subspace J (ω) is called the fiber space associated to ω. For a given range function J , we associate to each ω ∈ Ω the orthogonal projection onto J (ω), Pω : 2 () → J (ω).

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A range function J is measurable if for each a ∈ 2 () the function ω → Pω a, from Ω into 2 (), is measurable. That is, for each a, b ∈ 2 (), ω → Pω a, b is measurable in the usual sense. 2 2 Remark 3.7. Note that J is a measurable range function if and only if for all Φ ∈ L (Ω, ()), 2 the function ω → Pω (Φ(ω)) is measurable. That is, ∀b ∈ (), ω → Pω Φ(ω) , b is measurable in the usual sense.

Given a range function J (not necessarily measurable) we define the subset MJ as

MJ = Φ ∈ L2 Ω, 2 () : Φ(ω) ∈ J (ω) a.e. ω ∈ Ω . Lemma 3.8. The subset MJ is closed in L2 (Ω, 2 ()). Proof. Let {Φj }j ∈N ⊆ MJ such that Φj → Φ when j → ∞ in L2 (Ω, 2 ()). Let us consider the functions gj : Ω → R0 defined as gj (ω) := Φj (ω) − Φ(ω)22 () . Then, gj is measurable for all j ∈ N and ∀α > 0 it holds that

1 1

Φj (ω) − Φ(ω) 22 dmΓ (ω) → 0, mΓ {gj > α} gj (ω) dmΓ (ω) = () α α Ω

Ω

when j → ∞. So, gj → 0 in measure and therefore, there exists a subsequence {gjk }k∈N of {gj }j ∈N which goes to zero a.e. ω ∈ Ω. Then, Φjk (ω) → Φ(ω) in 2 () a.e. ω ∈ Ω and hence, since Φjk (ω) ∈ J (ω) a.e. ω ∈ Ω and J (ω) is closed, Φ(ω) ∈ J (ω) a.e. ω ∈ Ω. Therefore Φ ∈ MJ . 2 The following proposition is a generalization to the context of groups of a lemma of Helson, (see [7] and also [1]). Proposition 3.9. Let J be a measurable range function and Pω the associated orthogonal projections. Denote by P the orthogonal projection onto MJ . Then,

(PΦ)(ω) = Pω Φ(ω) , a.e. ω ∈ Ω, ∀Φ ∈ L2 Ω, 2 () . Proof. Let Q : L2 (Ω, 2 ()) → L2 (Ω, 2 ()) be the linear mapping Φ → QΦ, where

(QΦ)(ω) := Pω Φ(ω) . We want to show that Q = P. Since J is a measurable range function, due to Remark 3.7, QΦ is measurable for each Φ ∈ L2 (Ω, 2 ()). Furthermore, since Pω is an orthogonal projection, it has norm one, and therefore

2

2 QΦ22 = (QΦ)(ω) 2 () dmΓ (ω) = Pω Φ(ω) 2 () dmΓ (ω) Ω

Ω

Φ(ω) 22

()

dmΓ (ω) = Φ22 < ∞.

Ω

Then, Q is well defined and it has norm less or equal to 1.


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From the fact that Pω is an orthogonal projection, it follows that Q2 = Q and Q∗ = Q. So, Q is also an orthogonal projection. To complete our proof let us see that M = MJ , where M := Ran(Q). By definition of Q, M ⊆ MJ . If we suppose that M is properly included in MJ ,then there exists Ψ ∈ MJ such that Ψ = 0 and Ψ ⊥ M. Then, ∀Φ ∈ L2 (Ω, 2 ()), 0 = QΦ, Ψ = Φ, QΨ . Hence, QΨ = 0 and therefore Pω (Ψ (ω)) = 0 a.e. ω ∈ Ω. Since Ψ ∈ MJ , Ψ (ω) ∈ J (ω) a.e. ω ∈ Ω, thus Pω (Ψ (ω)) = Ψ (ω) a.e. ω ∈ Ω. Finally, Ψ = 0 a.e. ω ∈ Ω and this is a contradiction. 2 We now give a characterization of H -invariant spaces using range functions. Theorem 3.10. Let V ⊆ L2 (G) be a closed subspace and T the map defined in Proposition 3.3. Then, V is H -invariant if and only if there exists a measurable range function J such that V = f ∈ L2 (G): T f (ω) ∈ J (ω) a.e. ω ∈ Ω . Identifying range functions which are equal almost everywhere, the correspondence between H -invariant spaces and measurable range functions is one to one and onto. Moreover, if V = S(A) for some countable subset A of L2 (G), the measurable range function J associated to V is given by J (ω) = span T ϕ(ω): ϕ ∈ A ,

a.e. ω ∈ Ω.

For the proof, we need the following results. Lemma 3.11. If J and K are two measurable range functions such that MJ = MK , then J (ω) = K(ω) a.e. ω ∈ Ω. That is, J and K are equal almost everywhere. Proof. Let Pω and Qω be the projections associate to J and K respectively. If P is the orthogonal projection onto MJ = MK , by Proposition 3.9 we have that, for each Φ ∈ L2 (Ω, 2 ())

(PΦ)(ω) = Pω Φ(ω)

and (PΦ)(ω) = Qω Φ(ω)

a.e. ω ∈ Ω.

So, Pω (Φ(ω)) = Qω (Φ(ω)) a.e. ω ∈ Ω, for all Φ ∈ L2 (Ω, 2 ()). In particular, if eλ ∈ is defined by (eλ )δ = 1 if δ = λ and (eλ )δ = 0 otherwise, Pω (eλ ) = Qω (eλ ) a.e. ω ∈ Ω, for all λ ∈ . Hence, since {eλ }λ∈ is a basis for 2 (), it follows that Pω = Qω a.e. ω ∈ Ω. Thus J (ω) = K(ω) a.e. ω ∈ Ω. 2 2 ()

Remark 3.12. Note that for f ∈ L2 (G) and for h ∈ H , T th f (ω) = (h, −ω)T f (ω), since ∀y ∈ G, t y f (γ ) = (y, −γ )f(γ ) and, as we showed in Remark 2.4, the character (h, .) is -periodic.

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Proof of Theorem 3.10. Let us first suppose that V is H -invariant. Since L2 (G) is separable, V = S(A) for some countable subset A of L2 (G). We define the function J as J (ω) = span{T ϕ(ω): ϕ ∈ A}. Note that since A is a countable set, J is well defined a.e. ω ∈ Ω. We will prove that J satisfies: (i) V = {f ∈ L2 (G): T f (ω) ∈ J (ω) a.e. ω ∈ Ω}, (ii) J is measurable. To show (i) it is sufficient to prove that M = MJ , where M := T V . Let Φ ∈ M. Then, T −1 Φ ∈ V = span{th ϕ: h ∈ H, ϕ ∈ A}. Therefore, there exists a sequence {gj }j ∈N ⊆ span{th ϕ: h ∈ H, ϕ ∈ A} such that T gj := Φj converges in L2 (Ω, 2 ()) to Φ, when j → ∞. Due to the definition of J and Remark 3.12, Φj (ω) ∈ J (ω) a.e. ω ∈ Ω. Thus, in the same way that in Lemma 3.8, we can prove that Φ(ω) ∈ J (ω) a.e. ω ∈ Ω and therefore Φ ∈ MJ . So, M ⊆ MJ . Let us suppose that there exists Ψ ∈ L2 (Ω, 2 ()), such that Ψ = 0 and Ψ is orthogonal to M. Then, for each Φ ∈ M, Φ, Ψ = 0. In particular, if Φ ∈ T A ⊆ T V = M and h ∈ H , we have that (h, .)Φ(.) ∈ T V = M since (h, .)Φ(.) = T (t−h T −1 Φ)(.) and t−h T −1 Φ ∈ V . So, as (h, .) is -periodic, 0 = (h, .)Φ(.), Ψ = (h, ω) Φ(ω), Ψ (ω) 2 () dmΓ (ω). Ω

Hence, by Proposition 2.19, Φ(ω), Ψ (ω) 2 () = 0 a.e. ω ∈ Ω, and this holds ∀Φ ∈ T (A). Therefore Ψ (ω) ∈ J (ω)⊥ a.e. ω ∈ Ω. Now, if M is properly included in MJ , there exists Ψ ∈ MJ , with Ψ = 0 and orthogonal to M. Hence, Ψ (ω) ∈ J (ω)⊥ a.e. ω ∈ Ω. On the other hand since Ψ ∈ MJ , Ψ (ω) ∈ J (ω) a.e. ω ∈ Ω. Thus, Ψ (ω) = 0 a.e. ω ∈ Ω and this is a contradiction. Therefore M = MJ . It remains to prove that the range function J is measurable. For this we must show that, for all a, b ∈ 2 (), ω → Pω a, b is measurable, where Pω : 2 () → J (ω) are the orthogonal projections associated to J (ω), Let I be the identity mapping in L2 (Ω, 2 ()) and P : L2 (Ω, 2 ()) → M the orthogonal projection associated to M. If Ψ ∈ L2 (Ω, 2 ()), the function (I − P)Ψ is orthogonal to M and, by the above reasoning, (I − P)Ψ (ω) ∈ J (ω)⊥ , a.e. ω ∈ Ω. Then,

Pω (I − P)Ψ (ω) = Pω Ψ (ω) − PΨ (ω) = 0 a.e. ω ∈ Ω and therefore Pω (Ψ (ω)) = Pω (PΨ (ω)) = PΨ (ω) a.e ω ∈ Ω. In particular, Pω a = Pa(ω) a.e. ω ∈ Ω, ∀ a ∈ 2 (). Thus, since ω → Pa(ω)a, b is measurable ∀b ∈ 2 (), ω → Pω a, b is measurable as well. Conversely, if J is a measurable range function, let us see that the closed subspace in L2 (G), defined by V := T −1 (MJ ) is H -invariant. For this, let us consider f ∈ V and h ∈ H and let us prove that th f ∈ V . Since T (th f )(ω) = (h, −ω)T f (ω) a.e. ω ∈ Ω and T f ∈ MJ , we have that (h, −ω)T f (ω) ∈ J (ω) a.e. ω ∈ Ω. Then, T (th f ) ∈ MJ and therefore th f ∈ V . Furthermore, V = S(A) for some countable set A of L2 (G). Then, K(ω) = span T ϕ(ω): ϕ ∈ A a.e. ω ∈ Ω,


2049

defines a measurable range function which satisfies V = T −1 (MK ). Thus, MK = T V = MJ . Since J and K are both measurable range functions, Lemma3.11 implies that J = K a.e. ω ∈ Ω. This also shows that the correspondence between V and J is onto and one to one. 2 4. Frames and Riesz basis for H -invariant spaces Let H be a Hilbert space and {ui }i∈I a sequence of H. The sequence {ui }i∈I is a Bessel sequence in H with constant B if f, ui 2 Bf 2 ,

for all f ∈ H.

i∈I

The sequence {ui }i∈I is a frame for H with constants A and B if Af 2

f, ui 2 Bf 2 ,

for all f ∈ H.

i∈I

The frame {ui }i∈I is a tight frame if A = B, and the frame {ui }i∈I is a Parseval frame if A = B = 1. The sequence {ui }i∈I is a Riesz sequence for H if there exist positive constants A and B such that A

2

|ai | ai ui B |ai |2

2

i∈I

i∈I

H

i∈I

for all {ai }i∈I with finite support. Moreover, a Riesz sequence that in addition, is a complete family in H, is a Riesz basis for H. We are now ready to prove a result which characterizes when EH (A) is a frame of L2 (G) in terms of the fibers {T ϕ(ω): ϕ ∈ A}. It generalizes Theorem 2.3 of [1] to the context of groups. Theorem 4.1. Let A be a countable subset of L2 (G), J the measurable range function associated to S(A) and A B positive constants. Then, the following propositions are equivalent: (i) The set EH (A) is a frame for S(A) with constants A and B. (ii) For almost every ω ∈ Ω, the set {T ϕ(ω): ϕ ∈ A} ⊆ 2 () is a frame for J (ω) with constants A and B. Proof. Since f, g L2 (G) = T f, T g L2 (Ω,2 ()) , by Remark 3.12 we have that th ϕ, f

L2 (G)

2 = T (th ϕ), T f

L2 (Ω,2 ())

h∈H ϕ∈A

h∈H ϕ∈A

=

2

2 (h, −ω) T ϕ(ω), T f (ω) 2 dmΓ (ω) . ()

ϕ∈A h∈H Ω

2050


Let us define for each ϕ ∈ A, the following, 2 R(ϕ) = (h, −ω) T ϕ(ω), T f (ω) 2 () dmΓ (ω) h∈H Ω

and T (ϕ) =

T ϕ(ω), T f (ω)

2 ()

2 dmΓ (ω).

Ω

(i) ⇒ (ii) If EH2 (A) is a frame for S(A), in particular it holds that ∀f ∈ S(A), h∈H ϕ∈A | th ϕ, f | < ∞. Then, for each ϕ ∈ A, we have that R(ϕ) < ∞. Therefore, the sequence {ch }h∈H , with ch :=

(h, ω) T ϕ(ω), T f (ω) 2 () dmΓ (ω),

Ω

belongs to 2 (H ). Let us consider the function F (ω) := h∈H ch ηh (ω), where ηh are the functions defined in Lemma 2.16. Then, since {ch }h∈H ∈ 2 (H ) and {ηh }h∈H is an orthogonal basis of L2 (Ω), we have that F ∈ L2 (Ω) ⊆ L1 (Ω) (recall that mΓ (Ω) < ∞). On the other hand, the function ψ(ω) := T ϕ(ω), T f (ω) 2 () belongs to L1 (Ω). So, ψ − F ∈ L1 (Ω) and moreover

(h, −ω) ψ(ω) − F (ω) dmΓ (ω) = c−h − c−h = 0

Ω

for all h ∈ H . Thus, Proposition 2.19 yields that F = ψ a.e. ω ∈ Ω. Therefore ψ ∈ L2 (Ω) and ψ(ω) =

ch ηh (ω),

h∈H

a.e. ω ∈ Ω. As a consequence of Proposition 2.18, we obtain that R(ϕ) = T (ϕ) holds for all ϕ ∈ A. We will now prove that, for almost every ω ∈ Ω, {T ϕ(ω): ϕ ∈ A} is a frame with constants A and B for J (ω). Let us suppose that APω d22 ()

T ϕ(ω), Pω d 2 BPω d22 ϕ∈A

()

(7)

a.e. ω ∈ Ω, for each d ∈ D, where D is a dense countable subset of 2 () and Pω are the orthogonal projections associated to J . Then, for each d ∈ D, let Zd ⊆ Ω be a measurable set


2051

with mΓ (Zd ) = 0 such that (7) holds for all ω ∈ Ω \ Zd . So the set Z = d∈D Zd has null mΓ measure. Therefore for ω ∈ Ω \ Z and a ∈ J (ω), using a density argument it follows from (7) that Aa2

T ϕ(ω), a 2 Ba2 . ϕ∈A

Thus, it is sufficient to show that (7) holds. For this, we will suppose that this is not so and we will prove that there exist d0 ∈ D, a measurable set W ⊆ Ω with mΓ (W ) > 0, and ε > 0 such that T ϕ(ω), Pω d0 2 > (B + ε)Pω d0 2 ,

∀ω ∈ W

ϕ∈A

or T ϕ(ω), Pω d0 2 < (A − ε)Pω d0 2 ,

∀ω ∈ W.

ϕ∈A

So, let us take d0 ∈ D for which (7) fails. Then at least one of this sets ω ∈ Ω: K(ω) − BPω d0 2 > 0 , has positive measure, where K(ω) := generality, that mΓ

ϕ∈A | T

ω ∈ Ω: K(ω) − APω d0 2 < 0 ϕ(ω), Pω d0 |2 . Let us suppose, without loss of

ω ∈ Ω: K(ω) − BPω d0 2 > 0

> 0.

Since

1 2 ω ∈ Ω: K(ω) − BPω d0 > 0 = ω ∈ Ω: K(ω) − B + Pω d0 > 0 , j 2

j ∈N

there exists at least one set in the union, in the right-hand side of this equality, with positive measure and this proves our claim. Then, we can suppose that T ϕ(ω), Pω d0 2 > (B + ε)Pω d0 2 ,

∀ω ∈ W

(8)

ϕ∈A

holds. Now take f ∈ S(A) such that T f (ω) = χW (ω)Pω d0 . Note that this is possible since, by Theorem 3.10, χE (ω)Pω d0 is a measurable function. As EH (A) is a frame for S(A) and th ϕ, f

L2 (G)

h∈H ϕ∈A

2 = T ϕ(ω), T f (ω) 2 2 dmΓ (ω), () ϕ∈A Ω

2052


we have that Af 2

T ϕ(ω), T f (ω) 2 dmΓ (ω) Bf 2 .

(9)

ϕ∈A Ω

Using Proposition 3.3, we can rewrite (9) as 2 T ϕ(ω), T f (ω) 2 dmΓ (ω) BT f 2 . AT f

(10)

ϕ∈A Ω

Now,

T f =

χW (ω)Pω d0 2 dmΓ (ω)

2

Ω

and if we integrate in (8) over W , we obtain T ϕ(ω), χW (ω)Pω d0 2 dmΓ (ω) (B + ε)T f 2 . ϕ∈A Ω

This is a contradiction with inequality (10). Therefore, we proved inequality (7). (ii) ⇒ (i) If now {T ϕ(ω): ϕ ∈ A} is a frame for J (ω) a.e. ω ∈ Ω with constants A and B, we have that T ϕ(ω), a 2 Ba2 Aa2 ϕ∈A

for all a ∈ J (ω). In particular, if f ∈ S(A), by Theorem 3.10, T f (ω) ∈ J (ω) a.e. ω ∈ Ω and then,

2

T ϕ(ω), T f (ω) 2 B T f (ω) 2 (11) A T f (ω) ϕ∈A

a.e. ω ∈ Ω. Thus, integrating (11) over Ω, we obtain T ϕ(ω), T f (ω) 2 dmΓ (ω) BT f 2 . AT f 2

(12)

Ω ϕ∈A

So, T ϕ(.), T f (.) belongs to L2 (Ω) for each ϕ ∈ A and the equality R(ϕ) = T (ϕ), can be obtained in a similar way as we did before. Finally, since T f 22 = f 22 and th ϕ, f 2 2 = T ϕ(ω), T f (ω) 2 2 dmΓ (ω), L (G) () h∈H ϕ∈A

ϕ∈A Ω

inequality (12) implies that EH (A) is a frame for S(A) with constants A and B.

2


2053

Theorem 4.1 reduces the problem of when EH (A) is a frame for S(A) to when the fibers {T ϕ(ω): ϕ ∈ A} form a frame for J (ω). The advantage of this reduction is that, for example, when A is a finite set, the fiber spaces {T ϕ(ω): ϕ ∈ A} are finite dimensional while S(A) has infinite dimension. If A = {ϕ}, Theorem 4.1 generalizes a known result for the case G = Rd to the context of groups. This is stated in the next corollary, which was proved in [12]. We give here a different proof. ϕ (ω + δ)|2 = 0}. Then, the following Corollary 4.2. Let ϕ ∈ L2 (Ω) and Ωϕ = {ω ∈ Ω: δ∈ | are equivalent: (i) The set for S(ϕ) with constants A and B. EH (ϕ) is a frame ϕ (ω + δ)|2 B, a.e. ω ∈ Ωϕ . (ii) A δ∈ | Proof. Let J be the measurable range function associated to S(ϕ). Then, by Theorem 3.10, J (ω) = span{T ϕ(ω)} a.e ω ∈ Ω. Thus, each a ∈ J (ω) can be written as a = λT ϕ(ω) for some λ ∈ C. Therefore, by Theorem 4.1, (i) holds if and only if, for almost every ω ∈ Ω and for all λ ∈ C,

2

4

2

(13) A λT ϕ(ω) |λ|2 T ϕ(ω) B λT ϕ(ω) . Then, since T ϕ(ω)2 =

ϕ (ω + δ)|2 , δ∈ |

(13) holds if and only if 2 ϕ (ω + δ) B, a.e. ω ∈ Ωϕ . 2 A δ∈

For the case of Riesz basis, we have an analogue result to Theorem 4.1. Theorem 4.3. Let A be a countable subset of L2 (G), J the measurable range function associated to S(A) and A B positive constants. Then, they are equivalent: (i) The set EH (A) is a Riesz basis for S(A) with constants A and B. (ii) For almost every ω ∈ Ω, the set {T ϕ(ω): ϕ ∈ A} ⊆ 2 () is a Riesz basis for J (ω) with constants A and B. For the proof we will need the next lemma. Lemma 4.4. For each m ∈ L∞ (Ω) there exists a sequence of trigonometric polynomials {Pk }k∈N such that: (i) Pk (ω) → m(ω), a.e. ω ∈ Ω, (ii) There exists C > 0, such that Pk ∞ C, for all k ∈ N. Proof. By Lemma 2.12, taking into account Remark 2.11, we have that the trigonometric polynomials are dense in C(Ω). By Lusin’s Theorem, for each k ∈ N, there exists a closed set Ek ⊆ Ω such that mΓ (Ω \Ek ) < 2−k and m|Ek is a continuous function where m|Ek denotes the function m restricted to Ek . Since Ω is compact, Ek is compact as well. Therefore, m|Ek is bounded. Let m1 , m2 : Ek → R be continuous function such that m|Ek = m1 + im2 . As a consequence of Tietze’s Extension Theorem, it is possible to extend m1 and m2 , continuously to all Ω keeping

2054


their norms in L∞ (Ek ). Let us call the extensions m1 and m2 and let mk = m1 + im2 . Then, we have: (1) mk |Ek = m|Ek , (2) mk ∞ m1 ∞ + m2 ∞ m1 ∞ + m2 ∞ 2m∞ . Now, by Lemma 2.12, there exists a trigonometric polynomial Pk such that Pk − mk ∞ < So,

2−k .

(a) |Pk (ω) − m(ω)| < 2−k , for all ω ∈ Ek , (b) Pk ∞ Pk − mk ∞ + mk ∞ 2−k + 2m∞ 1 + 2m∞ . Repeating this argument for each k ∈ N, we obtain a sequence {Pk }k∈N of trigonometric polynomials and a sequence ∞ {Ek }k∈N of sets, which satisfy conditions (a) and (b). Let E = ∞ j =1 k=j Ek . It is a straightforward to see that mΓ (Ω \ E) = 0. Let us prove that if ω ∈ E, Pk (ω) → m(ω), for k → ∞. Since ω ∈ E, there exists k0 ∈ N for which ω ∈ Ek , ∀k k0 . Then, for all k k0 , we obtain that |Pk (ω) − m(ω)| = |Pk (ω) − mk (ω)| < 2−k → 0, when k → ∞. This proves part (i) of this lemma and taking C := 1 + 2m∞ we have that (ii) holds. 2 Proof of Theorem 4.3. Since S(A) = span EH (A) and, by Theorem 3.10, J (ω) = span{T ϕ(ω): ϕ ∈ A}, we only need to show that EH (A) is a Riesz sequence for S(A) with constants A and B if and only if for almost every ω ∈ Ω, the set {T ϕ(ω): ϕ ∈ A} ⊆ 2 () is a Riesz sequence for J (ω) with constants A and B. For the proof of the equivalence in the theorem, we will use the following reasoning. Let {aϕ,h }(ϕ,h)∈A×H be a sequence of finite support and let Pϕ be the trigonometric polynomials defined by Pϕ (ω) =

aϕ,h ηh (ω),

h∈H

with ω ∈ Ω and ηh as in Proposition 2.16. Note that, since {aϕ,h }(ϕ,h)∈A×H has finite support, only a finite number of the polynomials Pϕ are not zero. Now, as a consequence of Proposition 3.3 we have

(ϕ,h)∈A×H

2

aϕ,h th ϕ

2

L (G)

=

(ϕ,h)∈A×H

=

Ω

=

2

aϕ,h T th ϕ

2

(ϕ,h)∈A×H

L (Ω,2 ())

2

aϕ,h (−h, ω)T ϕ(ω)

2

P (ω)T ϕ(ω) ϕ

2 Ω

ϕ∈A

dmΓ (ω)

2 ()

()

dmΓ (ω).

(14)


2055

Furthermore, by Lemma 2.18,

2 |aϕ,h |2 = {aϕ,h }h∈H 2 (H ) = Pϕ 2L2 (Ω) ,

h∈H

and adding over A, we obtain

|aϕ,h |2 =

(ϕ,h)∈A×H

ϕ∈A

Pϕ 2L2 (Ω) .

(15)

(ii) ⇒ (i) If we suppose that for almost every ω ∈ Ω, {T ϕ(ω): ϕ ∈ A} ⊆ 2 () is a Riesz sequence for J (ω) with constants A and B, A

ϕ∈A

2

|aϕ | aϕ T ϕ(ω)

2

()

ϕ∈A

B

2

|aϕ |2

(16)

ϕ∈A

for all {aϕ }ϕ∈A with finite support. In particular, the above inequality holds for {aϕ }ϕ∈A = {Pϕ (ω)}ϕ∈A . Now, in (16), we can integrate over Ω with {aϕ }ϕ∈A = {Pϕ (ω)}ϕ∈A , in order to obtain A

ϕ∈A

Pϕ 2L2 (Ω)

2

P (ω)T ϕ(ω) ϕ

2 Ω

B

ϕ∈A

ϕ∈A

dmΓ (ω)

()

Pϕ 2L2 (Ω) .

(17)

Using Eqs. (14) and (15) we can rewrite (17) as A

|aϕ,h |

2

(ϕ,h)∈A×H

(ϕ,h)∈A×H

2

aϕ,h th ϕ

2

B

L (G)

|aϕ,h |2 .

(ϕ,h)∈A×H

Therefore EH (A) is a Riesz sequence of S(A) with constants A and B. (i) ⇒ (ii) We want to prove that, for every a = {aϕ }ϕ∈A ∈ 2 (A) with finite support, we have a.e. ω ∈ Ω A

ϕ∈A

2

|aϕ | aϕ T ϕ(ω)

2

B

2

()

ϕ∈A

|aϕ |2 .

(18)

ϕ∈A

Let us suppose that (18) fails. Then, using a similar argument as in Theorem 4.1, we can see that there exist a = {aϕ }ϕ∈A ∈ 2 (A) with finite support, a measurable set W ⊆ Ω with mΓ (W ) > 0 and ε > 0 such that

2

aϕ T ϕ(ω)

2 ϕ∈A

()

> (B + ε)

ϕ∈A

|aϕ |2 ,

∀ω ∈ W

(19)

2056


or

2

a T ϕ(ω) ϕ

2

< (A − ε)

()

ϕ∈A

|aϕ |2 ,

∀ω ∈ W.

(20)

ϕ∈A

With a = {aϕ }ϕ∈A and W , we define for each ϕ ∈ A, mϕ := aϕ χW . Thus, mϕ ∈ L∞ (Ω) and only finitely many of these functions are not null. ϕ By Lemma 4.4, for each ϕ ∈ A there exists a polynomial sequence {Pk }k∈N such that ϕ

(i) Pk → mϕ , ϕ (ii) Pk ∞ 1 + 2mϕ ∞ , ∀k ∈ N. Since EH (A) is a Riesz sequence for S(A) with constants A and B,

A

|aϕ,h |

2

(ϕ,h)∈A×H

(ϕ,h)∈A×H

B

2

aϕ,h th ϕ

2

L (G)

|aϕ,h |2 ,

(ϕ,h)∈A×H

for each sequence {aϕ,h }(ϕ,h)∈A×H with finite support. Now, for each k ∈ N take {aϕ,h }(ϕ,h)∈A×H to be the sequence formed with the coefficients of ϕ the polynomials {Pk }ϕ∈A . Then, using (14) and (15), we have for each k ∈ N A

ϕ 2

P 2 k

ϕ∈A

L

(Ω)

2

ϕ

P (ω)T ϕ(ω) k

2 Ω

dmΓ (ω) B

()

ϕ∈A

ϕ 2

P 2 k

ϕ∈A

L (Ω)

.

(21)

Therefore, since mΓ (Ω) < ∞ and by the Dominated Convergence Theorem, inequality (21) can be extended to mϕ as A

ϕ∈A

mϕ 2L2 (Ω)

2

mϕ (ω)T ϕ(ω)

2 Ω

dmΓ (ω) B

()

ϕ∈A

ϕ∈A

mϕ 2L2 (Ω) .

(22)

So, if (19) occurs, integrating over Ω we obtain

2

m (ω)T ϕ(ω) ϕ

2 Ω

ϕ∈A

()

2 dmΓ (ω) > (B + ε) mϕ (ω)2 dmΓ , Ω ϕ∈A

which contradicts inequality (22). We can proceed analogously if (20) occurs. Hence, (18) holds. 2 For the case of principal H -invariant spaces we have the following corollary.


2057

Corollary 4.5. Let ϕ ∈ L2 (Ω). Then, the following are equivalent: (i) The set basis for S(ϕ) with constants A and B. EH (ϕ) is a Riesz ϕ (ω + δ)|2 B, a.e. ω ∈ Ω. (ii) A δ∈ | Proof. The proof is a straightforward consequence of Theorem 4.3 and Theorem 3.10.

2

We now want to give another characterization of when the set EH (A) is a frame (Riesz sequence) for S(A) with constants A and B. For this we will introduce what in the classical case are the synthesis and analysis operators. For an LCA group G and for a subgroup H as in (3.1) let us consider a subset A = {ϕi : i ∈ I } of L2 (G) where I is a countable set. Let Ω be a Borel section of Γ /. Fix ω ∈ Ω and let D be the set of sequences in 2 (I ) with finite support. Define the operator Kω : D → 2 () as Kω (c) = ci T ϕi (ω). (23) i∈I

The proof of the following proposition can be found in [2, Theorem 3.2.3]. Proposition 4.6. The operator Kω defined above is bounded if and only the set {T ϕi (ω): i ∈ I } is a Bessel sequence in 2 (). In that case the adjoint operator of Kω , Kω∗ : 2 () → 2 (I ), is given by

Kω∗ (a) = T ϕi (ω), a 2 () i∈I . The operator Kω is called the synthesis operator and Kω∗ the analysis operator. Definition 4.7. Let {ϕi : i ∈ I } ⊆ L2 (G) be a countable subset and Kω and Kω∗ the synthesis and analysis operators. We define the Gramian of {T ϕi (ω): i ∈ I } as the operator Gω : 2 (I ) → 2 (I ) given by Gω = Kω∗ Kω , and we also define the dual Gramian of {T ϕi (ω): i ∈ I } as the operator G˜ω : 2 () → 2 () given by G˜ω = Kω Kω∗ . The Gramian Gω can be associated with the (possible) infinite matrix Gω = ϕi (ω + δ)ϕj (ω + δ) δ∈

i,j ∈I

since Gω ei , ej = T ϕi (ω), T ϕj (ω) , where {ei }i∈I be the standard basis of 2 (I ). In a similar way, considering the basis {eδ }δ∈ of 2 (), we can associate the dual Gramian G˜ω with the matrix

˜ Gω = ϕi (ω + δ) ϕi ω + δ . i∈I

δ,δ ∈

Remark 4.8. The operator Kω (Kω∗ ) is bounded if and only if Gω (G˜ω ) is bounded. In that case we have Kω 2 = Kω∗ 2 = Gω = G˜ω .

2058


Now we will give a characterization of when EH (A) is a frame (Riesz sequence) for S(A) in terms of the Gramian Gω and the dual Gramian G˜ω . Proposition 4.9. Let A = {ϕi : i ∈ I } ⊆ L2 (G) be a countable set. Then, (1) The following are equivalent: (a1 ) EH (A) is a Bessel sequence with constant B. (b1 ) supessω∈Ω Gω B. (c1 ) supessω∈Ω G˜ω B. (2) The following are equivalent: (a2 ) EH (A) is a frame for S(A) with constants A and B. (b2 ) For almost every ω ∈ Ω, Aa2 G˜ω a, a Ba2 , for all a ∈ span{T ϕi (ω): i ∈ I }. (c2 ) For almost every ω ∈ Ω, σ (G˜ω ) ⊆ {0} ∪ [A, B]. (3) The following are equivalent: (a3 ) EH (A) is a Riesz sequence for S(A) with constants A and B. (b3 ) For almost every ω ∈ Ω, Ac2 Gω c, c Bc2 , for all c ∈ 2 (I ). (c3 ) For almost every ω ∈ Ω σ (Gω ) ⊆ [A, B].

Proof. It follows easily from Theorem 4.1, Theorem 4.3, Proposition 4.6 and Remark 4.8.

2

Note that Corollary 4.2 and Corollary 4.5 can also be obtained from the previous proposition. Definition 4.10. For an H -invariant space V ⊆ L2 (G) we define the dimension function of V as the map dimV : Ω → N0 given by dimV (ω) = dim J (ω), where J is the range function associated to V . We also define the spectrum of V as s(V ) = {ω ∈ Ω: J (ω) = 0}. As in the Rd case, every H -invariant space can be decomposed into an orthogonal sum of principal H -invariant spaces. This can be easily obtained as a consequence of Zorn’s Lemma as in [12]. The next theorem establishes a decomposition of H -invariant space with additional properties as in [1]. We do not include its proof since it follows readily from the Rd case (see [1, Theorem 3.3]).


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Theorem 4.11. Let us suppose that V is an H -invariant space of L2 (G). Then V can be decomposed as an orthogonal sum V=

S(ϕn ),

n∈N

where EH (ϕn ) is a Parseval frame for S(ϕn ) and s(S(ϕn+1 )) ⊆ s(S(ϕn )) for all n ∈ N. Moreover, dimS(ϕn ) (ω) = T ϕn (ω) for all n ∈ N, and dimV (ω) =

T ϕn (ω) ,

a.e. ω ∈ Ω.

n∈N

Acknowledgments We are grateful to the referees for their useful comments which have led to a significant improvement to the presentation of the article. References [1] Marcin Bownik, The structure of shift-invariant subspaces of L2 (Rn ), J. Funct. Anal. 177 (2) (2000) 282–309. [2] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Inc., Boston, MA, 2003. [3] C. de Boor, R. De Vore, A. Ron, Approximation from shift-invariant subspaces of L2 (Rd ), Trans. Amer. Math. Soc. 341 (2) (1994) 787–806. [4] C. de Boor, R. De Vore, A. Ron, The structure of finitely generated shift-invariant subspaces of L2 (Rd ), J. Funct. Anal. 119 (1) (1994) 37–78. [5] M.M. Dodson, Groups and the sampling theorem, in: A. Jerry (Ed.), Sampling Theory in Signal and Image Processing, vol. 6, Sampling Publishing, 2007, pp. 1–2. [6] J. Feldman, F.P. Greenleaf, Existence of Borel transversal in groups, Pacific J. Math. 25 (3) (1964). [7] H. Helson, Lectures on Invariat Subspaces, Academic Press, New York/London, 1964. [8] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 1, Springer-Verlag, New York, 1963. [9] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, vol. 2, Springer-Verlag, New York, 1970. [10] M. Hasumi, T.P. Srinivasan, Doubly invariant subspaces II, Pacific J. Math. 14 (1964) 525–535. [11] E. Kanuith, G. Kutyniok, Zeros or the Zak transform on locally compact abelian groups, Amer. Math. Soc. 126 (12) (1998) 3561–3569. [12] R.A. Kamyabi Gol, R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl. 340 (2008) 219–225. [13] I. Kluvánek, Sampling theorem in abstract harmonic analysis, Mat.-Fyz. Casopis Sloven. Akad. Vied. 15 (1965) 43–48. [14] A. Ron, Z. Shen, Frames and stable bases for shift invariant subspaces of L2 (Rd ), Canad. J. Math. 47 (1995) 1051–1094. [15] W. Rudin, Fourier Analysis on Groups, John Wiley, 1962. [16] T.P. Srinivasan, Doubly invariant subspaces, Pacific J. Math. 14 (1964) 701–707.


Forward and inverse scattering on manifolds with asymptotically cylindrical ends Hiroshi Isozaki a , Yaroslav Kurylev b , Matti Lassas c,∗ a Institute of Mathematics, University of Tsukuba, Tsukuba, 305-8571, Japan b Department of Mathematics, University College of London, United Kingdom c Department of Mathematics and Statistics, University of Helsinki, Finland

Received 11 May 2009; accepted 7 November 2009 Available online 2 December 2009 Communicated by J. Coron

Abstract We study an inverse problem for a non-compact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a short-range perturbation of the metric of the form (dy)2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy)2 + h(x, dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energies, we show that these two manifolds are isometric. © 2009 Elsevier Inc. All rights reserved. Keywords: Scattering theory; Inverse problems; Riemannian manifolds; Cylindrical ends

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . A priori estimates in half-cylinders . . 2.1. Besov type spaces on cylinder 2.2. A priori estimates . . . . . . . . . Manifolds with cylindrical ends . . . . 3.1. Resolvent equation . . . . . . . .

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E-mail address: [email protected] (M. Lassas). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.009

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H. Isozaki et al. / Journal of Functional Analysis 258 (2010) 2060–2118

3.2. Essential spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Radiation condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Limiting absorption principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Forward problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Unperturbed spectral representations . . . . . . . . . . . . . . . . . . . . . . . 4.2. Perturbed spectral representations . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Time-dependent scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. S-matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. From scattering data to boundary data . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Non-physical scattering amplitude . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Splitting the manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Interior boundary value problem . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Boundary control method for manifolds with asymptotically cylindrical ends 6.1. Blagovešˇcenskii’s identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2. Finite propagation property of waves . . . . . . . . . . . . . . . . . . . . . . . 6.3. Boundary distance functions and reconstruction of topology . . . . . . . 6.4. Continuation of the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5. Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1. Local reconstruction of Riemannian structure . . . . . . . . . . . 6.5.2. Iteration of local reconstruction . . . . . . . . . . . . . . . . . . . . 6.5.3. Maximal reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6. Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2074 2075 2082 2084 2084 2087 2091 2094 2095 2095 2097 2099 2102 2103 2105 2109 2111 2114 2114 2114 2115 2116 2116

1. Introduction The aim of this paper is to study spectral properties and related inverse problems for a connected, non-compact Riemannian manifold Ω of dimension n 2 with or without boundary. We assume that Ω is split into N + 1 parts Ω = K ∪ Ω1 ∪ · · · ∪ ΩN ,

(1.1)

where K is an open, relatively compact set, and Ωi , called an end of Ω, is diffeomorphic to Mi × (0, ∞), Mi being a compact manifold of dimension n − 1.(See Fig. 1.) More precisely, we assume that Ωi ∩ Ωj = ∅ if i = j , and we put K = Ω \ ( N i=1 Ωi ). Denoting the local coordinates on M by x, we assume that M is equipped with a Riemannian metric hi (x, dx) = i i n−1 p q p,q=1 hi,pq (x) dx dx . Letting y be the coordinate on (0, ∞), we denote the local coordinates on Ωiby X = (x, y). We assume that the Riemannian metric G on Ω, which is denoted by Gi = np,q=1 gi,pq (X) dX p dX q on Ωi , has the following property α ∂ gi,pq (X) − hi,pq (x) Cα (1 + y)−1−0 , X

∀α,

(1.2)

where hi,pn (x) = hi,np (x) = 0 if 1 p n − 1 and hi,nn (x) = 1, and Cα is a constant. The metric hi (x, dx) on Mi is allowed to be different for different ends. We shall assume either Ω has no boundary or each Mi , consequently Ω itself, has a boundary. In the latter case, we impose Dirichlet or Neumann boundary condition on ∂Ω. Let H = −G , where G is the

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Fig. 1. Manifold Ω has ends Ωj , j = 1, 2, . . . , N .

Laplace–Beltrami operator associated with the metric G. One can then define a scattering matrix S(λ) = ( Sij (λ)), which is a bounded operator on L2 (M1 ) ⊕ · · · ⊕ L2 (MN ), where λ ∈ (E0 , ∞) \ E(H ) is the energy parameter, E0 = inf σess (H ), and E(H ) is the set of exceptional points to be defined in (3.34). Our goal is the following. Theorem 1.1. Suppose we are given two manifolds Ω (r) , r = 1, 2, of the form (1.1) having Nr (r) ends, Ωi , i = 1, . . . , Nr , equipped with the metric G(r) satisfying the assumption (1.2). Assume (1) (2) that Ω1 = Ω1 and (1) G1

(2) = G1

= (dy) + h1 (x, dx), 2

h1 (x, dx) =

n−1

h1,j k (x) dx j dx k

(1.3)

j,k=1 (1) (2) (1) (2) on Ω1 = Ω1 , moreover S11 (λ) = S11 (λ) for all λ ∈ (E , ∞) \ (E (1) ∪ E (2) ), where E (r) is the (1) (2) set of exceptional points for H (r) , and E = max(E0 , E0 ). Then Ω (1) and Ω (2) are isometric (1) (2) as Riemannian manifolds with metrics G , G .

This means that if we observe waves coming in and going out of one end Ω1 , which is assumed to be non-perturbed, we can identify the whole manifold Ω. Note that in Theorem 1.1, neither the (r) number of ends of each Ω (r) nor the metric on the manifold Mi are assumed to be known a priori. The key idea of the proof is to introduce generalized eigenfunctions of the Laplace–Beltrami operator which are exponentially growing at infinity, and define the associated non-physical scattering amplitude. The crucial fact is that this non-physical scattering amplitude is the analytic continuation of the physical scattering amplitude. Then the physical scattering amplitude determines the non-physical scattering amplitude, which further determines the Neumann–Dirichlet map of the interior domain. By the boundary control method (see [3,8,9,47,52,53]), one can determine the metric inside. In this paper, we exclusively deal with the Neumann boundary condition. The other cases are treated similarly and in fact more easily. The forward problem of scattering is well known for short-range perturbations (see e.g. [30,31,60,61,75,33,76,44], see also [63]). The new issue we have to discuss in this paper is the difference of conormal derivatives on the boundary associated with unperturbed and perturbed metrics. Therefore, focusing on this point, we only explain the outline of the proof of the forward problem under the assumption (1.2) following the approach in [41], where spectral theory and inverse problems on hyperbolic spaces are developed in an elementary way.


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In the Euclidean space, the first work on the multi-dimensional inverse problem was done by Faddeev in the case of potential scattering [27]. This was extended by Saito [69] for shortrange potentials, and by Isozaki–Kitada [40] for long-range potentials. The determination of the obstacle from the scattering matrix of the wave equation was done by Schiffer and Lax– Phillips [56]. As for the metric perturbation problem in Rn , we should stress that it is still unknown for the general short-range perturbations. However, although there seems to be no literature, it is known that, given the scattering matrices for all energies, one can compute the Dirichlet–Neumann map for a bounded domain for all energies, which enables us to recover the local perturbation of the metric by virtue of the boundary control method. In recent years, inverse scattering problems have been generalized for some non-compact Riemannian manifolds, see e.g. [32,41,43,68]. In the cylindrical ends, the physical generalized eigenfunction of the Laplace–Beltrami operator admits the analytic continuation with respect to the energy parameter, and this analytically continued eigenfunction is exponentially growing as y → ∞. This sort of non-physical exponentially growing generalized eigenfunction was first introduced by Faddeev to develop the multi-dimensional Gel’fand–Levitan theory [26]. The exponentially growing solutions of Schrödinger equation was rediscovered in 1980s and were used to solve the inverse problem for the isotropic conductivity equation in dimensions n > 2 for C 2 -smooth conductivities [72], even in a reconstructive way [64], and in dimension two for C 2 -conductivities in [65] and finally for the L∞ -conductivities in [5], see review [28]. Later, also the anisotropic inverse conductivity problem has been solved by applying the exponentially growing solutions in dimension two [6,71]. These solutions have also been crucial in the study of multi-dimensional inverse scattering problem in the Euclidean space [66,35]. The interesting fact is that this apparently mysterious exponentially growing generalized eigenfunctions appear naturally in the cylindrical domain. Using these exponentially growing eigenfunctions, it is possible to obtain, from S11 (λ), the entry of the scattering matrix corresponding to Ω1 , the Gel’fand spectral data on a part of the boundary Γ = M1 × {1} of the non-compact manifold Ω 1 = Ω \ (M1 × (1, ∞)). The Gel’fand boundary data for this case is the family of the Neumann–Dirichlet map, Λ(z), Λ(z)f = u|Γ , where u is the solution to the boundary value problem ⎧ 1 ⎪ ⎨ (−G − z)u = 0 in Ω , 1 ∂ν u = 0 on ∂Ω ∩ ∂Ω , ⎪ ⎩ ∂ν u = ∂ν y = f on Γ. To solve this problem, we use the boundary control (BC) method (see [8] for the pioneering work and [9,47] for the detailed exposition). We note that typically the BC method deals with inverse problem on compact manifolds. The case of non-compact manifold considered here requires substantial modifications into the method, since the spectrum is no more discrete and it is also impossible to use eigenfunctions as coordinate functions. A short description of the BC-method for non-compact manifolds was given in [49]. Here we provide detailed constructions for the considered case of a manifold with asymptotically cylindrical ends. The structure of this paper is as follows. Sections 2, 3, 4 are devoted to a detailed analysis of scattering on manifolds with asymptotically cylindrical ends. After some preliminary estimates for the case of a half-cylinder with a product metric in Section 2, we discuss the spectral properties of the Laplacian in Ω in Section 3. Using these properties, we develop the scattering

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theory for such manifolds in Section 4. The remaining part of the paper, Sections 5, 6 are devoted to the inverse scattering. In Section 4, we show that S11 (λ) determines the Neumann–Dirichelt map Λ(z). An important step, which at the moment requires the product structure of the metric on S11 (λ), the non-physical scattering M1 × (0, ∞), is the recovery, from physical scattering matrix amplitude. At last, Section 6 is devoted to the development of the BC-method for non-compact manifolds. For the convenience of the reader, interested predominantly in the inversion methods, we make this section independent of the previous ones. Our manifold Ω is a mathematical model of compound waveguides, e.g. settings of optical and electric cables, oil, gas and water pipelines, etc., which are the most typical geometric constructions encountered in the every-day life. As for the inverse problem, many works have been devoted so far to the distribution of resonances for the waveguides [15,7,21,22,4,16]. Identification or reconstruction of the domain or the medium for grating, layers or waveguides are studied by [19,39,67,25]. In particular, a similar inverse problem for waveguides was considered by Eskin–Ralston–Yamamoto [25] when Ω is a slab, (0, B) × R, with the variable sound speed c(x, y), where c(x, y) = c(x) for large |y|. Christiansen [17] proved that in the planar waveguide R × (−γ , γ ) \ O, one can determine the obstacle O from one or two entries of the scattering matrix for high energies, provided O is strictly convex, compact with analytic boundaries. The present paper deals with the forward and inverse scattering problems for waveguide in a full generality. The notation in this paper is standard. For a self-adjoint operator A, σ (A), σp (A) and σess (A) mean its spectrum, point spectrum and essential spectrum, respectively. For two Banach spaces H1 , H2 , B(H1 ; H2 ) means the space of all bounded operators from H1 to H2 . For an operator A on a Hilbert space H, D(A) denotes its domain of definition. For a Riemannian manifold M, H m (M) denotes the usual Sobolev space of order m on M. For a domain D and a Hilbert pace H, L2 (D; H; dμ) means the space of H-valued L2 -functions on D with respect to the measure dμ. If H = C, we omit it. For a differentiable manifold M and p ∈ M, Tp (M) denotes the tangent space of M at p. A simplified version of our results is given in [42]. 2. A priori estimates in half-cylinders The forward problem of scattering has a long history, and has been brought into a satisfactory stage in the case of short-range perturbations. For example, an early statement of the limiting absorption principle, which is the first important step for the study of the continuous spectrum, can be found in [36]. For the case of waveguides, it was proved by [70]. Assuming, roughly speaking, that the ends are purely Euclidean cylinders outside a compact set, the limiting absorption principle, eigenfunction expansion theorem, completeness of wave operators, representation of S-matrices have been studied by Eidus [23], Goldstein [30,31], Lyford [60,61], Wilcox [75], Guillot–Wilcox [33], Edward [44]. Christiansen [16], and Christiansen–Zworski [18] studied the waveguide problem in the framework of b-metric due to Melrose [62,63]. Assuming that the ends, whose manifolds at infinity do not have boundaries, are not necessarily Euclidean allowing exponentially decaying perturbations, they derived the trace formula and spectral asymptotics. Our assumptions on the ends are similar to those of [15,16,18]. The difference is that we allow general short-range perturbations and also deal with boundary conditions for the manifolds at infinity. Although this is a folklore result, we feel it necessary to add the proof, since the main techniques have now been scattered in many papers. As the method of the proof of limiting absorption principle, we employ integration by parts due to Eidus. This is an elementary


2065

tool, however, gives no less deeper result than modern machineries. We show the completeness of wave operators by observing the behavior at infinity of solutions to the wave equation. This will give an intuition for the propagation of waves in the waveguide. We also deduce the eigenfunction expansion theorem from the behavior of the resolvent at infinity. This is an important intermediate step between the forward problem and the inverse problem. As a preliminary, let us begin with proving some a priori estimates for the operator −∂y2 − h on Ω0 = M × R+ with Neumann boundary condition, where y ∈ R+ = (0, ∞), M is a compact Riemannian manifold, and h is the Laplace–Beltrami operator associated with metric h(x, dx) equipped on M. 2.1. Besov type spaces on cylinder We define an abstract Besov type space, which was introduced by Hörmander [1] in the case of Rn . Let M be the above mentioned compact manifold, and ( , )M , · L2 (M) be inner product and norm of L2 (M), respectively. We define intervals In by

In =

(2n−1 , 2n ], n 1, (0, 1], n = 0.

Let B be the Banach space of L2 (M)-valued functions on (0, ∞) equipped with norm

f B =

∞

2

1/2

f (y)2 2

n/2

L (M)

n=0

dy

.

In

Its dual space is the set of L2 (M)-valued functions u(y) satisfying

u B∗ = sup 2−n/2

n0

v(y)2 2 L

1/2 < ∞.

dy (M)

In

It is easy to see that there exists a constant C > 0 such that

C −1 sup 2−n/2 n0

v(y)2 2

1/2

L (M)

dy

In

1 sup R R>1

C sup 2

R

u(y)2 2

L (M)

−n/2

v(y)2 2

L (M)

In

Therefore, we identify B ∗ with the space equipped with norm

u B∗ =

1 sup R>1 R

R

u(y)2 2

L (M)

0

dy

0

n0

1/2

1/2 dy

.

1/2 dy

.

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We also use the following weighted L2 space and weighted Sobolev space: For s ∈ R, L

2,s

f

⇔

f 2s

∞ 2 = (1 + y)2s f (y)L2 (M) dy < ∞, 0

H

u

m,s

u H m,s = (1 + y)s uH m (M×(0,∞)) < ∞.

⇔

In the following, · means · 0 and (·,·) denotes the inner product of L2 (M × R+ ). It often denotes the coupling of two functions f ∈ L2,s and g ∈ L2,−s or f ∈ B and g ∈ B ∗ . The following inclusion relations can be shown easily, and the proof is omitted. Lemma 2.1. For s > 1/2, we have L2,s ⊂ B ⊂ L2,1/2 ⊂ L2 ⊂ L2,−1/2 ⊂ B ∗ ⊂ L2,−s . We often make use of the following lemma, whose proof is also elementary and omitted. Lemma 2.2. Suppose u ∈ B ∗ . Then 1 lim R→∞ R

R

u(y)2 2

L (M)

dy = 0,

(2.1)

0

if and only if 1 lim R→∞ R

∞ y u(y)2 2 ρ dy = 0, L (M) R

∀ρ ∈ C0∞ (0, ∞) .

(2.2)

0

2.2. A priori estimates Let us consider the following equation in Ω0 = M × R+ :

2 −∂y − h − z u = f ∂ν u = 0 on ∂Ω0 ,

in Ω0 ,

(2.3)

z being a complex parameter, and ∂ν conormal differentiation on the boundary. In the following, we often denote by ∂xα u the norm of derivatives of |α|-th order of u without mentioning local coordinates. Lemma 2.3. Let z ∈ C be given. Then: (1) If u, f ∈ L2,s for some s ∈ R, we have ∂ α ∂ l u C u s + f s . x y s

|α|+l2


2067

(2) If u, f ∈ B ∗ , then we have

∂x u B∗ + ∂y u B∗ C u B∗ + f B∗ . Proof. We shall prove (2). Pick χ(y) ∈ C0∞ (R) such that χ(y) = 1 (|y| < 1), χ(y) = 0 (|y| > 2) and put χR (y) = χ(y/R). We take the inner product in L2 (Ω0 ) of (2.3) and χR2 (y)u. We then have y 2

χR ∂y u 2 + χR ∂y u, χ u + χR ∂x u 2 − z χR u 2 = f, χR2 u , R R which implies 2 1 y 2 2 χ u + χR u + χR f .

χR ∂y u + χR ∂x u C R R2

2

2

Then we have for R > 1 R

∂y u 2L2 (M) dy + 0

2R

R

2R

u 2L2 (M) dy + f 2L2 (M) dy .

∂x u 2L2 (M) dy C 0

0

0

Dividing by R and taking the supremum with respect to R, we obtain (2). Let us prove (1). The 1st order derivatives are dealt with in the same way as above. We put v = (1 + y)s u. Then v satisfies (−∂y2 − h − z)v = g, where g ∈ L2 (Ω). By the a priori estimates for elliptic operators, we have v ∈ H 2 (Ω), which proves (1). 2 Let λ1 < λ2 · · · → ∞ be the eigenvalues of −h , and Pn the associated eigenprojection. Then

z + h =

∞

z − λn Pn ,

n=1

√ √ where for ζ = reiθ (r > 0, 0 < θ < 2π), we define ζ = reiθ/2 . Our next aim is to derive some a priori estimates for solutions to Eq. (2.3). We use the method of integration by parts due to Eidus [23]. We put P (z) =

z + h ,

D± (z) = ∂y ∓ iP (z). Then Eq. (2.3) is rewritten as ∂y D± (z)u = ∓iP (z)D± (z)u − f.

(2.4)

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Lemma 2.4. Let ϕ(y) ∈ C ∞ (R) be such that ϕ(y) 0. For a solution u of Eq. (2.3), we put w = D+ (z)u. Then if Im z 0 we have for any 0 < a < b < ∞ b

2 ϕ (y)w(y)L2 (M) dy 2

a

b

y=b ϕ(y)(f, w)L2 (M) dy + ϕ w 2L2 (M) y=a .

a

Proof. Since w satisfies ∂y w = −iP (z)w − f , we have b

b ϕ(y)(∂y w, w)L2 (M) dy = −i

a

ϕ(y) P (z)w, w L2 (M) dy −

a

b ϕ(y)(f, w)L2 (M) dy. a

Taking the real part and integrating by parts, we have b ϕ w 2L2 (M) a −

b

2 ϕ (y)w(y)L2 (M) dy

a

b =2

ϕ(y) Im P (z)w, w L2 (M) dy − 2 Re

a

b ϕ(y)(f, w)L2 (M) dy. a

Taking notice of Im P (z) 0 for Im z 0, we get the lemma.

2

Let C+ = {z ∈ C; Im z 0}. Lemma 2.5. Let w be as in Lemma 2.4 and suppose that 1 lim R→∞ R

R

w(y)2 2

L (M)

dy = 0.

(2.5)

1

Then there exists a constant C > 0 independent of z ∈ C+ such that w(y)2 2

L (M)

C f B w B∗ ,

∀y ∈ R.

Proof. Taking ϕ(y) = 1 in Lemma 2.4, we have w(a)2 2

L (M)

2 w(b) 2 L

2 w(b) 2

b +2 (M)

L (M)

(f, w)

L2 (M)

dy

a

+ C f B w B∗ .

The assumption of the lemma implies lim infb→∞ w(b) L2 (M) =0, which proves the lemma.

2


2069

Corollary 2.6. Under the assumption of Lemma 2.5, there exists a constant C > 0 such that

w B∗ C f B ,

∀z ∈ C+ .

Proof. Lemma 2.5 implies that 1

w B∗ = sup R R>1

R

2

which proves this corollary.

w(y)2 2

L (M)

dy C f B w B∗ ,

0

2

Theorem 2.7. For a small δ > 0, let Jδ = z ∈ C+ ; dist Re z, σ (−h ) > δ . Let u be a solution to (2.3) such that w = D+ (z)u satisfies (2.5). Then there exists a constant C > 0 such that

u B∗ C f B holds uniformly for z ∈ Jδ . Proof. Let A(z) = Re P (z) = (P (z) + P (z)∗ )/2. By Eq. (2.4), we have ∂y (w, u)L2 (M) = −i P (z)w, u L2 (M) − (f, u)L2 (M) + (w, ∂y u)L2 (M) . In view of the formula −i P (z)w, u L2 (M) = −2i A(z)w, u L2 (M) + i P (z)∗ w, u L2 (M) = −2i w, A(z)u L2 (M) + i w, P (z)u L2 (M) , we then have ∂y (w, u)L2 (M) = −2i w, A(z)u L2 (M) − (f, u)L2 (M) + w 2L2 (M) . Using w = ∂y u − iP (z)u, we compute 2 2i w, A(z)u L2 (M) = 2i ∂y u, A(z)u L2 (M) + P (z)uL2 (M) + P (z)2 u, u L2 (M) . Summing up, we have arrived at 2 ∂y (w, u)L2 (M) = −2i ∂y u, A(z)u L2 (M) − P (z)uL2 (M) − (z + h )u, u L2 (M) − (f, u)L2 (M) + w 2L2 (M) .

2070


Taking the imaginary part and integrating in y, we have y=b Im (w, u)L2 (M) y=a = −2 Re

b

∂y u, A(z)u L2 (M)

a

b − Im z

b

u 2L2 (M) dy

− Im

a

(f, u)L2 (M) dy. a

Since A(z) is self-adjoint, we have by integration by parts b 2 Re

y=b ∂y u, A(z)u L2 (M) dy = A(z)u, u L2 (M) y=a .

a

Using Im z 0, we obtain y=b y=b Im (w, u)L2 (M) y=a + A(z)u, u L2 (M) y=a C f B u B∗ ,

(2.6)

where C is independent of z ∈ C+ . We renumber the eigenvalues of −h in the increasing order μ1 < μ2 < · · · without counting multiplicities and put μ0 = −∞, i.e. {λn ; n = 1, 2, . . .} and {μn ; n = 1, 2, . . .} are the same as subsets of R. For a sufficiently small δ > 0, we put Jn,δ = {z ∈ C+ ; μn−1 + δ < Re z < μn − δ}. Assume z ∈ Jn,δ and split u as u = u< + u> , where u< =

Pj u,

λj μn−1

u> =

Pj u.

λj μn

Recall that Pj is the eigenprojection associated with λj . We also define w< , w> , f< , f> similarly. Note that w< = D+ (z)u< . Let us remark that (2.3) and therefore (2.6) hold with w, u, f replaced by w√ < , u< , f< and w> , u> , f> , respectively. For eigenvalues λj μn−1 , we have Re z − λj δ. Therefore √ A(z)u< , u< L2 (M) δ u< 2L2 (M) . Since ∂y u(0) = 0, we have w< (0) = −iP (z)u< (0). Therefore − Im w< (0), u< (0) L2 (M) = Re P (z)u< (0), u< (0) L2 (M) = A(z)u< (0), u< (0) L2 (M) . Letting a = 0, b = t in (2.6), we then have Im w< (t), u< (t) L2 (M) + A(z)u< (t), u< (t) L2 (M) C f B u B∗ .

(2.7)


2071

Using (2.7), we have u< (t)2 2

L (M)

2 C w< (t)L2 (M) + f 1 1 R

R

u< (y)2 2

L (M)

dy C f< 2B + f = −∂y2 + Bz − i Im z u> = f> ,

(2.9)

where Bz is a uniformly, with respect to z, strictly positive operator on L2 (M). Hence, we have

u> L2 C f> L2 ,

(2.10)

u> B∗ C f> B .

(2.11)

which by Lemma 2.1 implies

The above two inequalities (2.8) and (2.11) prove the theorem.

2

3. Manifolds with cylindrical ends 3.1. Resolvent equation We return to the manifold Ω = K ∪ Ω1 ∪ · · · ∪ ΩN introduced in Section 1. Fix a point P0 ∈ K arbitrarily, and let dist(P , P0 ) be the geodesic distance with respect to the metric G from P0 to P . We put Ω0 (R) = P ∈ Ω; dist(P , P0 ) < R ,

Ω∞ (R) = P ∈ Ω; dist(P , P0 ) R .

For R > 0 large enough, take χ0 ∈ C0∞ (Ω) such that χ0 = 1 on Ω0 (R), χ0 = 0 on Ω∞ (R + 1). Define χj = 1 − χ0 on Ωj , χj = 0 on Ω \ Ωj . Then {χj }N j =0 is a partition of unity on Ω. Let G be the Laplace–Beltrami operator for the metric G on Ω endowed with Neumann boundary condition on ∂Ω. The conormal differentiation with respect to G is denoted by ∂ν . We put H = −G ,

R(z) = (H − z)−1 .

2072


As in Section 1, we identify Ωj with Mj × (0, ∞), and let hj (x, dx) be the metric on Mj . (0) We compare G with the unperturbed metric Gj = (dy)2 + hj (x, dx) on Ωj . Let G(0) be the j

(0)

Laplace–Beltrami operator for Gj with Neumann boundary condition on ∂Ωj . The associated conormal differentiation is denoted by ∂ν (0) . We put j

(0)

Hj

(0) −1 (0) Rj (z) = Hj − z .

= −G(0) , j

(0)

Our next concern is the difference between the boundary conditions for H and Hj . We put for large R > 0 ∂Ωj (R) = ∂Ω ∩ Ωj ∩ Ω∞ (R). Lemma 3.1. There exists a real function w(x, y) ∈ C ∞ (Ωj ) such that

∂ν w(x, y) = 0

on ∂Ωj (R), as y → ∞. w(x, y) = y + O y −1−0

(3.1)

= Proof. By the decay assumption (1.2), letting w(x, y) = y + w (x, y), we should have ∂ν w −∂ν y = O(y −1−0 ) on ∂Ωj (R). Extending the vector field ν near the boundary and integrating along it, we get w = O(y −1−0 ). 2 For m 0 and s ∈ R, we define the weighted Sobolev space on the boundary by ψ ∈ H m,s ∂Ωj (R)

⇔

(1 + y)s ψ ∈ H m ∂Ωj (R) .

Lemma 3.2. There exists an operator of extension Ej such that for m 1/2 and ψ ∈ H m (∂Ωj (R)) ∂ν Ej ψ =

ψ on ∂Ωj (R), 0 on Ω \ (Ωj ∩ Ω∞ (R − 1/2)),

supp(Ej ψ) ⊂ Ωj ∩ Ω∞ (R − 1).

(3.2)

(3.3)

For m 1/2 and s 0, it satisfies Ej ∈ B H m,s ∂Ωj (R) ; H m+3/2,s (Ωj ) .

(3.4)

Proof. Let M = Ωj ∩ Ω∞ (R − 2). We smoothly modify the corner of M , i.e. {P ∈ Ωj ∩ ∂Ω; dist(P , P0 ) = R − 2}, and let M be the resulting manifold. Let νM be the unit outer normal to M. By solving the elliptic boundary value problem

(−G + i)u = 0 in M, ∂νM u = ψ on ∂M,

(3.5)


2073

we define Ej ψ = χ j u, where χ j ∈ C ∞ (Ωj ) is such that χ j = 1 on Ωj ∩ Ω∞ (R − 1/4), χ j = 0 on Ω \ (Ωj ∩ Ω∞ (R − 1/2)). It then satisfies (3.2), (3.3). The property (3.4) for s = 0 follows from the standard estimate for the elliptic boundary value problem. Let 0 < s 1 + 0 and take ψ ∈ H m,s (∂M). For the solution u to the boundary value problem (3.5), we define u1 = (1 + w(x, y))s u and ψ1 = (1 + w(x, y))s ψ , where w(x, y) is constructed in Lemma 3.1. Then u1 is a solution to the boundary value problem

(−G + L1 + κ)u1 = 0 in M, ∂νM u1 = ψ2

on ∂M,

where κ > 0 is sufficiently large, and L1 is a 1st order differential operator with bounded coefficients, and ψ2 = ψ1 on ∂Ωj (R). Since the mapping ψ2 → u1 is bounded from H m (∂M) to H m+3/2 (M), we get (3.4) with 0 < s 1 + 0 . Repeating this procedure, we can prove (3.4) for all s > 0. 2 For u ∈ H 2 (Ωj ) satisfying ∂ν (0) u = 0 on ∂Ωj (R), we have j

∂ν (χj u) = w(x, y)−1−0 Bj u on ∂Ωj (R),

(3.6)

Bj = w(x, y)1+0 χj (∂ν − ∂ν (0) ) + (∂ν χj )

(3.7)

where

j

is a 1st order differential operator on ∂Ωj (R) with bounded coefficients. We put Ej = w(x, y)−1−0 Ej .

(3.8)

Then by (3.1), (3.2) and (3.6), for u ∈ H 2 (Ω) satisfying ∂ν (0) u = 0 on ∂Ωj (R) the following j

formula holds ∂ν Ej Bj u = ∂ν (χj u)

on ∂Ωj (R).

(3.9)

Moreover y 1+0 Ej Bj ∈ B H 2 (Ω); H 2 (Ω) ∩ B H 3/2 (Ω); H 3/2 (Ω) .

(3.10)

Suppose u satisfies ⎧ ⎨ (−G(0) − z)u = f j

in Ωj ,

⎩ ∂ν (0) u = 0 on Ωj ∩ ∂Ω. j

Then by (3.9), vj = χj u − Ej Bj u satisfies

(−G − z)vj = χj f + Vj (z)u ∂ν vj = 0 on ∂Ω,

in Ω,

(3.11)

2074


where Vj (z) = [−G , χj ] + χj (G(0) − G ) + (G + z)Ej Bj .

(3.12)

j

Lemma 3.3. Let χ j ∈ C ∞ (Ω) be such that χ j = 1 on Ωj ∩ Ω∞ (R − 1) and χ j = 0 outside Ωj ∩ Ω∞ (R − 2). Then for z ∈ / R, the following resolvent equations hold: R(z)χj = χj − Ej Bj − R(z)Vj (z) Rj(0) (z) χj , (0) j Jj−1 Rj (z)Jj χj − (Ej Bj )∗ − Vj (z)∗ R(z) , χj R(z) = χ

(3.13) (3.14)

(0)

where Jj = (det G/ det Gj )1/2 , and the adjoint ∗ is taken with respect to the inner product of L2 (Ω) with volume element from the metric G. Moreover Rj (z)Jj (Ej Bj )∗ and (0)

Rj (z)Jj Vj (z)∗ R(z) are compact on L2 (Ω). (0)

(0)

Proof. Let u = Rj (z) χj f for z ∈ / R. Then checking the boundary condition by (3.9), we have (0)

(0)

χj f − Ej Bj Rj (z) χj f ∈ D(H ), and by (3.11) (H − z)vj = χj χ j f + Vj (z)u = vj = χj Rj (z) χj f + Vj (z)u, which implies (3.13). By extending f ∈ L2 (Ωj ) to be 0 outside Ωj , we regard L2 (Ωj ) as a closed subspace of (0) L2 (Ω). The volume elements dV and dVj(0) of G and G(0) j satisfy dV = Jj dVj . For A ∈ B(L2 (Ωj ); L2 (Ωj )), let A∗ and A∗(j ) denote their adjoint operators with respect to the volume (0) elements dV and dVj , respectively. Then it is easy to show that A∗ = Jj−1 A∗(j ) Jj . Taking A = Rj (z), and noting that R(z)∗ = R(z) and Rj (z)∗(j ) = Rj (z), we prove (3.14). (0)

(0)

(0)

By (3.10) and (3.12), Ej Bj Rj(0) (z) and R(z)Vj (z)Jj Rj(0) (z) are compact on L2 (Ω), which implies the last assertion of the lemma. 2 3.2. Essential spectrum Lemma 3.4. σess (H ) = [0, ∞). Proof. Lemma 3.3 implies χj R(z) − χ j Jj−1 Rj (z)Jj χj is compact. Therefore (0)

R(z) =

N

χ j Jj−1 Rj (z)Jj χj + K(z), (0)

(3.15)

j =1

where K(z) is a compact operator and satisfies K(z) C|Im z|−2 1 + |z| ,

(3.16)


2075

where · denotes the operator norm in L2 (Ω) and the constant C is independent of z. For f (λ) ∈ C0∞ (R), there exists F (z) ∈ C0∞ (C), called an almost analytic extension of f , such that F (λ) = f (λ) for λ ∈ R and |∂z F (z)| Cn |Im z|n , ∀n 0, and the following formula holds for any self-adjoint operator A: 1 f (A) = 2πi

∂z F (z)(z − A)−1 dz dz.

(3.17)

C

(See e.g. [34] or [41].) We replace (z − A)−1 by −R(z) and plug (3.15). The inequality (3.16) implies ∂z F (z)K(z) C, and the integral over C converges in the operator norm, hence it gives a (0) compact operator. We then see that ϕ(H ) − N j Jj−1 ϕ(Hj )Jj χj is compact for any ϕ(λ) ∈ j =1 χ C0∞ (R). Since σ (Hj ) = [0, ∞), we have ϕ(Hj ) = 0 if ϕ(λ) ∈ C0∞ ((−∞, 0)). Therefore ϕ(H ) is compact if ϕ(λ) ∈ C0∞ ((−∞, 0)), which implies that (−∞, 0) ∩ σess (H ) = ∅. For λ ∈ (0) (0) (0) (0, ∞) = σ (Hj ), one can construct un ∈ D(Hj ) such that un = 1, (Hj − λ)un → 0, and supp un ⊂ {y > Rn } with Rn → ∞. Then letting vn = χj un − Ej Bj un , we have vn ∈ D(H ),

(H − λ)vn → 0, vn → 0 weakly and vn > C uniformly in n with a constant C > 0. This implies λ ∈ σess (H ). 2 (0)

(0)

The set of thresholds for H is defined by T (H ) =

N

σp (−hj ),

(3.18)

j =1

where hj is the Laplace–Beltrami operator on Mj . Replacing Ω0 in Section 2 by Ωj with j = 1, . . . , N , we define the Besov type spaces Bj , Bj∗ . We put

f B = χ0 f L2 (Ω) +

N

χj f Bj ,

j =1

u B∗ = χ0 f L2 (Ω) +

N j =1

χj f Bj∗ .

The weighted L2 space L2,s and the weighted Sobolev space H m,s are defined similarly. 3.3. Radiation condition A solution u ∈ B ∗ of the reduced wave equation

(H − λ)u = f ∂ν u = 0

in Ω, λ > 0,

on ∂Ω,

is said to satisfy the outgoing radiation condition if

2076


1 lim R→∞ R

R

χj ∂y − iPj (λ) u2 2

L (Mj )

dy = 0,

1 ∀j N,

(3.19)

0

where Pj (z) =

z + hj .

If ∂y −iPj (λ) is replaced by ∂y +iPj (λ), we say that u satisfies the incoming radiation condition. In the following, u is always assumed to satisfy the boundary condition ∂ν u = 0 on ∂Ω. Lemma 3.5. Let λ ∈ (0, ∞) \ T (H ). If u ∈ B ∗ satisfies (H − λ)u = 0 and the outgoing (or incoming) radiation condition, it also satisfies 1 lim R→∞ R

R

χj u 2L2 (M ) dy = 0,

1 j N.

j

0

Proof. We take ρ(t) ∈ C0∞ ((0, ∞)) such that ρ(t) 0, supp ρ(t) ⊂ (1, 2) and and put y ϕR (y) = χ , R

∞ 0

ρ(t) dt = 1,

∞ χ(t) =

ρ(s) ds. t

Then ϕR (y) = 1 for y < R and ϕR (y) = 0 for y > 2R. We next construct ψR ∈ C0∞ (Ω) such that ψR = 1 on K and ψR = ϕR on Ωj for 1 j N . Then we have i[H, ψR ]u, u = i[H − λ, ψR ]u, u = 0. By the construction of ψR , [H, ψR ] = 0 on K. By the assumption (1.2), on Ωj the commutator has the form y 2i i[H, ψR ] = ρ (3.20) ∂y + Lj,R , R R where Lj,R is a 1st order differential operator whose coefficients have the form y 1 χ O y −0 R R and χ (y) is either ρ(y) or ρ (y). Let v = (1 + y)−0 u. Then by Lemma 2.1 and Lemma 2.3(1) (which also holds for Gj ), ∂x v, ∂y v ∈ L2,−δ for some 0 < δ < 1/2. Therefore 1 R

R

∂x v 2L2 (M ) + ∂y v 2L2 (M ) dy j

0

j

C R 1−2δ

∂x v 2−δ + ∂y v 2−δ ,


2077

which tends to 0 as R → ∞. Therefore by Lemma 2.2 lim (Lj,R u, u)L2 (Ωj ) = 0.

R→∞

Hence we have by using (3.20), ∞ N y 1 ρ lim (∂y χj u, χj u)L2 (Mj ) dy = 0. R→∞ R R j =1

(3.21)

0

Assume that u satisfies the outgoing radiation condition. Using the inequality ∞ 1 y ∂y − iPj (λ) χj u, χj u L2 (M ) dy ρ j R R 0

C u B∗

1 R

1/2 ∞ 2 y ∂y − iPj (λ) χj u 2 ρ dy L (Mj ) R 0

and (3.21), we then have ∞ N 1 y Pj (λ)χj u, χj u L2 (M ) dy = 0. ρ lim j R→∞ R R j =1

(3.22)

0

As in the proof of Theorem 2.7, we split χj u into two parts, χj u< = Ej (−∞, λ) χj u,

χj u> = Ej (λ, ∞) χj u,

where Ej (·) is the spectral projection associated with −hj . Then by the short-range decay assumption of the metric, 2 −∂y − hj − λ χj u> =: fj ∈ L2 (Ωj ). Since λ ∈ / σ (−Mj ), arguing in the same way as in the proof of (2.9), 2 −∂y − hj − λ χj u> = −∂y2 + Bj χj u> , where Bj is a self-adjoint operator on L2 (Mj ) such that Bj δ(1 − Mj ), δ > 0 being a constant. Therefore, Pj (λ)χj u> ∈ L2 (Ωj ), hence 1 lim R→∞ R

∞ y Pj (λ)χj u> , χj u> L2 (M ) dy = 0. ρ j R 0

(3.23)

2078


Since Pj (λ)χj u> = iCj (λ)χj u> , where Cj (λ) is a strictly positive operator on L2 (Mj ), this also implies 1 lim R→∞ R

∞ y

χj u> 2L2 (M ) dy = 0. ρ j R

(3.24)

0

We show that N 1 y Pj (λ)χj u< , χj u< L2 (M ) dy = 0. ρ j R→∞ R R lim

(3.25)

j =1

In fact, in view of (3.22), splitting u = u< + u> and using (3.23), to prove (3.25) we have only to show that 1 lim R→∞ R

y Pj (λ)χj u> , χj u< L2 (M ) dy = 0, ρ j R

(3.26)

and the same assertion with u< and u> exchanged. Let us note that Pj (λ)χj u> , χj u
, χj Pj (λ)∗ u

χj u< . Therefore 1 R

y Pj (λ)χj u> , χj u< L2 (M ) dy j R 1/2 1/2 C 1 y y 2 2

χj u> L2 (M ) dy

χj u< L2 (M ) dy ρ ρ j j R R R R 1/2 1 y

χj u> 2L2 (M ) dy C , ρ j R R

ρ

since χj u< ∈ B ∗ . By (3.24), this converges to 0. Similarly, we can prove (3.26) with u< and u> exchanged. On the other hand, (Pj (λ)χj u< , χj u< ) C χj u< 2L2 (Ω ) for a constant C > 0, which dej pends on λ. Therefore by (3.25) 1 lim R→∞ R

∞ y

χj u< 2L2 (M ) dy = 0. ρ j R 0

By (3.24) and (3.27), we complete the proof of the lemma.

2

(3.27)


2079

Lemma 3.6. Suppose u ∈ B ∗ satisfies (H − λ)u = f for λ ∈ (0, ∞) \ T (H ) and ∂ν u = 0 on ∂Ω. Assume also for some 1 j N , f ∈ L2,s (Ωj ) for any s > 0, and 1 lim R→∞ R

R

χj u 2L2 (M ) dy = 0.

(3.28)

j

0

Then u ∈ L2,s (Ωj ) for any s > 0. Moreover for any s > 0 and any compact interval I ⊂ (0, ∞) \ T (H ), there exists a constant Cs,I > 0 such that

χj u L2.s (Ωj ) Cs,I u B∗ (Ωj ) + f L2,s+1 (Ωj ) ,

∀λ ∈ I.

(3.29)

(0)

Proof. We construct counterparts of Ej and Bj when the roles of G and Gj are interchanged. (0) Namely, there exists an operator of extension Ej such that for m 1/2 and ψ ∈ H m (∂Ωj (R))

ψ on ∂Ωj (R), 0 on Ω \ (Ωj ∩ Ω∞ (R − 1/2)), (0) supp Ej ψ ⊂ Ωj ∩ Ω∞ (R − 1), (0) Ej ∈ B H m,s ∂Ωj (R) ; H m+3/2,s (Ωj ) , m 1/2, s 0. (0) ∂ν (0) Ej ψ = j

If ∂ν u = 0 on ∂Ω, we have ∂ν (0) (χj u) = y −1−0 Bj u on ∂Ωj (R), (0)

j

where (0)

Bj = y 1+0 χj (∂ν (0) − ∂ν ). j

We put Ej(0) = y −1−0 Ej(0) . Then (0)

(0)

∂ν (0) Ej Bj u = ∂ν (0) (χj u). j

j

Suppose u ∈ B ∗ satisfies (H − λ)u = f , λ ∈ (0, ∞) \ T (H ), and ∂ν u = 0 on ∂Ω. We put (0) (0) vj = χj u − Ej Bj u. Then vj satisfies ⎧ ⎨ −∂ 2 − h − λ vj = χj f + Lj u + ∂ 2 + h + λ E (0) B (0) u in Ωj , y y j j j j ⎩ ∂ν (0) vj = 0 on ∂Ωj . j

(3.30)

2080


Here Lj is a 2nd order differential operator with coefficients decaying like O(y −1−0 ). Note that (0) (0) fj := χj f + Lj u + (∂y2 + hj + λ)Ej Bj u ∈ L2,1+0 (Ωj ). Let vj,n = (vj (·, y), ψn (·))L2 (Mj ) , where ψn (x) is the normalized eigenvector associated with the eigenvalue λn of −hj . Then we have 2 −∂y − μn vj,n = gj,n ,

μ n = λ − λn ,

(3.31)

where gj,n ∈ L2,(1+0 )/2 ((−∞, ∞)), and vj,n (y) = gj,n (y) = 0 for y < 0. Let r0 (z) = (−∂y2 − z)−1 in L2 (R), i.e. i r0 (z)g (y) = √ 2 z

∞ ei

√ z|y−y |

g y dy ,

−∞

√ where Im z 0. Then as can be checked easily for any s > 0 and δ > 0, there exists a constant Cs,δ > 0 such that 1 + |y| s r0 (−a) 1 + |y| −s

B(L2 (R);L2 (R))

Cs,δ ,

∀a > δ.

Therefore by (3.31), one can show that Ej (λ, ∞) vj ∈ L2,(1+0 )/2 (Ωj ),

(3.32)

where Ej (·) is the spectral projection associated with −hj . For λn < λ, we study vj,n separately. By (3.28), 1 R→∞ R

R

lim

vj,n (y)2 dy = 0.

(3.33)

0

In view of (3.33), we see that vj,n satisfies both of the outgoing and incoming radiation conditions. Adopting the outgoing radiation condition, we see that vj,n is written as vj,n = r0 (μn + i0)gj,n , i.e. i vj,n (y) = √ 2 μn

y e

√ i μn (y−y )

gj,n y dy +

∞ e

√ i μn (y −y)

gj,n y dy .

y

0

Note that gj,n ∈ L1 ((0, ∞)), since gj,n ∈ L2,(1+0 )/2 ((0, ∞)). Therefore i lim vj,n (y) = √ y→∞ 2 μn

∞ ei

√

μn (y−y )

gj,n y dy .

0

The condition (3.33) implies that this limit is equal to 0, which implies


2081

∞ ∞ √ √ i i μn (y−y ) i μn (y −y) vj,n (y) = √ gj,n y dy + e gj,n y dy . − e 2 μn y

y

Using the following Lemma 3.7 (Hardy’s inequality), we have (1 + y)(0 −1)/2 vj,n ∈ L2 ((0, ∞)). Using (3.32), we then have vj ∈ L2,(−1+0 )/2 ((0, ∞)). By Lemma 3.2 and the formula χj u = (0) (0) vj + Ej Bj u, we have u ∈ L2,(−1+0 )/2 (Ωj ). Thus we have seen that u gains the decay of order 0 in Ωj . Then in (3.31), gj,n ∈ L2,(1+20 )/2 ((0, ∞)). Hence vj ∈ L2,(−1+3/2) (Ωj ). Repeating this procedure, we obtain χj u ∈ L2,m0 (Ωj ), ∀m > 0. The estimate (3.29) can be proven by re-examining the above arguments. 2 Lemma 3.7. Let f (y) ∈ L1 ((0, ∞)) and put ∞ u(y) =

f (t) dt. y

The for s > 1/2 ∞ y

2(s−1)

2 u(y) dy

0

4 (2s − 1)2

For the proof, see [41, Chapter 3, Lemma 3.3].

∞

2 y 2s f (y) dy.

0

2

Lemma 3.8. Let σrad (H ) be the set of λ ∈ / T (H ) for which there exists a non-trivial solution u ∈ B ∗ of the equation (H − λ)u = 0 satisfying the radiation condition. Then σrad (H ) = σp (H ) \ T (H ). Moreover it is a discrete subset of R \ T (H ) with possible accumulation points in T (H ) and ∞. Proof. The first part of the lemma is proved by Lemmas 3.5 and 3.6. Let I be a compact interval in R \ T (H ), and suppose there exists an infinite number of eigenvalues (counting multiplicities) in I . Let un , n = 1, 2, . . . be the associated orthonormal eigenvectors. Take any R > 0 and let χ0 = χ0R be the function introduced in the beginning of this section. We decompose un = χ0R un +

N 1 − χ0R χj un . j =0

Then by (3.29), for any > 0 there exists R > 0 such that (1 − χ0R )un L2 (Ω) < uniformly in n. 1 (Ω) to L2 (Ω), {χ u } is compact in L2 (Ω). By the compactness of the imbedding of Hloc R n n loc Therefore {un }n contains a convergent subsequence, which is a contradiction. 2 It is known that the eigenvalues embedded in σess (H ) can accumulate at τ (H ) only from below, see [45].

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The set of exceptional points E(H ) is now defined by E(H ) = T (H ) ∪ σp (H ).

(3.34)

Weyl’s formula for the asymptotic distribution of eigenvalues on compact manifolds and Lemma 3.8 imply that T (H ) is discrete and E(H ) has only finite number of accumulation points on any compact interval in R. 3.4. Limiting absorption principle For a self-adjoint operator H defined in a Hilbert space H, the limit lim (H − λ ∓ i)−1 =: (H − λ ∓ i0)−1 ,

→0

λ ∈ σ (H ),

does not exist as a bounded operator on H. However if λ is in the continuous spectrum of H , it is sometimes possible to guarantee the existence of the above limit in B(X ; X ∗ ), where X , X ∗ are Banach spaces such that X ⊂ H = H ∗ ⊂ X ∗ , and H is identified with its dual space via Riesz’ theorem. This fact, called the limiting absorption principle, is central in studying the absolutely continuous spectrum, and many works are devoted to it. We employ in this paper the classical method of integration by parts pioneered by Eidus [23]. The crucial step is to establish a priori estimates as in Section 2 of this paper, and to show the uniqueness of solutions to the reduced wave equation satisfying the radiation condition. After this hard analysis part, the remaining arguments are almost routine. We take any compact interval I ⊂ (0, ∞) \ E(H ) and let J = {z ∈ C; Re z ∈ I, Im z = 0}. We first note that Lemma 2.3 also holds for the solution to the equation

(H − z)u = f in Ω, ∂ν u = 0 on ∂Ω, (0)

(0)

by the standard elliptic regularity estimates. We put u = R(z)f and vj = χj u − Ej Bj u as in the proof of Lemma 3.6. Then vj satisfies (3.30) with λ replaced by z. We can then apply Theorem 2.7 to see that

χj u B∗ Cs f B + u −s ,

(3.35)

for any s > 1/2, where C is independent of z ∈ J . Once (3.35) is proved, we can repeat the arguments in Chapter 2, Section 2 of [41] or those of Ikebe–Saito [37] without any essential change. Note that here and in the sequel, we use ( , ) to denote the inner product (u, v) =

uv dV Ω

of L2 (Ω) as well as the coupling between B and B ∗ , or L2,s and L2,−s .


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Lemma 3.9. Take s > 1/2 sufficiently close to 1/2. (1) There exists a constant C > 0 such that sup R(z)f −s C f B . z∈J

(2) For any λ ∈ I and f ∈ B, the strong limit lim→0 R(λ ± i)f exists in L2,−s . (3) I λ → R(λ ± i0)f ∈ L2,−s is continuous. Sketch of the proof. Suppose the uniform bound (1) is not true. Then there exist a sequence zn ∈ J and fn ∈ B such that un = R(zn )fn satisfies un −s = 1 and fn B → 0. Without loss of generality, we can assume that zn → λ ∈ I . Using (3.35) with 0 < s < s and the compactness 2 into L2 , one can assume that u converges to some u ∈ B ∗ , and u of the embedding of Hloc n loc satisfies the equation (H − λ)u = 0 and the radiation condition (see Corollary 2.6). Therefore u = 0 by Lemma 3.8. However this contradicts un −s = 1. The assertions (2) and (3) are proved in a similar manner. 2 Using this lemma one can prove the following theorem. Theorem 3.10. (1) For any λ ∈ I , lim→0 R(λ ± i)f exists in the weak-∗ sense: ∃ lim R(λ ± i)f, g =: R(λ ± i0)f, g , →0

∀f, g ∈ B.

(2) There exists a constant C > 0 such that R(λ ± i0)f

B∗

C f B ,

λ ∈ I.

Moreover R(λ ± i0)f satisfies the outgoing radiation condition for λ + i0 and incoming radiation condition for λ − i0. (3) For any f, g ∈ B, I λ → (R(λ ± i0)f, g) is continuous. (4) Let E(·) be the spectral decomposition of H . Then E([0, ∞) \ E(H ))L2 (Ω) = Hac (H ), and we have the following orthogonal decomposition L2 (Ω) = Hac (H ) ⊕ Hp (H ). Sketch of the proof. Since L2,−s (s > 1/2) is dense in B ∗ , (1) follows from Lemma 3.9(2) and (3.35). The assertion (2) follows from Lemma 3.9(1) and (3.35). The remaining assertions are proved in the same way as in Chapter 2, Section 2 of [41] or Ikebe–Saito [37]. 2 ∞ Let us recall that for a self-adjoint operator H = −∞ λ dE(λ), the absolutely continuous subspace for H , Hac (H ), is the set of u such that (E(λ)u, u) is absolutely continuous with respect to dλ, and the point spectral subspace, Hp (H ), is the closure of the linear hull of eigenvectors of H .

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4. Forward problem 4.1. Unperturbed spectral representations ∗ Let {χj }N j =0 be the partition of unity defined in Section 3. Recall the spaces B and B introduced in Section 2. For two functions f , g on Ω, f g means that

1 lim R→∞ R

R

χj (y) f (·, y) − g(·, y) 2 2

L (Mj )

dy = 0,

1 ∀j N.

0

We also use the same notation f g for f , g defined on Ωj . Green’s function of −d 2 /dy 2 − ζ on (0, ∞) with Neumann boundary condition at y = 0 is i G y, y ; ζ = √ ζ

√ √ cos( ζ y)ei ζ y , 0 < y < y , √ √ ei ζ y cos( ζ y ), 0 < y < y.

Let λj,1 < λj,2 · · · be the eigenvalues of −hj with normalized eigenvectors ϕj,n (x), (0)

n = 1, 2, . . . . Without loss of generality, we assume that ϕj,n (x)’s are real-valued. Let Hj −∂y2

− hj with Neumann boundary condition. Then ∞ (0) Rj (z)f (x, y) =

∞

(0) (0) Rj (z) = (Hj

− z)−1

=

is written as

G y, y ; z − λj,n (Pj,n f ) x, y dy ,

n=1 0

(Pj,n f )(x, y) = f (·, y), ϕj,n ϕj,n (x),

(4.1)

(0) , being the inner product of L2 (Mj ; det(hj ) dx). Note that det(hij ) = det Gj . For f (x, y) ∈ 1/2 dx dy), we define its cosine transform by L2 (Mj × (0, ∞); (det G(0) j )

Fcos (λ)f (x) = π

−1/2 −1/4

∞

λ

√ cos(y λ )f (x, y) dy.

0

Lemma 4.1. For f ∈ B, and λ ∈ (0, ∞) \ σp (−hj ), we have √ √ (0) (λ − λj,n )−1/4 e±iy λ−λj,n Fcos (λ − λj,n )Pj,n f (x). Rj (λ ± i0)f ±i π λj,n λ is rewritten as Aj (λ)f :=

λj,n >λ

1 2kn

∞ −k |y−y | e n + e−kn (y+y ) fj,n x, y dy 0


where fj,n = Pj,n f and kn =

2085

λj,n − λ. Then we have

Aj (λ)f (·, y)2 2 = L (M) λj,n >λ

∞ 2 1 −kn |y−y | −kn (y+y ) e f ·, y , ϕ + e dy j,n 2 L (Mj ) 4kn2 0

∞ f (·, y), ϕj,n 2 dy

Cλ

λj,n >λ 0

Cλ f 2L2 (M

j ×(0,∞))

.

Hence Aj (λ) ∈ B(L2 ; L∞ (R+ ; L2 (Mj ))) ⊂ B(B; B ∗ ). To estimate the term in which λj,n < λ, we put ∞ uj,n (x) =

G y, y ; λ ± i0 − λj,n fj,n x, y dy .

0

Then we have uj,n (x) Cλ

∞

fj,n (x, y) dy.

0

Since

uj,n B∗ C uj,n L∞ ,

fj,n L1 C fj,n B .

We have proven that Rj (λ ± i0) ∈ B(B; B ∗ ). Now the assertion of the lemma is easy to prove if there exists n0 > 0 such that fj,n = 0 for n n0 , and fj,n is compactly supported for n < n0 . Since such an f is dense in B, we have proven the lemma. 2 (0)

(0)

The generalized eigenfunction of Hj

is defined for λ > λj,n

(0) Ψj,n (x, y; λ) = π −1/2 (λ − λj,n )−1/4 cos(y λ − λj,n )ϕj,n (x). (0)

(4.2)

(0)

This Ψj,n (x, y; λ) is often denoted by Ψj,n (λ) in the sequel. It satisfies ⎧ (0) ⎪ ⎨ (−G(0) − λ)Ψj,n (λ) = 0 in Ωj , j

(0) ⎪ ⎩ ∂ν (0) Ψj,n (λ) = 0 on ∂Ωj . j

(0)

The Fourier transformation associated with Hj

is defined by

(4.3)

2086


(0)

Fj (λ)f =

∞

(0)

(4.4)

χλj,n (λ)Fj,n (λ)f,

n=1

where χλj ,n is the characteristic function of the interval (λj,n , ∞), and (0) Fj,n (λ)f (x) =

(0) Ψj,n (λ)f

(0) dVj

ϕj,n (x)

Ωj

= Fcos (λ − λj,n )Pj,n f (x), (0)

where dVj

(4.5)

(0)

= (det Gj )1/2 dx dy. Define a subspace of L2 ((0, ∞); L2 (Mj ); dλ) by j = H

∞

L2 (λj,n , ∞); dλ ⊗ ϕj,n (x)

n=1

=

∞

fn (λ)ϕj,n (x); fn ∈ L2 (λj,n , ∞); dλ .

(4.6)

n=1

Then Fj(0) defined by (Fj(0) f )(λ) = Fj(0) (λ)f for f ∈ C0∞ (Ωj ) is uniquely extended to a unitary operator (0) j . Fj : L2 (Ωj ) → H

We put h=

N !

L2 (Mj ),

(4.7)

j =1

where L2 (Mj ) = L2 (Mj ; det(hj ) dx), and also (0) (0) F (0) = F1 , . . . , FN .

(4.8)

By the computation similar to the one to be given in the proof of Lemma 4.3 below, one can show that 2 (0) 1 (0) (0) Rj (λ + i0) − Rj (λ − i0) f, f = Fj (λ)f L2 (M ) . j 2πi Therefore, Fj(0) (λ) ∈ B(B; L2 (Mj )), and Fj(0) (λ)∗ ∈ B(L2 (Mj ); B ∗ ). (0)

Here we must pay attention to the following remarks. The first one is that in (4.4), Fj (λ) is a finite sum: Fj(0) (λ) =

λj,n 1/2.

hence

4.2. Perturbed spectral representations Using Ej , Bj and Vj (z) in Section 3.1, for λ > λj,n we define the generalized eigenfunction for H by (0) (0) (λ) − R(λ ∓ i0)Vj (λ)Ψj,n (λ). Ψj,n,± (λ) = (χj − Ej Bj )Ψj,n

(4.13)

Here putting s = (1 + 0 )/2, we regard Ej Bj and Vj (λ) in B(H 2,−s ; L2,s ). Note that Ψj,n,± (λ) ∈ B ∗ . This definition easily implies

(−G − λ)Ψj,n,± (λ) = 0

in Ω,

(4.14)

∂ν Ψj,n,± (λ) = 0 on ∂Ω. (0)

The generalized Fourier transformation for H is defined by perturbing Fj . We put for λ > λj,n (0) Fj,n,± (λ) = Fj,n (λ)Jj χj − (Ej Bj )∗ − Vj (λ)∗ R(λ ± i0),

(4.15)

where Jj = (det G/ det Gj )1/2 . Note that (Ej Bj )∗ , Vj (λ)∗ ∈ B(L2,−s ; H −2,s ), and R(λ ± i0) ∈ B(L2,s ; H 2,−s ) ∩ B(H −2,s ; L2,−s ), hence (4.15) is well-defined. (0)

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Lemma 4.2. For f ∈ C0∞ (Ω), we have Fj,n,± (λ)f (x) =

Ψj,n,± (λ)f dV ϕj,n (x),

(4.16)

Ω

where dV = (det(G))1/2 dx dy. Proof. We put u = (χj − (Ej Bj )∗ − Vj (λ)∗ )R(λ ± i0)f . Then by using (4.10) (0) Fj,n,± (λ)f, h L2 (M ) = Fj,n (λ)Jj u, h L2 (M ) j j (0) (0) = uFj,n (λ)∗ hJj dVj . Ωj

We then use (4.12) to see that the right-hand side is equal to

(0)

uΨj,n (λ) dV (h, ϕj,n )L2 (Mj ) Ωj

(0) χj − (Ej Bj )∗ − Vj (λ)∗ R(λ ± i0)f, Ψj,n (λ) (h, ϕj,n )L2 (Mj ) = f, Ψj,n,± (λ) (ϕj,n , h)L2 (Mj ) , =

which proves the lemma.

2

The adjoint operator Fj,n,± (λ)∗ is defined by the following formula:

Fj,n,± (λ)f, h L2 (M ) = f, Fj,n,± (λ)h∗ L2 (Ω) , j

h ∈ L2 (Mj ).

(4.17)

Lemma 4.3. The adjoint operator Fj,n,± (λ)∗ has the following expression: (0) Fj,n,± (λ)∗ = χj − Ej Bj − R(λ ∓ i0)Vj (λ) Fj,n (λ)∗ ,

(4.18)

where the adjoint Fj,n (λ)∗ is taken in the sense of (4.10). (0)

Proof. Let u = (χj − (Ej Bj )∗ − Vj (λ)∗ )R(λ ± i0)f . Then as is shown in the proof of Lemma 4.2, Fj,n,± (λ)f, h L2 (M ) =

j

(0)

(0)

uFj,n (λ)∗ hJj dVj

Ωj

(0) = u, Fj,n (λ)∗ h L2 (Ω) . Plugging the form of u, we see that the right-hand side is equal to


2089

(0) f, χj − Ej Bj − R(λ ∓ i0)Vj (λ) Fj,n (λ)∗ h L2 (Ω) , which proves the lemma.

2

We define Fj,± (λ) =

∞

χλj ,n (λ)Fj,n,± (λ) =

Fj,n,± (λ),

(4.19)

λj,n 0 ∞ ∞ (A − k ∓ i)−1 e±ikt f (k) dk f± (s) ds. 0

t

Proof. This is proved in [41, Chapter 2, Lemma 8.10]. For the reader’s convenience, we reproduce the proof. By virtue of the identity

2094


−1

(A − k ∓ i)

∞ = ±i

e∓is(A−k∓i) ds,

0

we have ∞ ∞ −1 ±ikt (A − k ∓ i) e f (k) dk = ±i e∓is(A∓i) f± (s + t) ds, 0

0

2

which proves the lemma. 4.4. S-matrix

S= The scattering operator is defined by S = (W+ )∗ W− . We consider its Fourier transform:

F (0) S(F (0) )∗ .

Lemma 4.9. We have a direct integral representation: ( Sf )(λ) = S(λ)f (λ),

∀λ > 0, ∀f ∈ H,

where S(λ) = ( Sj k (λ))1j,kN is a bounded operator on h called the S-matrix, and is written as follows (0) ∗ Sj k (λ) = δj k − 2πiFj,+ (λ)Vk (λ) Fk (λ) . Proof. Lemma 4.5 implies 1 R(λ + i0) − R(λ − i0) = F± (λ)∗ F± (λ). 2πi By Lemma 4.3, we then have Fk,+ (λ)∗ − Fk,− (λ)∗ = 2πiF+ (λ)∗ F+ (λ)Vk (λ)Fk (λ)∗ . (0)

Then we have by Theorem 4.6(2), for f, g ∈ H

N (0) ∗ (F+ − F− )(F+ ) f, g = −2πi f (λ), F+ (λ)Vk (λ) Fk (λ) g(λ) h dλ. ∗

∞

k=1 0

By (4.23), S = F+ (F− )∗ . Hence the lemma follows.

2

Let hj (λ) be the linear subspace of L2 (Mj ) spanned by ϕj,n such that λj,n < λ and put h(λ) =

N ! j =1

hj (λ).


2095

Then S(λ) is a partial isometry on h with initial and final set h(λ). The scattering amplitude is defined by ∗ Aj k (λ) = Fj,+ (λ)Vk (λ) Fk(0) (λ) . Let Aj m,kn (λ) : L2 (Mk ) → L2 (Mj ) be given by (0) ∗ Aj m,kn (λ) = Fj,m,+ (λ)Vk (λ) Fk,n (λ) .

(4.30)

We then have Sj k (λ) − δj k Ij = −2πi

Aj m,kn (λ),

λj,m max{λ1,m , λ1,n } is analytically continued to the upper half plane C+ = {Im λ > 0}. This analytic continuation can be extended to a continuous function on C+ ∪ (R \ E(H )). We denote (nph) the obtained function for {λ < max{λ1,m , λ1,n }} \ E(H ) by A1m,1n (λ) and call it the non-physical scattering amplitude. These functions can be represented by (5.2), where Fcos (λ − λ1,m ) and Fcos (λ − λ1,n ) are replaced by their analytic continuations. Let (0) Φ1,n (x, y; λ) = π −1/2 e−πi/4 (λ1,n − λ)−1/4 cosh(y λ1,n − λ )ϕ1,n (x),

(5.3)

and put, similarly to (4.5) Fcosh (λ1,n − λ)P1,n f (x) =

(0)

(0)

Φ1,n (λ)f dV1

ϕ1,n (x).

Ω1 (nph)

In the following, we always assume that λ ∈ / E(H ). The explicit form of A1m,1n (λ) is given by (nph)

the following lemma. Recall that the non-physical scattering amplitude A1m,1n (λ) coincides with the physical scattering amplitude A1m,1n (λ) for λ > max{λ1,m , λ1,n }. Lemma 5.1. (1) If λ1,m < λ < λ1,n , (nph) A1m,1n (λ) = Fcos (λ − λ1,m )P1,n J1 χ1 − V1 (λ)∗ R(λ + i0) ∗ · V1 (λ) Fcosh (λ1,n − λ) P1,n . (2) If λ1,n < λ < λ1,m , (nph) A1m,1n (λ) = Fcosh (λ1,m − λ)P1,m J1 χ1 − V1 (λ)∗ R(λ + i0) ∗ · V1 (λ) Fcos (λ − λ1,n ) P1,n . (3) If λ < min{λ1,m , λ1,n }, (nph) A1m,1n (λ) = Fcosh (λ1,m − λ)P1,m J1 χ1 − V1 (λ)∗ R(λ + i0) ∗ · V1 (λ) Fcosh (λ1,n − λ) P1,n . In accordance with (4.13), we define non-physical eigenfunction by (0) (0) (λ) − R(λ ∓ i0)V1 (λ)Φ1,m (λ). Φ1,m,± (λ) = χ1 Φ1,m

(5.4)

Note that the physical eigenfunction Ψ1,m,− (λ) defined for λ > λ1,m is analytically continued through the upper half space C+ to the non-physical eigenfunction Φ1,m,− (λ) defined for


2097

λ < λ1,m . The non-physical scattering amplitude is computed from the asymptotic behavior of non-physical eigenfunction in the following way. We put (nph) (nph) A1m,1n (λ) = A1m,1n (λ)ϕ1,n , ϕ1,m L2 (M ) . 1

Then we have for h ∈ L2 (M1 ) (nph)

(nph)

A1m,1n (λ)h = A1m,1n (λ)(h, ϕ1,n )L2 (M1 ) ϕ1,m . Lemma 5.2. (1) If λ1,m < λ < λ1,n , we have as y → ∞, √ √ i π eiy λ−λ1,m (nph) (0) A (λ)ϕ1,n . P1,m Φ1,n,− (λ) − Φ1,n (λ) − (λ − λ1,m )1/4 1m,1n (2) If λ < max{λ1,m , λ1,n }, we have as y → ∞, √ √ eπi/4 π e−y λ1,m −λ (nph) (0) P1,m Φ1,n,− (λ) − Φ1,n (λ) ∼ − A1m,1n (λ)ϕ1,n , (λ1,m − λ)1/4 with a super exponentially decreasing error, that is, with the error r(y) satisfying |r(y)| CN e−Ny for any N > 0. Proof. The assertion (1) is proved in the same way as in Lemma 4.8. By (4.1), letting ζ = λ − λ1,m , we have as y → ∞ √

iei ζ y P1,m R1 (λ + i0)f (x, y) ∼ √ ζ

∞

cos ζ y P1,m f x, y dy

0

with a super exponentially decaying error. This, together with (3.14) and Lemma 5.1, proves (2). 2 5.2. Splitting the manifold We take a compact hypersurface Γ ⊂ Ω1 having the following property. (C-1) Γ splits Ω into a union: Ω = Ωext ∪ Ωint so that Ωext ∩ Ωint = Γ , Ωint is a manifold with smooth boundary, and Ωext ⊂ Ω1 . (See Fig. 2.) Let O ⊂ Ωint be an open, relatively compact set such that it has a smooth boundary not intersecting ∂Ωint and that Ωint \ O is connected. Denote ΩO = Ωint \ O and

ΓO =

Γ ∂O

if O = ∅, if O = ∅.

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Fig. 2. Surface Γ splits Ω to two parts, manifold Ωint with a smooth boundary and its complement Ωext ⊂ Ω1 .

We put for f, g ∈ L2 (ΓO ) (f, g)ΓO =

f (x)g(x) dSx ,

ΓO

dSx being the measure induced from the metric G on ΓO . We put HO = −G in ΩO endowed with the Neumann boundary condition: ∂ν v = 0

on ∂ΩO ,

(5.5)

ν being the unit normal to the boundary. If Ω has only one end, Ωint is a bounded region. If Ω has more than one end, Ωint is unbounded and the spectral theory developed for H applies also to HO . To see this, we have only to replace K by K ∪ ((Ω1 ∩ Ωint ) \ O), and to argue in the same way as in Sections 3 and 4. Let E(HO ) be σp (HO ) when ΩO is bounded, and the set of exceptional points for HO when ΩO is unbounded. Next we consider the case O = ∅ so that ΓO = Γ . Lemma 5.3. Suppose λ ∈ / E(H ) ∪ E(H∅ ), and let Ψ1,n,− (λ) and Φ1,n,− (λ) be physical and non-physical eigenfunctions for H . Then the linear subspace spanned by ∂ν Ψ1,n,− (λ)|Γ , ∂ν Φ1,n,− (λ)|Γ , n = 1, 2, . . . , is dense in L2 (Γ ). Proof. We show that, if f ∈ L2 (Γ ) satisfies f, ∂ν Ψ1,n,− (λ) Γ = f, ∂ν Φ1,n,− (λ) Γ = 0,

∀n 1,

(5.6)

then f = 0. We define an operator δΓ ∈ B((H 1/2 (Γ )) ; H −2 (Ω)), where (H 1/2 (Γ )) is the dual space of H 1/2 (Γ ), by δΓ f, w = (f, ∂ν w)Γ ,

∀w ∈ H 2 (Ω),

and put u = R(λ − i0)δΓ f by duality. This means that, if G− (λ; X, X ) is Green’s function, i.e. the integral (Schwartz) kernel of R(λ − i0), u(X) = R(λ − i0)δΓ f (X) =

Γ

∂ν G− λ; X, X f X dSX ,


2099

where ∂ν means the conormal differentiation with respect to the variable X . Then u ∈ B ∗ , and by (3.14), we have the following asymptotic expansion on Ω1 u

Cn (λ)e−iy

√

λ−λ1,n

f, ∂ν Ψ1,n,− (λ) Γ ϕ1,n (x).

λ1,n λ (u, ϕ1,n ) ∼ Cn (λ)e−y

√

λ1,n −λ

f, ∂ν Φ1,n,− (λ) Γ

(5.8)

modulo a super exponentially decaying term. Note that un = (u, ϕ1,n ) satisfies the equation (−∂y2 + λ1,n − λ)un = 0 for y > a, a being a sufficiently large constant. In view of the assumption of (5.6) and (5.7), (5.8), we then have (u, ϕ1,n ) = 0 for y > a, hence u(x, y) = 0 for y > a. The unique continuation theorem then implies u = 0 on Ωext . By the property of classical double layer potential, ∂ν u is continuous across Γ , so that ∂ν u|Γ = 0. Next we show that u = 0 in Ωint . In the region Ωint , we have (−G − λ)u = 0. If Ωint is bounded, then u = 0 since λ is not a Neumann eigenvalue. If Ωint is not bounded, u satisfies the incoming radiation condition, since so does u in Ω. Then u = 0 in Ωint by Lemma 3.4. As u = R(λ − i0)δΓ f ∈ L2loc (Ω), it follows from the above that u = 0 in Ω. Applying H − λ, we have δΓ f = 0 as a distribution, hence f = 0 on Γ . 2 5.3. Interior boundary value problem For z ∈ C \ E(HO ), we consider the following boundary value problem ⎧ ⎪ ⎨ (HO − z)u = 0 in ΩO , ∂ν u = 0 on ∂ΩO \ ΓO , ⎪ ⎩ 1/2 ∂ν u = f ∈ H0 (ΓO ) on ΓO .

(5.9)

The incoming radiation condition is also imposed, if Ωint is unbounded and z ∈ R. The Neumann–Dirichlet map (N-D map) is then defined by ΛO (z)f = u|ΓO ,

(5.10)

where u is the solution to (5.9). When O = ∅, we use for the N-D map of the operator H∅ the notation ΛO (z) = Λ(z). Now we consider the operator theoretical meaning of the N-D map. Note that from now on O may be a non-empty set. We put F = (Fc , Fp ), where Fc is the generalized Fourier transform for HO (which is absent when Ωint is bounded) and Fp is defined by Fp : Hp (HO ) u → (u, ψ1 ), (u, ψ2 ), . . . ,

2100


and where Hp (HO ) is the point spectral subspace for HO and ψi is the eigenfunction associated with the eigenvalue λi of HO . There are two kinds of generalized Fourier transformation, F+ and F− . Both choices will do as Fc . Then F is a unitary ⊕ Cd , F : L2 (Ωint ) → H

(5.11)

where d = dimHp (HO ). If d = ∞, Cd is replaced by 2 . Moreover, we have −1

(HO − z)

∞ = 0

Pi Fc (λ)∗ Fc (λ) dλ + , λ−z λi − z d

(5.12)

i=1

where Pi are the eigenprojections associated with eigenvalues λi , numbered counting multiplicities by i = 1, 2, . . . , d, and the right-hand side converges in the sense of strong limit in L2 (ΩO ). Let rΓO ∈ B(H 1 (ΩO ); H 1/2 (ΓO )) be the trace operator to ΓO , rΓO : H 1 (ΩO ) f → f |ΓO ∈ H 1/2 (ΓO ). We define δΓO ∈ B((H 1/2 (ΓO )) ; (H 1 (ΩO )) ) as the adjoint of rΓO : (δΓO f, w)L2 (Ωint ) = (f, rΓO w)L2 (ΓO ) ,

f ∈ H 1/2 (ΓO ) , w ∈ H 1 (ΩO ).

With this in mind we write rΓO = δΓ∗ O . Lemma 5.4. For z ∈ / E(HO ), the N-D map has the following representation ΛO (z) = δΓ∗ O (HO − z)−1 δΓO ∞ = 0

δΓ∗ O Fc (λ)∗ Fc (λ)δΓO λ−z

dλ +

d δΓ∗ O Pi δΓO i=1

λi − z

.

1/2 Proof. For f ∈ H0 (ΓO ), take f ∈ H 2 (ΩO ) such that ∂ν f = f on ΓO and f has compact support in Ωint . Then the solution u of (5.9) is written as u = f− (HO − z)−1 (−G − z)f. Let int , where H int is defined by (4.22) with j = 2, . . . , N . g = Fc (λ)(G + z)f. Then for any h ∈ H

Fc (λ)(G + z)f, h = (G + z)f, Fc (λ)∗ h = ∂ν f, rΓO Fc (λ)∗ h L2 (Γ ) + f, (G + z)Fc (λ)∗ h O = f, rΓO Fc (λ)∗ h L2 (Γ ) + f, (−λ + z)Fc (λ)∗ h . O

This implies


2101

Fc (λ)(G + z)f= Fc (λ)δΓO f + (−λ + z)Fc (λ)f. Hence ∞ 0

Fc (λ)∗ Fc (λ)(G + z)f Fc (λ)∗ Fc (λ)δΓO f dλ = dλ − Fc∗ Fc f. λ−z λ−z ∞

0

Similarly, d Pi (G + z)f

λi − z

i=1

Since Fc∗ Fc f+

d

=

d d Pi δΓO f − Pi f . λi − z i=1

i=1

= f, by (5.12), these imply that

i=1 Pi f

u = (HO − z)−1 δΓO f, which proves the lemma.

2

Let us call the set d λ, δΓ∗ O Fc (λ)∗ Fc (λ)δΓO ; λ ∈ (0, ∞) \ E(HO ) ∪ λi , δΓ∗ O Pi δΓO i=1 ,

(5.13)

where d = dim Hp (HO ), the boundary spectral projection (BSP) for HO on ΓO . On the other hand, the set

d λ, Fc (λ)δΓO ; λ ∈ (0, ∞) \ E(HO ) ∪ λi , ψi (x)|ΓO i=1

(5.14)

is called the boundary spectral data (BSD) on ΓO . By using the formula (3.17), we have the following lemma. Lemma 5.5. For a bounded Borel function ϕ(λ) with support in R \ T (HO ), where T (HO ) is defined by (3.18) with j = 2, . . . , N , we have δΓ∗ O ϕ(HO )δΓO

∞ = 0

ϕ(λ)δΓ∗ O Fc (λ)∗ Fc (λ)δΓO

dλ +

d

ϕ(λi )δΓ∗ O Pi δΓO .

i=1

Proof. By the formulae (3.17) and (5.12), this lemma holds for any ϕ(λ) ∈ C0∞ (R \ T (HO )). The general case the follows from the approximation. 2 Usually BSD is referred as given data in the inverse boundary value problems. What is actually used in our reconstruction for the manifold is the BSP. Lemma 5.6. Let O ⊂ Ωint . Then knowing the N-D map ΛO (z) for all z ∈ / σ (HO ) is equivalent to knowing the BSP for HO .

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Proof. By Lemma 5.4, one can compute the N-D map by using BSP. Taking ϕ(λ) as the characteristic function of the interval [a, t) and taking note of the remark after (3.34), we differentiate the formula in Lemma 5.5 with respect to t to recover δΓ∗ O Fc (t)∗ Fc (t)δΓO for t ∈ R \ E(HO ). Since d δΓ∗ O Pi δΓO i=1

λi − z

∞ = ΛO (z) − 0

δΓ∗ O Fc (λ)∗ Fc (λ)δΓO λ−z

dλ,

one can obtain eigenvalues λi as the poles of the right-hand side. The residues in these poles provide us with δΓ∗ O λj =λi Pj δΓO . This determines the terms δΓ∗ O Pj δΓO for indexes j such that λj = λi , up to an orthogonal transformation of the eigenspace associated to the eigenvalue λi , see [47, Lemma 4.9] or [48]. Thus we can determine the BSP for HO . 2 We complete this section by the following result used later to prove Theorem 1.1. Let Ω (r) , r = 1, 2, be as in Theorem 1.1. We take Γ as above, which moreover has the following property: (1) (2) (1) (2) G1 = G1 on Ωext = Ωext = Ωext . We put the superscript (r) for all relevant operators and (r) functions explained above. Let Λ(r) (λ), r = 1, 2 be the N-D map for H∅ , that is, when O = ∅. The basic idea of the following lemma is due to Eidus [24]. Lemma 5.7. Under the assumptions of Theorem 1.1, we have Λ(1) (λ) = Λ(2) (λ) for λ ∈ (0, ∞) \ (r) (1) (2) (r) r=1,2 (E(H ) ∪ E(H∅ )), and BSP’s for H∅ and H∅ coincide on Γ . S11 (λ), the physical scattering amplitudes coincide, hence so do nonProof. Since S11 (λ) = (1) (2) physical scattering amplitudes by analytic continuation. Let u = Ψ1,n,− (λ) − Ψ1,n,− (λ) and (1)

(1)

(2)

(2)

v = Φ1,n,− (λ) − Φ1,n,− (λ). Then since H (1) = H (2) = −∂y2 − h1 on Ωext , u and v satisfy (−∂y2 − h1 − λ)u = 0 and (−∂y2 − h1 − λ)v = 0 in Ωext . Using Lemma 5.2 and arguing in the (r) (r) and Φ1,n,− same way as in the proof of Lemma 5.3, we have u = v = 0 in Ωext . Therefore, Ψ1,n,− as well as their normal derivatives coincide for r = 1, 2 and for all n ∈ Z+ . Since they satisfy (r) Eq. (5.9) for H∅ = H∅ , we have Λ(1) (λ) = Λ(2) (λ) due to Lemma 5.3. The last statement now follows immediately from Lemma 5.6. 2

6. Boundary control method for manifolds with asymptotically cylindrical ends In this section we reconstruct the isometry type of the manifold (Ωint , G) using given data (see Fig. 3). Theorem 6.1. Assume that we are given the set Γ as a differentiable manifold, the metric G on Γ , and the BSP for H∅ . These data determine the manifold (Ωint , G) up to an isometry. For proving this theorem, we use the boundary control (BC) method for inverse problems. The method goes back to [8] where it was used to recover the isotropic wave velocity in the acoustic equation in a domain in Rn . In [11] it was developed to prove the analog of Theorem 6.1 for compact manifolds when BSD is given on ∂Ω. The method was then extended to a large class of elliptic (and associated hyperbolic) operators on compact manifolds in e.g. [9,46,52,48,51], see


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Fig. 3. We will construct the manifold Ωint by iterating local constructions. First, a neighborhood U1 ⊂ Ωint of Σ ⊂ Γ is reconstructed. Next, a ball O = B(X1 , ρ) ⊂ U1 is removed from the manifold and data analogous to measurements on ∂O are constructed. After that, the metric is reconstructed in a larger ball B(X1 , τ ), and the procedure is iterated to reconstruct the whole manifold Ωint .

also [47]. Later it was also extended to a number of inverse problems for systems on compact manifolds, e.g. [10,55,54]. The BC method combines Tataru’s uniqueness results in the control theory for PDE’s with Blagovešˇcenskii’s identity that gives the inner product of the solutions of the wave equation in terms of the boundary data. This identity was originally used in the study of one-dimensional inverse problems, see [12,13]. The reconstruction of non-compact manifolds is considered previously in the conference proceedings [49] and in [14] (see also [20]) with different kind of data, using iterated time reversal for solutions of the wave equation. The reconstruction of (Ωint , G) below is based on matching local reconstructions. Geometrically, this procedure is similar to the one described in [47, Section 4.4]. However, the analytic technique used here is different. In [47] (see also [50]), the reconstruction is based on the combination of the use of Gaussian beams and the continuation of the eigenfunctions. In this section we develop a technique based on the continuation of Green’s function and BSP which is suitable for the non-compact (as well as compact) manifolds. The proof of Theorem 6.1 is divided into a series of lemmas. Our reconstruction of (Ωint , G) is of recurrent nature. We will begin with the case when O = ∅ so that we are given just the set ΓO = Γ as a differentiable manifold, the metric on it, and the BSP for the operator H∅ on Γ . We apply the boundary control method to reconstruct the metric G on some neighborhood U1 of Γ . Then, we will take a point X1 ∈ U1 \ Γ and ρ > 0 such that B(X1 , 2ρ) ⊂ U1 , where B(X1 , r) denotes the ball of radius r with center at X1 . We take O = B(X1 , ρ) and show that we can find the BSP for the operator HO on ΓO = ∂O. Then we apply the boundary control method starting from ΓO , which would allow us to recover (Ωint , G) in a larger neighborhood U2 ⊃ U1 of Γ . Proceeding in this way, we will eventually recover the whole of (Ωint , G). Therefore, our further considerations deal with arbitrary O ⊂ Ωint including the case O = ∅. 6.1. Blagovešˇcenskii’s identity Let us first consider the initial boundary value problem ⎧ 2 ⎪ ⎨ ∂t u = G u, in ΩO × R+ , u|t=0 = ∂t u|t=0 = 0, in ΩO , ⎪ ⎩ ∂ν u = f, in ∂ΩO × R+ , supp f ⊂ ΓO × R+ .

(6.1)

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Lemma 6.2. Assume that we are given the set ΓO as a differentiable manifold, the metric G on ΓO and the BSP for HO on ΓO . Then for any given f, h ∈ C0∞ (ΓO × R+ ) and t, s > 0 these data uniquely determine f u (t), uh (s) =

uf (X, t)uh (X, s) dVX ,

ΩO

where uf (t) and uh (t) are solutions of (6.1) with boundary data f and h, correspondingly. Proof. Let √ sin( λt) S(t, λ) = √ . λ Then the solution uf (t) is written as t u (t) = f

∞ ds

0

dλ S(t − s, λ)Fc (λ)∗ Fc (λ)δΓO f (s)

0

t +

ds 0

d

S(t − s, λi )Pi δΓO f (s).

i=1

Using the similar decomposition for uh (s) and the fact that Fc (μ)Fc (λ)∗ = δ(μ − λ), we obtain the following formula: f u (t), uh (s) t =

dt 0

s ds 0

t +

dt 0

∞

dλ S t − t , s − s , λ δΓ∗ O Fc (λ)∗ Fc (λ)δΓO f t , h s L2 (Γ

O)

0

s 0

d ds S t − t , s − s , λi δΓ∗ O Pi δΓO f t , h s L2 (Γ ) ,

i=1

O

(6.2)

where S(t, s, λ) = S(t, λ)S(s, λ). Observe that the right-hand side depends only on BSP and the metric on ΓO . 2 Above, the formula (6.2) is a generalization of Blagovešˇcenskii identity (see [47, Theorem 3.7]) for non-compact manifolds.


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6.2. Finite propagation property of waves Let us next introduce some notations. For t > 0 and Σ ⊂ ΓO arbitrary, let ΩO (Σ, t) = X ∈ ΩO ; dO (X, Σ) t be the domain of influence of Σ at time t. Here, dO (X, Y ) is the distance between X and Y in ΩO . We use also the notation ΩO (Y, t) = ΩO ({Y }, t). More generally, when I = {(Σj , tj )}Jj=1 is a finite collection of pairs (Σj , tj ), where Σj ⊂ ΓO and tj > 0, we denote ΩO (I ) =

J

ΩO (Σj , tj ) = X ∈ ΩO ; dO (X, Σj ) tj for some j = 1, . . . , J .

j =1

For any measurable set B ⊂ ΩO , we denote L2 (B) = {v ∈ L2 (ΩO ); v|ΩO \B = 0}, identifying functions and their zero continuations. Lemma 6.3. Assume that we are given the set ΓO as a differentiable manifold, the metric on ΓO and the BSP for HO on ΓO . Then, for any given f ∈ C0∞ (ΓO × R+ ), T > 0, and I = {(Σj , tj )}Jj=1 , where Σj ⊂ ΓO are open sets or single points, and tj < T , we can determine aI,T (f ) =

f u (T )2 dV .

(6.3)

ΩO \ΩO (I )

Proof. When Σ ⊂ ΓO is an open set and h ∈ C0∞ (Σ × R+ ), it follows from the finite velocity of wave propagation (see e.g. [57, Section 4.2], see also [41, Chapter 6]) that the wave uh (t) = uh (· , t) is supported in the domain ΩO (Σ, t) at time t > 0. It follows from Tataru’s seminal unique continuation result, see [73,74], that the set

uh (t); h ∈ C0∞ (Σ × R+ )

(6.4)

is dense in L2 (ΩO (Σ, t)), see e.g. [47, Theorem 3.10]. This clearly implies that, when T > 0 and I = {(Σj , tj )}Jj=1 , where Σj are open and tj < T , the set XIT := uh (T ); h = h1 + · · · + hJ , hj ∈ C0∞ Σj × [T − tj , T ] = spanj =1,...,J uh (tj ); h ∈ C0∞ Σj × [0, tj ] is dense in L2 (ΩO (I )). Next, we consider the non-linear functional 2 aI,T (f ) = inf uf −h (T )L2 (Ω ) ; h = h1 + · · · + hJ , hj ∈ C0∞ Σj × [T − tj , T ] , O

where f ∈ C0∞ (ΓO × R+ ), T > 0, and I = {(Σj , tj )}Jj=1 , Σj ⊂ ΓO are open, and tj < T . By the formula (6.2), the BSP and the metric on ΓO determine the value aI,T (f ) for any f . Moreover, as uf −h (T ) = uf (T ) − uh (T ) and XIT is dense in L2 (ΩO (I )), we see that

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2 aI,T (f ) = (1 − χΩO (I ) )uf (T )L2 (Ω ) ,

(6.5)

O

where χΩO (I ) (x) is the characteristic function of the set ΩO (I ) on ΩO . This proves the lemma for the case when all Σj are open. (k) If for some j , the set Σj is just a point Xj ∈ ΓO , we define for those j ’s Σj ⊂ ΓO , k = # (k) (k+1) (k) 1, 2, . . . to be open neighborhoods of Xj such that Σj ⊂ Σj and k Σj = {Xj }. For (k)

those j ’s for which Σj is open, we define Σj = Σj . Denote the corresponding finite collection (k)

of (Σj , tj ) by I (k). Then ΩO I (k + 1) ⊂ ΩO I (k) ,

ΩO (I ) =

∞ $

ΩO I (k) ,

k=1

and for any b ∈ L2 (ΩO ), (1 − χΩO (I (k)) )b → (1 − χΩO (I ) )b,

a.e. as k → ∞.

As |(1 − χΩO (I (k)) )b(·)| |(1 − χΩO (I ) )b(·)|, a.e., using the monotone convergence theorem, we see that 2 aI (k),T (f ) → (1 − χΩO (I ) )uf (T )L2 (Ω

O)

= aI,T (f ).

Thus, the BSP and the metric on ΓO determine aI,T (f ) for such I s.

2

Definition 6.4. Let I = {(Σj , tj )}Jj=1 , I = {(Σj , tj )}Jj=1 and T > 0, where Σj , Σj ⊂ ΓO and tj , tj < T . We say that the relation I I is valid on manifold ΩO if ΩO I \ ΩO (I )

has measure zero.

(6.6)

Lemma 6.5. Let I = {(Σj , tj )}Jj=1 , I = {(Σj , tj )}Jj=1 and T > 0, where Σj , Σj ⊂ ΓO are open sets or single points and tj , tj < T . Assume that we are given the set ΓO as a differentiable manifold, the metric on ΓO , the BSP for HO on ΓO , and the collections I and I . Then we can determine whether the relation I I is valid on manifold ΩO or not. Proof. The relation I I is valid on manifold ΩO if and only if aI,T (f ) aI ,T (f )

for all f ∈ C0∞ (ΓO × R+ ).

(6.7)

Indeed, the equivalence of (6.6) and (6.7) follows from (6.5) and the fact that, by Tataru’s density result (6.4), the functions uf (T ), f ∈ C0∞ (ΓO × R+ ), are dense in L2 (ΩO (ΓO , T )). As for given f , by Lemma 6.3, we can evaluate both sides of (6.7), using the BSP and the metric on ΓO , these data determine, for any pair (I, I ), if the relation I I is valid or not. 2


2107

For any X0 ∈ Ωint \ ∂Ωint , introduce the exponential map expX0 : (ξ, t) → γ(X0 ,ξ ) (t), where ξ ∈ SX0 (Ωint ) = {η ∈ TX0 (Ωint ); |η| = 1} and 0 t s(X0 , ξ ). Here γ(X0 ,ξ ) (t) is the geodesic on Ω, parametrized by the arclength, with γ(X0 ,ξ ) (0) = X0 , γ˙(X0 ,ξ ) (0) = ξ , and [0, s(X0 , ξ )) is the maximal interval of t, when γ(X0 ,ξ ) (t) stays in Ωint , that is, s(X0 , ξ ) = sup{t; γ(X0 ,ξ ) ([0, t)) ⊂ Ωint \ ∂Ωint }. Denote by s(X0 ) =

inf

ξ ∈SX0 (Ω)

s(X0 , ξ )

(6.8)

so that B X0 , s(X0 ) ⊂ Ωint \ ∂Ωint . Define now τ (X0 , ξ ) =

sup

0 0; γ(Z,ν) (0, t) ⊂ ΩO \ ∂ΩO .

(6.10)

For any Z ∈ ΓO , let τO (Z) =

sup

0tsO (Z)

t; dO γ(Z,ν) (t), ΓO = t .

(6.11)

In the following, we impose the following condition (C-2) on Σ . (C-2) For O = ∅, Σ is an open subset of Γ such that d∅ (Σ, ∂Γ ) > 0, and for O = ∅, Σ = ∂O.

2108


Fig. 4. In the figure O = ∅ and s is small enough so that the set ΩO (Y, t) \ ΩO (ΓO , t − ε) is not contained in ΩO (Z, s). This is the situation when Iε (t, s) I (t).

We define τO (Σ) = inf τO (Z).

(6.12)

Z∈Σ

In geometric terms, the above definition of τO (Σ) means that, in the set L Σ, τO (Σ) = γ(Z,ν) (t); Z ∈ Σ, 0 t < τO (Σ) ⊂ (ΩO \ ∂ΩO ) ∪ Σ, it is possible to introduce the boundary normal coordinates X → (Z, t),

Z ∈ Σ, 0 t < τO (Σ)

satisfying X = γ(Z,ν) (t). Observe that when O = B(X, ρ), X ∈ Ωint \ ∂Ωint and ρ > 0 is small enough, then τO (∂O) = τ (X) − ρ. Lemma 6.6. Assume that Σ ⊂ ΓO satisfies condition (C-2). Let Y ∈ Σ, Z ∈ ΓO , t < τO (Σ), and X = γ(Y,ν) (t). Assume that we are given the set ΓO as a differentiable manifold, the metric on ΓO and the BSP for HO on ΓO . Then we can determine the distance dO (X, Z) on ΩO . Proof. Note that as t < τO (Σ), the set ΩO (Y, t) \ ΩO (ΓO , t − ε) contains a non-empty open set for all ε > 0. For s, ε > 0, let us denote (see Fig. 4) I (t) = (Y, t), (ΓO , t − ε) ,

Iε (t, s) = (Z, s), (ΓO , t − ε) .

Let us next show that for any r > 0 there is ε0 > 0 such that ΩO (Y, t) \ ΩO (ΓO , t − ε) ⊂ B(X, r),

when ε < ε0 .


2109

If this is not true, there are r > 0, a sequence εj → 0, and Xj ∈ ΩO (Y, t) \ ΩO (ΓO , t − εj ) such that dO (Xj , X) r. As ΩO (Y, t) is compact, by considering a subsequence, we can assume that ∈ ΩO (Y, t). Then Xj converge to X Y ) = lim dO (Xj , Y ) t, dO (X, j →∞

ΓO ) = lim dO (Xj , ΓO ) t, dO (X, j →∞

and dO (X, Y ) = t. Let us recall that the shortest implying that Y is a closest point of ΓO to X curve from a point in ΩO to ΓO , which end point is an interior point of ΓO , is a normal geodesic = γ(Y,ν) (t) = X, which is in contradiction to d(X, X) r. Thus, the to ΓO . Thus, we see that X existence of ε0 for any r is proven. The above implies that when s > d(X, Z), the set ΩO (Y, t) \ ΩO (ΓO , t − ε) is contained in ΩO (Z, s) for all sufficiently small ε > 0 and therefore, there is ε1 > 0 such that Iε (t, s) Iε (t) for all 0 < ε < ε1 .

(6.13)

On the other hand, for s < d(X, Z), the set ΩO (Y, t) \ ΩO (ΓO , t − ε) = ∅ do not intersect with ΩO (Z, s) at any ε > 0 small enough and thus (6.13) does not hold. Thus, by Lemma 6.5, we can find dO (X, Z) for any Z ∈ ΓO as the infimum of all s > 0 for which (6.13) hold. 2 For Σ ⊂ ΓO satisfying (C-2) and 0 < T < τO (Σ), let NΣ,T and MΣ,T be the sets

MΣ,T

NΣ,T = X ∈ ΩO ; X = γ(Y,ν) (t), 0 t T , Y ∈ Σ , = X ∈ ΩO ; X = γ(Y,ν) (t), 0 < t < T , Y ∈ Σ ⊂ NΣ,T = MΣ,T .

(6.14)

Note that MΣ,T is open in ΩO . 6.3. Boundary distance functions and reconstruction of topology Let us next consider the collection of the boundary distance functions associated with ΓO . For each X ∈ ΩO , the corresponding restricted boundary distance function, rX ∈ C(ΓO ) (note that Γ O is compact) is given by rX : ΓO → R+ ,

rX (Z) = dO (X, Z),

Z ∈ ΓO .

The restricted boundary distance functions define the boundary distance map RO : ΩO → C(ΓO ), RO (X) = rX . The boundary distance representation of NΣ,T ⊂ ΩO is the set RO (NΣ,T ) = rX ∈ C(ΓO ); X ∈ NΣ,T , that is, the image of NΣ,T in RO . Clearly RO : ΩO → C(ΓO ) is continuous.

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Lemma 6.7. Assume that we are given the set ΓO as a differentiable manifold, the metric on ΓO , the BSP for HO on ΓO , an open set Σ ⊂ ΓO satisfying condition (C-2), and 0 < T < τO (Σ). Then we can determine the set RO (NΣ,T ) = RO γ(Y,ν) (t); Y ∈ Σ, 0 t T . Proof. By Lemma 6.6, for Y ∈ Σ , t < T and Z ∈ ΓO , we can find dO (X, Z) where X = γ(Y,ν) (t) from BSP. This gives us the function rX (Z), Z ∈ ΓO , and for such X’s. Thus, BSP and the metric on ΓO determine the set RO (MΣ,T ). Using (6.14), we obtain RO (NΣ,T ) by closure of RO (MΣ,T ) in C(ΓO ). 2 Consider properties of RO . Assume that rX = rY for some X, Y ∈ NΣ,T . Let Z ∈ ΓO be the point where the function rX attains its minimum. Then, it is the closest point of ΓO to X. Thus, the shortest geodesic from X to Z is normal to ΓO , i.e. X = γ(Z,ν) (t) with t = rX (Z). The same arguments show that Z is also the closest point of ΓO to Y and t = rY (Z), and hence Y = γ(Z,ν) (t). Thus X = Y and RO is injective on NΣ,T . Thus, map RO : NΣ,T → RO (NΣ,T ) is a bijective continuous map defined on a compact set, implying that it is a homeomorphism. This implies that the map RO : MΣ,T → RO (MΣ,T ) is a homeomorphism. As BSP and the metric on ΓO determine the manifold RO (MΣ,T ) with its topological structure inherited from C(ΓO ), we see that these data determine the manifold MΣ,T as a topological space. Lemma 6.8. The set RO (MΣ,T ) ⊂ C(ΓO ) can be endowed, in a constructive way, with a differ so that (RO (MΣ,T ), G) becomes a manifold which is entiable structure and a metric tensor G, isometric to (MΣ,T , G) with RO being an isometry. For compact manifolds, the result analogous to Lemma 6.8 is presented in detail in [47, Section 3.8]. Since the proof is based on local constructions, it works for non-compact manifolds without any change. However, for the convenience of the reader, we present this construction. Proof of Lemma 6.8. Let us define the evaluation functions, EZ , Z ∈ ΓO , EZ : RO (MΣ,T ) → R,

EZ (rX ) = rX (Z) = dO (X, Z).

For r(·) ∈ RO (MΣ,T ) corresponding to a point X ∈ MΣ,T , i.e. r(·) = rX (·), we can choose points Z1 , . . . , Zn ∈ ΓO close to the nearest point of ΓO to X so that X → (dO (X, Zj ))nj=1 forms a system of coordinates on ΩO near X, see [47, Lemma 2.14]. Similarly, the functions EZj , j = 1, . . . , n, form a system of coordinates in RO (MΣ,T ) near rX . These coordinates provide for RO (MΣ,T ) a differential structure which makes it diffeomorphic to manifold MΣ,T . the metric on RO (MΣ,T ) which makes it isometric to (MΣ,T , G), that Let us denote by G −1 ∗ is, G = ((RO ) ) G. Let r ∈ RO (MΣ,T ) and X ∈ MΣ,T be such that r = rX . Let Z0 is a point where r obtains its minimum, that is, the closest point of ΓO to X. When Z is close to Z0 , the see [47, Lemma 2.15]. differentials of functions EZ are covectors of length 1 on (RO (MΣ,T ), G), This is equivalent to the fact that the gradients of the distance functions X → dO (X, Z) have length one. By this observation, it is possible to find infinitely many covectors dEZ , Z ∈ ΓO of length 1 at any point r of RO (MΣ,T ). Using such vectors, one can reconstruct the metric


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at r. By the above considerations, BSP determines the manifold (MΣ,T , G) up to an tensor G isometry. 2 6.4. Continuation of the data Let us now consider the case when O = ∅ and we are given the set Γ as a differentiable (1) manifold, the metric G on Γ , and the BSP for H∅ . Assume that there are two manifolds Ωint (j ) (j ) (2) (j ) and Ωint such that Γ is isometric to subsets Γ ⊂ ∂Ωint for j = 1, 2 and that the BSP for H∅ , j = 1, 2, coincides with the given data. Let now Σ ⊂ Γ satisfy condition (C-2) and (1) (2) 0 < T < min τ∅ (Σ), τ∅ (Σ) . Then the above constructions show that the manifolds (j ) (j ) MΣ,T = X ∈ Ωint ; X = γ(Y,ν) (t), 0 < t < T , Y ∈ Σ (1)

with j = 1 and j = 2, are isometric. Thus, we can consider the set MΣ,T , denoted by U1 as a (1)

(2)

subset of both manifolds Ωint and Ωint , and, by the previous considerations, we can construct a on it which makes (U1 , G) isometric to (M (j ) , G(j ) ), j = 1, 2. metric G Σ,T We continue the construction by continuation of the data using Green’s functions, cf. [58,59]. To this end, let z ∈ C \ R+ and consider the Schwartz kernel GO (z; Y, Y ) of the operator (HO − z)−1 . It satisfies the equation (HO − z)GO z; · , Y = δY , Y, Y ∈ ΩO = Ωint \ O, ∂ν GO z; · , Y ∂Ω = 0. O

(6.15)

We denote G(z; Y, Y ) = GO (z : Y, Y ) when O = ∅. Lemma 6.9. Let U ⊂ Ωint be a connected neighborhood of an open set Σ ⊂ Γ , where Σ satisfies condition (C-2) with O = ∅. Let X0 ∈ U \ ∂Ωint and ρ > 0 be such that O = B(X0 , ρ) ⊂ U \ ∂Ωint . Assume that we are given the metric tensor G in U . Then BSP on Γ for the operator H∅ determines G(z; Y, Y ) for Y, Y ∈ U and z ∈ C \ E(H∅ ). Moreover, these data determine BSP on ΓO for the operator HO . Proof. By Lemma 5.6, BSP on Γ determines the N-D map Λ(z) at Γ × Γ . By Lemma 5.4, the Schwartz kernel of the N-D map Λ(z) at Γ × Γ coincides with G(z; Y, Y ). Thus we know the function G(z; Y, Y ) for Y, Y ∈ Σ. As the Neumann boundary values of Y → G(z; Y, Y ) on Γ \ {Y } vanish, using the Unique Continuation Principle for the elliptic equation (6.15) in the Y variable, we see that the values of G(z; Y, Y ) are uniquely determined for Y ∈ Σ and Y ∈ U \ {Y }. Using the symmetry G(z; Y, Y ) = G(z; Y , Y ) and again the Unique Continuation Principle, now in the Y variable, we can determine the values of G(z; Y, Y ) in {(Y, Y ) ∈ U × U ; Y = Y }. Considering G(z; Y, Y ) as a locally integrable function, we see that it is defined a.e. in U × U. For Y ∈ (ΩO ∩ U ) \ ∂ΩO , denote by Gext O (z; Y, Y ) a smooth extension of GO (z; Y, Y ) into O. Then

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∞ (−G − z)Gext O z; Y, Y − δ Y, Y = F Y, Y ∈ C (Ωint ), where supp F (·, Y ) ⊂ O. Therefore, GO z; Y, Y = G z; Y, Y +

G z; Y, Y F Y , Y dVY .

O

In particular, ∂ν(Y ) G z; Y, Y +

∂ν(Y ) G z; Y, Y F Y , Y dVY = 0,

Y ∈ ∂O,

(6.16)

O

where ν(Y ) is the unit normal to O at Y . On the other hand, if F (· , Y ) ∈ C ∞ (U ), supp F (· , Y ) ⊂ O, satisfies (6.16), the function G z; Y, Y +

G z; Y, Y F Y , Y dVY ,

Y, Y ∈ U \ O,

(6.17)

O

is GO (z; Y, Y ). As we have in our disposal G(z; Y, Y ) for Y, Y ∈ U , we can verify for a given F , condition (6.16). (1) (2) Now, we return to Ωint , Ωint with Γ and BSP on Γ being the same. We denote the associated functions appearing above by adding the superscript (j ). Let (6.16) hold with G(z; Y, Y ), F (Y , Y ) replaced by G(1) (z; Y, Y ), F (1) (Y , Y ), respectively. Since G(1) (z; Y, Y ) = G(2) (z; Y, Y ) on U × U , (6.16) also holds with G(z; Y, Y ), F (Y , Y ) replaced by G(2) (z; Y, Y ), F (1) (Y , Y ), respectively. Thus, for Y, Y ∈ U \ O, we have (j ) GO z; Y, Y = G(j ) z; Y, Y +

G(j ) z; Y, Y F (1) Y , Y dVY ,

j = 1, 2,

O

so that (2) G(1) O z; Y, Y = GO z; Y, Y , (1)

(2)

z ∈ C \ R, Y, Y ∈ U \ O. (1)

In particular, this implies that ΛO (z) = ΛO (z), z ∈ C \ R. Then by Lemma 5.6, BSP’s for HO (2) and HO coincide. 2

Next we show that we can use these data to determine the critical distance which we use in the step-by-step construction of the manifold. Lemma 6.10. Let X0 ∈ Ωint \ ∂Ωint and 0 < ρ < τ (X0 )/2. Let O = B(X0 , ρ) and ΓO = ∂O. Assume that we are given the set ΓO as a differentiable manifold, the metric G|ΓO on ΓO , and the BSP for HO on ΓO . Then these data determine τO (ΓO ) = τ (X0 ) − ρ.


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Proof. Let us assume that t0 < τ (X0 ) − ρ. Then, for any Y ∈ ΓO , the set ΩO (Y, t0 ) \ ΩO (ΓO , t0 − ε) contains an open neighborhood of γ(Y,ν) (t0 − ε/2) and, therefore, has positive measure. Hence, if t < τ (X0 ) − ρ, then the condition ∀Y ∈ ΓO , ∀ε > 0: I,t := (ΓO , t − ε) IY,t := (Y, t)

(6.18)

is valid. Let us next assume that condition (6.18) is valid and consider its consequences. First, observe that by (6.8) and (6.9), we have either (a) s(X0 ) = τ (X0 ) and there is Y ∈ ΓO such that X = γ(Y,ν) (τ (X0 ) − ρ) ∈ ∂Ωint , or (b) s(X0 ) > τ (X0 ) and there are Y ∈ ΓO and s such that s(X0 ) > s > τ (X0 ) − ρ and dO (γ(Y,ν) (s), ΓO ) < s. Let us consider these two cases separately. (a) It follows from (6.8) and (6.9) that X is a closest point to X0 on ∂Ωint . Therefore, the geodesic γ(Y,ν) intersects ∂Ωint normally at X = γ(Y,ν) (s), s = τ (X0 ) − ρ. Assume next that t > 0 is such that . ∀ε > 0: I,t IY,t

(6.19)

Xε ∈ ΩO (Y, t) \ ΩO (ΓO , t − ).

(6.20)

Then for any ε > 0 there is

As ΩO (Y, t) is relatively compact, there are εn → 0 and Xn = Xεn such that Xn → X ∈ Ωint as n → ∞. Then dO X , Y = t,

dO X , ΓO = t.

(6.21)

This shows that Y is the closest point of ΓO to X in ΩO . Consider a shortest curve μ(s) from Y to X . By [2], a shortest curve between two points on a manifold with boundary is a C 1 -curve. Moreover, it is a geodesic on ΩO \ ∂ΩO . Since μ(s) is a shortest curve from X to ∂ΩO , it is normal to ∂ΩO at Y . Thus μ(s) = γY,ν (s), s τ (X0 ) − ρ. However, γ(Y,ν) (s) hits ∂Ωint normally at s = τ (X0 ) − ρ. Therefore, by the short-cut arguments, we see that the curve γ(Y,ν) ([0, τ (X0 ) − ρ]) ⊂ ΩO cannot be extended to a longer curve which is a shortest curve between Y and its other end point. Thus μ ⊂ γ(Y,ν) ([0, τ (X0 ) − ρ]), implying that t = dO (Y, X ) τ (X0 )−ρ. Hence in the case (a) the condition (6.18) implies that t τ (X0 )−ρ. (b) In this case arguments are similar but slightly simpler. Again, assume that t > 0 is such that (6.19) is satisfied. Again, there are n > 0 and Xn = Xεn satisfying (6.20), such that Xn → X and X ∈ Ωint satisfies (6.21). Moreover, a shortest curve μ(s) from Y to X coincides with the normal geodesic γ(Y,ν) (s) for small values of s. Since the geodesic γ(Y,ν) ([0, s ]) is a shortest

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curve between its end points for s τ (X0 ) − ρ but not for s(X0 ) − ρ > s > τ (X0 ) − ρ, we see that μ ⊂ γ(Y,ν) ([0, τ (X0 ) − ρ]) and thus t τ (X0 ) − ρ. Therefore, in both cases (a) and (b), the condition (6.18) implies that t τ (X0 ) − ρ. Combining these facts, we see that τ (X0 ) − ρ = sup t > 0; condition (6.18) is satisfied for t . The lemma then follows from this and Lemma 6.5.

2

6.5. Proof of Theorem 6.1 We are now in a position to complete the proof of Theorem 6.1. 6.5.1. Local reconstruction of Riemannian structure We start our considerations with O = ∅. Let Σ ⊂ Γ satisfies condition (C-2) and T > 0 be sufficiently small. In fact, we can consider any 0 < T < τ∅ (Σ). Using Lemma 6.7 we see that the set R∅ (MΣ,T ) ⊂ C(Γ ) is uniquely determined. On this set we introduce the boundary normal coordinates, r(·) → (Z, t),

t = min r Z , Z ∈Σ

where Z is the unique point on Σ on which r(·) attains its minimum. Observe that these coordinates on R∅ (MΣ,T ) coincide with the boundary normal coordinates of the point X ∈ Ωint such that r(·) = rX (·). Thus, R∅ (MΣ,T ) with the above coordinates is diffeomorphic to MΣ,T . so that Next we use Lemma 6.8 to endow R∅ (MΣ,T ) with Riemannian metric, G, (R∅ (MΣ,T ), G) is isometric to the manifold (MΣ,T , G). Remark. For the inverse scattering problem considered in the introduction, Section 6.5.1 is not necessary, because we know a priori the Riemannian structure of the open set (Ωint \∂Ωint )∩Ω1 . However, to make the results of Section 6 appropriate for general non-compact manifolds with asymptotically cylindrical ends, we have included this step. 6.5.2. Iteration of local reconstruction To describe the procedure which we will iterate, let us assume that U1 ⊂ Ωint is a connected neighborhood Σ ⊂ Γ which satisfies condition (C-2) with O = ∅ and that we know the Rieman is already determined, nian manifold (U1 , G) up to an isometry. Since the set (R∅ (MΣ,T , G)) we can take U1 = MΣ,T , where T > 0 is sufficiently small. Choose X1 ∈ U1 and ρ > 0 such that O = B(X1 , ρ) ⊂ U1 . By Lemma 6.9 we can determine G(z; Y, Y ) for all Y, Y ∈ U1 and z ∈ C \ R. Moreover, it gives us BSP on ∂O. Therefore by Lemma 6.10, these data determine τO (ΓO ), hence τ (X1 ) = τO (ΓO ) + ρ. Take any X ∈ B(X1 , τ ) \ O, where τ = τ (X1 ), and let Y be the intersection of ∂O and the geodesic with end points X1 and X. Taking any Z ∈ ∂O and applying Lemma 6.6, we can then find dO (X, Z).


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Using, similarly to the above, Lemmas 6.7 and 6.8, we can find the image of the embedding RO : B(X1 , τ ) \ O → C(∂O). We then recover, in the boundary normal coordinates associated with ∂O, i.e. the Riemannian normal coordinates centered at X1 , the metric tensor G on B(X1 , τ ) \ B(X1 , ρ), and, since G on B(X1 , ρ) is known, on the whole B(X1 , τ ). This construction makes it possible to introduce the structure of the differentiable manifold on U1 B(X1 , τ ) which we considered, by now, as a disjoint union of two Riemannian manifolds. Next we glue these two components together. To this end we observe that, since O ⊂ U1 , we have in our disposal Green’s function G(z; Y, Y ) for Y, Y ∈ O and z ∈ C \ R. The set O can be considered also as the subset B(X1 , ρ) of B(X1 , τ ), and thus we know the function G(z; Y, Y ) for Y, Y ∈ B(X1 , ρ) e.g. in the Riemannian normal coordinates centered at X1 . Thus, using the Unique Continuation Principle, we can determine, in the Riemannian normal coordinates, the function G(z; Y, Y ) for all Y ∈ B(X1 , τ ) and Y ∈ B(X1 , ρ). Since Y → G(z; Y, Y ) is a smooth function in Ωint \ {Y } and G(z; Y, Y ) → ∞ as Y → Y , we see that for Y1 , Y2 ∈ Ωint , we have Y1 = Y2 if and only if Gz (Y1 , Y ) = Gz (Y2 , Y ) for all Y ∈ Ωint , z ∈ C \ R. Using the Unique Continuation Principle, this is equivalent to G(z; Y1 , Y ) = G(z; Y2 , Y ) for all Y ∈ B(X1 , ρ), z ∈ C \ R. Next, let us define that the points XU ∈ U1 and XB ∈ B(X1 , τ ) are equivalent and denote XU ∼ XB if G(z; XU , Y ) = G(z; XB , Y ) for all Y ∈ B(X1 , ρ), z ∈ C \ R. Then the manifold U2 = U1 ∪ B(X1 , τ ) ⊂ Ωint is diffeomorphic to manifold (U1 B(X1 , τ ))/ ∼, which is obtained by glueing together the equivalent points on U1 and B(X1 , τ ). As we know the metric tensor on both U1 and B(X1 , ρ), we have reconstructed a Riemannian manifold (U2 , G) ⊂ (Ωint , G) up to an isometry. 6.5.3. Maximal reconstruction Let us iterate the above process, that is, we start from an open set Σ ⊂ Γ satisfying condition (C-2) with O = ∅, construct its neighborhood U1 , and iterate the construction by choosing at each step j = 1, 2, . . . a point Xj ∈ Uj and constructing a Riemannian manifold isometric to Uj +1 = Uj ∪ B(Xj , τ (Xj )) ⊂ Ωint . Consider the open sets in Ωint \ ∂Ωint which can be reconstructed, with the metric, when we are given the set Γ with its metric and the BSP on Γ . As the collection of these sets is closed with respect to taking the union, consider maximal open set Umax ⊂ Ωint \ ∂Ωint which can be reconstructed, with its metric, from the set Γ with its metric and the BSP on Γ . Let us show that Umax = Ωint \ ∂Ωint . Since Ωint \ ∂Ωint is connected, it suffices to show that Umax is open and closed in Ωint . By construction, Umax is open. Let now X ∈ / ∂Ωint be a limit point of Umax , i.e., X = limn→∞ Xn , Xn ∈ Umax . Denote a = d(X, ∂Ωint ) so that if Y ∈ B(X, a/4), then s(Y ) 3a/4, see (6.8). Since the cut locus distance of the Riemannian normal coordinates is continuous with respect to the center, see e.g. [49, Section 2.1] or [29], there is δ > 0 such that τ (Y ) δ for all Y ∈ B(X, a/4). Let now Xn ∈ Umax satisfy the inequality d(Xn , X) < σ = min(a/4, δ/4). Let us assume that Xn has a neighborhood B(Xn , ρn ), with a sufficiently small ρn < d(Xn , X), which can be reconstructed using N (n) iteration steps, that is, B(Xn , ρn ) ⊂ UN (n) . Then τ (Xn ) > 4σ so that X ∈ B(Xn , τ (Xn )). By Lemma 6.9, we can find the BSP for the operator HO with O = B(Xn , τ (Xn )) and, using one more iteration step, reconstruct the Riemannian structure on UN (n) ∪ B(Xn , τ (Xn )) which includes the point X. Therefore, the point X is in Umax . This shows that Umax is relatively open and closed in Ωint \ ∂Ωint . Thus, Umax = Ωint \ ∂Ωin . The above shows that using an enumerable number of iteration steps we can construct a Riemannian manifold isometric to (Ωint \ ∂Ωint , G). Thus we have reconstructed the Riemannian manifold (Ωint \ ∂Ωint , G) up to an isometry.

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It remains to identify the differentiable and Riemannian structures near ∂Ωint . Observe that Ωint is just the closure of Ωint \ ∂Ωint with respect to the distance function generated by the metric G on Ωint \ ∂Ωint . Moreover, for any open relatively compact set Σ ⊂ ∂Ωint , there exists δ > 0 such that τ∅ (Σ) δ > 0. Let 0 < t < δ and consider the set Σt = X ∈ Ωint \ ∂Ωint ; d(X, ∂Ωint ) = t, d(X, Z) = t, for some Z ∈ Σ . This implies that for X ∈ Σt the closest point Z ∈ Ωint is in Σ and X = γZ,ν (t). Therefore, Σt is a smooth (n − 1)-dimensional open submanifold in Ωint of points having the form X = γ(Z,ν) (t), Z ∈ Σ . This makes it possible to introduce the boundary normal coordinates in MΣ,δ which provides the differentiable structure near Σ. Writing the metric tensor G in these coordinates and extending this tensor continuously on Σ, we find the metric tensor in Ωint in the boundary normal coordinates associated to Σ . 6.6. Proof of Theorem 1.1 Having Theorem 6.1 in our disposal, Theorem 1.1 follows immediately from Lemma 5.7.

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AT structure of AH algebras with the ideal property and torsion free K-theory ✩ Guihua Gong a,b,∗ , Chunlan Jiang b , Liangqing Li a , Cornel Pasnicu a a Department of Mathematics, University of Puerto Rico, Rio Piedras, PR 00931, USA b Department of Mathematics, Hebei Normal University, Shijiazhuang, China

Received 29 May 2009; accepted 17 November 2009 Available online 2 December 2009 Communicated by D. Voiculescu Dedicated to Professor George Elliott on the occasion of his sixtieth birthday

Abstract Let A be an AH algebra, that is, A is the inductive limit C ∗ -algebra of φ1,2

φ2,3

A1 −−→ A2 −−→ A3 −→ · · · −→ An −→ · · · tn with An = i=1 Pn,i M[n,i] (C(Xn,i ))Pn,i , where Xn,i are compact metric spaces, tn and [n, i] are positive integers, and Pn,i ∈ M[n,i] (C(Xn,i )) are projections. Suppose that A has the ideal property: each closed two-sided ideal of A is generated by the projections inside the ideal, as a closed two-sided ideal. Suppose that supn,i dim(Xn,i ) < +∞. (This condition can be relaxed to a certain condition called very slow dimension growth.) In this article, we prove that if we further assume that K∗ (A) is torsion free, then A is an approximate circle algebra (or an AT algebra), that is, A can be written as the inductive limit of

B1 −→ B2 −→ · · · −→ Bn −→ · · · , ✩ The first author was supported by NSF grants DMS 0200739, 0701150 and Chinese NSF grant 10628101, the second author was supported by Chinese NSF grant 10628101, the third author was supported by NSF grants DMS 0200739 and 0701150, and the fourth author was supported by NSF grant DMS 0101060 and by a FIPI grant from University of Puerto Rico. * Corresponding author at: Department of Mathematics, PO Box 23355, University of Puerto Rico, Rio Piedras, San Juan, PR 00931-3355, USA. E-mail address: [email protected] (G. Gong).


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s n where Bn = i=1 M{n,i} (C(S 1 )). One of the main technical results of this article, called the decomposition theorem, is proved for the general case, i.e., without the assumption that K∗ (A) is torsion free. This decomposition theorem will play an essential role in the proof of a general reduction theorem, where the condition that K∗ (A) is torsion free is dropped, in the subsequent paper Gong et al. (preprint) [31]—of course, in that case, in addition to space S 1 , we will also need the spaces TII,k , TIII,k , and S 2 , as in Gong (2002) [29]. © 2009 Elsevier Inc. All rights reserved. Keywords: C ∗ -algebras; Ideal property; AH algebras; AT algebras; Reduction theorem; Classification

1. Introduction An AH algebra is a nuclear C ∗ -algebra of the form A = lim(An , φn,m ) with An = −→ tn P M (C(X ))P , where X are compact metric spaces, tn and [n, i] are posin,i [n,i] n,i n,i n,i i=1 tive integers, M[n,i] (C(Xn,i )) are algebras of [n, i] × [n, i] matrices with entries in C(Xn,i )—the algebra of complex-valued continuous functions on Xn,i —, and finally Pn,i ∈ M[n,i] (C(Xn,i )) are projections (see [1]). For the special case that Xn,i are the single point space {pt}, the algebra A is called an AF algebra, approximately finite dimensional algebra. AF algebras were completely classified by G. Elliott [13] after the work of Glimm [24] and Bratteli [3]. Nowadays the classification of AH algebras is an important ingredient of Elliott’s classification project. Successful classification results have been obtained for the two classes of AH algebras. On one hand is the classification of real rank zero AH algebras with slow dimension growth (see [14,50,37,36,38,22,21,17,18,6,7,25,28,26,27,23,9,12,8]; the final paper [8] contains the general classification result, and [18] contains perhaps the most important breakthrough in this direction). On the other hand is the classification obtained in [29,20] for the class of simple AH algebras with very slow dimension growth (also, see [15,16,33–35,39,52]). (There are also some interesting classification results of AI algebras involving other invariants (see [51,4]).) As a unification of the AH algebras of real rank zero and of the simple AH algebras, it seems most natural to study the AH algebras with the ideal property: each closed two-sided ideal of the algebra is generated by the projections inside the ideal, as closed two-sided ideals. There are many C ∗ -algebras naturally arising from C ∗ -dynamical systems which have the ideal property. For example, if (A, G, α) is a C ∗ -dynamical system such that G is a discrete is essentially free, that is, for every G-invariant closed amenable group and the action of G on A subset F ⊂ A, the subset {x ∈ F : gx = x for all g ∈ G\{1}} is dense in F , then A α G has the ideal property provided that A has the ideal property (use [48, Theorem 1.16]). Many of these C ∗ -algebras are neither simple nor of real rank zero. Even if we assume, in the above class of examples, that A is the commutative C ∗ -algebra C(X) with dim(X) = 0 and G = Z, it is not known whether A α G is of real rank zero (it is known that A α G is not simple if α is not minimal). However, it follows from the above result or from [45] that C(X) α Z has the ideal property. Hence, it is important and natural to extend the classification of simple C ∗ -algebras and the classification of real rank zero C ∗ -algebras to C ∗ -algebras with the ideal property. In a sequence of papers [43,44,42,41] (notably [43]), the fourth author Pasnicu obtained several important results (including the characterization theorem) for AH algebras with the ideal property (see also [46,47]). In the study of simple AH algebras [34], the third author Li proved


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that at the level of spectra, a connecting homomorphism (in a simple inductive system) can be approximately factored through a direct sum of matrix algebras over C[0, 1]. This factorization is essential for the classification of simple AH algebras (see [35,29,20]). Fortunately, Li’s result can be generalized to the case of the AH algebras with the ideal property, by combining the characterization theorem in [43] and techniques in [34]. Furthermore, the decomposition theorem of Gong (the first author) for simple AH algebras (see [29]) is based on such factorization. Therefore, we can generalize Gong’s decomposition theorem to the case of AH algebras with the ideal property (see Section 2 of this paper). Based on this decomposition theorem, we are able to prove the reduction theorem for AH algebras with the ideal property which unifies and generalizes the reduction theorem for real rank zero AH algebras due to Dadarlat and Gong (see [6,7,26,27,8]) and the reduction theorem for simple AH algebras due to Gong (see [29]). To make the main idea of our proof clear and to keep this article at a reasonable length, we will only prove the reduction theorem for the case when K∗ (A) is torsion free. The proof of the reduction theorem in the general case will appear in a subsequent paper [31], which essentially uses the decomposition theorem of this paper. For the special case when K∗ (A) is torsion free, we will prove that A can be written as an inductive limit of finite direct sums of matrix algebras over C(S 1 ), C[0, 1] or C. That is, all the spaces Xn,i can be replaced by S 1 , [0, 1] or {pt}. (Note that, the spaces [0, 1] and {pt} can be replaced by S 1 , if one does not require the connecting homomorphisms φn,m to be injective.) This result is certainly an important step in the classification of AH algebras with the ideal property. The paper is organized as follows. In the rest of this section, we will introduce some notation and collect some known results. Most of them can be found in Sections 1.1 and 1.2 of [29]. In Section 2, we will prove the decomposition theorem. In Section 3, we will prove the main theorem. 1.1. If A and B are two C ∗ -algebras, we use Map(A, B) to denote the space of all linear, completely positive ∗-contractions from A to B. If both A and B are unital, then Map(A, B)1 will denote the subset of Map(A, B) consisting of unital maps. By word “map”, we shall mean a linear, completely positive ∗-contraction between C ∗ -algebras, or else we shall mean a continuous map between topological spaces, which one will be clear from the context. By a homomorphism between C ∗ -algebras, will be meant a ∗-homomorphism. Let Hom(A, B) denote the space of all the homomorphisms from A to B. Similarly, if both A and B are unital, let Hom(A, B)1 denote the subset of Hom(A, B) consisting of all the unital homomorphisms. Definition 1.2. Let G ⊂ A be a finite set and δ > 0. We shall say that φ ∈ Map(A, B) is G–δ multiplicative if φ(ab) − φ(a)φ(b) < δ for all a, b ∈ G. 1.3. In the notation for an inductive system (An , φn,m ), we understand that φn,m = φm−1,m ◦ φm−2,m−1 ◦ · · · ◦ φn,n+1 , where all φn,m : An → Am are homomorphisms. n Ain , necessarily, We shall assume that, for any summand Ain in the direct sum An = ti=1 i φn,n+1 (1Ain ) = 0, since, otherwise, we could simply delete An from An without changing the limit algebra.

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1.4. If An =


i

Ain and Am =

j

j

i,j

Am , we use φn,m to denote the partial map of φn,m from the j

i-th block Ain of An to the j -th block Am of Am . 1.5. By 2.3 of [1] and Theorem 2.1 [19], we know that any AH algebra can be written as of n an inductive limit A = lim(An = ti=1 Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ), where Xn,i are finite −→ simplicial complexes and φn,m are injective. In this article, we will always assume that Xn,i are (path) connected finite simplicial complexes and φn,m are injective in any inductive system. It is well known that, for any connected finite simplicial complex X, there is a metric d on X with the following property: for any x ∈ X and η > 0, the η-ball centered in x, Bη (x) = {x ∈ X | d(x , x) < η} is path connected. So in this article, we will always assume that the metric on a connected simplicial complex has this property. (The metric with such property can be defined by identifying each simplex with the regular simplex of sides of length 1, and define the distance between two points to be the length of the shortest path connecting these two points, where a path should be broken into several pieces with each piece being inside a single simplex and then the length of the path is measured as the sum of the lengths of pieces of the path.) 1.6. In this article, we will assume that the inductive system satisfies the very slow dimension growth condition: for any n, there is a positive integer M such that lim

m−→∞

min

dim(Xm,j )>M rank(φn,m (Pn,i ))=0

i,j rank φn,m (Pn,i ) = +∞. (dim Xm,j + 1)3

(We use the convention that the minimum of an empty set is +∞.) This is a strengthened form of the condition of slow dimension growth in [27] (see also [2])—we replace dim Xm,j + 1 in the condition of slow dimension growth by (dim Xm,j + 1)3 . Without this condition, the main theorem of this article does not hold as shown by a counterexample of Villadsen [53]. We leave the following problem open: Does the main theorem of this article hold if one replaces the very slow dimension growth by the slow dimension growth? 1.7. By Lemma 1.3.3 of [29], any AH algebra A = lim An = −→

tn

Pn,i M[n,i] C(Xn,i ) Pn,i , φn,m

i=1

is isomorphic nto a limit corner subalgebra (see Definition 1.3.2 of [29] for this concept) of A = M{n,i} (C(Xn,i )), φ˜ n,m )—an inductive limit of full matrix algebras over Xn,i . lim(An = ti=1 −→ is an AT algebra, then A itself is also an AT algebra. Therefore, in Once we prove that A this article, we will assume A itself is an inductive limit of finite direct sums of full matrix algebras over Xn,i . That is Pn,i = 1M[n,i] (C(Xn,i )) . Even in this case, one is still forced to consider the cut-down of M[n,i] (C(Xn,i )) by a projection, since the image of a trivial projection under homomorphism φn,m may not be trivial (see 1.8 below for the definition of a trivial projection).


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1.8. Let Y be a compact metrizable space. Let P ∈ Mk1 (C(Y )) be a projection with rank(P ) = k k1 . For each y, there is a unitary uy ∈ Mk1 (C) (depending on y) such that ⎛1 ⎜ ⎜ ⎜ P (y) = uy ⎜ ⎜ ⎜ ⎝

⎞ ..

⎟ ⎟ ⎟ ∗ ⎟u , ⎟ y ⎟ ⎠

. 1 0 ..

. 0

where there are k 1’s on the diagonal. If the unitary uy can be chosen to be continuous in y, then P is called a trivial projection. It is well known that any projection P ∈ Mk1 (C(Y )) is locally trivial. That is, for any y0 ∈ Y , there is an open set Uy0 y0 , and there is a continuous unitary-valued function u : Uy0 → Mk1 (C) such that the above equation holds for u(y) (in place of uy ) for any y ∈ Uy0 . If P is trivial, then P Mk1 (C(X))P ∼ = Mk (C(X)). 1.9. Let X be a compact metrizable space and ψ : C(X) → P Mk1 (C(Y ))P be a unital homomorphism. For any given point y ∈ Y , there are points x1 (y), x2 (y), . . . , xk (y) ∈ X, and a unitary Uy ∈ Mk1 (C) such that ⎛ ⎜ ⎜ ⎜ ⎜ ψ(f )(y) = P (y)Uy ⎜ ⎜ ⎜ ⎝

⎞

f (x1 (y)) ..

⎟ ⎟ ⎟ ⎟ ∗ ⎟ Uy P (y) ∈ P (y)Mk1 (C)P (y) ⎟ ⎟ ⎠

. f (xk (y)) 0 ..

. 0

for k all f ∈ C(X). Equivalently, there are k rank one orthogonal projections p1 , p2 , . . . , pk with i=1 pi (y) = P (y) and x1 (y), x2 (y), . . . , xk (y) ∈ X, such that ψ(f )(y) =

k

f xi (y) pi (y),

∀f ∈ C(X).

i=1

Let us denote the set {x1 (y), x2 (y), . . . , xk (y)}, counting multiplicities, by SP ψy . In other words, if a point is repeated in the diagonal of the above matrix, it is included with the same multiplicity in SP ψy . We shall call SP ψy the spectrum of ψ at the point y (see also [40]). Let us define the spectrum of ψ , denoted by SP ψ , to be the closed subset

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SP ψ :=

SP ψy = SP ψy ⊂ X.

y∈Y

y∈Y

Alternatively, SP ψ is the complement of the spectrum of the kernel of ψ, considered as a closed ideal of C(X). The map ψ can be factored as

ψ1 i∗ C(X) − → C(SP ψ) −→ P Mk1 C(Y ) P with ψ1 an injective homomorphism, where i denotes the inclusion SP ψ → X. Also, if A = P Mk1 (C(Y ))P , then we shall call the space Y the spectrum of the algebra A, and write SP A = Y (= SP(id)). 1.10. In 1.9, if we group together all the repeated points in {x1 (y), x2 (y), . . . , xk (y)}, and sum their corresponding projections, we can write

ψ(f )(y) =

l

f λi (y) Pi

(l k),

i=1

where {λ1 (y), λ2 (y), . . . , λl (y)} is equal to {x1 (y), x2 (y), . . . , xk (y)} as a set, but λi (y) = λj (y) if i = j ; and each Pi is the sum of the projections corresponding to λi (y). If λi (y) has multiplicity m (i.e., it appears m times in {x1 (y), x2 (y), . . . , xk (y)}), then rank(Pi ) = m. 1.11. Set P k (X) = X ×X× · · · × X / ∼, where the equivalence relation ∼ is defined by k

(x1 , x2 , . . . , xk ) ∼ (x1 , x2 , . . . , xk ) if there is a permutation σ of {1, 2, . . . , k} such that xi = xσ (i) , for each 1 i k. A metric d on X can be extended to a metric on P k (X) by

d [x1 , x2 , . . . , xk ], x1 , x2 , . . . , xk = min max d xi , xσ (i) , σ

1ik

where σ is taken from the set of all permutations, and [x1 , . . . , xk ] denotes the equivalence class in P k (X) of (x1 , . . . , xk ). 1.12. Let X be a metric space with metric d. Two k-tuples of (possibly repeating) points {x1 , x2 , . . . , xk } ⊂ X and {x1 , x2 , . . . , xk } ⊂ X are said to be paired within η if there is a permutation σ such that

d xi , xσ (i) < η,

i = 1, 2, . . . , k.

This is equivalent to the following. If one regards (x1 , x2 , . . . , xk ) and (x1 , x2 , . . . , xk ) as two points in P k (X), then

d [x1 , x2 , . . . , xk ], x1 , x2 , . . . , xk < η.


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1.13. Let ψ : C(X) → P Mk1 (C(Y ))P be a unital homomorphism as in 1.9. Then ψ ∗ : y → SP ψy defines a map Y → P k (X), if one regards SP ψy as an element of P k (X). This map is continuous (compare with 3.2 and 3.3 in [40]). In terms of this map and the metric d, let us define the spectral variation of ψ : SPV(ψ) := the diameter of the image of ψ ∗ . Definition 1.14. We shall call the projection Pi in 1.10 the spectral projection of φ at y with respect to the spectral element λi (y). For a subset X1 ⊂ X, we shall call

Pi

λi (y)∈X1

the spectral projection of φ at y corresponding to the subset X1 (or with respect to the subset X1 ). In general, for an open set U ⊂ X, the spectral projection P (y) of φ at y corresponding to U does not depend on y continuously. But the following two lemmas are well-known facts (see 1.3 in [11] and 1.2.9 and 1.2.10 of [29]). Lemma 1.15. Let X be a finite simplicial complex, let X1 ⊂ X be a closed subset, and let φ : C(X) → Mn (C(Y )) be a homomorphism. For any y0 ∈ Y , if SPφ y0 ∩ X1 = ∅, then there is an open set W y0 such that SPφ y ∩ X1 = ∅ for any y ∈ W . Another equivalent statement is the following. Let U ⊂ X be an open subset. For any y0 ∈ Y , if SPφ y0 ⊂ U , then there is an open set W y0 such that SPφ y ⊂ U for any y ∈ W . Lemma 1.16. Let U ⊂ X be an open subset. Let φ : C(X) → Mn (C(Y )) be a homomorphism. Suppose that W ⊂ Y is an open subset such that SPφ y ∩ (U \U ) = ∅,

∀y ∈ W.

Then the function y → spectral projection of φ at y corresponding to U is a continuous function on W . Furthermore, if W is connected then #(SPφ y ∩ U ) (counting multiplicity) is the same for any y ∈ W , and the map y → SPφ y ∩ U ∈ P l X is a continuous map on W , where l = #(SPφ y ∩ U ). Here #(S) denotes the number of elements in the set S counting multiplicity. 1.17. Let φ : Mk (C(X)) → P Ml (C(Y ))P be a unital homomorphism. Set φ(e11 ) = p, where e11 is the canonical matrix unit corresponding to the upper left corner. Set

φ1 = φ|e11 Mk (C(X))e11 : C(X) −→ pMl C(Y ) p.

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Then P Ml (C(Y ))P can be identified with pMl (C(Y ))p ⊗ Mk in such a way that φ = φ1 ⊗ idk . Let us define SP φy := SP(φ1 )y , SP φ := SP φ1 , SPV(φ) := SPV(φ1 ). Suppose that X and Y are connected. Let Q be a projection in Mk (C(X)) and φ : QMk (C(X))Q → P Ml (C(Y ))P be a unital map. By the Dilation lemma (2.13 of [18]), there are an n, a projection P1 ∈ Mn (C(Y )), and a unital homomorphism

φ˜ : Mk C(X) −→ P1 Mn C(Y ) P1 such that ˜ QMk (C(X))Q . φ = φ| (Note that this implies that P is a subprojection of P1 .) We define: SP φy := SP φ˜ y , ˜ SP φ := SP φ, ˜ SPV(φ) := SPV(φ). ˜ (Note that these definitions do not depend on the choice of the dilation φ.) 1.18. Let φ : Mk (C(X)) −→ P Ml (C(Y ))P be a (not necessarily unital) homomorphism, where X and Y are connected finite simplicial complexes. Then #(SP φy ) =

rank φ(1k ) , rank(1k )

for any y ∈ Y,

where again #(.) denotes the number of elements in the set counting multiplicity. It is also true φ(p) that for any nonzero projection p ∈ Mk (C(X)), #(SP φy ) = rank rank(p) . 1.19. Let X be a compact connected space and let Q be a projection of rank n in MN (C(X)). The weak variation of a finite set F ⊂ QMN (C(X))Q is defined by ω(F ) = sup

inf max uΠ1 (a)u∗ − Π2 (a),

Π1 ,Π2 u∈U (n) a∈F

where Π1 , Π2 run through the set of irreducible representations of QMN (C(X))Q into Mn (C).


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Let X i be compact connected spaces and Qi ∈ Mni (C(Xi )) be projections. For a finite set F⊂ i Qi Mni (C(Xi ))Qi , define the weak variation ω(F ) to be maxi ω(πi (F )), where πi : j Qj Mnj (C(Xj ))Qj → Qi Mni (C(Xi ))Qi is the natural projection map onto the i-th block. The set F is said to be weakly approximately constant to within ε if ω(F ) < ε. Another description of this concept can be found in [18, 1.4.11] (see also [7, 1.3]). 1.20. The following notations will be frequently used in this article. (a) As in 1.16 and 1.18 above, we use the notation #(.) to denote the cardinal number of the set counting multiplicity. (b) For any metric space X, any x0 ∈ X and any c > 0, let Bc (x0 ) := x ∈ X d(x, x0 ) < c denote the open ball with radius c and centre x0 . (c) Suppose that A is a C ∗ -algebra, B ⊂ A is a sub-C ∗ -algebra, F ⊂ A is a (finite) subset and let ε > 0. If for each element f ∈ F , there is an element g ∈ B such that f − g < ε, then we shall say that F is approximately contained in B to within ε, and denote this by F ⊂ε B. (d) Let X be a compact metric space. For any δ > 0, a finite set {x1 , x2 , . . . , xn } is said to be δ-dense in X, if for any x ∈ X, there is xi such that dist(x, xi ) < δ. (e) We shall use • to denote any possible positive integer. To save notation, a1 , a2 , . . . may be used for a finite sequence if we do not care how many terms are in the sequence. Similarly, A1 ∪ A2 ∪ · · · or A1 ∩ A2 ∩ · · · may be used for a finite union or a finite intersection. If there is a danger of confusion with an infinite sequence, union or intersection, we will write them as a1 , a2 , . . . , a• , A1 ∪ A2 ∪ · · · ∪ A• , A1 ∩ A2 ∩ · · · ∩ A• . (f) In this paper, we often use 1 to denote the units of different unital C ∗ -algebras. In particular, if 1 appears in φ(1), where φ is a homomorphism, then 1 is the unit of the domain algebra. For example for a homomorphism φ : ri=1 Ai −→ B, then 1 in φ(1) means 1ri=1 Ai and 1 in φ i (1) means 1Ai . (g) For any two projections p, q ∈ A, we use the notation [p] [q] to denote that p is unitarily equivalent to a sub projection of q. And we use p ∼ q to denote that p is unitarily equivalent to q. 2. The decomposition theorem 2.1. Let X be a finite simplicial complex with simplicial structure σ . In the usual sense, a subcomplex Y is a union of a collection of simplices of σ such that if a simplex is in the collection, then all its faces are in the collection. In this paper, we will call a closed subspace Y ⊂ X a subcomplex of X if there is a subdivision τ of the simplicial structure σ such that Y is a subcomplex of (X, τ ) as in the above usual sense (in such a way, we will not always mention the subdivision τ of σ ). First, let us use the characterization theorem of the AH algebras with the ideal property (see [43]) to prove the following theorem.

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n Theorem 2.2. Let A = lim(An = ti=1 M[n,i] (C(Xn,i )), φn,m ) be an AH algebra with the −→ ideal property. Then for any An and η > 0, there is an m and there are subcomplexes 1 , X 2 , . . . , X tm ⊂ X Xn,i n,i for each Xn,i such that the following hold: n,i n,i i,j

j

(1) SP(φn,m ) ⊂ Xn,i . j

i,j

j

(2) For any point x0 ∈ Xn,i and any y ∈ Xm,j , SP(φn,m )y ∩ B η (x0 , Xn,i ) = ∅, where j

2

j

B η (x0 , Xn,i ) = {x ∈ Xn,i | dist(x, x0 ) < η2 }. 2

Proof. Applying Lemma 2.9 of [43] (with η2 in place of δ), there is an m such that for any closed i,j set F ⊂ Xn,i and any j ∈ {1, 2, . . . , tm }, the map φn,m satisfies either i,j

SP φn,m y ∩ F = ∅,

for all y ∈ Xm,j

or i,j

SP φn,m y ∩ B η (F ) = ∅, 4

for all y ∈ Xm,j .

! i,j i,j i,j Let Fj = {x: B η (x) ∩ SP(φn,m )y = ∅, ∀y ∈ Xm,j } ⊇ SP(φn,m ) = y∈Xm,j SP(φn,m )y . 4 We can find a subdivision τ of the simplicial complex Xn,i such that each simplex has diamj eter smaller than η4 . Define the subcomplex Xn,i as the union of all the simplices (and their j

faces) of (Xn,i , τ ) with ∩ B η (Fj ) = ∅ (consequently, Xn,i ⊇ B η (Fj )). It is straightforward to 4 4 verify the conditions (1) and (2) of the theorem. 2 j

2.3. Let Xn,i be the finite simplicial complex as defined in the proof of Theorem 2.2. We can write

j Xn,i

as disjoint union of Z1 ∪ Z2 ∪ · · · ∪ Z• , where Zk are connected subsimplicial comj

plexes of Xn,i . We will prove the following lemma. i,j

Lemma 2.4. Let η and Zk be as the above. For any x ∈ Zk , y ∈ Xm,j , SP(φn,m )y ∩ Bη (x, Zk ) = ∅. j

Proof. Note that Zk ⊂ Xn,i . If x ∈ Zk , then there is a simplex x (of Zk ) such that ∩ B η (Fj ) = ∅. 4

Let x ∈ ∩ B η (Fj ). Then x and x can be connected by a path in Xn,i (since ⊂ Xn,i is path 4 connected). Since x ∈ B η (Fj ), there is a point x ∈ Fj , such that dist(x , x ) < η4 . From the 4 general assumption in 1.5, Xn,i itself is a path connected simplicial complex whose open ball j B η (x ) is also path connected. Note that from the definition of Xn,i in the proof of Theorem 2.2, j

4

j

Xn,i ⊃ B η (Fj ). Hence B η (x ) ⊂ Xn,i . So x and x can be connected by a path in Xn,i . For j

j

4

4

j

any y ∈ Xm,j , since B η (x ) ∩ SP(φn,m )y = ∅, there is a point x ∈ B η (x ) ∩ SP(φn,m )y ⊂ Xn,i . 4

i,j

i,j

4

j


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x can be connected to x within B η (x ) ⊂ Xn,i . So x and x can be connected by a path in j

4

Xn,i . Hence x ∈ Zk . The lemma follows from j

3η dist x, x dist x, x + dist x , x + dist x , x < η. 2 4 n M[n,i] (C(Xn,i )), φn,m ) be an AH algebra with the ideal Lemma 2.5. Let A = lim(An = ti=1 −→ property. For any η > 0, there is m > n such that for each pair of indices i,j j i ∈ {1, 2, . . . , tn }, j ∈ {1, 2, . . . , tm }, φn,m : Ain −→ Am can be written as a composition s ⊕φ j j,s j,s π i,j,s − Ain − → −→ Am , with B i,j,s = M[n,i] (C(Yi )), where Yi ⊂ Xn,i are connected sB finite simplicial complexes, π is defined by π(f ) = (f |Y j,1 , f |Y j,2 , . . . , f |Y j,• ) ∈ i

i

i

B i,j,s ,

s

j

j,s

and φ s : B i,j,s −→ Am satisfies that for any y ∈ Ym,j , SP(φ s )y is η-dense in Yi , that is,

j,s

SP φ s y ∩ Bη x0 , Yi = ∅ j,s

for all y ∈ Ym,j and x0 ∈ Yi . i,j

Proof. For each i, j from Theorem 2.2, φn,m factors as j

π j M[n,i] C(Xn,i ) − → M[n,i] C Xn,i −→ Am . j,s

Let Yi

j

= Zs as in 2.3, then Xn,i =

i,j

φ s = φn,m |M

j,s [n,i] (C(Yi ))

! s

j,s

j

Yi . Hence M[n,i] (C(Xn,i )) =

. This ends the proof.

s

j,s

M[n,i] (C(Yi )). Let

2

2.6. The following terminology is quoted from 4.26 of [29] (see also [33,34]): For any η > 0, δ > 0, a unital homomorphism φ : P Mk (C(X))P −→ QMk (C(Y ))Q is said to have the property sdp(η, δ) (spectral distribution property with respect to η and δ) if for any η-ball Bη (x) and any point y ∈ Y ,

rank(Q) # SP φy ∩ Bη (x) δ#(SP φy ) = δ rank(P ) counting multiplicity. Any homomorphism φ:

i

Pi Mki C(Xi ) Pi −→ Qj Mlj C(Yj ) Qj j

is said to have the property sdp(η, δ) if each partial map

φ i,j : Pi Mki C(Xi ) Pi −→ φ i,j (Pi )Mli C(Yj ) φ i,j (Pi )

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has the property sdp(η, δ) as a unital homomorphism. (Note that this is the definition of the sdp property defined in [29] and it is weaker than the corresponding property defined in [20] (see 2.1 of [20]).) Note that by definition, a nonunital homomorphism φ : Mk (C(X)) −→ Ml (C(Y )) has the property sdp(η, δ) if and only if the corresponding unital map

φ : Mk C(X) −→ φ(1k )Ml C(Y ) φ(1k ) has the property sdp(η, δ). Lemma 2.7. Let φ : si=1 Mki (C(Xi )) −→ ti=1 Mli (C(Yi )) and ψ : ti=1 Mli (C(Yi )) −→ u i=1 Mmi (C(Zi )) be homomorphisms. If φ satisfies the following dichotomy condition: (∗): for each pair i, j , the homomorphisms

φ i,j : Mki C(Xi ) −→ Mlj C(Yj ) either has the property sdp(η, δ) or is the zero homomorphism, then the composition ψ ◦ φ also satisfies the dichotomy condition: for each pair i, j the partial map (ψ ◦ φ)i,j =

t

ψ i ,j ◦ φ i,i : Mki C(Xi ) −→ Mnj C(Zj )

i =1

either has the property sdp(η, δ) or is the zero homomorphism. Proof. For each z ∈ Zj , i,j

SP(ψ ◦ φ)z =

t

SP φyi,i .

i =1 y∈SP ψ i ,j z

Since φ satisfies the dichotomy condition, for any η-ball Bη (x) and i , we know that

rank φ i,i (1ki ) . # SP φ i,i y ∩ Bη (x) δ rank(1ki )

(Note that if φ i,i = 0, then both sides of the above inequality are zero, so the inequality still holds.) Also, from 1.18

i ,j

rank ψ i ,j (p) # SP ψz = rank(p)

for any nonzero projection p ∈ Mli (C(Yi )). For each triple i, i , j , with φ i,i (1ki ) = 0, let φ i,i (1ki ) = p, then


# SP ψ i ,j ◦ φ i,i z ∩ Bη (x) =

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# SP φyi,i ∩ Bη (x) i ,j

y∈SP ψz

rank ψ i ,j (φ i,i (1ki )) rank φ i,i (1ki ) δ rank(1ki ) rank φ i,i (1ki )

=δ

rank ψ i ,j (φ i,i (1ki )) . rank(1ki )

Hence

i ,j

i,j # SP(ψ ◦ φ)z ∩ Bη (x) = # SP ψ ◦ φ i,i z ∩ Bη (x) i

δ =δ

rank ψ i ,j (φ i,i (1ki )) rank(1ki )

rank(ψ ◦ φ)i,j (1ki ) . rank(1ki )

2

n Lemma 2.8. Let A = lim(An = ti=1 M[n,i] (C(Xn,i )), φn,m ) be an AH algebra with the ideal −→ property. For An and any η > 0, there exist a δ > 0, a positive integer m > n, connected finite simplicial complexes Zi1 , Zi2 , . . . , Zi• ⊂ Xn,i , i = 1, 2, . . . , tn , and a homomorphism φ:B =

tn

M[n,i] C Zis −→ Am

s

i=1

such that φ

π → B− → Am , where π is defined by (1) φn,m factors as An −

π(f ) = (f |Z 1 , f |Z 2 , . . . , f |Zi• ) ∈ i

i

M[n,i] C Zis ⊂ B,

s

for any f ∈ M[n,i] (C(Xn,i )). (2) The homomorphism φ satisfies the dichotomy condition (∗): for each Zis , the partial map

j φ (i,s),j : M[n,i] C Zis −→ Am η either has the property sdp( 32 , δ) or is the zero map. Furthermore for any m > m, each partial map of φm,m ◦ φ satisfies the dichotomy condition (∗): either it has the property η sdp( 32 , δ) or it is the zero map.

Proof. Once we prove the conclusion for m, the conclusion for m > m follows from Lemma 2.7. η For any η > 0, apply Lemma 2.5 to 32 (in place of η) to obtain m, connected finite simplicial j,s i,j complexes Yi (for each pair i, j with φn,m = 0) and homomorphisms

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j,s

j −→ Am φ s : M[n,i] C Yi as described in Lemma 2.5. Choose δ = min

1 rank(φ s (1

j,s

M[n,i] (C(Yi ))

))

.

From Lemma 2.5, φn,m : An −→ Am factors through as π

An − →

tm tn i=1 j =1

j,s

i j s φ s M[n,i] C Yi −−−−−−−−→ Am .

s

Order the spaces Yi1,1 , Yi1,2 , . . . , Yi2,1 , Yi2,2 , . . . , and rename them as Zi1 , Zi2 , . . . , Zit , . . . , Zi• . Let B=

tn i=1

and φ =

tn tm j =1

i=1

tn tm t

j,s

M[n,i] C Zi = M[n,i] C Yi i=1 j =1

t

s

j

s

φ s . Then for each Zit and Am , either the partial map

j φ (i,t),j : M[n,i] C Zit −→ Am

j ,s

η is zero (when Zit = Yi 1 with j1 = j ) or SP(φ (i,t),j )y is 32 dense in Zit for any y ∈ Xm,j when j,s η t Zi = Yi . In the second case, for any y ∈ Xm,j , the 32 -ball B η (x) ⊂ Zit contains at least one 32

element in SP(φ (i,t),j )y . So

# SP φ (i,t),j y ∩ B η (x) 1 δ # SP φ (i,t),j y 32

since (i,t),j (1 (i,t),j rank φ

M[n,i] (C(Zik )) ) rank φ (i,t),j (1M[n,i] (C(Z k )) ) # SP φ = y i rank(1M[n,i] (C(Z k )) ) i

η (see the definition of δ). Hence φ (i,t),j has the property sdp( 32 , δ).

2

The following lemma is a modification of Lemma 2.11 of [43]. Lemma 2.9. (Cf. [43].) Let A = lim(An =

tn

be an AH algebra with the ideal property and with very slow dimension growth. For any An , finite set Fn = Fni ⊂ An , ε > 0, and positive integers L1 and L2 , there is an Am , such that for each pair (i, j ), one of the following conditions holds: −→

i,j

(i)

rank(φn,m (1Ai )) n

rank(1Ai ) n

L1 (L2 + dim Xm.j )3 , or

i=1 M[n,i] (C(Xn,i )), φn,m )


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(ii) there is a homomorphism i,j

j

i,j

ψ : Ain −→ φn,m (1Ain )Am φn,m (1Ain ) i,j

with finite dimensional image such that φn,m (f ) − ψ(f ) < ε for all f ∈ Fni . Proof. From the definition of the very slow dimension growth condition 1.6, there is an M > 0, such that i,j

lim

m−→∞

rank(φn,m (1Ain ))

min

dim(Xm,j )>M rank(φn,m (1Ai ))=0

(dim Xm,j + 1)3

= +∞.

n

j

Therefore there is an N such that for m > N , for each block Am , either dim(Xm,j ) M or i,j

rank φn,m (1Ain ) L1 (L2 )3 (dim Xm,j + 1)3 rank(1Ain ) L1 (L2 + dim Xm,j )3 rank(1Ain ). Now applying Lemma 2.10 of [43] to the integer G = L1 (L2 + M + 1)3 maxi {rank(1Ain )}, one i,j

can obtain an m > N such that either φn,m is at least G-large (see [10] for the definition of L-large, where L is a positive integer) or i,j φn,m (f ) − ψ(f ) < ε,

∀f ∈ Fni

for a homomorphism i,j

j

i,j

ψ : Ain −→ φn,m (1Ain )Am φn,m (1Ain ) with finite dimensional image. Now we can verify that Am is as desired as the follows. If i,j

rank φn,m (1Ain ) < L1 (L2 + dim Xm,j )3 rank(1Ain ), then dim(Xm,j ) M. Therefore i,j

rank φn,m (1Ain ) < L1 (L2 + M + 1)3 rank(1Ain ) G. i,j

That is, φn,m cannot be G-large. Hence the condition (ii) of the lemma holds.

2

2.10. Suppose that F ⊂ Mk (C(X)) is a finite set and ε > 0. Let F ⊂ C(X) be the finite set consisting of all the entries of the elements in F and ε = kε , where k is the order of the matrix algebra Mk (C(X)). It is well known that, for any k × k matrix a = (aij ) ∈ Mk (B) with entries aij ∈ B, a k maxij aij . This implies the following two facts.

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Fact 1. If φ1 , ψ1 ∈ Map(C(X), B) are (completely positive) linear ∗-contraction (as the notation in 1.1) which satisfy φ1 (f ) − ψ1 (f ) < ε ,

∀f ∈ F ,

then φ := φ1 ⊗ idk ∈ Map(Mk (C(X)), Mk (B)) and ψ := ψ1 ⊗ idk ∈ Map(Mk (C(X)), Mk (B)) satisfy φ(f ) − ψ(f ) < ε,

∀f ∈ F.

Fact 2. Suppose that φ1 ∈ Map(C(X), M• (C(Y ))) is a (completely positive) linear ∗-contraction. If φ1 (F ) is weakly approximately constant to within ε , then φ1 ⊗ idk (F ) is weakly approximately constant to within ε. Suppose that a homomorphism φ1 ∈ Hom(C(X), B) has a decomposition as follows. There exist mutually orthogonal projections p1 , p2 ∈ B with p1 + p2 = 1B and ψ1 ∈ Hom(C(X), p2 Bp2 ) such that φ1 (f ) − p1 φ1 (f )p1 ⊕ ψ1 (f ) < ε ,

∀f ∈ F .

Then there is a decomposition for φ := φ1 ⊗ idk :

φ(f ) − P1 φ(f ) P1 ⊕ ψ(f ) < ε,

∀f ∈ F,

where ψ := ψ1 ⊗ idk and P1 = p1 ⊗ 1k . In particular, if B = M• (C(Y )) and ψ1 is described by ψ1 (f )(y) = where qi ∈ B are projections with be described by ψ(f )(y) =

f αi (y) qi (y),

∀f ∈ C(X),

qi = p2 and αi : Y → X are continuous maps, then ψ can

qi (y) ⊗ f αi (y) ,

∀f ∈ Mk C(X) ,

regarding Mk (M• (C(Y ))) as M• (C(Y )) ⊗ Mk . From the above, we know that to decompose φ ∈ Hom(Mk (C(X)), M• (C(Y ))), one only needs to decompose φ1 := φ|e11 Mk (C(X))e11 ∈ Hom(C(X), φ(e11 )M• (C(Y ))φ(e11 )). The following proposition is Theorem 4.35 of [29] (see also [5,30,18,35]). Proposition 2.11. (Cf. [29].) Let X be a connected finite simplicial complex, and ε > η > 0. For any δ > 0, there is an integer L > 0 such that the following holds. Suppose that F ⊂ C(X) is a finite set such that dist(x, x ) < 2η implies |f (x) − f (x )| < 3ε for all f ∈ F . η , δ), and rank(φ(1)) If φ : C(X) → Mk (C(Y )) is a homomorphism with the property sdp( 32 2J · L2 · 2L (dim X + dim Y + 1)3 , where Y is a connected finite simplicial complex and J is any fixed positive integer, then there are three mutually orthogonal projections Q0 , Q1 , Q2 ∈ Mk (C(Y )),


a map φ0 ∈ Map(C(X), Q0 Mk (C(Y ))Q0 )1 and two homomorphisms Hom(C(X), Q1 Mk (C(Y ))Q1 )1 and φ2 ∈ Hom(C(X), Q2 Mk (C(Y ))Q2 )1 such that

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φ1 ∈

(1) φ(1) = Q0 + Q1 + Q2 ; (2) φ(f ) − φ0 (f ) ⊕ φ1 (f ) ⊕ φ2 (f ) < ε for all f ∈ F ; (3) The homomorphism φ2 factors through C[0, 1] as

ξ1 ξ2 φ2 : C(X) −→ C[0, 1] −→ Q2 Mk C(Y ) Q2 . Furthermore, if Y = {pt}, then ξ2 is injective; (4) The set (φ0 ⊕ φ1 )(F ) is approximately constant to within ε; (5) Q1 = p1 + · · · + pn , with J [Q0 ] [pi ] (i = 1, 2, . . . , n), φ0 is defined by φ0 (f ) = Q0 φ(f )Q0 , and φ1 is defined by φ1 (f ) =

n

f (xi )pi ,

∀f ∈ C(X),

i=1

where p1 , . . . , pn are mutually orthogonal projections and {x1 , x2 , . . . , xn } ⊂ X is an εdense subset of X. (Again by J [p] [q], we mean that p ⊕ p ⊕ · · · ⊕ p is (unitarily) J

equivalent to a subprojection of q.) Furthermore, we can choose any two of the projections Q0 , Q1 , Q2 to be trivial, if we wish. If φ(1) is trivial, then all of them can be chosen to be trivial projections. The following theorem is one of the main results of this article. It will be used in Section 3 and in [31]. Theorem 2.12. Let A = lim(An =

tn

algebra with the ideal n property and with very slow dimension growth condition. For any An , finite set F = ti=1 Fni ⊂ An , positive integer J and ε > 0, there exists m and there exist projections Q0 , Q1 , Q2 ∈ Am with Q0 + Q1 + Q2 = φn,m (1An ), a unital map ψ0 ∈ Map(An , Q0 Am Q0 )1 and two unital homomorphisms ψ1 ∈ Hom(An , Q1 Am Q1 )1 , ψ2 ∈ Hom(An , Q2 Am Q2 )1 , such that the following statements are true. −→

i=1 M[n,i] (C(Xn,i )), φn,m ) be an AH

(1) φn,m (f ) − (ψ0 (f ) ⊕ ψ1 (f ) ⊕ ψ2 (f )) < ε, for all f ∈ F . (2) The set (ψ0 ⊕ ψ1 )(F ) is weakly approximately constant to within ε. (3) The homomorphism ψ2 factors through C—a finite direct sum of matrix algebras over C[0, 1] or C as ξ

ξ

1 2 C −→ Q2 Am Q2 , ψ2 : An −→

where ξ1 and ξ2 are unital homomorphisms. i,j i,j j i,j i,j i,j j i,j (4) Let ψ0 : Ain −→ ψ0 (1Ain )Am ψ0 (1Ain ) and ψ1 : Ain −→ ψ1 (1Ain )Am ψ1 (1Ain ) be the corresponding partial maps of ψ0 and ψ1 . For each pair (i, j ), one of the following is true:

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G. Gong et al. / Journal of Functional Analysis 258 (2010) 2119–2143 i,j

i,j

(i) Both ψ0 and ψ1 are zero. i,j (ii) ψ1 is a homomorphism with finite dimensional image and for each nonzero projection e ∈ An (including any rank one projection) i,j i,j ψ1 (e) > J ψ0 (1Ain ) (∈ Am ). j

j

j

Furthermore, we can assume that Q0 , Q1 are trivial projections in Am . Proof. Fixed An , a finite set F =

tn

i i=1 Fn

⊂ An , and ε > 0. Let ε =

ε i max1itn {[n,i]} . Let F ⊂ the elements in Fni (⊂ M[n,i] (C(Xn,i ))). 1, 2, . . . , tn ) and dist(x, x ) < 2η, then

C(Xn,i ) be the finite set consisting of all the entries of Let η > 0 (η ε) be such that if x, x ∈ Xn,i (i = |f (x) − f (x )| < ε3 , for all f ∈ F i . Applying Lemma 2.8 to An and η, there exist an m1 > n, connected simplicial complex finite Zi1 , Zi2 , . . . , Zi• ⊂ Xn,i , i = 1, 2, . . . , tn and a homomorphism φ : i s M[n,i] (C(Zis )) −→ Am1 satisfying the following conditions: (1) φn,m1 = φ ◦ π , where π : An −→ i s M[n,i] (C(Zis )) is the restriction map, and j (2) φ satisfies the dichotomy condition (∗): for each Zis and Am1 , the partial map j η M[n,i] (C(Zis )) −→ Am1 either has the property sdp( 32 , δ) or is the zero map. Let φ (i,s),j : B = i s M[n,i] (C(Zis )). Let L > 0 be the number in Proposition 2.11 corresponding to ε , η and δ. Set F (i,s) = π(Fni ) ⊂ M[n,i] (C(Zis )) and F (i,s) = π(F i ) ⊂ C(Z i ), where π is restriction map either from M[n,i] (C(Xn,i )) to M[n,i] (C(Z i )) or from C(Xn,i ) to C(Z i ). Set Λ = maxi,s dim Zis , T = maxi [n, i]. Now applying Lemma 2.9, there is m > m1 , such that each partial map

j φ (i,s),j : M[n,i] C Zis −→ Am1 of φ := φm1 ,m ◦ φ : B −→ Am satisfies one of the following two conditions. (a) There is a unital homomorphism with finite dimensional image (i,s),j

ψ2

j : M[n,i] C Zis −→ φ (i,s),j (1)Am φ (i,s),j (1) (i,s),j

(f ) < for all f ∈ F (i,s) . such that φ (i,s),j (f ) − ψ2 (b) φ (i,s),j is 2JTL2 2L (Λ + dim Xm,j + 1)3 -large. (i,s),j

(i,s),j

(i,s),j

= 0 and ψ1 = 0, and let ψ2 be For each partial map φ (i,s),j satisfying (a), let ψ0 the homomorphism appeared in the condition (a). (i,s),j (i,s),j For each partial map φ (i,s),j satisfying condition (b), we will define ψ0 , ψ1 and (i,s),j (i,s),j (i,s),j ψ2 as below. Let P = φ (1) and p = ψ (e11 ), where e11 is the matrix unit corresponding to the upper left corner in M[n,i] (C(Zis )). Set (i,s),j

φ1

:= φ (i,s),j e

s 11 M[n,i] (C(Zi ))e11

j : C Zis −→ pAm p.

G. Gong et al. / Journal of Functional Analysis 258 (2010) 2119–2143 (i,s),j

(i,s),j

Then φ (i,s),j = φ1 ⊗ id[n,i] . Furthermore φ1 η , δ). Recall has property sdp( 32 F (i,s) = f |Zis : f ∈ F i ,

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is 2JTL2 2L (Λ + dim Xm,j + 1)3 -large and

F (i,s) = f |Zis : f ∈ F i ⊂ M[n,i] C Zis .

Note that Zis is a subspace of Xn,i with the induced metric. So for any function f ∈ F (i,s) , if dist(x, x ) 2η, then |f (x) − f (x )| < ε3 . Applying Proposition 2.11 (and note that L is chosen as in the proposition) to (i,s),j

φ1 (i,s),j

we obtain ψ0 (1) (2) (3) (4)

(i,s),j

, ψ1

(i,s),j

(i,s),j

and ψ2

(i,s),j

j : C Zis −→ pAm p, as in the proposition. That is, the following are true. (i,s),j

φ1 (f ) − (ψ0 ⊕ ψ1 ⊕ ψ2 )(f ) < ε , for all f ∈ F (i,s) . (i,s),j ψ2 factors through C[0, 1] or C. (i,s),j (i,s),j The set (ψ0 ⊕ ψ1 )(F (i,s) ) is weakly approximately constant to within ε . (i,s),j (i,s),j ψ1 is a homomorphism with finite dimensional image and if ψ0 = 0, then (i,s),j

(i,s),j (i,s),j (1) JTψ0 (1), ψ1 where 1 is the unit of C(Zis ). Let A be the set of all s such that each φ (i,s),j satisfies condition (a), and B be the complement of A. Hence for any s ∈ B, φ (i,s),j satisfies condition (b). Let i,j

ψ0 =

(i,s),j

ψ0 ⊗ id[n,i] ◦ π i,s , s∈B

i,j ψ1

(i,s),j

ψ1 = ⊗ id[n,i] ◦ π i,s , s∈B

i,j

ψ2

(i,s),j

(i,s),j i,s i,s ψ2 = ⊗ id[n,i] ◦ π ψ2 ◦π ⊕ , s∈B

s∈A

where π i,s : M[n,i] (C(Xn,i )) −→ M[n,i] (C(Zis )) is the restriction map. Then by 2.10, i,j i,j (ψ0 ⊕ ψ1 )(F ) is weakly approximately constant to within ε and i,j

ψn,m (f ) − ψ i,j ⊕ ψ i,j ⊕ ψ i,j (f ) < ε, 0 1 2 for all f ∈ F (i,s) . Let ψ0 , ψ1 and ψ2 be the maps with partial maps ψ0 , ψ1 and ψ2 . Evidently, all the requirements of the theorem are satisfied. 2 i,j

i,j

i,j

Corollary We use the notation For any An , any projection P = 2.13. iin Theorem i i 2.12. Ain , any finite set F = F ⊂ P An P i = P An P , any positive integer J , and Pi ∈ any number ε > 0, there are an Am , mutually orthogonal projections Q0 , Q1 , Q2 ∈ Am with

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Q0 + Q1 + Q2 = φn,m (1An ) a unital map ψ0 ∈ Map(An , Q0 Am Q0 )1 and two unital homomorphisms ψ1 ∈ Hom(An , Q1 Am Q1 )1 , ψ2 ∈ Hom(An , Q2 Am Q2 )1 , such that for each pair i,j i,j (i, j ), ψ0 (Pi ) and ψ0 (1Ain − Pi ) are mutually orthogonal projections and the approximate decomposition of φn,m := φn,m |P An P as direct sum of ψ0 := ψ0 |P An P , ψ1 := ψ1 |P An P and ψ2 := ψ2 |P An P , satisfies the following conditions: (1) (2) (3) (4)

(f ) − (ψ ⊕ ψ ⊕ ψ )(f ) < , for all f ∈ F . ψn,m 0 1 2 ψ2 factors through a finite direct sum of matrix algebras over C[0, 1] or C. i,j i,j i,j j If ψ0 = 0, then for any nonzero projection e ∈ P i Ain P i , [ψ1 (e)] > J [ψ0 (P )](∈ Am ). ψ0 is F − ε multiplicative.

Proof. The proof of the corollary without (4) is completely the same as the proof of Corollary 4.39 of [29]. (4) follows from 4.40 of [29] combined with (1) with a modified set F and positive number ε. 2 3. The main theorem The following proposition is what was proved in the proof of Theorem 2.6 of [43] (see p. 168 of [43]), though the statement is a little different from the theorem in [43]. Proposition 3.1. (Cf. [43].) Let A = lim(An , φn,m ) be an AH algebra with the ideal property −→ and with slow dimension growth (see [27]). There exist a sequence of positive integers l1 < l2 < l3 < · · · and a sequence of homomorphisms ψn,n+1 : Aln −→ Aln+1 such that each φln ,ln+1 is homotopic to ψn,n+1 and such that B = lim(Aln , ψn,m ) is of real rank zero. −→

Remark 3.2. In Proposition 3.1, if we further assume that K∗ (A) is torsion free, then K∗ (B) is torsion free. By the classification of the real rank zero AH algebras in [8], B is isomorphic to an inductive limit of direct sums of matrix algebras over C(S 1 ). That is, B is an AT algebra. In particular by 4.3 of [14], for any finite set F ⊂ Aln , there are an m, a circle algebra A0 —a finite direct sum of matrix algebras over C(S 1 ) and a homomorphism ξ : A0 −→ Alm such that ψn,m (F ) ⊂ε ξ(A0 ). n M[n,i] (C(Xn,i )), φn,m ) be an AH algebra with the ideal Lemma 3.3. Let A = lim(An = ti=1 −→ property, with slow dimension growth, and with torsion free K-group K∗ (A). Then for any projection R ∈ An , finite set F ⊂ RAn R and ε > 0, there exist an Am , a unital homomorphism ψ : RAn R −→ φn,m (R)Am φn,m (R) and a circle algebra A0 with a homomorphism ξ : A0 −→ φn,m (R)An φn,m (R) such that (1) φn,m |RAn R ∼h ψ , and (2) ψ(F ) ⊂ ξ(A0 ). Proof. Apply Remark 3.2 and Proposition 3.1 to the inductive limit lim(A l = φn,l (R)Al φn,l (R), φl,m |A l ) beginning with A n = RAn R.

2

−→

The following proposition was stated in Theorem 1.6.9 of [29] (see also Theorem 2.29 of [18]).


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Proposition 3.4. (Cf. [29,18].) Suppose that A = si=1 Mki (C(Xi )) and F ⊂ A is weakly approximately constant to within ε, where Xi are finite CW complexes. Suppose that C is a C ∗ -algebra and that the homomorphisms φ and ψ ∈ Hom(A, C) are homotopic to each other. There are a finite set G ⊂ C, a number δ > 0, and a positive integer L > 0 such that the following is true: If B is a unital C ∗ -algebra, p ∈ B is a projection, λ0 ∈ Map(C, pBp) is G–δ multiplicative, λ1 ∈ Hom(C, (1 − p)B(1 − p)) is a homomorphism with finite dimensional image satisfying that i )] L · [p], where ei is the matrix unit (of upper left for each i ∈ {1, 2, . . . , s}, [(λ1 ◦ φ)(e11 11 corner) of the i-th block, Mki (C(Xi )), of A, then there is a unitary u ∈ B such that (λ ◦ φ)(f ) − u(λ ◦ ψ)(f )u∗ < 8ε,

∀f ∈ F,

where λ ∈ Map(A1 , B) is defined by λ = λ0 ⊕ λ1 . Lemma 3.5. Let C be a finite direct sum of matrix algebras over C(S 1 ), and ξ : C −→ A be a homomorphism, and F ⊂ A be a finite set with F ⊂a ξ(C), where a is a positive number. For any ε > 0, there exist a finite set G ⊂ A and δ > 0 such that if φ : A −→ A is a G–δ multiplicative map, then there is a homomorphism ξ : C −→ A such that φ(F ) ⊂a+ε ξ (C). Proof. Since F ⊂a ξ(C), there is a finite set F1 ⊂ C such that F ⊂a ξ(F1 ). It is well known (see [14] or Lemma 1.6.1 of [29]) that there exist a finite set G1 ⊂ C and δ > 0 such that if a map τ : C −→ A is a G1 –δ multiplicative, then there is a homomorphism τ : C −→ A such that τ (f ) − τ (f ) < ε for all f ∈ F1 . Let G = ξ(G1 ). If φ : A −→ A is G–δ multiplicative, then φ ◦ ξ is G1 –δ multiplicative, then there is a homomorphism ξ : C −→ A such that ξ (f ) − φ ◦ ξ(f ) < ε for all f ∈ F1 . It follows that φ ◦ ξ(F1 ) ⊂ε ξ (F1 ). Combining with φ(F ) ⊂a (φ ◦ ξ )(F1 ), we obtain φ(F ) ⊂a+ε ξ (F1 )(⊂ ξ (C)). 2 The following theorem is the main result of this article. n M[n,i] (C(Xn,i )), φn,m ) be an AH algebra with the ideal Theorem 3.6. Let A = lim(An = ti=1 −→ property and with very slow dimension growth. Suppose that K∗ (A) is torsion free. Then A is an AT algebra—i.e. an inductive limit of finite direct sums of matrix algebras over C(S 1 ). Proof. By Theorem 4.3 of [14], it is sufficient to prove that for any finite set F ⊂ An and ε > 0, there is a subalgebra A0 ⊂ A which is a finite direct sum of matrix algebras over C(S 1 ) (or quotients of C(S 1 )) such that F ⊂ε A0 . Note that both C and C[0, 1] are quotients of C(S 1 ). We will construct A0 in a certain Am for a large integer m. That is, we will construct A0 ⊂ Am such that φn,m (F ) ⊂ε A0 . ε , there exist m1 > n, orthogonal projections Step 1. Applying Theorem 2.12 to F ⊂ An , 32 Q0 , Q1 , Q2 ∈ Am1 with Q0 , Q1 trivial and Q0 + Q1 + Q2 = φn,m1 (1An ), a unital completely positive map ψ0 ⊕ ψ1 : An −→ (Q0 ⊕ Q1 )Am1 (Q0 ⊕ Q1 ) and a unital homomorphism ψ2 : An −→ Q2 Am1 Q2 such that ε < 4ε , for all f ∈ F . (i) φn,m1 (f ) − (ψ0 ⊕ ψ1 )(f ) ⊕ ψ2 (f ) < 32 ε in (Q0 ⊕ Q1 )Am1 (Q0 ⊕ Q1 ). (ii) (ψ0 ⊕ ψ1 )(F ) is weakly approximately constant to within 32

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(iii) ψ2 factors through B—a finite direct sum of matrix algebras over C[0, 1] or C as ψ

2 Q2 Am1 Q2 . An −→ B −→

It follows that

φn,m1 (F ) ⊂ 4ε (ψ0 ⊕ ψ1 )(F ) ⊕ ψ2 (F ) ⊂ (ψ0 ⊕ ψ1 )(F ) ⊕ ψ2 (B) .

(∗)

Step 2. By Lemma 3.3 applied to Am1 , 8ε , R := Q0 + Q1 and F1 := (ψ0 ⊕ ψ1 )(F ) ⊂ RAm1 R, there exist an m2 > m1 , a C ∗ -algebra C—a finite direct sum of matrix algebras over C(S 1 ), a homomorphism β : RAm1 R −→ φm1 ,m2 (R)Am2 φm1 ,m2 (R) and a homomorphism ξ : C −→ φm1 ,m2 (R)Am2 φm1 ,m2 (R), such that (i) β(F1 ) = β((ψ0 ⊕ ψ1 )(F )) ⊂ 8 ξ(C). (ii) φm1 ,m2 |RAm1 R ∼h β. Write W := φm1 ,m2 (R) as W =

tm2

j =1 W

j

∈

j

Am2 .

ε Step 3. Considering 32 , F1 ⊂ RAm1 R, which is weakly approximately constant to within and the homotopic homomorphisms

β

ε 32 ,

and φm1 ,m2 |RAm1 R : RAm1 R −→ W Am2 W,

there exist a finite set G ⊂ W Am2 W , a number δ > 0, and a positive integer J > 0 as in Proposition 3.4. That is, if A is a unital C ∗ -algebra, p ∈ A is a projection, λ0 ∈ Map(W Am2 W, pAp) is a G–δ multiplicative map, and λ1 ∈ Hom(W Am2 W, (1 − p)A(1 − p)) is a homomorphism with finite dimensional image, satisfying the following condition: λ0 (W i ) are projections, and for any nonzero projection e ∈ W i Aim2 W i , [λ1 (e)] J [λ0 (W i )], then there is a unitary u ∈ A such that

(λ0 ⊕ λ1 ) ◦ φm ,m (f ) − u (λ0 ⊕ λ1 ) ◦ β (f ) u∗ < 8 ε = ε , 1 2 32 4 for all f ∈ F1 ⊂ RAm1 R. Applying Lemma 3.5 to ξ : C −→ φm1 ,m2 (R)Am2 φm1 ,m2 (R), β(F1 ) ⊂a ξ(C) with a = 8ε and 8ε (in place of ε), by making G larger and δ smaller (if necessary), we can further assume that there is a homomorphism ξ : C −→ A such that

(λ0 ⊕ λ1 ) β(F1 ) ⊂ 8ε + 8ε ξ (C). Step 4. By Corollary 2.13 applied to W Am2 W , min( 4ε , δ) and the finite set F2 := φm1 ,m2 (F1 ) ∪ G ⊂ W Am2 W, there exist an m > m2 , orthogonal projections P0 , P1 , P2 ∈ φm2 ,m (W )Am φm2 ,m (W ) with P0 + P1 + P2 = φm2 ,m (W ) and λ0 ∈ Map(W Am2 W, P0 Am P0 )1 , λ1 ∈ Hom(W Am2 W, P1 Am P1 ), λ2 ∈ Hom(W Am2 W, P2 Am P2 )1 such that


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(i) φm2 ,m (f ) − λ0 (f ) ⊕ λ1 (f ) ⊕ λ2 (f ) < 4 for all f ∈ F2 . (ii) λ2 factors through D—a finite direct sum of matrix algebras over C and C[0, 1] as λ

2 P2 Am P2 . W Am2 W −→ D −→

(iii) λ1 has finite dimensional image and for any nonzero projection e ∈ W i Aim2 W i , λ1 (e) J λ0 W i (∈ Am ). (iv) λ0 is G–δ multiplicative. As a consequence of Step 3, there is a unitary u ∈ (P0 ⊕ P1 )Am (P0 ⊕ P1 ) such that (v) (λ0 ⊕ λ1 )(φm1 ,m2 (f )) − u(λ0 ⊕ λ1 )(β(f ))u∗ < 4ε , for all f ∈ F1 = (ψ0 ⊕ ψ1 )(F ). Also there is a homomorphism ξ : C −→ (P0 ⊕ P1 )Am (P0 ⊕ P1 ) (as in Step 3) such that

(λ0 ⊕ λ1 ) β(F1 ) ⊂ 4 ξ (C).

(∗∗)

Step 5. From (ii) of Step 4, (λ2 ◦ φm1 ,m2 ◦ (ψ0 ⊕ ψ1 ))(F ) ⊂ λ 2 (D). Combining this with (i) of Step 4, we have

φm2 ,m ◦ φm1 ,m2 ◦ (ψ0 ⊕ ψ1 ) (F ) ⊂ 4 (λ0 ⊕ λ1 ) ◦ φm1 ,m2 ◦ (ψ0 ⊕ ψ1 ) (F ) ⊕ λ 2 (D). That is,

φm1 ,m (ψ0 ⊕ ψ1 )(F ) ⊂ 4 (λ0 ⊕ λ1 ) ◦ φm1 ,m2 ◦ (ψ0 ⊕ ψ1 ) (F ) ⊕ λ 2 (D).

(∗∗∗)

From (∗∗), we have ((λ0 ⊕ λ1 ) ◦ β ◦ (ψ0 ⊕ ψ1 ))(F ) ⊂ 4ε ξ (C). Combining with (v) of Step 4,

(λ0 ⊕ λ1 ) ◦ φm1 ,m2 ◦ (ψ0 ⊕ ψ1 ) (F ) ⊂ 4ε + 4ε Ad u ◦ ξ (C).

(∗∗∗∗)

Combining (∗∗∗) and (∗∗∗∗), we have

φm1 ,m ◦ (ψ0 ⊕ ψ1 ) (F ) ⊂ 4ε + 2ε Ad u ◦ ξ (C) ⊕ λ 2 (D).

(∗∗∗∗∗)

From (∗) in Step 1, we have φn,m (F ) ⊂ 4ε (φm1 ,m ◦ (ψ0 ⊕ ψ1 ))(F ) ⊕ (φm1 ,m ◦ ψ2 )(B). Combining this with (∗∗∗∗∗), we have

φn,m (F ) ⊂ 3ε + ε Ad u ◦ ξ (C) ⊕ λ 2 (D) ⊕ φm1 ,m ◦ ψ2 (B). 4

4

Let A0 = (Ad u ◦ ξ )(C) ⊕ λ 2 (D) ⊕ (φm1 ,m ◦ ψ2 )(B) ⊂ Am . Since B, C, D are finite direct sums of matrix algebras over C(S 1 ), C[0, 1] and C and Ad u ◦ ξ , λ 2 and φm1 ,m ◦ ψ2 are homomorphisms, we know that A0 is a finite direct sum of matrix algebras over C(S 1 ) (or quotients of C(S 1 )). By Theorem 4.3 of [14], the limit algebra A is an inductive limit of finite direct sums of matrix algebras over C(S 1 ). 2

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The following theorem follows from 1.7 and the above main theorem. n Pn,i M[n,i] (C(Xn,i ))Pn,i , φn,m ) be an AH algebra with Theorem 3.7. Let A = lim(An = ti=1 −→ the ideal property and with very slow dimension growth. Suppose that K∗ (A) is torsion free. Then A is an AT algebra. Remark 3.8. If in addition to the condition that K∗ (A) is torsion free, we also assume that K1 (A) = 0, then A is an AI-algebra—an inductive limit of finite direct sums of matrix algebras over C[0, 1]. With two extra restrictions: (1) A is unital and (2) A is approximately divisible, this class of C ∗ -algebras A (with the ideal property) has been classified in Steven’s thesis [49] using an invariant consisting of K-theory and tracial state data. Recently, Kui Ji and Chunlan Jiang (the second author) have generalized this classification theorem to include all the AI-algebras with the ideal property—that is, both above restrictions (of being unital and approximately divisible) have been removed (see [32]). References [1] B. Blackadar, Matricial and ultra-matricial topology, in: R.H. Herman, B. Tanbay (Eds.), Operator Algebras, Mathematical Physics, and Low Dimensional Topology, A K Peters, Massachusetts, 1993, pp. 11–38. [2] B. Blackadar, M. Dadarlat, M. Rørdam, The real rank of inductive limit C ∗ -algebras, Math. Scand. 69 (1991) 211–216. [3] O. Bratteli, Inductive limits of finite dimensional C ∗ -algebras, Trans. Amer. Math. Soc. 171 (1972) 195–234. [4] A. Ciuperca, G.A. Elliott, A remark on invariants for C ∗ -algebras of stable rank one, Int. Math. Res. Not. IMRN 5 (2008), Art. ID rnm 158, 33 pp. [5] J. Cuntz, K-theory for certain C ∗ -algebras, Ann. of Math. 113 (1981) 181–197. [6] M. Dadarlat, Approximately unitarily equivalent morphisms and inductive limit C ∗ -algebras, K-Theory 9 (1995) 117–137. [7] M. Dadarlat, Reduction to dimension three of local spectra of real rank zero C ∗ -algebras, J. Reine Angew. Math. 460 (1995) 189–212. [8] M. Dadarlat, G. Gong, A classification result for approximately homogeneous C ∗ -algebras of real rank zero, Geom. Funct. Anal. 7 (1997) 646–711. [9] M. Dadarlat, T. Loring, Classifying C ∗ -algebras via ordered, mod-p K-theory, Math. Ann. 305 (4) (1996) 601–616. [10] M. Dadarlat, A. Nemethi, Shape theory and (connective) K-theory, J. Operator Theory 23 (2) (1990) 207–291. [11] M. Dadarlat, G. Nagy, A. Nemethi, C. Pasnicu, Reduction of topological stable rank in inductive limits of C ∗ algebras, Pacific J. Math. 153 (2) (1992) 267–276. [12] S. Eilers, A complete invariant for AD algebras with bounded torsion in K1 , J. Funct. Anal. 139 (1996) 325–348. [13] G.A. Elliott, On the classification of inductive limits of sequences of semisimple finite dimensional algebras, J. Algebra 38 (1976) 29–44. [14] G.A. Elliott, On the classification of C ∗ -algebras of real rank zero, J. Reine Angew. Math. 443 (1993) 179–219. [15] G.A. Elliott, A classification of certain simple C ∗ -algebras, in: H. Araki, et al. (Eds.), Quantum and NonCommutative Analysis, Kluwer, Dordrecht, 1993, pp. 373–385. [16] G.A. Elliott, A classification of certain simple C ∗ -algebras, II, J. Ramanujan Math. Soc. 12 (1997) 97–134. [17] G.A. Elliott, G. Gong, On inductive limits of matrix algebras over two-tori, Amer. J. Math. 118 (1996) 263–290. [18] G.A. Elliott, G. Gong, On the classification of C ∗ -algebras of real rank zero, II, Ann. of Math. 144 (1996) 497–610. [19] G.A. Elliott, G. Gong, L. Li, Injectivity of the connecting maps in AH inductive systems, Canad. Math. Bull. 26 (2004) 4–10. [20] G.A. Elliott, G. Gong, L. Li, On the classification of simple inductive limit C ∗ -algebras, II: The isomorphism theorem, Invent. Math. 168 (2) (2007) 249–320. [21] G.A. Elliott, G. Gong, H. Lin, C. Pasnicu, Homomorphisms, homotopies, and approximations by circle algebras, C. R. Math. Rep. Acad. Sci. Can. 16 (1994) 45–50. [22] G.A. Elliott, G. Gong, H. Lin, C. Pasnicu, Abelian C ∗ -subalgebras of C ∗ -algebras of real rank zero and inductive limit C ∗ -algebras, Duke Math. J. 83 (1996) 511–554.


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Solvability of parabolic equations in divergence form with partially BMO coefficients Hongjie Dong 1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA Received 16 October 2008; accepted 5 January 2010 Available online 15 January 2010 Communicated by J. Bourgain

Abstract 1 solvability of second order parabolic equations in divergence form with leading coeffiWe prove the Hp ij cients a measurable in (t, x 1 ) and having small BMO (bounded mean oscillation) semi-norms in the other variables. Additionally we assume a 11 is measurable in x 1 and has small BMO semi-norms in the other variables. The corresponding results for the Cauchy problem are also established. Parabolic equations in 1 with mixed norms are also considered under the same conditions of the coefficients. Sobolev spaces Hq,p © 2010 Elsevier Inc. All rights reserved.

Keywords: Second order equations; Vanishing mean oscillation; Partially BMO coefficients; Sobolev spaces; Mixed norms

1. Introduction Many authors have studied the Lp theory of second order parabolic and elliptic equations under various regularity assumptions on the coefficients. It is of particular interest not only because of its important applications in nonlinear equations, but also due to its subtle links with the theory of stochastic processes. With continuous leading coefficients, the Lp theory for both divergence and non-divergence form equations has been known for a long time; see, for example, [1] and [24]. In [7], Chiarenza, Frasca and Longo initiated the study the Wp2 -estimates for elliptic equations with VMO leadE-mail address: [email protected]. 1 The author was partially supported by a start-up funding from the Division of Applied Mathematics of Brown

University, an NSF grant number DMS-0635607 from IAS, and an NSF grant number DMS-0800129. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.003

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H. Dong / Journal of Functional Analysis 258 (2010) 2145–2172

ing coefficients. Their proof is based on the estimates of Calderón–Zygmund type and related commutators with BMO functions. This approach was later developed into a solvability theory of elliptic and parabolic equations in non-divergence form in Chiarenza, Frasca and Longo [8] and Bramanti and Cerutti [6] (see also [12], and references therein). For elliptic and parabolic equations in divergence form with small BMO coefficients, a different approach was given in Byun [3,4]. The main tools in [3,4] are the weak compactness, the Hardy–Littlewood maximal function, and a covering lemma originally due to Safonov. In [21], Krylov gave a unified approach of studying the Lp solvability of both divergence and non-divergence form parabolic equations with leading coefficients measurable in the time variable and VMO in spatial variables. This class is denoted as VMOx , which is wider than those considered in [6,4,12]. Unlike the arguments in [7,8,6,12], which are based on certain estimates of Calderón–Zygmund theorem and the Coifman–Rochberg–Weiss commutator theorem, the proofs in [21] rely mainly on pointwise estimates of sharp functions of spatial derivatives of solutions. It is worth noting that although the results in [21] are claimed for equations with VMO coefficients, the proofs there only require a ij to have small BMO semi-norms in small balls (or cylinders). We also remark that for divergence form equations a similar result was also obtained in Byun [5] by adapting the approach in [3,4]. The method in [21] was later improved and generalized in [15–19,22,23,9,10]. With the leading coefficients in the same class, Krylov [22] established the solvability of both divergence and non-divergence form parabolic equations in mixed norm Sobolev spaces. As pointed out in [22], the interest in results concerning equations in spaces with mixed Sobolev norms arises, for example, when one wants to get better regularity of traces of solutions for each time slide (see, for instance, [20,26,28,29]). For other results about the solvability of parabolic and elliptic equations in Sobolev spaces with discontinuous coefficients, we refer the reader to [2,11,14,25,27], and references therein. The theory of elliptic and parabolic equations with partially VMO coefficients was originated in Kim and Krylov [18]. In [18], the authors proved the Wp2 solvability of elliptic equations in non-divergence form with leading coefficients measurable in a fixed direction and VMO in the others. Very recently, their result was generalized by Krylov [23], where the leading coefficients are assumed to be measurable in one direction and VMO in the orthogonal directions in each small ball with the direction depending on the ball. For non-divergence form parabolic equations, 1,2 solvability for q p > 2 was established in Kim [15–17], in which most the Wp1,2 and Wq,p leading coefficients are measurable in the time variable and one spatial variable, and VMO in the other variables. We remark that the above mentioned results concerning the Lp solvability of elliptic and parabolic equations with partially VMO coefficients are all for non-divergence form. In this paper, we consider the Hp1 solvability of parabolic equations in divergence form: Pu − λu = div g + f,

(1.1)

where λ 0 is a constant and Pu = −ut + a ij ux i x j + bi u x i + bˆ i ux i + cu,

g = g1, g2, . . . , gd .

We assume all the coefficients are bounded and measurable, and a ij are uniformly elliptic. The objective of the paper is to extend the results in [21,22,5] to equations with partially BMO leading coefficients with small BMO semi-norms. More precisely, we assume the coefficients a ij are measurable in (t, x 1 ) and BMO in the other variables with small BMO semi-norms in small


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cylinders, except a 11 which is measurable in x 1 and BMO in the remaining variables with small semi-norms in small cylinders (see Assumptions 2.2 and 2.3 for a more rigorous definition). This is the same class of coefficients considered in [15] and [16], in which non-divergence form parabolic equations are studied. 1 solvability of divergence form Under these assumptions, we establish the Hp1 and Hq,p parabolic equations (cf. Theorems 2.5 and 2.7) and the solvability for the corresponding Cauchy problems (see Theorems 2.6 and 2.8). It is worth noting that, as in [21] and [22], one feature of these results is that the matrix {a ij } is not assumed to be symmetric. Our approach is based on the aforementioned method from [21] and [22]. Since a ij are merely measurable in x 1 , we are only able to estimate the sharp function of ux , not the full gradient ux as in [21] and [22]. This is the main difficulty of the problem. To overcome this difficulty, roughly speaking, we need to bound ux 1 by ux . One idea in the paper is to break the ‘symmetry’ of the coordinates by a scaling so that t and x 1 are distinguished from x . Another idea is to estimate the sharp function of a 11 ux 1 instead of ux 1 . This estimate together with a generalized Stein– Fefferman theorem proved in [23] enables us to bound ux 1 . An interesting corollary of Theorem 2.7 is the Wp1 solvability of divergence form elliptic equations with leading coefficients measurable in one direction and BMO in the others with small BMO semi-norms (Theorem 2.11). This further enables us to solve elliptic equations in a half space with partially small semi-norm BMO leading coefficients by using the method of even/odd extensions. Furthermore, via a partition of unity technique we can treat elliptic (or parabolic) equations with small semi-norm BMO coefficients in a Lipschitz domain (or cylindrical domain) with a small Lipschitz constant. In fact, we allow leading coefficients to be partially BMO with small semi-norms in the interior of the domain. We give two concrete examples as applications at the end of the next section. A similar result for equations with small BMO coefficients and without lower order terms was recently obtained in [3]. The main difference of the approaches is that here we can obtain the boundary estimate immediately from the estimate in the whole space by taking the advantage that the leading coefficients are allowed to be measurable in one direction. Note that for the Poisson equation in arbitrary Lipschitz domains but with a restricted range of p, the solvability result was established by Jerison and Kenig [13] (see also Shen [27]). In connection to [23], another interesting problem is the solvability of divergence form parabolic or elliptic equations with variably partially VMO/BMO coefficients. We will study these problems elsewhere. The paper is organized as follows. In the next section, we introduce the notation and state the main results: Theorems 2.5–2.8. Section 3 is devoted to several auxiliary results which will be used later. In Section 4, we estimate the Lp norm ux 1 by the Lp norm of ux (Theorem 4.1). Then in Section 5, we give an estimate of the sharp function of ux . By combining this with Theorem 4.1, we are able to prove Theorem 2.5. In Section 6, we generalize the results in Section 4 and estimate the Lq,p norm ux 1 by the Lq,p norm of ux (Theorem 6.5). Finally, in Section 7, we give an improved estimate of the sharp function of ux , and complete the proof of Theorem 2.7. 2. Notation and main results We begin the section by introducing some notation. Let d 1 be an integer. A typical point in Rd is denoted by x = (x 1 , . . . , x d ) = (x 1 , x ). We set ux j = Dj u,

ux j x k = Dj k u,

ut = Dt u.

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By Du and D 2 u we mean the gradient and the Hessian matrix of u. Throughout the paper, we always assume that 1 < p, q < ∞ unless explicitly specified otherwise. By N (d, p, . . .) we mean that N is a constant depending only on the prescribed quantities d, p, . . . . For a function f (t, x) in Rd+1 , we set 1 f (t, x) dx dt = − f (t, x) dx dt, (f )D = |D| D

D

where D is an open subset in Rd+1 and |D| is the (d + 1)-dimensional Lebesgue measure of D. For −∞ S < T ∞, we set Lq,p (S, T ) × Rd = Lq (S, T ), Lp Rd , i.e., f (t, x) ∈ Lq,p ((S, T ) × Rd ) if T f Lq,p ((S,T )×Rd ) = S

f (t, x)p dx

1/q

q/p dt

< ∞.

Rd

Denote Lp (S, T ) × Rd = Lp,p (S, T ) × Rd ,

1,2 (S, T ) × Rd = u: u, ut , Du, D 2 u ∈ Lq,p (S, T ) × Rd , Wq,p 1,2 Wp1,2 (S, T ) × Rd = Wp,p (S, T ) × Rd , 1 1,2 Hq,p (S, T ) × Rd = (1 − )1/2 Wq,p (S, T ) × Rd , 1 Hp1 (S, T ) × Rd = Hp,p (S, T ) × Rd , d 1/2 H−1 Lq,p (S, T ) × Rd , q,p (S, T ) × R = (1 − ) d −1 d H−1 p (S, T ) × R = Hp,p (S, T ) × R . We also use the abbreviations Lp = Lp (Rd+1 ), Hp1 = Hp1 (Rd+1 ), etc. For any T ∈ (−∞, ∞], we denote RT = (−∞, T ),

Rd+1 = RT × Rd . T

Let

Br x = y ∈ Rd−1 : x − y < r , Qr (t, x) = t − r 2 , t × Br (x),

Br (x) = y ∈ Rd : |x − y| < r , Qr (t, x) = t − r 2 , t × Br (x).

Set Br = Br (0),

Br = Br (0),

Qr = Qr (0, 0),

Qr = Qr (0, 0),


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and |Br |, |Br |, |Qr |, |Qr | to be the volume of Br , Br , Qr , Qr , respectively. Let Q = {Qr (t, x): (t, x) ∈ Rd+1 , r ∈ (0, ∞)}. For a function g defined on Rd+1 , we denote its (parabolic) maximal and sharp function, respectively, by − g(s, y) dy ds, Mg(t, x) = sup Q∈Q: (t,x)∈Q

g (t, x) = #

sup

Q∈Q: (t,x)∈Q

Q

− g(s, y) − (g)Q dy ds.

Q

For a function g defined in a set D ⊂ Rn+1 , we denote [g]C α,α/2 (D) :=

sup

(t,x)=(s,y) (t,x),(s,y)∈D

|g(t, x) − g(s, y)| , |x − y|α + |t − s|α/2

where α ∈ (0, 1].

Next we state our assumptions on the coefficients precisely. Assumption 2.1. There exist δ ∈ (0, 1] and K 1 such that for any unit vector ξ ∈ Rd and any (t, x) ∈ Rd+1 we have δ a ij (t, x)ξ i ξ j δ −1 , i i b (t, x) K, bˆ (t, x) K,

c(t, x) K,

(2.1) i = 1, 2, . . . , d.

For R > 0, we denote aR11,# =

sup

−

sup

11 a (t, x) − A11 x 1 dx dt,

(t0 ,x0 )∈Rd+1 rR Qr (t0 ,x0 )

where for each Qr (t0 , x0 ), A11 x 1 =

−

a 11 s, x 1 , y dy ds.

Qr (t0 ,x0 )

Assumption 2.2 (γ ). There exists a positive constant R0 such that aR11,# γ. 0 For R > 0 we denote aR#

=

sup

sup

sup

(t0 ,x0 )∈Rd+1 rR (i,j )=(1,1)

−

ij a (t, x) − Aij t, x 1 dx dt,

Qr (t0 ,x0 )

where for each Qr (t0 , x0 ) and (i, j ) = (1, 1), Aij t, x 1 = − a ij t, x 1 , y dy . Br (x0 )

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Assumption 2.3 (γ ). There exists a positive constant R0 such that aR# 0 γ . Remark 2.4. Clearly, {Aij } satisfies the ellipticity condition (2.1) and A11 takes value on [δ, δ −1 ]. Now we state the first two main results of the article. Theorem 2.5. Let p ∈ (1, ∞), T ∈ (−∞, ∞] and u ∈ Hp1 (Rd+1 T ). Then there exist constants γ0 = γ0 (d, p, δ) > 0, and λ0 0 and N > 0 depending only on d, p, R0 , δ and K, such that under Assumptions 2.1, 2.2 (γ0 ) and 2.3 (γ0 ) we have √ λuLp (Rd+1 ) + λux Lp (Rd+1 ) + ut H−1 (Rd+1 ) p T T T √ N ( λ + 1)Pu − λuH−1 (Rd+1 ) p

(2.2)

T

for all λ λ0 . Moreover, for any λ > λ0 and f, g ∈ Lp (Rd+1 T ), there exists a unique u ∈ d+1 1 Hp (RT ) solving Pu − λu = f + div g in Rd+1 T . And u satisfies the estimate λuLp (Rd+1 ) + T

√

√ λux Lp (Rd+1 ) + ut H−1 (Rd+1 ) N λgLp (Rd+1 ) + N f Lp (Rd+1 ) . T

p

T

T

T

Our next result is regarding the solvability of the Cauchy problem. For T > 0, de1 ((0, T ) × Rd )) to be the subspace of H1 ((0, T ) × Rd ) (or note H˚ p1 ((0, T ) × Rd ) (or H˚ q,p p 1 d Hq,p ((0, T ) × R ), respectively) consisting of functions satisfying u(0, ·) = 0. Theorem 2.6. Let p ∈ (1, ∞), T ∈ (0, ∞). Then there exists a constant γ0 > 0 depending only on d, p and δ, such that under Assumptions 2.1, 2.2 (γ0 ) and 2.3 (γ0 ), for any f, g ∈ Lp ((0, T ) × Rd ), there exists a unique u ∈ H˚ p1 ((0, T ) × Rd ) satisfying Pu = f + div g in (0, T ) × Rd . Moreover, there is a constant N depending only on d, p, T , R0 , δ and K such that uHp1 ((0,T )×Rd ) N f Lp ((0,T )×Rd ) + gLp ((0,T )×Rd ) . 1 solvability. Under the same assumptions, we also have the Hq,p 1 (Rd+1 ). Then there exist constants Theorem 2.7. Let p, q ∈ (1, ∞), T ∈ (−∞, ∞] and u ∈ Hq,p T γ1 = γ1 (d, p, q, δ) > 0, λ0 0 and N > 0, depending only on d, p, q, R0 , δ and K, such that under Assumptions 2.1, 2.2 (γ1 ) and 2.3 (γ1 ) we have


√ λuLq,p (Rd+1 ) + λux Lq,p (Rd+1 ) + ut H−1 (Rd+1 ) q,p T T T √ N ( λ + 1)Pu − λuH−1 (Rd+1 ) q,p

T

2151

(2.3)

for all λ λ0 . Moreover, for any λ > λ0 and f, g ∈ Lq,p (Rd+1 T ), there exists a unique u ∈ d+1 1 Hq,p (RT ) solving Pu − λu = f + div g in Rd+1 T . And u satisfies the estimate √ λuLq,p (Rd+1 ) + λux Lq,p (Rd+1 ) + ut H−1 (Rd+1 ) q,p T T T √ N λ + gLq,p (Rd+1 ) + N f Lq,p (Rd+1 ) . T

T

Theorem 2.8. Let p, q ∈ (1, ∞), T ∈ (0, ∞). Then there exists a constant γ0 > 0 depending only on d, p and δ, such that under Assumptions 2.1, 2.2 (γ1 ) and 2.3 (γ1 ), there exists a unique 1 ((0, T ) × Rd ) satisfying u ∈ H˚ q,p Pu = f + div g in (0, T ) × Rd . Moreover, there is a constant N depending only on d, p, q, R0 , T , δ and K such that uHq,p d + gL d . 1 ((0,T )×Rd ) N f L q,p ((0,T )×R ) q,p ((0,T )×R ) Remark 2.9. 1. By the method of continuity and a density argument, to prove Theorems 2.5 and 2.7 we only need to establish the a priori estimates (2.2) and (2.3) for u ∈ C0∞ . 2. Note that the case p = 2 is well known even without any regularity assumption on the coefficients; see, for instance, [24]. Moreover, if we can prove Theorem 2.5 for p ∈ (2, ∞), the case p ∈ (1, 2) follows from the standard duality argument. Due to the same reason, for Theorem 2.7 it suffices to consider the case q > p. 3. It suffices to prove (2.2) and (2.3) for T = ∞. This is because for general T we have u = w 1 ) solves in Rd+1 for t < T where w ∈ Hp1 (or Hq,p Pw − λw = χt 0 now we denote aR# = sup sup sup − a ij (x) − Aij x 1 dx, x0 ∈Rd rR i,j

Br (x0 )

where for each Br (x0 ) and (i, j ), 1 A x = − a ij x 1 , y dy . ij

Br (x0 )

Assumption 2.10 (γ ). There exists a positive constant R0 such that aR# 0 γ . Theorem 2.11. Let p ∈ (1, ∞) and u ∈ Wp1 (Rd ). Then there exist constants γ0 = γ0 (d, p, δ) > 0, and λ0 0, N > 0 depending only on d, p, K, δ and R0 such that under Assumptions 2.1 and 2.10 (γ0 ) we have λuLp (Rd ) +

√ √ λux Lp (Rd ) N ( λ + 1)Lu − λuH−1 (Rd ) p

for all λ λ0 . Moreover, for any λ > λ0 and f, g ∈ Lp (Rd ), there exists a unique u ∈ Wp1 (Rd ) solving (2.4) in Rd . And u satisfies the estimate λuLp (Rd ) +

√

√ λux Lp (Rd ) N λgLp (Rd ) + N f Lp (Rd ) .

Theorem 2.11 follows from Theorem 2.5 using the idea that solutions to elliptic equations can be viewed as steady state solutions to parabolic equations. We omit the details and refer the reader to the proof of Theorem 2.6 [21]. An application of Theorem 2.11 is the Wp1 solvability of elliptic equations with piecewise uniformly continuous leading coefficients, which have jump discontinuity at parallel hyperplanes in Rd . For non-divergence equations, similar results were obtained in [25] and [15]. The same theorem also implies the Wp1 solvability of the equation (a ij ux i )x j = div g in B1 with zero Dirichlet boundary condition if a ij are measurable in |x|, VMO with respect to the angular coordinates, and continuous near the origin. In both examples, the coefficients are neither in VMO nor in BMO with small semi-norms. So the results in [21] and [3] are not applicable in these two cases. 3. Preliminaries In this section we give a few auxiliary results which will be used later. The H21 solvability of (1.1) is well known. In particular, we recall the following lemma.


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Lemma 3.1. i) Let T ∈ (−∞, ∞], λ 0, b = bˆ = 0 and c = 0. Assume u ∈ H21 (Rd+1 T ) and Pu − λu = d+1 div g + f , where f, g ∈ L2 (RT ). Then just under the uniform ellipticity condition (with no regularity assumption on a ij ), there exists a constant N = N (d, δ) such that √ √ λux L2 (Rd+1 ) + λuL2 (Rd+1 ) N λgL2 (Rd+1 ) + N f L2 (Rd+1 ) . T

T

T

T

If λ = 0 and f = 0, we have ux L2 (Rd+1 ) N gL2 (Rd+1 ) . T

T

(3.1)

d+1 1 ii) For λ > 0 and any f, g ∈ L2 (Rd+1 T ), there exists a unique u ∈ H2 (RT ) solving Pu − λu = d+1 div g + f in RT .

In the sequel, by weak solutions we mean H21 solutions. We recall the following Caccioppoli-type inequality for parabolic equations in divergence form. 1 Lemma 3.2. Let r > 0, ν > 1, b = bˆ = 0 and c = 0. Assume u ∈ H2,loc and Pu = div g + f in Qνr , where f, g ∈ L2 (Qνr ). Then there exists a constant N = N (d, δ, ν) such that

ux L2 (Qr ) N gL2 (Qνr) + rf L2 (Qνr) + r −1 uL2 (Qνr ) . The next lemma is the local boundedness estimate and the Moser–Nash estimate for parabolic equations in divergence form with measurable coefficients. ∞ satisfies u − (a ij u ) = 0 in Q . Then for some α ∈ (0, 1) Lemma 3.3. Assume u ∈ Cloc t 2 xi xj depending on d and δ,

[u]C α/2,α (Q1 ) N uL1 (Q2 ) ,

uL∞ (Q1 ) N uL1 (Q2 ) .

A scaling argument yields the following result. ∞ satisfies u − (a ij u ) = 0 Corollary 3.4. Let ν ∈ [2, ∞) and r ∈ (0, ∞). Assume u ∈ Cloc t xi xj in Qνr . Then for some α ∈ (0, 1) depending on d and δ,

[u]C α/2,α (Qr ) [u]C α/2,α (Qνr/2 ) N (νr)−α |u| Q , νr uL∞ (Qr ) uL∞ (Qνr/2 ) N |u| Q . νr

∞ satisfies u − Lemma 3.5. Let a ij = a ij (x), i, j = 1, 2, . . . , d and a ij = a j i . Assume u ∈ Cloc t ij (a ux i )x j = 0 in Q2 . Then we have

ut L2 (Q1 ) N ux L2 (Q2 ) .

(3.2)

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Proof. Choose a nonnegative cutoff function η ∈ C0 satisfying η ≡ 1 in Q1 and vanishing outside the closure of Q2 ∩ (−Q2 ). After multiplying both sides of the equation by ut η2 and integrating on Q2 , we get

a ij ux i utx j η2 + 2ut ηηx j dx dt = 0.

u2t η2 dx dt + Q2

Q2

Integrating by parts and using Young’s inequality, we get

u2t η2 dx dt

Q2

Q2

a ij ux i (ux j ηηt − 2ut ηηx j ) dx dt |ux |2 dx dt +

N Q2

which yields (3.2).

1 2

u2t η2 dx dt, Q2

2

Combining Corollary 3.4 and Lemma 3.5, we have Corollary 3.6. Let ν ∈ [4, ∞), r ∈ (0, ∞), a ij = a ij (x), i, j = 1, 2, . . . , d and a ij = a j i . Assume ∞ satisfies u − (a ij u ) = 0 in Q . Then we have u ∈ Cloc t νr xi xj 1/2 ut L∞ (Qr ) N (νr)−1 |ux |2 Q .

(3.3)

νr

Proof. We only need to notice that ut satisfies the same equation in Qνr .

2

Lemma 3.7. Let ν ∈ [8, ∞), r ∈ (0, ∞), a ij = a ij (x), i, j = 1, 2, . . . , d and a ij = a j i . Assume ∞ satisfies u − (a ij u ) = 0 in Q . Then we have u ∈ Cloc t νr xi xj 1/2 utt L∞ (Qr ) N (νr)−3 |ux |2 Q . νr

(3.4)

If in addition a ij = a ij (x 1 ), i, j = 1, 2, . . . , d, then 1/2 ux t L∞ (Qr ) N (νr)−2 |ux |2 Q ,

(3.5)

1/2 ux x L∞ (Qr ) N (νr)−1 |ux |2 Q .

(3.6)

νr

νr

Proof. Since ut satisfies the same equation in Qνr , (3.4) follows from Corollary 3.6, Lemmas 3.2 and 3.5. If in addition a ij = a ij (x 1 ), then ux and ux x also satisfy the same equation in Qνr . In this case (3.5) follows from Corollary 3.6 and Lemma 3.2, while (3.6) follows from Corollary 3.4 and Lemma 3.2. The lemma is proved. 2


2155

We will also make use of a generalization of the Fefferman–Stein theorem proved recently in [23]. Let Cn = {Cn (i0 , i1 , . . . , id ), i0 , . . . , id ∈ Z}, n ∈ Z, be the filtration of partitions given by parabolic dyadic cubes, where Cn (i0 , i1 , . . . , id ) is −2n i0 2 , (i0 + 1)2−2n × i1 2−n , (i1 + 1)2−n × · · · × id 2−n , (id + 1)2−n . Theorem 3.8. Let p ∈ (0, 1), U, V , H ∈ L1 . Assume V |U |, H 0 and for any n ∈ Z and C ∈ Cn there exists a measurable function U C given on C such that |U | U C V on C and

U − (U )C dx dt, U C − U C dx dt H dx dt. min C C

C

C

Then we have p

p−1

U Lp N H Lp V Lp , provided that H, V ∈ Lp . 4. An estimate of ux 1 This section is devoted to proving the following key estimate. Theorem 4.1. Let p ∈ (2, ∞), b = bˆ = 0 and c = 0. Assume u ∈ C0∞ and Pu = div g, where g ∈ Lp . Then we can find μ ∈ (1, ∞), γ0 ∈ (0, ∞) depending on d, δ and p such that under Assumptions 2.1 and 2.3 (γ0 ) there exists a constant N = N (d, p, δ, μ) such that ux 1 Lp N ux Lp + gLp , provided that u vanishes outside Qμ−1 R for some R R0 . To prove this theorem we will make a few preparations. ∞ satisfies u − Lemma 4.2. Let ν ∈ [8, ∞), r ∈ (0, ∞) and a 11 = a 11 (x 1 ). Assume u ∈ Cloc t 11 (a ux 1 )x 1 − d−1 u = 0 in Qνr . Then we have

11 a u 1 1 x

x

L∞ (Qr )

1/2 N (νr)−1 |ux |2 Q . νr

(4.1)

Proof. From the equation we have 11 a ux 1 x 1 = ut − d−1 u. Now (4.1) follows immediately from (3.3) and (3.6).

2

Let the assumptions of Theorem 4.1 be satisfied. We make a change of variables to ‘break’ the symmetry of the coordinates. Denote u¯ t, x 1 , x = u μ−2 t, μ−1 x 1 , x

2156


with a sufficiently large constant μ to be chosen later. Clearly, u¯ satisfies −μ2 u¯ t + μ2 a¯ 11 u¯ x 1 x 1 + μa¯ 1j u¯ x 1 x j + μa¯ j 1 u¯ x j x 1 + a¯ ij u¯ x i x j = div(g), ¯ j >1

j >1

i>1, j >1

−1 x 1 , x ). where a¯ = a(μ−2 t, μ−1 x 1 , x ) and g¯ = (μg 1 , g 2 , . . . , g d )(μ−2 t, μ 11 We denote Pu = −ut + (a¯ ux 1 )x 1 + d−1 u, where d−1 u = dj =2 ux j x j . Then we have

Pu¯ = div g,

(4.2)

where g1 = μ−2 g¯ 1 − μ−1

a¯ j 1 u¯ x j ,

j >1

gk = μ−2 g¯ k − μ−1 a¯ 1k u¯ x 1 − μ−2

a¯ j k u¯ x j + u¯ x k ,

k 2.

j >1

Lemma 4.3. Let r ∈ (0, ∞), ν ∈ [32, ∞), b = bˆ = 0, c = 0 and a 11 = a 11 (x 1 ). Assume u ∈ 1 and Pu = div g, where g ∈ L2,loc . Then under Assumption 2.1 there exists a constant H2,loc N = N(d, δ) such that 11 a¯ u¯

x1

2 1/2 1/2 − a¯ 11 u¯ x 1 Q Q N ν −1 |u¯ x 1 |2 Q r

δ −2 νr

r

+ Nν

d+2 2

1/2 |u¯ x |2 Q

δ −2 νr

+ Nν

d+2 2

2 1/2 |g| Q

δ −2 νr

.

(4.3)

.

(4.4)

In particular, 11 a¯ u¯

x1

2 1/2 1/2 − a¯ 11 u¯ x 1 Q Q N ν −1 |u¯ x 1 |2 Q r

δ −2 νr

r

+ Nν

d+2 2

+ Nν

−1 1/2 μ |u¯ x 1 |2 Q

d+2 2

δ −2 νr

1/2 |u¯ x |2 Q

δ −2 νr

2 1/2 ¯ Q + μ−2 |g|

δ −2 νr

Proof. Without loose of generality, we may assume that a 11 and g are infinitely differentiable. Indeed, if not, we take the standard mollifications and prove the estimate for the mollifications. Then we take the limit because the concerned constants are independent of the smoothness of the functions involved. Choose η ∈ C0∞ such that η ≡ 1 in Qδ −2 νr/2 and η ≡ 0 outside the closure of Qδ −2 νr ∪ (−Qδ −2 νr ). Let w be a weak solution of Pw = div(ηg), and v = u¯ − w so that v is a weak solution of Pv = div (1 − η)g . Note that Pv = 0 in Qδ −2 νr/2 . By the classical theory of parabolic equations, both v and w are smooth.


2157

Estimates of v: Due to Lemma 4.2, we have √ 2 1/2 r 2 −|x |2 1/2 11 − a¯ 11 vx 1 dx 1 N ν −1 |vx |2 Q . a¯ vx 1 − δ −2 νr/2 √ Qr −

(4.5)

r 2 −|x |2

To estimate the oscillation of a¯ 11 vx 1 in t and x , we make another change of variables: y 1 = φ x 1 :=

x 1 0

1 a¯ 11 (s)

ds,

yj = xj ,

j 2.

It is easy to see that φ is a bi-Lipschitz function and δ y 1 /x 1 δ −1 ,

Dy 1 = a¯ 11 x 1 Dx 1 .

Denote v(t, y 1 , y ) = v(t, φ −1 (y 1 ), y ). We have √ 2 1/2 r 2−|x |2 − a¯ 11 vx 1 dx 1 − a¯ 11 vx 1 Q r √ Qr −

r 2 −|x |2

1/2 1/2 N r |vx 1 x |2 Q + N r 2 |vx 1 t |2 Q r r 2 1/2 2 2 1/2 N r |vy 1 y | Q + N r |vy 1 t | Q δ −1 r

δ −1 r

(4.6)

.

It is clear that in Qδ −1 νr/2 v satisfies −vt + vy 1 y 1 /a¯ 11 φ −1 y 1 + d−1 v = 0.

(4.7)

−vt + v = a¯ 11 φ −1 y 1 − 1 (vt − d−1 v).

(4.8)

Let us rewrite (4.7) as

Since vy satisfies the same Eq. (4.8), by using Lemma 3.2 with P = −Dt + , f = a¯ 11 φ −1 y 1 − 1 vy t , g 1 = 0, g j = − a¯ 11 φ −1 y 1 − 1 vy j y , j = 2, . . . , d, we get |vy y 1 |2 Q

δ −1 r

N |vy y |2 Q + N r 2 |vy t |2 Q + N r −2 |vy |2 Q −1 −1 2δ r 2δ r 2δ −1 r N |vx x |2 Q + N r 2 |vx t |2 Q + N r −2 |vx |2 Q . 2δ −2 r

2δ −2 r

2δ −2 r

2158


Thanks to Lemma 3.7, we obtain |vy y 1 |2 Q

δ −1 r

N ν −2 r −2 |vx |2 Q

δ −2 νr/2

+ N ν d+2 r −2 |vx |2 Q

δ −2 νr/2

(4.9)

.

Similarly, since vt also satisfies (4.8), by using Lemma 3.2 with f = a¯ 11 φ −1 y 1 − 1 vtt , P = −Dt + , g 1 = 0, g j = − a¯ 11 φ −1 y 1 − 1 vy j t , j = 2, . . . , d, we get |vty 1 |2 Q

δ −1 r

N |vty |2 Q + N r 2 |vtt |2 Q + N r −2 |vt |2 Q 2δ −1 r 2δ −1 r 2δ −1 r 2 2 2 2 −2 |vt | Q N |vtx | Q + N r |vtt | Q + Nr . 2δ −2 r

2δ −2 r

2δ −2 r

Thanks to Lemma 3.7 and Corollary 3.6, we obtain |vty 1 |2 Q

δ −1 r

N ν −2 r −4 |vx |2 Q

δ −2 νr/2

(4.10)

.

Combining (4.5), (4.6), (4.9) and (4.10) together yields 11 a¯ v

x1

2 1/2 1/2 − a¯ 11 vx 1 Q Q N ν −1 |vx |2 Q r

δ −2 νr/2

r

+ Nν

d+2 2

1/2 |vx |2 Q

δ −2 νr/2

.

(4.11)

Estimates of w: By the energy inequality (3.1), we have wx L2 (Rd+1 ) N ηgL2 (Rd+1 ) = N gL2 (Qδ−2 νr ) . 0

0

Therefore, 1/2 1/2 d+2 |wx |2 Q N ν 2 |g|2 Q r

1/2 |wx |2 Q

δ −2 νr/2

1/2 N |g|2 Q

δ −2 νr

δ −2 νr

(4.12)

,

(4.13)

.

Putting (4.11)–(4.13) together, we get 2 1/2 − a¯ 11 u¯ x 1 Q Q r r 1/2 11 11 2 1/2 N a¯ vx 1 − a¯ vx 1 Q Q + N |wx |2 Q r r r 1/2 d+2 d+2 1/2 1/2 Nν −1 |vx |2 Q + N ν 2 |vx |2 Q + N ν 2 |g|2 Q

11 a¯ u¯

x1

δ −2 νr/2

1/2 Nν −1 |u¯ x 1 |2 Q

δ −2 νr/2

δ −2 νr/2

+ Nν

d+2 2

1/2 |u¯ x |2 Q

δ −2 νr/2

δ −2 νr

+ Nν

d+2 2

2 1/2 |g| Q

δ −2 νr

,

which is (4.3). Finally, by the definition of g we conclude (4.4). The lemma is proved.

2


2159

Next we consider more general a 11 . Lemma 4.4. Let ν ∈ [32, ∞), β1 , β2 ∈ (1, ∞) such that 1/β1 + 1/β2 = 1, b = bˆ = 0 and c = 0. 1 vanishes outside Qμ−1 R for some R R0 and Pu = div g, where g ∈ L2,loc . Assume u ∈ H2,loc Then under Assumptions 2.1 and 2.2 (γ0 ) for any parabolic cube C ∈ Cn there exists a measurable function A¯ 11 (x 1 ) taking values on [δ, δ −1 ] such that for any (t, x) ∈ C, (4.14) − A¯ 11 u¯ x 1 − A¯ 11 u¯ x 1 C dx dt H (t, x), C

where 1/2 1/2 d+2 H (t, x) = N ν −1 M|u¯ x 1 |2 + N ν 2 M|u¯ x |2 1/(2β1 ) 1/(2β2 ) 1/2 d+2 M|u¯ x 1 |2β2 ¯2 , + N ν 2 μ−1 + μ3 γ0 + μ−2 M|g| and N is a positive constant depending only on d, β1 and δ. Proof. Let Q1 = Qr (t0 , x0 ) be the smallest parabolic cylinder containing C and Q2 = Qδ −2 νr (t0 , x0 ). By the triangle inequality and Hölder’s inequality, we have 11 11 ¯ ¯ − A u¯ x 1 − A u¯ x 1 C dx dt N − A¯ 11 u¯ x 1 − A¯ 11 u¯ x 1 Q1 dx dt C

Q1

2 1/2 N A¯ 11 u¯ x 1 − A¯ 11 u¯ x 1 Q Q . 1

1

(4.15)

We consider two cases separately. i) Case δ −2 νr < R. Let −

A11 x 1 =

a 11 s, x 1 , y dy ds,

Q −2 (μ−2 t0 ,x0 ) δ

νr

and let A¯ 11 (t, x 1 , x ) = A11 (μ−1 x 1 ). By Assumption 2.2 (γ0 ), it is easy to check that − a¯ 11 − A¯ 11 dx dt μ3 γ0 . Q2

Now we proceed as in Lemma 4.3 with a¯ 11 replaced by A¯ 11 and a¯ j 1 u¯ x j , g1 = A¯ 11 − a¯ 11 u¯ x 1 + μ−2 g¯ 1 − μ−1 gk = μ−2 g¯ k − μ−1 a¯ 1k u¯ x 1 − μ−2

j >1

to get (after a shift of the origin)

j >1

a¯ j k u¯ x j + u¯ x k ,

k 2,

(4.16)

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2 1/2 − A¯ 11 u¯ x 1 Q Q 1 1 1/2 1/2 d+2 d+2 −1 2 1/2 N ν |u¯ x 1 | Q2 + N ν 2 |u¯ x |2 Q2 + N ν 2 |g|2 Q2 .

11 A¯ u¯

x1

(4.17)

By the definition of g and Hölder’s inequality, we obtain 11 A¯ u¯ 1 − A¯ 11 u¯ 1 x x

Q1

2 1/2 1/2 1 N ν −1 |u¯ 1 |2 1/21 + N ν d+2 2 |u¯ x |2 Q2 x Q Q 1/2 d+2 + N ν 2 μ−1 |u¯ x 1 |2 Q2 1/(2β ) 1/(2β ) 2 1/2 ¯ Q2 . + A¯ 11 − a¯ 11 Q2 1 |u¯ x 1 |2β2 Q2 2 + μ−2 |g|

This together with (4.15) and (4.16) proves (4.14). ii) Case δ −2 νr R. Define 11 1 A x = − a 11 s, x 1 , y dy ds, QR

and let A¯ 11 (t, x) = A11 (μ−1 x 1 ). As in the first case, we have (4.17) and − a¯ 11 − A¯ 11 dx dt μ3 γ0 .

(4.18)

QR

Since u¯ vanishes outside QR , it holds that 11 a¯ − A¯ 11 u¯ 1 2 1/22 = χQ a¯ 11 − A¯ 11 u¯ 1 2 1/22 x x R Q Q 1/2β 11 1/(2β ) N a¯ − A¯ 11 Q 1 |u¯ x 1 |2β2 Q2 2 . R

(4.19)

By the definition of g and Hölder’s inequality, we combine (4.15), (4.17)–(4.19) to get (4.14). Bearing in mind Remark 2.4, the lemma is proved. 2 Now we are ready to prove Theorem 4.1. Fix a β2 in Lemma 4.4 such that p > 2β2 > 2. We set U = δ u¯ x 1 ,

V = δ −1 |u¯ x 1 |,

and for each C ∈ Cn set U C := |A¯ 11 u¯ x 1 | and H (t, x) as in Lemma 4.4. Since A¯ 11 takes value on [δ, δ −1 ], we have |U | U C V on C, and by Lemma 4.4 and the triangle’s inequality C C U − U dx dt 2 A¯ 11 u¯ 1 − A¯ 11 u¯ 1 dx dt 2 H (t, x) dx dt. x x C C C

C

C

Thanks to Theorem 3.8, we obtain p

p−1

δ p u¯ x 1 Lp N H Lp δ 1−p u¯ x 1 Lp .

(4.20)


2161

Since p > 2β2 , we can use the Hardy–Littlewood maximal function theorem to get from (4.20) that d+2

u¯ x 1 Lp NH Lp N ν −1 u¯ x 1 Lp + N ν 2 u¯ x Lp 1/(2β1 ) d+2 u¯ x 1 Lp + μ−2 g + N ν 2 μ−1 + μ3 γ0 ¯ Lp .

(4.21)

Now we first choose ν sufficiently large, then μ sufficiently large and finally γ0 sufficiently small in (4.21) such that N ν −1 + N ν

d+2 2

−1 3 1/(2β1 ) μ + μ γ0 1/2.

This together with (4.21) yields u¯ x 1 Lp N u¯ x Lp + g ¯ Lp . To finish the proof of Theorem 4.1, it suffices to go back to u and g by scaling. 5. Proof of Theorem 2.5 We shall prove Theorem 2.5 in this section. As is explained in Remark 2.9, to prove the theorem we only need to establish (2.2) for u ∈ C0∞ . We set P¯ u = −ut + a ij ux i x j where a 11 = a 11 (x 1 ), and a ij = a ij (t, x 1 ) when (i, j ) = (1, 1). First we need a Hölder estimate of ux which enable us to obtain a sharp function estimate of ux . ∞ and P¯ u − λu = 0 Lemma 5.1. Let ν ∈ [2, ∞), λ ∈ [0, ∞) and r ∈ (0, ∞). Assume u ∈ Cloc in Qνr . Then for some α ∈ (0, 1) and N depending only on d and δ, [ux ]C α/2,α (Qr ) N (νr)−α |ux | + λ|u| Q . νr

Proof. Because ux satisfies the same equation as u, the case λ = 0 thus follows immediately √ from Corollary 3.4. For λ > 0, we use an idea of S. Agmon by considering u(t, x) cos( λy) instead of u; cf. Lemma 5.9 of [22]. The lemma is proved. 2 ∞ and P¯ u − λu = 0 Corollary 5.2. Let ν ∈ [2, ∞), λ ∈ [0, ∞) and r ∈ (0, ∞). Assume u ∈ Cloc in Qνr . Then for the same α ∈ (0, 1) as in Lemma 5.1 we have

ux − (ux )Q 2 r

Qr

2 N ν −2α |ux | + λ|u| Q , νr

where N depends only on d and δ. Proof. It follows immediately from Lemma 5.1 since in Qr ux − (ux )Q r α [ux ] α/2,α C (Qr ) . r

2

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1 Theorem 5.3. Let ν ∈ [4, ∞), r ∈ (0, ∞), α be the constant taken from Lemma 5.1, u ∈ H2,loc ¯ and g ∈ L2,loc . Assume that P u = div g in Qνr . Then there exists a constant N depending only on d, δ such that,

ux − (ux )Q 2 r

Qr

N ν −2α |ux |2 Q + N ν d+2 |g|2 Q . νr

νr

Proof. The proof is similar to that of Lemma 4.3 by using Corollary 5.2 and Lemma 3.1; see also Theorem 7.1 [22]. 2 Lemma 5.4. Let b = bˆ = 0, c = 0, β1 , β2 ∈ (1, ∞), 1/β1 + 1/β2 = 1 and R ∈ (0, R0 ]. Assume 1 u ∈ H2,loc vanishing outside QR and Pu = div g, where g ∈ L2,loc . Then under Assumptions 2.1, 2.2 (γ0 ) and 2.3 (γ0 ) there exists a positive constant N depending only on d, β1 and δ such that ux − (ux )Q

r (t0 ,x0 )

2

Qr (t0 ,x0 )

N ν −2α |ux |2 Q (t ,x ) + N ν d+2 |g|2 Q (t ,x ) νr 0 0 νr 0 0 1/β1 2β2 1/β2 |ux | , (5.1) + γ0 Q (t ,x ) νr 0

0

for any r ∈ (0, ∞), ν 4 and (t0 , x0 ) ∈ Rd+1 . Proof. The estimate is derived from Theorem 5.3 by using the technique of freezing the coefficients; cf. the proof of Lemma 4.4, or Theorem 5.3 [21]. 2 Proof of Theorem 2.5. Now we are in the position to prove Theorem 2.5 bearing in mind that we may assume p > 2 and that we only have to show (2.2) for u ∈ C0∞ and T = ∞. Fix a β2 such that p > 2β2 > 0. Let μ and γ0 be the constants in Theorem 4.1. Step 1. First we suppose u ∈ C0∞ vanishes outside Qμ−1 R0 , b = bˆ = 0,

λ = c = 0,

(5.2)

and Pu = div g,

g ∈ Lp .

Then by using (5.1), the Fefferman–Stein theorem on sharp functions, and the Hardy–Littlewood maximal function theorem, we get 1/(2β1 ) ux Lp N ν −α + ν (d+2)/2 γ0 ux Lp + N ν (d+2)/2 gLp .

(5.3)

Due to Theorem 4.1 and (5.3), we get 1/(2β1 ) ux Lp N ν −α + ν (d+2)/2 γ0 ux Lp + N ν (d+2)/2 gLp , where N is independent of γ0 and ν. By taking ν sufficiently large, and γ0 even smaller such that 1/(2β1 ) N ν −α + ν (d+2)/2 γ0 1/2,


2163

we get ux Lp N gLp . Step 2. We now remove the restriction (5.2). We assume Pu − λu = div g + f,

g, f ∈ Lp ,

(5.4)

and still u vanishing outside Qμ−1 R0 . This case follows from the previous one by again using Agmon’s idea; see, for example, Lemma 5.5 of [21]. In particular, we get ut H−1 + p

√

√ λux Lp + λuLp N λgLp + λf Lp .

(5.5)

Step 3. For general u ∈ C0∞ satisfying (5.4), we can use a partition of unity to obtain (5.5). To get (2.2), it suffices to define f = (1 − )−1 (Pu − λu),

g i = −(1 − )−1 (Pu − λu)x i ,

and use boundedness of the operators (1 − )−1/2 and (1 − )−1/2 Dx i in Lp . The theorem is proved. 2 A simple scaling argument gives the following corollary. d+1 Corollary 5.5. Let p ∈ (1, ∞), T ∈ (−∞, ∞], u ∈ Hp1 (Rd+1 T ) and f, g ∈ Lp (RT ). Assume b = bˆ = 0, c = 0, a 11 = a 11 (x 1 ) and a ij = a ij (t, x 1 ) for ij > 1. Then there exists a constant N > 0, depending only on d, p and δ, such that under Assumption 2.1 we have

λuLp (Rd+1 ) + T

√ √ λux Lp (Rd+1 ) N λgLp (Rd+1 ) + N f Lp (Rd+1 ) , T

T

T

provided that λ 0 and Lu − λu = f + div g. In particular, when λ = 0 and f = 0, we have ux Lp (Rd+1 ) N gLp (Rd+1 ) . T

T

6. A mixed norm estimate of ux 1 1 solvability of (1.1), we need a mixed norm estimate of u in terms In order to prove the Hq,p x1 of ux , which is the objective of this section. Our estimate relies on the following lemma.

Lemma 6.1. Let p, q ∈ (1, ∞), b = bˆ = 0 and c = 0. Then there exists a constant γ0 = γ0 (d, p, q, δ) > 0 such that under Assumptions 2.1, 2.2 (γ0 ) and 2.3 (γ0 ), for any r ∈ (0, R0 ] 1 and u ∈ Hq,loc satisfying Pu = 0 in Q2r we have ux ∈ Lp (Qr ) and 1/p 1/q |ux |p Q N |ux |q Q , r

where N depends only on d, p, q, δ.

2r

2164


Proof. First we assume R0 = 1. In this case, the lemma is proved in Corollary 8.4 of [22] with the only difference that the coefficients a ij are assumed to be in VMOx in that paper. The proof of Corollary 8.4 [22] uses the Lp solvability of equations with VMOx coefficients. Since the solvability is already established with coefficients satisfying Assumptions 2.1, 2.2 (γ0 ) and 2.3 (γ0 ) with a γ0 depending on d, p, q and δ, we can just reproduce the proof with almost no change. For general R0 ∈ (0, 1], we use a scaling (t, x) → (R02 t, R0 x) and notice that the new coefficients satisfy Assumptions 2.2 (γ0 ) and 2.3 (γ0 ) with R0 replaced by 1. The lemma is proved. 2 The next lemma is an analogue of Lemma 4.3. Lemma 6.2. Let p ∈ (1, ∞), r ∈ (0, ∞), ν ∈ [64, ∞) and a¯ 11 = a¯ 11 (x 1 ). Assume u¯ ∈ C0∞ and Pu¯ = div g, where g ∈ Lp,loc and ¯ Pu¯ = −u¯ t + a¯ 11 u¯ x 1 x 1 + d−1 u. Then under Assumption 2.1 there exists a constant N = N (d, p, δ) such that 11 a¯ u¯

x1

2 1/2 1/p − a¯ 11 u¯ x 1 Q Q N ν −1 |u¯ x 1 |p Q r

δ −2 νr

r

+ Nν

d+2 p

1/p |u¯ x |p Q

δ −2 νr

+ Nν

d+2 p

p 1/p |g| Q

δ −2 νr

.

(6.1)

Proof. We proceed as in the proof of Lemma 4.3. Without loose of generality, we may assume that a 11 and g are infinitely differentiable. Choose η ∈ C0∞ such that η ≡ 1 in Qδ −2 νr/2 and η ≡ 0 outside the closure of Qδ −2 νr ∪ (−Qδ −2 νr ). Let w be a weak solution of Pw = div(ηg), and v = u¯ − w so that v is a weak solution of Pv = div((1 − η)g). By the classical theory of parabolic equations, both v and w are smooth. Define v(t, y 1 , y ) = v(t, φ −1 (y 1 ), y ). First we estimate v. Since v satisfies the homogeneous equation in Qδ −2 νr/2 , from (4.5) and Lemma 6.1 we get √ 2 1/2 r 2 −|x |2 11 11 1 − a¯ vx 1 dx a¯ vx 1 − √ Qr r 2 −|x |2

−

1/2 N ν −1 |vx |2 Q

δ −2 νr/4

1/p N ν −1 |vx |p Q

δ −2 νr/2

(6.2)

.

Observe that, by local estimates and Lemma 6.1, similar to (4.9) and (4.10) we have 1/2 |vy y 1 |2 Q

1/p N ν −1 r −1 |vx |p Q

+ Nν

1/2 |vty 1 |2 Q

1/p N ν −1 r −2 |vx |p Q

.

δ −1 r

δ −1 r

δ −2 νr/2

δ −2 νr/2

d+2 p

1/p r −1 |vx |p Q

δ −2 νr/2

(6.3)

,

(6.4)

Combining (6.2), (4.6), (6.3) and (6.4) together yields 11 a¯ v

x1

2 1/2 1/p − a¯ 11 vx 1 Q Q N ν −1 |vx |p Q r

r

δ −2 νr/2

+ Nν

d+2 p

1/p |vx |p Q

δ −2 νr/2

.

(6.5)


2165

To estimate w, we use Corollary 5.5 and get wx Lp (Rd+1 ) N ηgLp (Rd+1 ) = N gLp (Qδ−2 νr ) . 0

0

Therefore, d+2 1/p 1/p |wx |p Q N ν p |g|p Q r

1/p |wx |p Q

δ −2 νr/2

1/p N |g|p Q

δ −2 νr

δ −2 νr

,

.

(6.6) (6.7)

Putting (6.5)–(6.7) together, we conclude (6.1) in exactly the same way as in the proof of Lemma 4.3. 2 Now we prove a lemma which improves Lemma 4.4. 1 Lemma 6.3. Let p ∈ (1, ∞), ν ∈ [256, ∞), b = bˆ = 0 and c = 0. Assume u ∈ H2,loc and Pu = div g, where g ∈ Lp,loc . Let γ1 ∈ (0, γ0 ] where γ0 = γ0 (d, p, 2p, δ) is taken from Lemma 6.1. Then under Assumptions 2.1 and 2.2 (γ1 ) for any parabolic cylinder Qr (t0 , x0 ) with r ∈ (0, R0 δ 2 ν −1 ) there exists a measurable function A¯ 11 (x 1 ) taking values on [δ, δ −1 ] such that, d+2 11 1/p 1 |A¯ u¯ x 1 − A¯ 11 u¯ x 1 Q1 Q1 N ν −1 + ν p μ−1 + μ3 γ 2p |u¯ x 1 |p Q4

+ Nν

d+2 p

d+2 1/p p 1/p |u¯ x |p Q4 + N ν p μ−2 |g| ¯ Q4 ,

(6.8)

where Q1 = Qr (t0 , x0 ),

Q4 = Qδ −2 νr (t0 , x0 ),

and N is a positive constant depending only on d, p and δ. Proof. As in the proof of Lemma 6.1, by a scaling we may assume R0 = 1. Recall (4.2), where a¯ ij = a ij (μ−2 t, μ−1 x 1 , x ) and g are defined before Lemma 4.3. Define v, w as in the proof of Lemma 4.3. Choose − a 11 s, x 1 , y dy ds, A11 x 1 = Q −2 (μ−2 t0 ,x0 ) δ

νr

and let A¯ 11 = A¯ 11 (x 1 ) = A11 (μ−1 x 1 ). We denote Q2 = Qδ −2 νr/4 (t0 , x0 ),

Q3 = Qδ −2 νr/2 (t0 , x0 ).

Then in Q3 we have ˆ x1 , −vt + A¯ 11 vx 1 x 1 + d−1 v = (g) where gˆ = (A¯ 11 − a¯ 11 )vx 1 .

2166


By Lemma 6.2 with v, A¯ 11 and gˆ in place of u, ¯ a¯ 11 and g, respectively, we get 11 A¯ v

x1

2 1/2 1/p − A¯ 11 vx 1 Q1 Q1 N ν −1 |vx 1 |p Q2 + Nν

d+2 p

1/p p 1/p |vx |p Q2 + |g| ˆ Q2 .

(6.9)

We apply Hölder’s inequality and Lemma 6.1 to bound the last term in (6.9) by 1/(2p) 1/(2p) 1/(p) 11 A¯ − a¯ 11 2p 1/(2p) |vx 1 |2p 2 N μ3 γ1 |vx 1 |p 3 . 2 Q

Q

Q

(6.10)

Here we also used an inequality like (4.16), and N only depends on d, p and δ. To estimate w, we use Theorem 2.6 and get wx Lp ((t0 −1,t0 )×Rd ) N ηgLp ((t0 −1,t0 )×Rd ) = N gLp (Q4 ) . Therefore, d+2 1/p 1/p |wx |p Q1 N ν p |g|p Q4 , 1/p 1/p 1/p |wx |p Q2 N |wx |p Q3 N |g|p Q4 .

As before, (6.9)–(6.12) lead to (6.8).

(6.11) (6.12)

2

Corollary 6.4. Under the same assumptions of Lemma 6.3, for any r ∈ (0, R0 δ 2 ν −1 ), we have −

p − u¯ x 1 (t, ·)Lp (Rd ) − u¯ x 1 (s, ·)Lp (Rd ) dt ds

(−r 2 ,0) (−r 2 ,0)

1/2 N ν −p + ν d+2 μ−p + μ3 γ1

+ N ν d+2

−

u¯ 1 (t, ·)p p d dt x L (R )

(−δ −2 νr 2 ,0)

−

u¯ x (t, ·)p p

L (Rd )

p ¯ ·)Lp (Rd ) dt. + μ−2p g(t,

(6.13)

(−δ −2 νr 2 ,0)

Proof. By the triangle inequality, the left-hand side of (6.13) is less than I :=

−

−

(−r 2 ,0) (−r 2 ,0) Rd

=

−

−

u¯ 1 (t, x) − u¯ 1 (s, x)p dx dt ds x x p − u¯ x 1 (t, y) − u¯ x 1 (s, y) dy dx dt ds.

(−r 2 ,0) (−r 2 ,0) Rd Br (x)

For each parabolic cylinder Qr (0, x) we find A¯ 11,(x) as in Lemma 6.3, which takes values on [δ, δ −1 ]. Therefore, I is less than


δ

−p

−

2167

p − − A¯ 11,(x) y 1 u¯ x 1 (t, y) − A¯ 11,(x) y 1 u¯ x 1 (s, y) dy dx dt ds

(−r 2 ,0) (−r 2 ,0) Rd Br (x)

11,(x) A¯ u¯ x 1 − A¯ 11,(x) u¯ x 1

N

Qr (0,x)

p

Qr (0,x)

dx.

Rd

Due to Lemma 6.3, we get I N

−p 1/2 ν + ν d+2 μ−p + μ3 γ1 |u¯ x 1 |p Q

Rd

+ ν d+2 |u¯ x |p Q

δ −2 νr (0,x)

δ −2 νr (0,x)

p ¯ Q + N ν d+2 μ−2p |g|

δ −2 νr (0,x)

dx.

(6.14)

Note that the right-hand side of (6.14) is exactly the right-hand side of (6.13). The corollary is proved. 2 Now we state and prove the main result of this section. Theorem 6.5. Let 1 < p < q < ∞, b = bˆ = 0 and c = 0. Assume u ∈ C0∞ and Pu = div g, where g ∈ Lq,p . Let γ0 = γ0 (d, p, 2p, δ) be the constant from Lemma 6.1. Then we can find γ1 ∈ (0, γ0 ], μ ∈ (1, ∞) and R1 ∈ (0, R0 ] depending on d, δ, p and q such that under Assumptions 2.1 and 2.2 (γ1 ) there exists a constant N = N (d, p, q, δ, μ) such that ux 1 Lq,p N ux Lq,p + gLq,p ,

(6.15)

provided that u vanishes outside (−μ−2 R14 , 0) × Rd . Proof. Denote U (t) = u¯ x 1 (t, ·)Lp (Rd ) ,

V (t) = u¯ x (t, ·)Lp (Rd ) ,

¯ ·)Lp (Rd ) . G(t) = g(t,

Let γ1 ∈ (0, γ0 ], μ ∈ (1, ∞), R1 ∈ (0, R0 ] and ν ∈ [256, ∞) be numbers to be chosen later. Assume u vanishes outside (−μ−2 R14 , 0) × Rd so that u¯ vanishes outside (−R14 , 0) × Rd . We claim d+2 1/(2p) p 1/p 2(1−1/p) M U U # N R1 δ −2 ν + ν −1 + ν p μ−1 + μ3 γ1 d+2 d+2 1/p 1/p + Nν p M V p + N ν p μ−2 M Gp .

(6.16)

Indeed, fix a point t0 ∈ R and consider an interval (S, T ) ⊂ R containing t0 . When T − S R12 δ 4 ν −2 , by a shift of the origin we get from Corollary 6.4 that (U − (U )(S,T ) )(S,T ) is less than the right-hand side of (6.16) at t0 . One the other hand, when T − S > R12 δ 4 ν −2 we then have U − (U )(S,T ) (S,T ) 2 − χ(−R 4 ,0) U (t) dt 1

(S,T )

2168


1−1/p 1/p p − U (t) dt 2 − χ(−R 4 ,0) dt 1

(S,T )

(S,T )

2(1−1/p) p 1/p M U (t0 ) N R1 δ −2 ν . By taking supremum over all intervals (S, T ) containing t0 , we obtain (6.16) at point t0 . Since t0 ∈ R is arbitrary, the claim is proved. Now we use the Fefferman–Stein theorem and the Hardy–Littlewood theorem (recall that q > p) to get d+2 1 2(1− 1 ) p + ν −1 + ν p μ−1 + μ3 γ1 2p U Lq (R) U Lq (R) N R1 δ −2 ν

+ Nν

d+2 p

V Lq (R) + N μ−2 ν

d+2 p

GLq (R) .

(6.17)

To finish the proof of the theorem, we choose a large ν, then a large μ and small γ1 and R1 in (6.17) such that d+2 1/(2p) 2(1−1/p) 1/2. + ν −1 + ν p μ−1 + μ3 γ1 N R1 δ −2 ν

2

7. Proof of Theorem 2.7 We recall that, by the duality argument and a density argument, to prove Theorem 2.7 it suffices to prove (2.3) for any u ∈ C0∞ and q > p. As in Section 5, we define P¯ u = −ut + a ij ux i x j , where the coefficients a 11 = a 11 (t), and a ij = a ij (t, x 1 ) when (i, j ) = (1, 1). By using Corollary 5.2 and the Wp1 solvability of L¯ proved in Section 5, we get an analogue of Theorem 5.3. Theorem 7.1. Let α be the constant in Lemma 3.3, p ∈ (1, ∞), ν ∈ [4, ∞), r ∈ (0, ∞), u ∈ 1 Hp,loc and g ∈ Lp,loc . Assume that P¯ u = div g in Qνr . Then there exists a constant N depending only on d, δ, p such that, ux − (ux )Q r

Qr

d+2 p1 1 p p p |g| N ν −α |ux |p Q + N ν Qνr . νr

The following lemma is an improvement of Lemma 5.4. Lemma 7.2. Let α be the constant in Lemma 3.3, p ∈ (1, ∞), b = bˆ = 0 and c = 0. Let γ1 ∈ 1 and Pu = div g, (0, γ0 ], where γ0 = γ0 (d, p, 2p, δ) is taken from Lemma 6.1. Assume u ∈ Hp,loc where g ∈ Lp,loc . Then under Assumptions 2.1, 2.2 (γ1 ) and 2.3 (γ1 ) there exists a positive constant N depending only on d, δ and p such that ux − (ux )Q r

Qr

1 d+2 d+2 p1 1 p p p |g| N ν −α + ν p γ12p |ux |p Q + N ν Qνr , νr

for any ν ∈ [16, ∞) and r ∈ (0, R0 ν −1 ).

(7.1)


2169

Proof. As in the proof of Lemma 6.1, it suffices to consider the case R0 = 1. By Theorem 2.6 there exists a unique solution w ∈ H˚ p1 ((−1, 0) × Rd ) of Pw = div(χQνr g). Let v = u − w so that Pv = div (1 − χQνr )g . Clearly, Pv = 0 in Qνr , and therefore −vt + Aij vx i x j = div(g), ˆ where A11 x 1 = − a 11 s, x 1 , y dy ds, Qνr

1 A t, x = − a ij t, x 1 , y dy , ij

(i, j ) = (1, 1),

Bνr

and gˆ j = Aij − a ij vx i ,

j = 1, 2, . . . , d.

Bearing in mind Remark 2.4, {Aij } satisfies (2.1). By Theorem 7.1, we have d+2 p1 p1 −α vx − (vx )Q |vx |p Q ˆp Q + N ν p |g| r Q Nν νr /4 νr/4 r

1 d+2 p1 1 2p 2p Aij − a ij 2p 2p p |v N ν −α |vx |p Q + N ν | x Qνr /4 Qνr /4 . νr/4

(7.2)

Due to Lemma 6.1, we get from (7.2) vx − (vx )Q r

Qr

1 d+2 p1 N ν −α + ν p γ12p |vx |p Q . νr /2

(7.3)

To estimate w, we use Corollary 5.5 and get wx Lp (Rd+1 ) N χQνr gLp (Rd+1 ) = N gLp (Qνr ) . 0

0

Therefore, d+2 1/p 1/p |wx |p Q N ν p |g|p Q , r νr 1/p 1/p 1/p |wx |p Q |wx |p Q N |g|p Q . νr/4

Combining (7.3)–(7.5) together gives (7.1).

νr/2

2

νr

(7.4) (7.5)

2170


The following corollary can be derived from Lemma 7.2 as Corollary 6.4 is derived from Lemma 6.3. Corollary 7.3. Under the assumptions of Lemma 7.2, for any ν ∈ [16, ∞) and r ∈ (0, R0 ν −1 ) we have p − ux (t, ·)Lp (Rd ) − ux (s, ·)Lp (Rd ) dt ds − (−r 2 ,0) (−r 2 ,0) 1 N ν −αp + ν d+2 γ12

−

ux (t, ·)p p

L (Rd )

dt + N ν

(−(νr)2 ,0)

−

d+2

g(t, ·)p p

L (Rd )

dt.

(−(νr)2 ,0)

Theorem 7.4. Let 1 < p < q < ∞, b = bˆ = 0 and c = 0. Let γ0 = γ0 (d, p, 2p, δ) be the constant from Lemma 6.1. Assume u ∈ C0∞ and Pu = div g, where g ∈ Lq,p . Then we can find γ1 ∈ (0, γ0 ], R2 ∈ (0, R0 ] depending on d, δ, p, q such that under Assumptions 2.1–2.3 there exists a constant N = N(d, p, q, δ) such that ux Lq,p N gLq,p , provided that u vanishes outside (−R24 , 0) × Rd . Proof. Denote U (t) = ux (t, ·)Lp (Rd ) ,

V (t) = ux (t, ·)Lp (Rd ) ,

G(t) = g(t, ·)Lp (Rd ) .

Let γ1 be less than the constant of the same notation in Theorem 6.5. Let ν ∈ [16, ∞) and R2 ∈ (0, R0 ) be numbers to be chosen later. Assume u vanishes outside (−R24 , 0) × Rd . We claim 1 d+2 d+2 1 1 2(1− p1 ) U # N (R2 ν) + ν −α + ν p γ12p M V p p + N ν p M Gp p .

(7.6)

Indeed, fix a point t0 ∈ R and consider an interval (S, T ) ⊂ R containing t0 . When T − S ν −2 R22 , by a shift of the origin we get from Corollary 7.3 that (U − (U )(S,T ) )(S,T ) is less than the right-hand side of (7.6) at t0 . When T − S > ν −2 R22 , we then have U − (U )(S,T ) (S,T ) 2 − χ(−R 4 ,0) U (t) dt 2

(S,T )

2

1−1/p 1/p p − χ(−R 4 ,0) dt − U (t) dt 2

(S,T )

(S,T )

1/p N (R2 ν)2(1−1/p) M U p (t0 ) . By taking supremum over all intervals (S, T ) containing t0 , we obtain (7.6) at point t0 . Since t0 ∈ R is arbitrary, the claim is proved.


2171

Recall that q > p. It then follows from the Fefferman–Stein theorem and the Hardy– Littlewood theorem that 1 d+2 d+2 2(1− p1 ) + ν −α + ν p γ12p ux Lq,p + N ν p gLq,p . ux Lq,p N (R2 ν)

(7.7)

Take R2 to be less than μ−1/2 R1 with μ and R1 in Theorem 6.5 so that Theorem 6.5 is applicable. From (6.15) and (7.7) we then get 1 d+2 d+2 2(1− p1 ) ux Lq,p N (R2 ν) + ν −α + ν p γ12p ux Lq,p + N ν p gLq,p .

(7.8)

To complete the proof of Theorem 7.4, it suffices to take ν sufficiently large, then R2 small and γ1 even smaller in (7.8) so that 1 d+2 2(1− p1 ) + ν −α + ν p γ12p 1/2. N (R2 ν)

2

Proof of Theorem 2.7. The proof is similar to that of Theorem 2.5 in Section 5 by using Theorem 6.5 and Theorem 7.4 instead of Theorem 4.1 and Lemma 5.4. We leave the details to interested readers. 2 Acknowledgments The author is sincerely grateful to Nicolai V. Krylov for very helpful suggestions and comments, in particular for his enlightening suggestion of Lemma 4.3. The author also thanks Doyoon Kim and the referee for valuable comments. References [1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959) 623–727; II, Comm. Pure Appl. Math. 17 (1964) 35–92. [2] P. Auscher, M. Qafsaoui, Observations on W 1,p estimates for divergence elliptic equations with VMO coefficients, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. 5 (2002) 487–509. [3] S. Byun, Elliptic equations with BMO coefficients in Lipschitz domains, Trans. Amer. Math. Soc. 357 (3) (2005) 1025–1046. [4] S. Byun, Parabolic equations with BMO coefficients in Lipschitz domains, J. Differential Equations 209 (2) (2005) 229–265. [5] S. Byun, Optimal W 1,p regularity theory for parabolic equations in divergence form, J. Evol. Equ. 7 (3) (2007) 415–428. [6] M. Bramanti, M. Cerutti, Wp1,2 solvability for the Cauchy–Dirichlet problem for parabolic equations with VMO coefficients, Comm. Partial Differential Equations 18 (9–10) (1993) 1735–1763. [7] F. Chiarenza, M. Frasca, P. Longo, Interior W 2,p estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat. 40 (1991) 149–168. [8] F. Chiarenza, M. Frasca, P. Longo, W 2,p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (2) (1993) 841–853. [9] H. Dong, D. Kim, Parabolic and elliptic systems with VMO coefficients, Methods Appl. Anal., in press. [10] H. Dong, N.V. Krylov, Second-order elliptic and parabolic equations with B(R2 , VMO) coefficients, Trans. Amer. Math. Soc., in press, arXiv:0810.2739 [math.AP]. [11] G. Di Fazio, Lp estimates for divergence form elliptic equations with discontinuous coefficients, Boll. Unione Mat. Ital. A (7) 10 (2) (1996) 409–420 (Italian summary).

2172


[12] R. Haller-Dintelmann, H. Heck, M. Hieber, Lp –Lq -estimates for parabolic systems in non-divergence form with VMO coefficients, J. London Math. Soc. (2) 74 (3) (2006) 717–736. [13] D. Jerison, C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. 130 (1) (1995) 161–219. [14] D. Kim, Second order elliptic equations in Rd with piecewise continuous coefficients, Potential Anal. 26 (2) (2007) 189–212 (English summary). [15] D. Kim, Parabolic equations with measurable coefficients. II, J. Math. Anal. Appl. 334 (1) (2007) 534–548 (English summary). [16] D. Kim, Elliptic and parabolic equations with measurable coefficients in Lp -spaces with mixed norms, Methods Appl. Anal. 15 (4) (2008) 437–468. [17] D. Kim, Parabolic equations with partially VMO coefficients and boundary value problems in Sobolev spaces with mixed norms, Potential Anal. (2010), doi:10.1007/s11118-009-9158-0. [18] D. Kim, N.V. Krylov, Elliptic differential equations with coefficients measurable with respect to one variable and VMO with respect to the others, SIAM J. Math. Anal. 39 (2) (2007) 489–506. [19] D. Kim, N.V. Krylov, Parabolic equations with measurable coefficients, Potential Anal. 26 (4) (2007) 345–361. [20] N.V. Krylov, Parabolic equations in Lp -spaces with mixed norms, Algebra i Analiz 14 (4) (2002) 91–106 (in Russian); English translation: St. Petersburg Math. J. 14 (4) (2003) 603–614. [21] N.V. Krylov, Parabolic and elliptic equations with VMO coefficients, Comm. Partial Differential Equations 32 (3) (2007) 453–475. [22] N.V. Krylov, Parabolic equations with VMO coefficients in spaces with mixed norms, J. Funct. Anal. 250 (2) (2007) 521–558. [23] N.V. Krylov, Second-order elliptic equations with variably partially VMO coefficients, J. Funct. Anal. 257 (2009) 1695–1712. [24] O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1967. [25] A. Lorenzi, On elliptic equations with piecewise constant coefficients. II, Ann. Sc. Norm. Super. Pisa (3) 26 (1972) 839–870. [26] P. Maremonti, V.A. Solonnikov, On the estimates of solutions of evolution Stokes problem in anisotropic Sobolev spaces with mixed norm, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 222 (1995) 124–150 (in Russian); English translation: J. Math. Sci. (N. Y.) 87 (5) (1997) 3859–3877. [27] Z. Shen, Bounds of Riesz transforms on Lp spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble) 55 (1) (2005) 173–197. [28] L. Softova, P. Weidemaier, Quasilinear parabolic problems in spaces of maximal regularity, J. Nonlinear Convex Anal. 7 (3) (2006) 529–540. [29] P. Weidemaier, Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed Lp -norm, Electron. Res. Announc. Amer. Math. Soc. 8 (2002) 47–51.


Littlewood–Paley–Stein gk -functions for Fourier–Bessel expansions ✩ Óscar Ciaurri ∗ , Luz Roncal Departamento de Matemáticas y Computación, Universidad de La Rioja, Edificio J.L. Vives, Calle Luis de Ulloa s/n, 26004 Logroño, Spain Received 20 February 2009; accepted 23 December 2009 Available online 13 January 2010 Communicated by C. Kenig

Abstract gk -Functions related to the Poisson semigroup of Fourier–Bessel expansions are defined for each k 1. It is proved that these gk -functions are Calderón–Zygmund operators in the sense of the associated space of homogeneous type, hence their mapping properties follow from the general theory. © 2010 Elsevier Inc. All rights reserved. Keywords: Fourier–Bessel expansions; gk -Functions; Weighted inequalities; Ap weights

1. Introduction Given α > −1, we consider the differential operator Lα = − −

2α + 1 d . x dx

(1.1)

Let {sn,α }n1 denote the sequence of successive positive zeros of the Bessel function Jα . The functions φnα (x) = dn,α sn,α Jα (sn,α x)x −α , 1/2

✩

n = 1, 2, . . .

Research of the first author supported by grant MTM2009-12740-C03-03 of the DGI. The second author was supported by a FPI grant of the University of La Rioja. * Corresponding author. E-mail addresses: [email protected] (Ó. Ciaurri), [email protected] (L. Roncal). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.014

2174

with dn,α =

Ó. Ciaurri, L. Roncal / Journal of Functional Analysis 258 (2010) 2173–2204 √

2

1/2 |sn,α Jα+1 (sn,α )|

2 . , are eigenfunctions of Lα with the corresponding eigenvalue sn,α

Furthermore, the system {φnα }n∈N is a complete orthonormal basis in L2 ((0, 1), dμα (x)) (for instance, see [7] for details) where dμα (x) = x 2α+1 dx. The Fourier–Bessel expansion of a function f is f=

∞

an (f )φnα ,

n=1

1

where an (f ) = 0 f (y)φnα (y) dμα (y) provided the integrals exist. For t > 0, the corresponding Poisson semigroup, Ptα f is defined by Ptα f =

∞

e−tsn,α an (f )φnα ,

f ∈ L2 (0, 1), dμα .

(1.2)

n=1

We can write (1.2) in the form 1 Ptα f (x) =

Ptα (x, y)f (y) dμα (y)

(1.3)

0

where the kernel Ptα (x, y) is given by Ptα (x, y) =

∞

e−sn,α t φnα (x)φnα (y).

(1.4)

n=1

In order to simplify our computations, we make a change of variable e−t = r and, using that sn,α ∼ n, from now on we will take e−tsn,α as r n , 0 < r < 1; in this way, we write (1.2) in the form 1 Prα f (x) =

Prα (x, y)f (y) dμα (y)

(1.5)

0

where the kernel Prα (x, y) is given by Prα (x, y) =

∞

r n φnα (x)φnα (y).

(1.6)

n=1

We now define the gk -functions, for each integer k 1, by 2 gk (f, x) =

2 2k−1 1 ∂ k α dr r log 1 , P f (x) r ∂r r r 0

(1.7)

Ó. Ciaurri, L. Roncal / Journal of Functional Analysis 258 (2010) 2173–2204

2175

∂ k ∂ where (r ∂r ) means that we are applying k times the operator r ∂r . Note that our definition of gk -function is inspired by the definition of Thangavelu in [16, Chapter 4] after the change of variable e−t = r (to be precise, Thangavelu works with the heat semigroup for Hermite expansions). Square functions or gk -functions were first developed through the thirties of the last century by Littlewood and Paley, Zygmund, and Marcinkiewicz. In these first approaches, the gk -functions are defined for the Fourier series. They are non-linear operators which allow us to give a useful characterization of the Lp norm of a function in terms of the behavior of its Poisson integral. Mainly, the gk -functions are applied to obtain results for multipliers with “Hörmander conditions” and estimates for the Riesz transform. Theorems for multipliers using gk -functions have been proved in [14] for the n-dimensional Fourier transform, in [10] for the ultraspherical expansions, in [15] for Hermite expansions, in [9] for general semigroups, and in [4] for Laguerre expansions and including potential weights. The gk -functions appear in results related to the Riesz transform in [5] for the Ornstein–Uhlenbeck semigroup, in [6] for the Hermite semigroup, and in [11] for the Laguerre expansions. We prove that with the assumption k 1, the gk -functions are vector-valued Calderón– Zygmund operators in the sense of the underlying space of homogeneous type, hence their mapping properties follow by applying results from the general theory. In particular, we are interested in weighted inequalities for weights in an appropriate Ap Muckenhoupt class. Similar techniques have been used, for example, in [1] and [12]. For 1 < p < ∞, p is its adjoint, 1/p + 1/p = 1 and we denote by Aαp = Aαp ((0, 1), dμα ) the Muckenhoupt class of Ap weights on the space ((0, 1), dμα , | · |). More precisely, Aαp is the class of all nonnegative functions w ∈ L1loc ((0, 1), dμα ) such that w −p /p ∈ L1loc ((0, 1), dμα ) and

1 I ∈I μα (I )

sup

I

w(x) dμα (x)

1 μα (I )

p/p w(x)−p /p dμα (x) 0, μ(B(x, r)) < ∞, and such

2176


that there exists 0 < C < ∞ verifying that μ(B(x, 2r)) Cμ(B(x, r)); i.e., μ is a doubling measure. Given α > −1, we shall work on the space (0, 1), equipped with the measure dμα (x) and with the Euclidean distance |·|. Since dμα possesses the doubling property, the triple ((0, 1), dμα , |·|) forms a space of homogeneous type. For B a Banach space, we say that a kernel K : X × X \ {(x, x)} → B is a standard kernel if there exist ε > 0 and C < ∞ such that for all x, y, z ∈ X (x = y), with ρ(x, z) ερ(x, y), then

K(x, y) B

C μ(B(x, ρ(x, y)))

(2.1)

and

K(x, y) − K(z, y) + K(y, x) − K(y, z) C B B

ρ(x, z) ρ(x, y)μ(B(x, ρ(x, y)))

(2.2)

holds. Thus, a vector-valued Calderón–Zygmund operator with associated kernel K is a linear operator T bounded from L2 (X, dμ) into L2B (X, dμ) such that, for every f ∈ L2 (X, dμ) and x outside the support of f , Tf (x) = K(x, y)f (y) dμ. X

It is known that any vector-valued Calderón–Zygmund operator as above is bounded from p Lp (X, w dμ) into LB (X, w dμ), for 1 < p < ∞ and any weight w in the Muckenhoupt type class Ap (X, dμ) (see [13]). The weights in Ap (X, dμ) are nonnegative functions w ∈ L1loc (X, dμ) p /p

such that w ∈ Lloc (X, dμ) and

1 sup B∈B μ(B)

p/p 1 −p /p w(x) dμ(x) w(x) dμ(x) < ∞, μ(B)

B

B

where B is the class of all the balls in (X, ρ). The gk -functions can be seen as vector-valued operators taking values in a Banach space. For each integer k 1, we consider the space 1 2k−1 dr Bk = L (0, 1), log r r 2

and the vector-valued kernel

∂ Gr,k (x, y) = r ∂r

k Prα (x, y).

Defining 1 Gr,k f (x) =

Gr,k (x, y)f (y) dμα (x), 0


2177

clearly, the identity

gk (f, x) = Gr,k f (x) B

k

holds. To prove that the operator Gr,k f (x) is bounded from L2 ((0, 1), dμα ) into the space L2Bk ((0, 1), dμα ) it is enough the result contained in the following lemma where it is established that gk -functions are isometries in L2 ((0, 1), dμα ). Lemma 1. For each k 1 and f ∈ L2 ((0, 1), dμα ) one has

gk (f ) 2 2

L ((0,1),dμα )

= 2−2k Γ (2k)f 2L2 ((0,1),dμ ) .

(2.3)

α

Proof. The proof works the same as in [16, Theorem 4.1.1] after a change of variable.

2

In this way, showing the inequalities

Gr,k (x, y)

Bk

C μα (B(x, |x − y|))

(2.4)

and

∇x,y Gr,k (x, y)

C |x − y|μα (B(x, |x − y|))

(2.5)

(note that (2.4) and (2.5) imply (2.1) and (2.2)) we have the following proposition. Proposition 1. For each k 1, Gr,k f (x) is a vector-valued Calderón–Zygmund operator taking values in Bk . 3. Preliminaries The Bessel function Jα satisfies Jα (t) =

α Jα (t) − Jα+1 (t). t

(3.1)

We will use the fact that sn,α = O(n),

dn,α = O(1).

(3.2)

The following asymptotics will be used (see [8, p. 122]): M √ Bα,j Aα,j zJα (z) = sin z + cos z + HM (z), zj zj

(3.3)

j =0

where M = 0, 1, . . . and |HM (z)| Cz−(M+1) , z → ∞. At z = 0 Jα (z) = O zα , z → 0+ .

(3.4)

2178


Poisson’s integral formula 1 Jα (z) = Cα z

α

α−1/2 1 − t2 cos(zt) dt,

α > −1/2,

(3.5)

0

will also be helpful. Lemma 2. Let α > −1, be a nonnegative integer and γ be a real number. Then each of the four functions γ 2 dn,α sn,α

sin sn,α (x ± y) , cos

n = 1, 2, . . . ,

(3.6)

can be written as the sum of terms of the form nγ

sin πn(x ± y) Eγ , (n, x, y), cos

where Eγ , (n, x, y) =

Ak (x, y) k=0

nk

+ qn( ) (x, y),

( )

and Ak (x, y), k = 0, 1, . . . , , qn (x, y), n = 1, 2, . . . , are functions such that |Ak (x, y)| C, ( ) |qn (x, y)| Cn− −1 , 0 < x, y < 1, with a constant C = Cα, ,γ . The lemma follows by taking μ = ν, m = j = 0 in [3, Lemmas 4.1 and 4.2] (the functions Ak (x, y) now incorporate some bounded functions that appear in those lemmas). The following estimate will be used from now on with no additional comments 1 2k−1 1, 0 < r < 1/2, r log C (1 − r)2k−1 , 1/2 r < 1. r By Pr and Qr , 0 < r < 1, we denote the usual Poisson and conjugate Poisson kernels, ∞

Pr (x) =

1 n 1 − r2 + , r cos(nx) = 2 2(1 − 2r cos x + r 2 ) n=1

Qr (x) =

∞ n=1

r n sin(nx) =

r sin x . 1 − 2r cos x + r 2

Notice that for x = 2kπ , k ∈ Z, limr→1− Pr (x) = 0, limr→1− Qr (x) = 12 cot( π2 x).

(3.7)


2179

Lemma 3. For n = 1, . . . , 0 < r < 1 and 0 < |x| < 3π/2, we have ∂ n (1 − r)n+1 + |sin x2 |n+1 r Cn r P (x) , r ∂r ((1 − r)2 + 4r sin2 x )n+1 2

∂ n (1 − r)n+1 + |sin x2 |n+1 r ∂r Qr (x) Cn r ((1 − r)2 + 4r sin2 x )n+1 . 2

∂ n ∂ n ) Pr (x)| or |(r ∂r ) Qr (x)| has the form Proof. We have that |(r ∂r

rSn (r, x) ((1 − r)2

+ 4r sin2 x2 )n+1

,

where x dSn (r, x) Sn+1 (r, x) = r (1 − r)2 + 4r sin2 2 dr 2 2 x 2 x − 2(n + 1)r 2 sin − (1 − r) Sn (r, x) + (1 − r) + 4r sin 2 2 and S0 (r, x) = 12 (1 − r 2 ) in the case of Pr or S0 (r, x) = r sin x in the case Qr . It follows inductively that Sn (r, x) can be written as the sum of terms of the form f (r, x)(1 − r)j sink x2 , where f (r, x) is a bounded function and j + k n + 1. From this, |Sn (r, x)| C((1 − r)n+1 + |sin x2 |n+1 ). 2 ∂ n Remark. The result in the previous lemma allows us to obtain estimates for |(r ∂r ) Pr (x)| and ∂ n |(r ∂r ) Qr (x)| depending on r or x separately. The exact bounds are

∂ n −(n+1) r , ∂r Pr (x) Cn |x|

∂ n −(n+1) r , ∂r Qr (x) Cn |x|

and ∂ n r Cn r(1 − r)−(n+1) , P (x) r ∂r

∂ n r Cn r(1 − r)−(n+1) . Q (x) r ∂r

These results follow immediately using the previous lemma and the estimates (1 − r)n+1 + |sin x2 |n+1 ((1 − r)2 + 4r sin2 x2 )n+1

C|x|−(n+1) ,

for 1 − r |x| < 3π/2, and (1 − r)n+1 + |sin x2 |n+1 ((1 − r)2 for 0 < |x| < 1 − r.

+ 4r sin2 x2 )n+1

, C(1 − r)−(n+1) ,

2180


Lemma 4. For k 1, m = 1, 2, . . . , and 0 < |x| < 3π/2, we have 1

1 log r

2k−1 ∞

n

k+m−1 n

r

n=1

0

2 dr sin (nx) Cx −2m cos r

with a constant C independent of x. Proof. Our task reduces to proving that 1 log

1 r

2k−1 r

∂ ∂r

k+m−1

2 dr Cx −2m , Pr (x) r

0 ∂ k+m−1 and the analogous for (r ∂r ) Qr (x). We show the proof for Pr , the other one is obtained following the same reasoning. By Lemma 3, the left-hand side of the previous inequality is bounded by

1 (1 − r)k+m + |sin x2 |k+m 2 dr 1 2k−1 log r . r r ((1 − r)2 + 4r sin2 x2 )k+m

(3.8)

0

To estimate the integral (3.8) we decompose it into the intervals 0 < r 1/2 and 1/2 < r < 1. For the first one, using (3.7) we obtain that 1/2 0

1 log r

2k−1 1/2 (1 − r)k+m + |sin x2 |k+m 2 dr C r C dr C. x 2 r (1 − r)2(k+m) ((1 − r)2 + 4r sin 2 )k+m 0

For the interval 1/2 < r < 1, using (3.7), taking into account that sin x/2 ∼ x/2, and applying the change of variable (1 − r)/|x| = w, we have 1 log

1 r

1/2

2k−1 (1 − r)k+m + |sin x2 |k+m 2 dr r r ((1 − r)2 + 4r sin2 x2 )k+m

1 C

(1 − r)

2k−1

((1 − r)2 + 4r sin2 x2 )k+m

1/2

1 ∼ 1/2

(1 − r)

2k−1

(1 − r)k+m + |sin x2 |k+m

(1 − r)k+m + |x|k+m ((1 − r)2 + x 2 )k+m

2 dr

2 dr


1 = C 2m |x|

1/(2|x|)

w 2k−1 0

where in the last step we use that

∞ 0

w k+m + 1 (w 2 + 1)k+m

2

2181

dw C|x|−2m

2 w 2k−1 ( (ww2 +1)+1 k+m ) dw < ∞. k+m

2

Lemma 5. Let 0 < γ < 1 and f be a 2π -periodic function such that |f (x)| C|x|−γ for 0 < |x| π . Then, for 0 < |x| < 3π/2, it is verified that (1 − r)γ + |sin x2 |γ Pr − 1 ∗ f (x) Cr 2 ((1 − r)2 + 4r sin2 x2 )γ when 0 < r 12 , and (1 − r)γ + |sin x2 |γ Pr ∗ f (x) Cr ((1 − r)2 + 4r sin2 x2 )γ when

1 2

< r < 1. The constants in both inequalities depend on γ but are independent of r and x.

Proof. To prove our estimates, we will use that Pr (x) −

1−r 1 Cr 2 2 (1 − r) + 4r sin2

x 2

for 0 < r 12 , and Pr (x) Cr

1−r (1 − r)2 + 4r sin2

x 2

for 12 < r < 1. With the inequality for |Pr (x) − 12 | we can prove the first estimate in the lemma, the second one is a consequence of the inequality for |Pr (x)|. The proof in both cases follows in the same way, so we will check the first estimate and omit the details for the second one. (1−r)γ +|sin x2 |γ It is sufficient to check that |(Pr − 12 ) ∗ f (x)| Cr 2 x γ for 0 < |x| < π/2 and 2 ((1−r) +4r sin

2)

|(Pr − 12 ) ∗ f (x)| Cr for π/2 |x| 3π/2 with constants independent of r and x. Since the periodicity of f allows the hypothesized estimate |f (x)| C|x|−γ to hold for 0 < |x| 3π/2, the proof of the first bound reduces to showing that π Pr (y) − 0

(1 − r)γ + (sin x2 )γ 1 −γ |x ± y| dy Cr , 2 ((1 − r)2 + 4r sin2 x2 )γ

0 < x < π/2.

(3.9)

2182


To show (3.9) we split the region of integration (0, π) into three parts, A = (0, x/2), B = (2x, π) and C = (x/2, 2x). For the region A, we separately consider 1−r x and 0 < x < 1−r. For the first case, the fact that π −π

r(1 − r) (1 − r)2

dy =

+ 4r sin2 y2

2πr , 1+r

easily gives Pr (y) − A

where we have used that

(1 − r)γ + (sin x2 )γ 1 −γ −γ |x ± y| dy Crx Cr , 2 ((1 − r)2 + 4r sin2 x2 )γ

(1−r)γ +(sin x2 )γ ((1−r)2 +4r sin2 x2 )γ

1 1 ∼ ( 1−r+x )γ , and 1 − r x implies that ( 1−r+x )−γ

Cx γ . For the second case, 0 < x < 1 − r we have that x/2 < (1 − r + x)/4, hence we enlarge the region of integration to get Pr (y) −

1−r+x

4 1 −γ |x ± y| dy 2

A

Pr (y) −

1 |x ± y|−γ dy 2

0

r 1−r C

1−r+x 4

|x ± y|−γ dy C

0

r (1 − r + x)1−γ 1−r

r , (1 − r + x)γ

and the last expression is equivalent to Cr

(1−r)γ +(sin x2 )γ ((1−r)2 +4r sin2 x2 )γ

.

For the region B we also consider separately the cases 1 − r x and 0 < x < 1 − r. For the first one, we make the change of variable t = π − y to obtain the integral π−2x

r 0

−γ 1−r dt t x ± (π − t) 2 + 4r cos 2

(1 − r)2

and the bound is obtained as in the case 1 − r x in the region A, since π −π

r(1 − r) (1 − r)2 + 4r cos2

t 2

dt =

2πr . 1+r

For the case 0 < x < 1 − r, we split the integral into two parts


Pr (y) −

2183

1 |x ± y|−γ dy 2

B x+(1−r)

Pr (y) −

=

1 |x ± y|−γ dy + 2

2x

π

Pr (y) −

1 |x ± y|−γ dy =: J1 + J2 . 2

x+(1−r)

First, we have r J1 1−r

1−r+x

|x ± y|−γ dy C

2x

r r (1 − r + x)1−γ C . 1−r (1 − r + x)γ

On the other hand, π

|x ± y|−γ dy Cr(1 − r + x)−γ ,

J2 Cr x+(1−r)

and the result is proved for the entire region B because (1 − r + x)−γ ∼ In the region C, we have Pr (y) − C

(1−r)γ +(sin x2 )γ ((1−r)2 +4r sin2 x2 )γ

.

1 1−r −γ |x ± y| dy r |x ± y|−γ dy 2 (1 − r)2 + 4r sin2 y2 C

Cr

Cr

1−r (1 − r)2 + 4r sin2 1−r (1 − r)2 + 4r sin2

x 2

|x ± y|−γ dy

C

x 1−γ Cr x 2

γ

1−r (1 − r)2 + 4r sin2

x 2

,

1−r where we have used that Cx −1 . This bound for the integral in C is enough for (1−r)2 +4r sin2 x2 our purpose. The proof for the region π/2 |x| 3π/2 reduces to showing that

π

y −γ Pr (y ± x) −

1 dy Cr, 2

π/2 x 3π/2.

0

The observation Pr (x) −

1−r 1 Cr 2 2 (1 − r) + 4r sin2

x 2

=C

2r Pr (x), 1+r

2184


implies that π y

Pr (y ± x) −

−γ

π 1 dy Cr y −γ Pr (y ± x) dy, 2

0

0

and the estimate follows since this last integral is bounded by a constant, as we see in [3, Lemma 3.2]. 2 Lemma 6. For k 1, γ > 0, and 0 < |x| < 3π/2, we have 1

1 log r

2k−1 ∞

n k+γ −1

r n

n=1

0

2 dr sin (nx) Cx −2γ cos r

(3.10)

with a constant C independent of x. Proof. For γ = m, m = 1, 2, . . . , the result is contained in Lemma 4. To prove (3.10) for other values of γ we are going to prove that for each β > 1, the expression ∞ sin (nx) r n nβ−1 cos n=1

can be written as a finite sum of terms of the kind r

(1 − r)δ + |sin x2 |δ ((1 − r)2 + 4r sin2 x2 )δ

,

δ β.

With this estimate, proceeding as in Lemma 4, the result follows. The following estimate is stated in [17, (13), p. 70, vol. I]: for any 0 < β < 1 and 0 < |x| < 3π/2, the function f (x) =

∞

n

β−1

n=1

sin (nx) cos

satisfies that f (x) Cγ |x|−β . Now, using that

π

−π

f (x) dx = 0, it is clear that 1 Pr ∗ f (x) = Pr − ∗ f (x). 2


2185

Moreover, taking f with sines, we have Pr ∗ f (x) = −π

∞

r n nβ−1 cos(nx),

n=1

and when f involves the cosines, the identity Pr ∗ f (x) = π

∞

r n nβ−1 sin(nx)

n=1

holds. This, together with Lemma 5, shows that ∞ (1 − r)β + |sin x2 |β sin (nx) Cr r n nβ−1 cos ((1 − r)2 + 4r sin2 x )β

(3.11)

2

n=1

for 0 < β < 1. Suppose now that m < β < m + 1 with m 1. To simplify the notation write Sβ (x) = Sβ (r, x) =

∞

Cβ (x) = Cβ (r, x) =

r n nβ−1 sin(nx),

n=1

∞

r n nβ−1 cos(nx).

n=1

In [3, (3.3), p. 4449] we can find the identity Sβ (x) = Sm (x) +

m m

as,p,β Ss (x)Cβ−p (x) + Cs (x)Sβ−p (x)

s=1 p=s

+

m

Ss (x)As,β (r, x) + Cs (x)Bs,β (r, x) ,

(3.12)

s=1

where as,p,β are constants and As,β (r, x), Bs,β (r, x) are bounded functions. An analogous formula holds for Cβ (to be precise, on the right side of (3.12), Sm , Ss , Cs and the second plus sign have to be replaced by Cm , Cs , Ss and the minus sign). So, with (3.12), we finish by using the bounds in Lemma 3 and (3.11). 2 Lemma 7. For k 1, 0 < x < 1, and N = [1/x], we have 1 0

for β > −1, and

1 log r

2k−1 N n=1

2 n k+β

r n

dr Cx −(2β+2) , r

(3.13)

2186


1

1 log r

2k−1 ∞

2 n k+β

r n

n=N +1

0

dr Cx −(2β+2) , r

(3.14)

for β < −1. Proof. We begin by proving (3.13). First, we will think of the case k = 1; we are going to estimate 1

2

N 1 n 1+β dr . log r n r r n=1

0

Integrating by parts, this expression equals 1 r N 0

0

2 s n−1/2 n1+β

n=1

dr = ds r

1 r N 0

0

s n+j −1 (nj )1+β ds

n,j =1

dr r

1 N r n+j 1+β dr = (nj ) n+j r n,j =1

0

1 N dr r n+j C √ (nj )1+β r nj n,j =1 0

=C

1 N

r n+j −1 (nj )1/2+β dr C

0 n,j =1

=C

N

N (nj )1/2+β √ nj n,j =1

2 n

CN 2β+2 Cx −(2β+2) .

β

n=1

For k > 1, we integrate by parts 2k − 2 times to obtain 1 0

1 log r

2k−1 N

2 r n nk+β

n=1

and then apply the result for k = 1. The proof of (3.14) is analogous.

dr C r

1 0

2

N 1 n β+1 dr log r n r r n=1

2

The next lemma analyzes a situation similar to the case γ = 0 in Lemma 6. This case has to be investigated separately because in this situation the proof of Lemma 4 does not work.


2187

Lemma 8. For k 1, 0 < |x| < 3π/2, and N = [1/|x|], we have 2

∞ 1 1 2k−1 n k−1 sin dr log (πnx) C, r n cos r r

(3.15)

n=N

0

with a constant C independent of x. Proof. We will prove the result for the case of sines. The other case is analogous. Applying the summation by parts formula, ∞

an (bn+1 − bn ) = −aN bN −

n=N

∞

bn+1 (an+1 − an ),

n=N

with the sequences an = n and bn = −

∞

j =n j

k−2 r j

sin(πj x), we have

∞ ∞ ∞ ∞ n k−1 k−2 j k−2 j r n sin(πnx) N j r sin(πj x) + j r sin(πj x). j =N

n=N

n=N j =n+1

The last sum, after a translation j − n = m and using the formula for sin(a + b), equals k−2 k−2 s=0

+

s ∞

∞

nk−s−2 r n cos(πnx)

n=N

n

k−s−2 n

ms r m sin(πmx)

m=1

r sin(πnx)

n=N

∞

∞

s m

m r cos(πmx) ,

m=1

hence ∞ ∞ n k−1 k−2 j r n sin(πnx) N j r sin(πj x) j =N

n=N

k−2 ∞ ∞ k−2 + nk−s−2 r n cos(πnx) ms r m sin(πmx) s n=N s=0 m=1 ∞ ∞ k−s−2 n s m + n r sin(πnx) m r cos(πmx) . n=N

m=1

Concerning the first summand, using (3.14) with β = −2, the expression 1 0

1 log r

2k−1 ∞ n=N

2 n

k−2 n

r sin(πnx)

dr r

2188


is bounded by x 2 , so the result follows, since N 2 ∼ x −2 . To estimate the second summand, by s m sin −2(s+1) , and, on the other hand, Lemma 3.3 in [3], we have that ∞ m=1 m r { cos }(πmx) Cx 1

1 log r

2k−1 ∞

2 n

k−s−2 n

r

n=N

0

dr r

is bounded by x 2(s+1) , after applying (3.14) with β = −s − 2, which finishes the proof.

2

To conclude this section, we give estimates for the measure of the balls B(x, |x − y|) in the space ((0, 1), dμα ). Lemma 9. For α > −1 and x = y, the inequality ⎧ 2α+2 , ⎨x μα B x, |x − y| C (xy)α+1/2 |x − y|, ⎩ 2α+2 y ,

0 < y x/2, x/2 < y < min{1, 3x/2}, min{1, 3x/2} y < 1,

(3.16)

holds. Proof. (3.16) follows easily by studying separately the three considered regions. Thus, for the case 0 < y x/2, μα B x, |x − y| =

2x−y

dμα (t) =

t 2α+1 dt y

B(x,|x−y|)

= C (2x − y)2α+2 − y 2α+2 Cx 2α+2 . The case min{1, 3x/2} y < 1 follows analogously. Concerning the case x/2 < y < min{1, 3x/2} we distinguish between α −1/2 and −1 < α < −1/2. For the first one, we obtain, taking into account that x ∼ y, μα B x, |x − y| =

x+|x−y|

dμα (t) =

t 2α+1 dt

x−|x−y|

B(x,|x−y|)

2α+1 |x − y| C(xy)α+1/2 |x − y|. 2C x + |x − y| For the second case, we need to consider two regions separately. For the points (x, y) such that x/2 < y < x, we have μα B x, |x − y| =

B(x,|x−y|)

2x−y

dμα (t) =

t 2α+1 dt y

Cx 2α+1 |x − y| C(xy)α+1/2 |x − y|.


2189

In the case in which x < y < min{1, 3x/2}, μα B x, |x − y| =

y dμα (t) =

t 2α+1 dt

2x−y

B(x,|x−y|)

C(2x − y)

2α+1

|x − y| Cy 2α+1 |x − y| C(xy)α+1/2 |x − y|.

2

4. Proof of the estimate (2.4) The estimate (2.4) is an immediate consequence of Lemma 9 and the following proposition. Proposition 2. Let α > −1. For k 1, 2 1 ∂ k α 1 2k−1 dr r ∂r Pr (x, y) log r r 0

⎧ −4(α+1) , ⎨x C (xy)−2(α+1/2) |x − y|−2 , ⎩ −4(α+1) y ,

0 < y x/2, x/2 < y < min{1, 3x/2}, min{1, 3x/2} y < 1,

with C independent of 0 < r < 1, x and y. ∂ k α Proof. Case 1: 0 < y x/2. We split the series defining (r ∂r ) Pr (x, y) into

A=

N −1

nk r n φnα (x)φnα (y)

n=1

=

N −1

2 nk r n dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα (sn,α x) · (sn,α y)1/2 Jα (sn,α y)

n=1

and B=

∞

nk r n φnα (x)φnα (y)

n=N

=

∞

2 nk r n dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα (sn,α x) · (sn,α y)1/2 Jα (sn,α y)

n=N

where N = [1/x]. Using (3.2) and (3.4) we write |A|

N −1 n=1

2 nk r n dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα (sn,α x)(sn,α y)1/2 Jα (sn,α y)

(4.1)

2190


C(xy)−α

N −1

nk+1 r n Jα (sn,α x)Jα (sn,α y)

n=1

C

N −1

n2α+k+1 r n .

n=1

Using (3.13) with β = 2α + 1 we obtain N −1 2 1 1 2k−1 2α+k+1 n dr log Cx −4(α+1) . n r r r n=1

0

To get the same estimate for |B| it is enough to show that for 0 < r < 1, 0 < x < 1, 0 < y x/2 and α > −1/2 2

∞ 1 1 2k−1 k n α dr α+1/2 log Cx −4(α+1) , n r φn (x)dn,α sn,α cos(sn,α y) r r

(4.2)

n=N

0

and the analogous estimate with the exponents α + 1/2 and −4(α + 1) replaced by (α + 2) + 1/2 and −4((α + 2) + 1) correspondingly (this is needed in the case −1 < α −1/2 only). Indeed, using (4.2), Poisson’s integral formula (3.5) applied to Jα (sn,α y), and Minkowsky’s inequality give, for α > −1/2, 2

∞ 1 1 1 2k−1 k n α dr α+1/2 2 α−1/2 log 1−t n r φn (x)dn,α sn,α cos(sn,α yt) dt r r n=N

0

1 C

log

1 r

0

∞ 2 dr α+1/2 2 α−1/2 k n α 1−t n r φn (x)dn,α sn,α cos(sn,α yt) dt r n=N

0

1 C

α−1/2 1 − t2

0

×

0

2k−1 1

1 log

1 r

2k−1

0 ∞

2

α+1/2 nk r n φnα (x)dn,α sn,α cos(sn,α yt)

n=N

dr r

2

1/2 dt

Cx −4(α+1) . In the case −1 < α −1/2, applying the identity Jα (z) = −Jα+2 (z) + gives

2(α + 1) Jα+1 (z), z


B =−

∞

2191

nk r n φnα (x)dn,α (sn,α y)1/2 y −(α+1/2) Jα+2 (sn,α y)

n=N

+ 2(α + 1)

∞

nk r n φnα (x)dn,α (sn,α y)−1/2 y −(α+1/2) Jα+1 (sn,α y).

n=N

Now, using Poisson’s formula (3.5) for Jα+1 (sn,α y) and Jα+2 (sn,α y) (together with the assumption y x/2 in the first summand) and applying (4.2) we obtain the result. Proving (4.2) (the proof of its counterpart with aforementioned replacements in exponents is completely analogous hence we do not treat it separately) we use (3.3) to expand (sn,α x)1/2 Jα (sn,α x) and choose M to be the unique nonnegative integer satisfying M − 1 α + 1/2 < M. It is then clear that ∞ M α+1/2 k n α n r φn (x)dn,α sn,α cos(sn,α y) C x −j −(α+1/2) |Cj | + |Sj | + GM ,

(4.3)

j =0

n=N

where

Sj Cj

=

∞

−j +α+1/2

2 nk r n dn,α sn,α

n=N


j = 0, 1, . . . , M, and GM = x −(α+1/2)

∞

α+1/2 2 HM (sn,α x)sn,α . nk r n dn,α

n=N

Then, using (3.2), GM Cx −(α+M+3/2)

∞

nk+α−M−1/2 r n .

n=N

Since M > α + 1/2 we use (3.14) with β = α − M − 1/2 and we obtain

x

−2(α+M+3/2)

1 0

1 log r

2 2k−1 ∞ k+α−M−1/2 n dr Cx −2(α+M+3/2) x −2(α−M+1/2) n r r n=N

Cx −4(α+1) . We now take into account (4.3), to finish the proof of (4.2). It follows from Lemma 2 that for given j = 0, 1, . . . , M, Sj and Cj are sums of series of the form ∞ n=N

sin nk−j +α+1/2 r n Ek−j +α+1/2,M−j (n, x, y) πn(x ± y) . cos

(4.4)

2192


It is therefore clear that our task is reduced to dealing with the absolute value of each of the series in (4.4). Given j = 0, . . . , M, we use the expression for Ek−j +α+1/2,M−j (n, x, y) from Lemma 2 to estimate (4.4) with the sum of the absolute value of ∞

Rj,m =

n

k−j −m+α+1/2 n

r

n=N

sin πn(x ± y) , cos

(4.5)

for m = 0, . . . , M − j , and ∞ sin (M−j ) nk−j +α+1/2 r n qn (x, y) πn(x ± y) . cos n=N

(M−j )

For the term involving qn

(x, y), it is verified that

∞ ∞ sin (M−j ) nk−j +α+1/2 r n qn (x, y) πn(x ± y) C nk+α−M−1/2 r n . cos n=N

n=N

Then, using that −M − 1/2 + α < −1, we can apply (3.14) with β = α − M − 1/2 to obtain

x

−2j −2(α+1/2)

1

1 log r

2k−1 ∞

2 n

k+α−M−1/2 n

r

n=N

0

dr Cx −4(α+1) . r

The hypothesis made on M shows that α + 1/2 − j − m > −1 for j = 0, . . . , M and m = 0, . . . , M − j when M − 1 < α + 1/2 and the same is true for j = 0, . . . , M − 1 and m = 0, . . . , M − j − 1 when M − 1 = α + 1/2. Hence, in these cases, N −1 N −1 k+α+1/2−j −m+ n sin πn(x ± y) C n r nk+α+1/2−j −m r n , cos n=1

n=1

and using (3.13) with β = α + 1/2 − j − m,

x

−2j −2(α+1/2)

N −1 2 1 1 2k−1 k+α+1/2−j −m n dr log Cx −4(α+1)+2m Cx −4(α+1) . n r r r n=1

0

In consequence, in (4.5) we can extend the sum to start from n = 1 and then use Lemma 6 to get 2 2k−1 1 ∞ 1 dr sin x −2j −2(α+1/2) log πn(x ± y) nk+α+1/2−j −m r n cos r r 0

Cx −4(α+1)+2m .

n=1


2193

This completes the estimate involving the terms Rj,m , m = 0, 1, . . . , M − j , except for the cases of Rj,M−j when M − 1 = α + 1/2 for j = 0, . . . , M. In these exceptional cases we have to show that

x

−2j −2(α+1/2)

1 1 2k−1 dr log Cx −2M−2(α+1/2) . (Rj,M−j )2 r r 0

Since Rj,M−j takes the form of the series in (3.15), then Lemma 8 and the fact that N = [1/x] ∼ [1/x + y] ∼ [1/|x − y|] give the desired bound. Case 2: x/2 < y < min{1, 3x/2}. We use (3.3) with M = 1 to expand the functions (sn,α x)1/2 Jα (sn,α x) and (sn,α y)1/2 Jα (sn,α y). Then, taking N = [1/x] ∼ [1/y], we write ∂ k α ) Pr (x, y) as the sum (r ∂r F (r, x, y) +

1

x −j y −l Oj,l (r, x, y) + J1 (r, x, y) + J2 (r, x, y) + G1 (r, x, y),

j,l=0

where F (r, x, y) =

N −1

2 nk r n dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα (sn,α x) · (sn,α y)1/2 Jα (sn,α y),

n=1

and, for the remainder sum that starts from n = N , the Oj,l terms capture the part that comes from the main parts of the aforementioned expansions and are sums of terms of the form Dj,l

∞

−j −l

2 nk r n dn,α (xy)−(α+1/2) sn,α

n=N

sin sin (sn,α x) (sn,α y) cos cos

(Dj,l is a product of Aα+1,j or Bα+1,j and Aα,l or Bα,l depending on the choice of the sine or cosine), J1 gathers the part that comes from the main parts of the second expansion and the remainder of the first one, hence its absolute value is bounded by 2 ∞ sin k n 2 −(α+1/2) J1 (r, x, y) C n r dn,α (xy) H1 (sn,α x) (sn,α y) cos 1 n=N 2 ∞ sin 2 −1 + Cy −1 nk r n dn,α (xy)−(α+1/2) sn,α H1 (sn,α x) (sn,α y) cos 1 n=N

(the sign 21 indicates that we add two series, one for the choice of the sine and another for the cosine), J2 acts as J1 but with the position of the both expansions switched, and its absolute value is controlled by 2 ∞ k n 2 −(α+1/2) sin J2 (r, x, y) C (sn,α x)H1 (sn,α y) n r dn,α (xy) cos 1 n=N

2194


+ Cx

2 ∞ k n 2 −(α+1/2) −1 sin n r dn,α (xy) sn,α (sn,α x)H1 (sn,α y) cos

−1

1 n=N

and, eventually, G1 captures the part that comes from the remainders, ∞

G1 (r, x, y) =

2 nk r n dn,α (xy)−(α+1/2) H1 (sn,α x)H1 (sn,α y).

n=N

For F (r, x, y), using (3.4) and (3.2) we have N −1 F (r, x, y) C nk+2α+1 r n , n=1

then, applying (3.13) with β = 2α + 1, we obtain the estimate 1 2 dr 1 2k−1 log F (r, x, y) Cx −4(α+1) C(xy)−2(α+1/2) |x − y|−2 . r r 0

For J1 (r, x, y) (the same reasoning works for J2 (r, x, y)), using H1 (z) = O(z−2 ), z 1, and again (3.4) and (3.2), shows that J1 (r, x, y) Cx −2

∞

n

−(α+1/2)

k−2 n

r (xy)

+y

−1

n=N

∞

n

−(α+1/2)

k−3 n

r (xy)

.

n=N

Then, the required bound in this case boils down to estimating

x

−4

−2(α+1/2)

1

(xy)

1 log r

2k−1 ∞

0

2 n k−2

r n

n=N

dr r

and

x

−4 −2

y

−2(α+1/2)

1

(xy)

0

1 log r

2k−1 ∞

2 n k−3

r n

n=N

dr . r

Applying (3.14) with β = −2 and β = −3, these expressions are bounded, respectively, by a constant times x −2 (xy)−2(α+1/2) and y −2 (xy)−2(α+1/2) , and the task is done. We show, in the same manner, that ∞ G1 (r, x, y) C(xy)−2 (xy)−(α+1/2) nk−4 r n n=N


2195

and −4

(xy)

−2(α+1/2)

(xy)

∞ 2 1 1 2k−1 n k−4 dr log r n Cx −2 (xy)−2(α+1/2) , r r n=N

0

by (3.14) with β = −4. The remainder part of the proof is concerned with a more delicate analysis of the x −j y −l Oj,l (r, x, y) terms. We start with the x −1 y −1 O1,1 (r, x, y) term. It is clear that ∞ −1 −1 x y O1,1 (r, x, y) Cx −2 nk−2 r n (xy)−(α+1/2) , n=N

and, using (3.14) with β = −2,

x

−4

−2(α+1/2)

1

(xy)

1 log r

2k−1 ∞

2 n k−2

r n

n=N

0

dr x −2 (xy)−2(α+1/2) . r

Lemma 2 with γ = −1 and = 0 yields 1 log

1 r

2k−1

−1 x O1,0 (r, x, y)2 dr Cx −2 (xy)−2(α+1/2) r

0

once we show that 1

1 log r

2 2k−1 ∞ dr sin k−1 n C. n r E−1,0 (n, x, y) πn(x ± y) cos r n=N

0

The form of E−1,0 reduces the task to showing the estimates 1

1 log r

2k−1 ∞ n=N

0

n

k−1 n

r

2 dr sin πn(x ± y) C, cos r

(4.6)

and 1 0

1 log r

2k−1 ∞ n=N

nk−1 r n qn(0) (x, y)

2 dr sin C, πn(x ± y) cos r

(4.7)

where |qn (x, y)| Cn−1 . (4.6) follows immediately from Lemma 8. The estimate (4.7) is k−2 r n . So, using (3.14) n proved taking into account that the inner series is bounded by ∞ n=N 2 with β = −2, the left term in (4.7) is bounded by x , therefore controlled by a constant. The estimate for y −1 O0,1 (r, x, y) follows analogously. (0)

2196


It remains to consider the case of O0,0 (r, x, y). Using Lemma 2 with γ = 0 and = 1 shows that each of the four terms of O0,0 (r, x, y) is a sum of terms of the form ∞

(xy)−(α+1/2)

n=N

A1 (x, y) sin nk r n A0 + πn(x ± y) . + qn(1) (x, y) cos n

(4.8)

The estimate for the remainder term is immediate since, using |qn (x, y)| Cn−2 for 0 < x, y < 1, gives ∞ ∞ sin k n (1) n r qn (x, y) πn(x ± y) C nk−2 r n . cos (1)

n=N

n=N

Then, using (3.14) with β = −2, −2(α+1/2)

(xy)

2

∞ 1 dr 1 2k−1 k n (1) sin log n r qn (x, y) πn(x ± y) cos r r n=N

0 −2(α+1/2)

1

C(xy)

1 log r

0

2k−1 ∞

2 n

k−2 n

r

n=N

dr C(xy)−2(α+1/2) . r

Concerning the term involving A1 (x, y)n−1 , note that A1 (x, y) is a bounded function on 0 < x, y < 1, hence our task reduces to estimating (4.6). Finally, consider the term involving A0 . It is possible to extend the summation of the series involving this term from n = 1 since 1 2

N −1 dr 1 2k−1 k n sin log πn(x ± y) n r Cx −2 , cos r r n=1

0

after using (3.13) with β = 0. Then, we have to prove that 1 2

∞ dr 1 2k−1 k n sin log πn(x ± y) n r C|x − y|−2 . cos r r

(4.9)

n=1

0

With this, we can apply Lemma 4 with m = 1 in the case |x − y| 3/2, to obtain (4.9) for the minus sign. When x + y 3/2 the result follows in the same manner taking into account that (x + y)−1 |x − y|−1 . For x + y > 3/2 we need an extra argument. Points (x, y) such that x + y > 3/2 are contained in the region where 3/8 < x, y < 1. Then, writing x = 1 − u and y = 1 − v, we have u + v 3/2 and ∞ n=1

k n

n r

∞ sin k n sin πn(x + y) = πn(u + v) . n r cos cos n=1

In this way, for x + y > 3/2, by Lemma 4 with m = 1, it is verified that


2197

1 2

∞ dr 1 2k−1 k n sin πn(x ± y) n r log C(u + v)−2 cos r r n=1

0

= C(2 − x − y)−2 C|x − y|−2 , and the proof of (4.9) is completed. Case 3: min{1, 3x/2} y < 1. This case is completely analogous to Case 1 so we omit the proof. 2 5. Proof of the estimate (2.5) ∂ means the partial derivative against eiThe result in the following proposition (where ∂x,y ther x or y) and Lemma 9 allow us complete the proof of (2.5).

Proposition 3. Let α > −1 and k 1. Then 1 0

1 log r

2 2k−1 ∂ dr ∂ k α ∂x, y r ∂r Pr (x, y) r

⎧ −(4α+6) , ⎨x C (xy)−2(α+1/2) |x − y|−4 , ⎩ −(4α+6) y ,

0 < y x/2, x/2 < y < min{1, 3x/2}, min{1, 3x/2} y < 1,

(5.1)

with C independent of 0 < r < 1, x and y. Proof. We use (3.1) to find that dφnα (x) 1/2 = −sn,α dn,α sn,α Jα+1 (sn,α x)x −α . dx In this way (exchanging summation with differentiation is easily seen to be possible) ∞ ∂ k α ∂ 1/2 r Pr (x, y) = − nk r n sn,α dn,α sn,α Jα+1 (sn,α x)x −α φnα (y) ∂x ∂r

(5.2)

∞ ∂ ∂ k α 1/2 r Pr (x, y) = − nk r n sn,α φnα (x)dn,α sn,α Jα+1 (sn,α y)y −α . ∂y ∂r

(5.3)

n=1

and

n=1

∂ ∂ k α ∂ ∂ k α (r ∂r ) Pr (x, y) only since treating ∂y (r ∂r ) Pr (x, y) is completely We shall consider the case ∂x analogous. Case 1: 0 < y x/2. This case is proved analogously to the case min{1, 3x/2} y < 1 that will be proved later.

2198


Case 2: x/2 < y < min{1, 3x/2}. We use the asymptotic expansion (3.3) with M = 2, to expand the functions (sn,α x)1/2 Jα+1 (sn,α x) and (sn,α y)1/2 Jα (sn,α y) and take N = [ x1 ] ∼ [ y1 ] to write

∂ ∂ k α ∂x (r ∂r ) Pr (x, y)

as the sum

2

F (r, x, y) +

x −j y −l Oj,l (r, x, y) + J1 (r, x, y) + J2 (r, x, y) + G2 (r, x, y).

j,l=0

Here F (r, x, y) =

N −1

2 nk r n dn,α (xy)−(α+1/2) sn,α (sn,α x)1/2 Jα+1 (sn,α x) · (sn,α y)1/2 Jα (sn,α y),

n=1

and, for the remainder sum that starts from n = N , the Oj,l terms capture the part that comes from the main parts of the aforementioned expansions and are the sums of terms of the form

Dj,l

∞

−j −l+1

2 nk r n dn,α (xy)−(α+1/2) sn,α

n=N


J1 gathers the part that comes from the main parts of the second expansion and the remainder of the first one, hence its absolute value is bounded by 2 ∞ sin k n 2 −(α+1/2) J1 (r, x, y) C n r dn,α (xy) sn,α H2 (sn,α x) (sn,α y) cos 1 n=N

+ Cy

2 ∞ sin k n 2 −(α+1/2) n r dn,α (xy) H2 (sn,α x) (sn,α y) cos

−1

1 n=N

2 ∞ sin −2 k n 2 −(α+1/2) −1 + Cy n r dn,α (xy) sn,α H2 (sn,α x) (sn,α y), cos 1 n=N

J2 acts as J1 but with the position of the both expansions switched, and its absolute value is controlled by 2 ∞ sin k n 2 −(α+1/2) J2 (r, x, y) C (sn,α x)H2 (sn,α y) n r dn,α (xy) sn,α cos 1 n=N

+ Cx

2 ∞ k n 2 −(α+1/2) sin (sn,α x)H2 (sn,α y) n r dn,α (xy) cos

−1

1 n=N

2 ∞ −2 k n 2 −(α+1/2) −1 sin + Cx (sn,α x)H2 (sn,α y) n r dn,α (xy) sn,α cos 1 n=N


2199

and, eventually, G2 captures the part that comes from the remainders, G2 (r, x, y) =

∞

2 nk r n dn,α (xy)−(α+1/2) sn,α H2 (sn,α x)H2 (sn,α y).

n=N

We will now analyze separately each of the summands in the above decomposition of ∂ ∂ k α −2(α+1/2) |x − y|−4 . ∂x (r ∂r ) Pr (x, y) and bound them by C(xy) For F (r, x, y), using (3.4) and (3.2), we have N −1 F (r, x, y) Cx n2α+k+3 r n . n=1

Then, using (3.13) with β = 2α + 3, we obtain that 1

1 log r

2k−1

2 dr F (r, x, y) Cx 2 r

0

1

1 log r

2k−1 N −1

2 n

2α+k+3 n

r

n=1

0

dr r

Cx −2(2α+3) C(xy)−2(α+1/2) |x − y|−4 . For J1 (r, x, y) (the same reasoning works for J2 (r, x, y)), using H2 (z) = O(z−3 ), z 1, and again (3.4) and (3.2), shows that J1 (r, x, y) Cx −3

∞

nk−2 r n (xy)−(α+1/2)

n=N

+y

−1

∞

n

−(α+1/2)

k−3 n

r (xy)

+y

−2

n=N

∞

n

−(α+1/2)

k−4 n

r (xy)

n=N

Then, the required bound comes down to estimating

x

−6

−2(α+1/2)

(xy)

∞ 2 1 1 2k−1 k−2 n dr , n r log r r n=N

0

x

−6 −2

y

−2(α+1/2)

1

(xy)

1 log r

2k−1 ∞

0

2 n

k−3 n

r

n=N

dr , r

and

x

−6 −4

y

−2(α+1/2)

1

(xy)

0

1 log r

2k−1 ∞ n=N

2 n

k−4 n

r

dr . r

.

2200


Applying (3.14) with β = −2, −3 and −4, these expressions are bounded, respectively, by a constant times x −4 (xy)−2(α+1/2) , x −2 y −2 (xy)−2(α+1/2) and y −4 (xy)−2(α+1/2) , and the task is done. In a similar way we show that ∞ G2 (r, x, y) C(xy)−3 (xy)−(α+1/2) nk−5 r n n=N

and −6

(xy)

−2(α+1/2)

(xy)

∞ 2 1 1 2k−1 k−5 n dr log Cx −4 (xy)−2(α+1/2) , n r r r 0

n=N

by (3.14) with β = −5. The remainder part of the proof is concerned with a more delicate analysis of the x −j y −l Oj,l (r, x, y) terms. We start with the x −2 y −2 O2,2 (r, x, y) term. It is clear that ∞ −2 −2 x y O2,2 (r, x, y) Cx −4 nk−3 r n (xy)−(α+1/2) , n=N

and, using (3.14) with β = −3,

x

−8

−2(α+1/2)

(xy)

∞ 2 1 1 2k−1 k−3 n dr x −4 (xy)−2(α+1/2) . n r log r r 0

n=N

Similarly, for |x −2 y −1 O2,1 (x, y)|, ∞ −2 −1 x y O2,1 (r, x, y) Cx −2 y −1 nk−2 r n (xy)−(α+1/2) n=N

holds, and

x

−4 −2

y

−2(α+1/2)

(xy)

∞ 2 1 1 2k−1 k−2 n dr log x −2 y −2 (xy)−2(α+1/2) n r r r 0

n=N

by using (3.14) with β = −2. We get the same bound for |x −1 y −2 O1,2 (x, y)| in a similar way. The estimate of |x −2 O2,0 (r, x, y)| by (xy)−2(α+1/2) C|x − y|−4 uses Lemma 2 with γ = −1 and = 0, and essentially is contained in the estimate of |x −1 O1,0 (r, x, y)| already discussed when proving (4.1) in the region x/2 < y < min{1, 3x/2}. The estimate of |y −2 O0,2 (r, x, y)| as well as |x −1 y −1 O1,1 (r, x, y)| follows analogously. The estimate of the term involving |x −1 O1,0 (r, x, y)| by C(xy)−2(α+1/2) |x − y|−4 uses Lemma 2 with γ = 0 and = 1, and essentially is contained in the estimate of |O0,0 (r, x, y)| already discussed when proving (4.1) in the considered region. The estimate for the term with |y −1 O0,1 (r, x, y)| follows analogously.


2201

It remains to consider the case of O0,0 (r, x, y). We use Lemma 2 with γ = 1 and = 2, to conclude that each of the terms of O0,0 (r, x, y) is a sum of series of the form ∞

n k+1

r n

n=N

A1 (x, y) A2 (x, y) sin (2) A0 + + + qn (x, y) πn(x ± y) . 2 cos n n

(5.4)

The estimate for the remainder follows from the bound |qn (x, y)| Cn−3 for 0 < x, y < 1. Indeed, in this case we have, using (3.14) with β = −2, (2)

−2(α+1/2)

(xy)

2

∞ 1 dr 1 2k−1 k n (2) sin log n r qn (x, y) πn(x ± y) cos r r n=N

0 −2(α+1/2)

1

C(xy)

1 log r

0

2k−1 ∞

2 n

k−2 n

r

n=N

dr C(xy)−2(α+1/2) r

which is enough for our purpose. The series resulting from taking into account either A1 or A2 were already discussed in Case 2 of Proposition 2 and are bounded by C(xy)−2(α+1/2) |x − y|−4 in the considered region. We are left with the series A0 . Note that 1

1 log r

0

2k−1 N −1

2 n

k+1 n

r

n=1

dr x −4 r

so it is possible to extend the summation in the series from n = 1. Now, we have to show 1

∞ 2 dr 1 2k−1 k+1 n sin n r log πn(x ± y) C|x − y|−4 . cos r r 0

n=1

In the case of the minus sign the estimate is a consequence of Lemma 4 with m = 2. For the plus sign we have to consider separately the cases x + y 3/2 and x + y > 3/2 and this can be done as in the previous proposition. ∂ ∂ k α Case 3: min{1, 3x/2} y < 1. We split the series defining ∂x (r ∂r ) Pr (x, y) into A and B, being

A=

N −1

nk r n sn,α dn,α sn,α Jα+1 (sn,α x)x −α φnα (y) 1/2

n=1

=

N −1 n=1

and

2 nk r n sn,α dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα+1 (sn,α x) · (sn,α y)1/2 Jα (sn,α y)

2202


B=

∞

nk r n sn,α dn,α sn,α Jα+1 (sn,α x)x −α φnα (y) 1/2

n=N

=

∞

2 nk r n sn,α dn,α (xy)−(α+1/2) (sn,α x)1/2 Jα+1 (sn,α x) · (sn,α y)1/2 Jα (sn,α y)

n=N

with N = [1/y]. Using (3.2) and (3.4) we get |A|

N −1

2 2 nk r n sn,α dn,α (xy)−α Jα+1 (sn,α x)Jα (sn,α y)

n=1

C(xy)−α

N −1

nk+2 r n Jα+1 (sn,α x)Jα (sn,α y)

n=1

Cx

N −1

n2α+k+3 r n .

n=1

Then, using (3.13) with β = 2α + 3, we obtain N −1 2 1 1 2k−1 2α+k+3 n dr log Cx 2 y −(4α+8) Cy −(4α+6) . n r x r r 2

n=1

0

To get the analogous estimate for |B| it is enough to show that for 0 < r < 1, 0 < x 2y/3, 0 < y < 1 and α > −1 2

∞ 1 1 2k−1 k n dr α+3/2 α x log Cx 2 y −(4α+8) . n r sn,α sn,α dn,α cos(sn,α x)φn (y) r r 2

0

(5.5)

n=N

Indeed, using (5.5), Minkowsky’s inequality and Poisson’s integral formula (3.5) applied to Jα (sn,α y), for α > −1, we obtain the result as in Case 1 in the proof of Proposition 2. To check (5.5) we can also proceed as in the proof of (4.2) in Proposition 2. 2 6. Proof of Theorem 1 The direct inequality. Observe that gk (f, x) = Gr,k f (x)Bk . Therefore, the boundedness of gk (f, x) in Lp ((0, 1), w dμα ) is equivalent to the boundedness of the operator Gr,k from p Lp ((0, 1), w dμα ) into LBk ((0, 1), w dμα ). By Proposition 1, Gr,k is a vector-valued Calderón– Zygmund operator. Hence, by the general theory, for any 1 < p < ∞ and w ∈ Aαp , Gr,k is p bounded from Lp ((0, 1), w dμα ) into LBk ((0, 1), w dμα ). The reverse inequality. By polarizing the isometry

gk (f ) 2 2

L ((0,1),dμα )

= 2−2k Γ (2k)f 2L2 ((0,1),dμ

α)


2203

of Lemma 1 we obtain 1 f1 (x)f2 (x) dμα (x) 0

22k = Γ (2k)

1 1 (log 1/r)2k−1

r

∂ ∂r

k

dr ∂ k α Prα f1 (x) r Pr f2 (x) dμα (x) ∂r r

0 0

which leads to the inequality 1 1 22k gk (f1 , x)gk (f2 , x) dμα (x). f1 (x)f2 (x) dμα (x) Γ (2k) 0

0

Taking h(x) = w(x)1/p f2 (x) we get 1 1 22k 1/p gk (f1 , x)w(x)1/p w(x)−1/p gk (h, x) dμα (x). f1 (x)w(x) f2 (x) dμα (x) Γ (2k) 0

0

By applying Holder’s inequality then 1

1/p f1 (x)w(x) f2 (x) dμα (x) C gk (f1 ) Lp ((0,1),w dμα ) gk (h) Lp ((0,1),w dμα ) 0 −1

−1

with w = w p−1 . Since if w ∈ Aαp then w p−1 ∈ Aαp and by the direct part of (1.9), we have

gk (h) p ChLp ((0,1),w dμα ) = Cf2 Lp ((0,1),dμα ) . L ((0,1),w dμ ) α

Therefore, we have the inequality 1

1/p f1 (x)w(x) f2 (x) dμα (x) C gk (f1 ) Lp ((0,1),w dμα ) f2 Lp ((0,1),dμα ) . 0

Taking supremum over all f2 with f2 Lp ((0,1),dμα ) 1, then 1 1/p f1 Lp ((0,1),w dμα ) = sup f1 (x)w(x) f2 (x) dμα (x) 0

and we get

f1 Lp ((0,1),w dμα ) C gk (f1 ) Lp ((0,1),w dμ ) . α

2204


References [1] D. Buraczewski, M.T. Martínez, J.L. Torrea, Calderón–Zygmund operators associated to ultraspherical expansions, Canad. J. Math. 59 (2007) 223–1244. [2] M. Christ, Lectures on Singular Integral Operators, CBMS Reg. Conf. Ser. Math., vol. 77, American Mathematical Society, Providence, RI, 1990. [3] Ó. Ciaurri, K. Stempak, Transplantation and multiplier theorems for Fourier–Bessel expansions, Trans. Amer. Math. Soc. 358 (2006) 4441–4465. [4] G. Garrigós, E. Harboure, T. Signes, J.L. Torrea, B. Viviani, A sharp weighted transplantation theorem for Laguerre function expansions, J. Funct. Anal. 244 (2007) 247–276. [5] C.E. Gutiérrez, On the Riesz transforms for Gaussian measures, J. Funct. Anal. 120 (1994) 107–134. [6] E. Harboure, L. de Rosa, C. Segovia, J.L. Torrea, Lp -dimension free boundedness for Riesz transforms associated to Hermite functions, Math. Ann. 328 (2004) 653–682. [7] H. Hochstadt, The mean convergence of Fourier–Bessel series, SIAM Rev. 9 (1967) 211–218. [8] N.N. Lebedev, Special Functions and Its Applications, Dover, New York, 1972. [9] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110 (1990) 639–647. [10] B. Muckenhoupt, E.M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965) 17–92. [11] A. Nowak, On Riesz transforms for Laguerre expansions, J. Funct. Anal. 215 (2004) 217–240. [12] A. Nowak, K. Stempak, Riesz transforms for multi-dimensional Laguerre function expansions, Adv. Math. 215 (2007) 642–678. [13] J.L. Rubio de Francia, F. Ruiz, J.L. Torrea, Calderón–Zygmund theory for operator-valued kernels, Adv. Math. 62 (1988) 221–243. [14] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1971. [15] S. Thangavelu, Multipliers for Hermite expansions, Rev. Mat. Iberoamericana 3 (1987) 1–24. [16] S. Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, NJ, 1993. [17] A. Zygmund, Trigonometric Series, vols. I and II, Cambridge University Press, Cambridge, 1959.


On Sobolev extension domains in Rn Pavel Shvartsman Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel Received 14 April 2009; accepted 4 January 2010 Available online 15 January 2010 Communicated by H. Brezis

Abstract We describe a class of Sobolev Wpk -extension domains Ω ⊂ Rn determined by a certain inner subhyperbolic metric in Ω. This enables us to characterize finitely connected Sobolev Wp1 -extension domains in R2 for each p > 2. © 2010 Elsevier Inc. All rights reserved. Keywords: Sobolev space; Extension; Domain; Inner metric

1. Introduction Let Ω be a domain in Rn . This paper is devoted to the problem of extendability of functions from the Sobolev space Wpk (Ω) to functions from Wpk (Rn ). We recall that, given k ∈ N and p ∈ [1, ∞], the Sobolev space Wpk (Ω), see e.g. Maz’ja [23], consists of all functions f ∈ L1,loc (Ω) whose distributional partial derivatives on Ω of all orders up to k belong to Lp (Ω). Wpk (Ω) is normed by f Wpk (Ω) :=

D α f

Lp (Ω)

: |α| k .

A domain Ω in Rn is said to be a Sobolev Wpk -extension domain if there exists a continuous linear extension operator EΩ : Wpk (Ω) → Wpk Rn , E-mail address: [email protected]. 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.002

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P. Shvartsman / Journal of Functional Analysis 258 (2010) 2205–2245

see, e.g. [23, §1.1], or [24, §1.6]. In other words, Ω is a Sobolev extension domain (for the space Wpk (Rn )) if every Sobolev function f ∈ Wpk (Ω) can be extended to a Sobolev Wpk -function F defined on all of Rn . Moreover, one can choose the function F in such a way so that it depends on f linearly and satisfies the inequality F Wpk (Rn ) Cf Wpk (Ω) , where C is a constant depending only on n, k, p, and Ω. For instance, Lipschitz domains (Calderón [9], 1 < p < ∞, Stein [31], p = 1, ∞) in Rn are k Wp -extension domains for every p ∈ [1, ∞] and every k ∈ N. Jones [20] introduced a wider class of (ε, δ)-domains and proved that every (ε, δ)-domain is a Sobolev Wpk -extension domain in Rn for every k 1 and every p 1. Burago and Maz’ya [8], [23, Chapter 6], described extension domains for the space BV(Rn ) of functions whose distributional derivatives of the first order are finite Radon measures. Our main result is the following Theorem 1.1. Let n 0 such that the following condition is satisfied: for every x, y ∈ Ω such that x − y θ , there exists a rectifiable curve γ ⊂ Ω joining x to y such that

1−n

p−n

dist(z, ∂Ω) p−1 ds(z) Cx − y p−1 .

(1.1)

γ

Here ∂Ω denotes the boundary of Ω and ds denotes arc length measure. Then Ω is a Sobolev Wqk -extension domain for every k 1 and every q > p˜ where p˜ ∈ (n, p) is a constant depending only on n, p and C. For k = 1 and q > p this result has been proved by Koskela [21]. Observe that this theorem is also known for the case p = ∞ (with p˜ = q = ∞). In that case every domain Ω satisfying inequality (1.1) is quasi-Euclidean, i.e., its inner metric is (locally) equivalent to the Euclidean distance. This case was studied by Whitney [33] who proved that k -extension domain for every k 1. every quasi-Euclidean domain is a W∞ Our next result, Theorem 1.2, relates to description of Sobolev extension domains in R2 . The first result in this direction was obtained by Gol’dshtein, Latfullin and Vodop’janov [15–17] who proved that a finitely connected bounded planar domain Ω is a Sobolev W21 -extension domain if and only if its boundary is a quasicircle, i.e., the image of a circle under a quasiconformal mapping of the plane onto itself. Maz’ja [23,24] gave an example of a simply connected domain Ω ⊂ R2 such that Ω is a Wp1 -extension domain for every p ∈ [1, 2), while R2 \ Ω is a Wp1 extension domain for all p > 2. However the boundary of Ω is not a quasicircle. Buckley and Koskela [4] showed that if a finitely connected bounded domain Ω ⊂ R2 is a Sobolev Wp1 -extension domain for some p > 2, then there exists a constant C > 0 such that for every x, y ∈ Ω there exists a rectifiable curve γ ⊂ Ω satisfying inequality (1.1) (with n = 2). Combining this result with Theorem 1.1, we obtain the following Theorem 1.2. Let 2 0 the following condition is


2207

satisfied: for every x, y ∈ Ω there exists a rectifiable curve γ ⊂ Ω joining x to y such that p−2 1 dist(z, ∂Ω) 1−p ds(z) Cx − y p−1 . (1.2) γ 1 can even be We note that this result is also true for the case p = ∞ and then the space W∞ k replaced by W∞ for arbitrary k 1. This follows from the aforementioned theorem of Whitney [33] combined with a result of Zobin [35] which states that every finitely connected bounded k -extension domain is quasi-Euclidean. Zobin [34] also showed that for every k 2 planar W∞ k -extension domain which is not quasi-Euclidean. there exists a bounded planar W∞ Buckley and Koskela [4] proved that a finitely connected bounded planar domain Ω satisfies the condition (1.2) if and only if the following imbedding

Wp1 (Ω) → C 0,α (Ω)

with α = 1 − 2/p

holds. (See also Koskela [21, Corollary 4.1].) Here C 0,α (Ω) denotes the Hölder space of bounded functions on Ω equipped with the norm f C 0,α (Ω) := sup f (x) + x∈Ω

sup

x,y∈Ω, x=y

|f (x) − f (y)| . x − yα

This result, Theorem 1.2 and Theorem 1.1 imply the following Corollary 1.3. Let 2 < p < ∞ and let Ω be a finitely connected bounded planar domain. Then (i) Ω is a Sobolev Wp1 -extension domain if and only if Wp1 (Ω) → C (ii) If Wp1 (Ω) → C

0,1− p2

0,1− p2

(Ω).

0,1− q2

(Ω)

(Ω), then Wq1 (Ω) → C

for some q ∈ (2, p). Let us briefly indicate the main ideas of our approach for the case k = 1, i.e., for the Sobolev space Wp1 (Rn ). Recall that, when p > n, it follows from the Sobolev embedding theorem that every function f ∈ Wp1 (Ω), p > n, can be redefined, if necessary, on a subset of Ω of Lebesgue measure zero so that it satisfies a local Hölder condition of order α := 1 − pn on Ω: i.e., for every ball B ⊂ Ω f (x) − f (y) C(n, p)f

Wp1 (Ω) x

− y

1− pn

,

x, y ∈ B.

(1.3)

We will identify each element of Wp1 (Ω) with its unique continuous representative. Thus we will be able to restrict our attention to the case of continuous Sobolev functions.

2208


Following Buckley and Stanoyevitch [6], given α ∈ [0, 1] and a rectifiable curve γ ⊂ Ω, we define the subhyperbolic length of γ by lenα,Ω (γ ) :=

dist(z, ∂Ω)α−1 ds(z). γ

Then we let dα,Ω denote the corresponding subhyperbolic metric on Ω given, for each x, y ∈ Ω, by dα,Ω (x, y) := inf lenα,Ω (γ )

(1.4)

γ

where the infimum is taken over all rectifiable curves γ ⊂ Ω joining x to y. The metric dα,Ω was introduced and studied by Gehring and Martio in [14]. See also [1,22,4] for various further results using this metric. Note also that len0,Ω and d0,Ω are the well-known quasihyperbolic length and quasihyperbolic distance, and d1,Ω is the inner (or geodesic) metric on Ω. The subhyperbolic metric dα,Ω with α = (p − n)/(p − 1) arises naturally in the study of Sobolev Wp1 (Ω)-functions for p > n. In particular, Buckley and Stanoyevitch [5] proved that the local Hölder condition (1.3) is equivalent to the following Hölder-type condition: for every x, y ∈ Ω 1− p1 1− n f (x) − f (y) C(n, p)f 1 + x − y p Wp (Ω) dα,Ω (x, y)

(1.5)

with α = (p − n)/(p − 1). In turn, since any extension F ∈ Wp1 (Rn ) of f satisfies the global Hölder condition F (x) − F (y) C(n, p)F

Wp1 (Rn ) x

− y

1− pn

,

x, y ∈ Rn ,

1− pn

,

x, y ∈ Ω.

we have f (x) − f (y) C(n, p)F

Wp1 (Rn ) x

− y

(1.6)

Of course the conditions (1.5) and (1.6) with f Wp1 (Ω) and F Wp1 (Rn ) replaced by unspecified constants are not equivalent to membership of f in Wp1 (Ω) or in Wp1 (Rn )|Ω respectively. However the preceding remarks suggest that a reasonable property which might perhaps be necessary or perhaps sufficient for a domain Ω to be a Sobolev extension domain could be this: Whenever a function f : Ω → R satisfies 1 n f (x) − f (y) dα,Ω (x, y)1− p + x − y1− p for all x, y ∈ Ω and α = (p − n)/(p − 1) then it also satisfies n f (x) − f (y) C(n, p)x − y1− p for all x, y ∈ Ω and for some constant C(n, p) depending only on n and p.


2209

One would like to have a simpler condition on Ω which would be sufficient to imply the above “reasonable property”. It is clear that the following property, which has already been considered and studied by other authors, namely 1− p1

dα,Ω (x, y)

Cx − y

1− pn

for all x, y ∈ Ω and α = (p − n)/(p − 1)

or, equivalently, dα,Ω (x, y) Cx − yα for all x, y ∈ Ω, is such a condition. These considerations lead us to work with a certain class of domains, essentially those which were introduced in [14]. In our context here, it seems convenient to use terminology different from that of [14] and other papers. Definition 1.4. For each α ∈ (0, 1], the domain Ω ⊂ Rn is said to be α-subhyperbolic if there exist constants Cα,Ω > 0 and θα,Ω > 0 such that dα,Ω (x, y) Cα,Ω x − yα for every x, y ∈ Ω satisfying x − y θα,Ω . We denote the class of α-subhyperbolic domains in Rn by Uα (Rn ). In [14] and also in [22] these domains are called “Lipα -extension domains”. (This name is derived from the fact that Ω ∈ Uα (Rn ) iff all functions which are locally Lipschitz of order α on Ω are Lipschitz of order α on Ω.) These domains have also been studied in [5,7,6] where they are called “(α − m)-cigar domains”, and in [4] where they are termed “local weak α-cigar domains”. Now Theorem 1.1 can be reformulated as follows: For each p > n and for each p−n p−1 subhyperbolic domain Ω in Rn , there exists a constant p˜ ∈ (n, p) depending only on n, p and ˜ Ω, such that Ω is a Sobolev Wqk -extension domain for every q p. In turn, Theorem 1.2 admits the following reformulation: For each p > 2, a finitely connected bounded domain Ω ⊂ R2 is a Sobolev Wp1 -extension domain if and only if Ω is a p−2 p−1 -subhyperbolic domain. The family {Uα (Rn ): α ∈ (0, 1]}

is an “increasing family”, i.e.,

Uα Rn ⊂ Uα Rn

whenever 0 < α < α 1,

see, e.g. [4]. Lappalainen [22] proved that

Uα Rn Uτ R n

for every α ∈ (0, 1).

α n and k 1): Let Ω be an α-subhyperbolic domain in Rn with α = (p − n)/(p − 1). Given f ∈ C k−1 (Ω) and x ∈ Ω we let Txk−1 (f ) denote the Taylor polynomial of f at x of degree at most k − 1. We prove that there exist p˜ ∈ (n, p) and constants θ, λ, C > 0 such that for every function f ∈ C k−1 (Ω) ∩ Wpk (Ω) and every x, y ∈ Ω, x − y θ , the following inequality


n f (y) − T k−1 (f )(y) Cx − yk− p˜ x

2211

1 p˜ k p˜ ∇ f du

B∩Ω

holds. Here B = B(x, λx − y) is the ball centered at x of radius r = λx − y. This inequality is a particular case of Theorem 3.1 which we prove in Section 3. In Section 4 we prove a corollary of this result related to the sharp maximal function fk,Ω (x) := sup r −k

r>0

inf

P ∈Pk−1

1 |B(x, r)|

|f − P | du,

x ∈ Ω.

B(x,r)∩Ω

Here Pk−1 is the space of polynomials of degree at most k − 1 defined on Rn and |B(x, r)| is the Lebesgue measure of the ball B(x, r). We show that for every f ∈ Wpk (Ω) and every x ∈ Ω the following inequality p˜ 1

(x) p˜ + M f (x) fk,Ω (x) C M ∇ k f

(1.12)

holds. Here M denotes the Hardy–Littlewood maximal function and the symbol g stands for the extension by zero of a function from Ω to all of Rn . The sharp maximal function is a useful tool in the study of Sobolev functions. In [10] Calderón

proved that, for p > 1, a function f is in Wpk (Rn ) if and only if f and fk,Rn are both in n Lp (R ). In [28] this description has been generalized to the case of the so-called regular subsets of Rn , i.e., the sets S such that |B ∩ S| ∼ |B| for all balls B centered in S of radius at most 1. We proved in [28] that if S is regular and f ∈ Lp (S), p > 1, then f can be extended

to a function F ∈ Wpk (Rn ) if and only if its sharp maximal function fk,S ∈ Lp (S). (For the case k = 1 see also [29,18,19].) Observe that every Sobolev Wpk -extension domain, 1 p < ∞, is a regular subset of Rn , see Hajlasz, Koskela and Tuominen [18]. Also note that Rychkov [27] proved that for every regular set S ⊂ Rn there exists a continuous linear extension operator ES : Wpk (Rn )|S → Wpk (Rn ); here Wpk (Rn )|S denotes the space of all restrictions F |S of the Sobolev functions F ∈ Wpk (Rn ) equipped with the standard quotient space norm. In [30] we present a description of the trace space Wp1 (Rn )|S , p > n, for an arbitrary set S ⊂ Rn via an L∞ -version of the sharp maximal function. Every subhyperbolic domain is a regular set, as shown in Lemma 2.3. So, in order to prove, for some given q > p, ˜ that a function f ∈ Wqk (Ω) extends to a Sobolev Wqk -function on Rn ,

it suffices to show that fk,Ω ∈ Lq (Ω). We do this by applying the Hardy–Littlewood maximal

theorem to inequality (1.12). This gives us the inequality fk,Ω Lq (Ω) Cf Wqk (Ω) which completes the proof of Theorem 1.1. 2. Subhyperbolic domains: Intrinsic metrics and self-improvement Throughout the paper C, C1 , C2 , . . . will be generic positive constants which depend only on parameters determining sets (say, n, α, the constants Cα,Ω or θα,Ω , etc.) or function spaces (p, q, etc.). These constants can change even in a single string of estimates. The dependence of a constant on certain parameters is expressed, for example, by the notation C = C(n, p).

2212


The Lebesgue measure of a measurable set A ⊂ Rn will be denoted by |A|. Given subsets A, B ⊂ Rn , we put diam A := sup{a − a : a, a ∈ A} and dist(A, B) := inf a − b: a ∈ A, b ∈ B . For x ∈ Rn we also set dist(x, A) := dist({x}, A). Let γ : [a, b] → Rn be a curve in Rn , and let u = γ (t1 ), v = γ (t2 ) where a t1 < t2 b. By γuv we denote the arc of γ joining u to v. We will be needed the following auxiliary lemma. Lemma 2.1. (i) Let x, y ∈ Ω and let max dist(x, ∂Ω), dist(y, ∂Ω) 2x − y.

(2.1)

Let γ be a rectifiable curve joining x to y in Ω. Assume that for some α ∈ (0, 1) and C > 0 the following inequality dist(z, ∂Ω)α−1 ds(z) Cx − yα

(2.2)

γ

holds. Then length(γ ) 2eC x − y. (ii) Let x, y ∈ Ω and let max dist(x, ∂Ω), dist(y, ∂Ω) > 2x − y.

(2.3)

Then the line segment [x, y] ⊂ Ω and for every β ∈ (0, 1] we have dist(z, ∂Ω)β−1 ds(z) x − yβ .

(2.4)

[x,y]

Proof. (i) Let us parameterize γ by arclength; thus we identify γ with a function γ : [0, ] → Ω satisfying γ (0) = x, γ () = y. Now (2.2) is equivalent to the inequality

α−1 dist γ (t), ∂Ω dt Cx − yα .

(2.5)

0

Since dist(·, ∂Ω) is a Lipschitz function on Rn , dist(u, ∂Ω) dist(v, ∂Ω) + u − v,

u, v ∈ Ω,

(2.6)


2213

so that for every t ∈ (0, ] dist γ (t), ∂Ω dist(x, ∂Ω) + x − γ (t). Since γ is parameterized by arclength, x − γ (t) length(γxγ (t) ) = t, so that dist γ (t), ∂Ω dist(x, ∂Ω) + t,

t ∈ [0, ].

This inequality and (2.5) imply Cx − y α

α−1 dist γ (t), ∂Ω dt

0

α−1 dist(x, ∂Ω) + t dt

0

α =α dist(x, ∂Ω) + − dist(x, ∂Ω)α α −1 α − dist(x, ∂Ω)α . −1

But 2x − y dist(x, ∂Ω) so that α Cx − yα α −1 α − 2x − y . Hence 1 αC + 2α α x − y 2eC x − y proving (i). (ii) Clearly, (2.3) implies [x, y] ⊂ Ω. Prove (2.4). We may assume that dist(x, ∂Ω) > 2x − y. Also note that x − z x − y for every z ∈ [x, y]. These inequalities and (2.6) imply the following: 1 dist(x, ∂Ω) dist(x, ∂Ω) − x − y dist(x, ∂Ω) − x − z dist(z, ∂Ω). 2 Hence,

β−1

dist(z, ∂Ω) [x,y]

ds(z)

21−β dist(x, ∂Ω)β−1 ds(z)

[x,y]

= 21−β x − y dist(x, ∂Ω)β−1 β−1 21−β x − y 2x − y = x − yβ proving the lemma.

2

2214


Lemma 2.2. Let x, y ∈ Ω and let γ ⊂ Ω be a rectifiable curve joining x to y. Suppose that for some α ∈ (0, 1) and C 1 the following inequality dist(z, ∂Ω)α−1 ds(z) C lengthα (γ )

(2.7)

γ

holds. Then (i) There exists a point z¯ ∈ γ such that 1

length(γ ) C 1−α dist(¯z, ∂Ω). (ii) We have 1 length(γ )

dist(z, ∂Ω)α−1 ds(z) 2C inf dist(z, ∂Ω)α−1 . z∈γ

γ

Proof. (i) Put := length(γ ). Let z¯ be a point on the curve γ such that max dist(z, ∂Ω): z ∈ γ = dist(¯z, ∂Ω). Then

α−1

dist(z, ∂Ω) γ

ds(z)

dist(¯z, ∂Ω)α−1 ds(z) = dist(¯z, ∂Ω)α−1 γ

so that, by (2.7), dist(¯z, ∂Ω)α−1 Cα . Hence 1

C 1−α dist(¯z, ∂Ω) proving (i). (ii) Put w(z) := dist(z, ∂Ω). Then, by (2.7), 1

w(z)α−1 ds(z) −1 Cα = Cα−1 .

γ

For every z1 , z2 ∈ γ we have w(z1 ) − w(z2 ) = dist(z1 , ∂Ω) − dist(z2 , ∂Ω) z1 − z2

(2.8)


2215

so that max w(z) min w(z) + .

(2.9)

z∈γ

z∈γ

Let us consider two cases. First suppose that maxz∈γ w(z) 2. Since α ∈ (0, 1), we obtain α−1 21−α min w(z)α−1 z∈γ

so that, by (2.8), 1

w(z)α−1 ds(z) 21−α C min w(z)α−1 2C min w(z)α−1 . z∈γ

z∈γ

γ

Now assume that 2 < maxz∈γ w(z). Then, by (2.9), max w(z) min w(z) + z∈γ

z∈γ

1 max w(z) 2 z∈γ

so that maxz∈γ w(z) 2 minz∈γ w(z). Hence max w(z)α−1 21−α min w(z)α−1 2 min w(z)α−1 . z∈γ

z∈γ

z∈γ

Finally, we have 1

w(z)α−1 ds(z) max w(z)α−1 2 min w(z)α−1 . z∈γ

z∈γ

γ

The lemma is proved.

2

This lemma implies the following important property of subhyperbolic domains. Lemma 2.3. Let α ∈ (0, 1) and let Ω be an α-subhyperbolic domain. There exist constants δ > 0 and σ ∈ (0, 1] depending only on n, α, Cα,Ω and θα,Ω such that every ball B centered in Ω of diameter at most δ contains a ball B ⊂ Ω of diameter at least σ diam B. Proof. Let δ := min{θα,Ω , 12 diam Ω}. Let B = B(x, r) be a ball with center in x ∈ Ω and radius r δ. Put r˜ := r/(8eCα,Ω ). Since r˜ δ

1 diam Ω, 2

there exists a point a ∈ Ω such that x − a > r˜ . Let Γ ⊂ Ω be a curve joining x to a. Since a∈ / B(x, r˜ ), we have Γ ∩ ∂(B(x, r˜ )) = ∅ so that there exists a point b ∈ Ω such that x − b = r˜ .

2216


If max dist(x, ∂Ω), dist(b, ∂Ω) > 2x − b = 2˜r , then either B(x, r˜ ) ⊂ B(x, r) ∩ Ω or B(b, r˜ ) ⊂ B(x, r) ∩ Ω, so that in this case the ball B exists. Suppose that max dist(x, ∂Ω), dist(b, ∂Ω) 2x − b = 2˜r . Since r˜ δ θα,Ω , there exists a curve γ ⊂ Ω joining x to b such that dist(z, ∂Ω)α−1 ds(z) Cα,Ω x − bα . γ

(We may assume that Cα,Ω 1.) By Lemma 2.1, part (i), length(γ ) 2eCα,Ω x − b = 2eCα,Ω r˜ = r/4. Moreover, by part (ii) of Lemma 2.2, there exists a point z¯ ∈ γ such that 1

1−α length(γ ) Cα,Ω dist(¯z, ∂Ω).

Hence, 1

1−α r˜ = x − b length(γ ) Cα,Ω dist(¯z, ∂Ω). 1

1−α ) and B := B(¯z, r ). Then, r r˜ r/4 (recall that Cα,Ω 1) and r < Put r := r˜ /(2Cα,Ω dist(¯z, ∂Ω). Hence, B = B(¯z, r ) ⊂ Ω. On the other hand,

x − z¯ length(γ ) r/4 so that B ⊂ B(x, r/2) ⊂ B. The lemma is proved.

2

Before to present the proof of Theorem 1.5 let us demonstrate its main ideas for a family of the so-called strongly subhyperbolic domains in Rn . Definition 2.4. Let α ∈ (0, 1]. A domain Ω ⊂ Rn is said to be strongly α-subhyperbolic if there exist constants C, θ > 0 such that every x, y ∈ Ω, x − y θ , can be joined by a rectifiable curve γ ⊂ Ω satisfying the following condition: for every u, v ∈ γ dist(z, ∂Ω)α−1 ds(z) Cu − vα . γuv

(2.10)


2217

Remark 2.5. Given x, y ∈ Ω a rectifiable curve γ ⊂ Ω joining x to y is said to be dα,Ω -geodesic if dα,Ω (x, y) = lenα,Ω (γ ) :=

dist(z, ∂Ω)α−1 ds(z). γ

(See definition (1.4).) Clearly, if Ω is α-subhyperbolic and for every x, y ∈ Ω there exists dα,Ω -geodesic, then Ω is strongly α-subhyperbolic. In fact, in this case every arc of dα,Ω -geodesic curve is dα,Ω -geodesic as well so that inequality (2.10) holds. However, for every α ∈ (0, 1] there exists a domain Ω ∈ Uα (Rn ) and x, y ∈ Ω such that dα,Ω geodesic for x, y does not exist. This is trivial for α = 1, i.e., for quasi-Euclidean domains. For the case α ∈ (0, 1) see [5]. Let us slightly generalize this example. Fix C 1. We say that a rectifiable curve γ ⊂ Ω joining x to y is (C, dα,Ω )-geodesic if for every u, v ∈ γ the following inequality lenα,Ω (γuv ) Cdα,Ω (u, v), holds. Clearly, a rectifiable curve γ is (1, dα,Ω )-geodesic iff it is dα,Ω -geodesic. Moreover, if Ω ∈ Uα (Rn ) and for every x, y ∈ Ω, x − y θ , there exists (C, dα,Ω )-geodesic joining x to y in Ω, then Ω is strongly α-subhyperbolic. This observation motivates the following question: Let Ω be a domain in Rn and let α ∈ (0, 1]. Does there exist a constant C = CΩ > 1 such that every two points x, y ∈ Ω can be joined by a (C, dα,Ω )-geodesic curve? Even for the quasi-Euclidean domains, i.e., for α = 1, we do not know the answer to this question. 2 Proposition 2.6. Let α ∈ (0, 1) and let Ω be a strongly α-subhyperbolic domain in Rn . Then Ω is τ -subhyperbolic for some τ ∈ (0, α). Proof. Since Ω is strongly α-subhyperbolic, there exist constants θ > 0 and C 1 such that every x, y ∈ Ω, x − y θ , can be joined by a rectifiable curve γ ⊂ Ω satisfying the following condition: for every u, v ∈ γ dist(z, ∂Ω)α−1 ds(z) Cu − vα . γuv

In particular, dist(z, ∂Ω)α−1 ds(z) Cx − yα .

(2.11)

γ

Let := length(γ ). We parameterize γ by arclength: thus γ : [0, ] → Ω, γ (0) = x, γ () = y. Let u = γ (t1 ), v = γ (t2 ) where 0 t1 < t2 . Recall that by γuv we denote the arc of γ joining u to v.

2218


Applying part (ii) of Lemma 2.2 to the arc γuv , we obtain 1 length(γuv )

dist(z, ∂Ω)α−1 ds(z) 2C inf dist(z, ∂Ω)α−1 , z∈γuv

γuv

or, in the parametric form, 1 t2 − t1

t2

α−1 α−1 dist γ (t), ∂Ω dt 2C inf dist γ (t), ∂Ω . t∈[t1 ,t2 ]

t1

(2.12)

Put w(t) := dist γ (t), ∂Ω .

(2.13)

By (2.12), this function possesses the following property: for every subinterval I ⊂ [0, ] 1 |I |

w(t)α−1 dt 2C inf w(t)α−1 . I

I

Thus the function h := w α−1 is a Muckenhoupt’s A1 -weight on [0, ], see, e.g., [12]. Recall that every A1 -weight satisfies the reverse Hölder inequality on [0, ] (see Muckenhoupt [26], Gehring [13], Coifman and Fefferman [11]) so that there exist constants q˜ > 1 and C1 1 (depending only on C) such that 1/q˜ 1 1 q˜ h (t) dt C1 h(t) dt. 0

0

Then, by (2.13) and (2.11), 1/q˜ 1 1 1 (α−1)q˜ w (t) dt C1 w α−1 (t) dt C1 C x − yα . 0

0

We put q := min{q, ˜ 1−α/2 1−α } and τ = q(α − 1) + 1. Clearly, 1 < q q˜ and 0 < τ < α. Hence, 1/q 1/q 1 1 τ −1 q(α−1) w (t) dt = w (t) dt 0

0

1/q˜ 1 1 q(α−1) ˜ w (t) dt C2 x − yα 0

where C2 := C1 C. Finally, we obtain


τ −1

dist(z, ∂Ω)

ds(z) =

γ

w τ −1 (t) dt

2219

q

C2 x − yαq q−1

0 q

q

C2 x − yαq−q+1 = C2 x − yτ proving that Ω is a τ -subhyperbolic domain.

2

Proof of Theorem 1.5. Let ε > 0 and let Ω ∈ Uα (Rn ). We will assume that the constant Cα,Ω 1. Put 1 θ := e−2Cα,Ω θα,Ω 2

(2.14)

and fix x, y ∈ Ω such that x − y θ . By part (ii) of Lemma 2.1, if inequality (2.3) is satisfied, then the statement of Theorem 1.5 is true with Γ = Γ = [x, y] and any α ∗ ∈ (0, α). Now suppose that x, y satisfy inequality (2.1), i.e., max dist(x, ∂Ω), dist(y, ∂Ω) 2x − y. Lemma 2.7. Let Ω ∈ Uα (Rn ) and let x, y ∈ Ω, x − y θ . Let 0 < δ dα,Ω (x, y) and let γ ⊂ Ω be a rectifiable curve joining x to y such that dist(z, ∂Ω)α−1 ds(z) < dα,Ω (x, y) + δ.

(2.15)

γ

Then: (i) We have dist(z, ∂Ω)α−1 ds(z) 2Cα,Ω x − yα γ

and length(γ ) 2e2Cα,Ω x − y. (ii) For every u, v ∈ γ such that 1

length(γuv ) δ α the following inequality dist(z, ∂Ω)α−1 ds(z) 2Cα,Ω lengthα (γuv ) γuv

holds.

(2.16)

2220


Proof. First prove that dist(z, ∂Ω)α−1 ds(z) < dα,Ω (u, v) + δ. γuv

In fact, assume that dα,Ω (u, v) + δ

dist(z, ∂Ω)α−1 ds(z).

γuv

Then

dist(z, ∂Ω)α−1 ds(z) = γ

dist(z, ∂Ω)α−1 ds(z) +

γxu

γuv

+

dist(z, ∂Ω)α−1 ds(z)

dist(z, ∂Ω)α−1 ds(z)

γvy

dα,Ω (x, u) + dα,Ω (u, v) + δ + dα,Ω (v, y) so that, by the triangle inequality for the metric dα,Ω , dist(z, ∂Ω)α−1 ds(z) dα,Ω (x, y) + δ γ

which contradicts inequality (2.15). Since 0 < δ dα,Ω (x, y), by (2.15), dist(z, ∂Ω)α−1 ds(z) < 2dα,Ω (x, y). γ

Since θ θα,Ω and Ω ∈ Uα (Rn ), we have dα,Ω (x, y) Cα,Ω x − yα so that dist(z, ∂Ω)α−1 ds(z) < 2Cα,Ω x − yα . γ

By Lemma 2.1, part (i), length(γ ) 2e2Cα,Ω x − y proving (i). This inequality and (2.14) imply

length(γ ) 2e2Cα,Ω θ = 2e2Cα,Ω

1 −2Cα,Ω e θα,Ω = θα,Ω 2

(2.17)


2221

so that for every u, v ∈ γ we have u − v length(γ ) θα,Ω . Since Ω ∈ Uα (Rn ), we have dα,Ω (u, v) Cα,Ω u − vα Cα,Ω lengthα (γuv ). Combining this inequality with (2.17) and (2.16), we obtain dist(z, ∂Ω)α−1 ds(z) Cα,Ω lengthα (γuv ) + δ γuv

Cα,Ω lengthα (γuv ) + lengthα (γuv ) 2

proving (ii) and the lemma. Put

1 m := 2(2Cα,Ω ) 1−α + 1.

(2.18)

Let k be a positive integer such that 2e2Cα,Ω x − y(1 − 1/m)k < ε.

(2.19)

Finally, we put δ := min dα,Ω (x, y), m−αk x − yα . Thus 0 < δ dα,Ω (x, y) and 1

x − ym−k δ α .

(2.20)

Let Γ ⊂ Ω be a rectifiable curve joining x to y such that dist(z, ∂Ω)α−1 ds(z) < dα,Ω (x, y) + δ. Γ

Then, by Lemma 2.7, dist(z, ∂Ω)α−1 ds(z) 2Cα,Ω x − yα

(2.21)

Γ

and length(Γ ) 2e2Cα,Ω x − y.

(2.22)

2222


Moreover, for every u, v ∈ Γ such that 1

length(Γuv ) δ α

(2.23)

the following inequality dist(z, ∂Ω)α−1 ds(z) 2Cα,Ω lengthα (Γuv )

(2.24)

Γuv

holds. Inequality (2.24) and part (ii) of Lemma 2.2 imply the following: 1 length(Γuv )

dist(z, ∂Ω)α−1 ds(z) 4Cα,Ω inf dist(z, ∂Ω)α−1 . z∈Γuv

(2.25)

Γuv

Put L := length(Γ ). Since x − y L, by (2.20) and (2.23), for every u, v ∈ Γ such that length(Γuv ) Lm−k

(2.26)

inequality (2.24) is satisfied. By Im we denote the family of all m-adic closed subintervals of the interval I0 := [0, L]. Recall that this family of intervals can be obtained by the standard iterative procedure: we start with the entire interval [0, L] and, at each level of the construction, we split every interval of the given level into m equally sized closed subintervals. Let Ij,m := {all m-adic intervals of the j -th level}. Thus I0,m := {[0, L]}, I1,m :=

Li/m, L(i + 1)/m : i = 0, 1, . . . , m − 1 ,

etc. Clearly, |I | = Lm−j for every interval I ∈ Ij,m . Put Sk,m :=

k

Ij,m = {all m-adic intervals of the level at most k}.

j =0

Let us parameterize Γ by arclength; thus we identify Γ with a function Γ : [0, L] → Ω satisfying Γ (0) = x, Γ (L) = y. Finally, put α−1 , g(t) := dist Γ (t), ∂Ω

t ∈ [0, L].


2223

Then, by (2.26) and (2.25), the following is true: For each m-adic interval I ∈ Sk,m we have 1 |I |

g(t) dt Cg inf g(t)

(2.27)

t∈I

I

with Cg := 4Cα,Ω . Following Melas [25] we say that g is a Muckenhoupt A1 -weight on [0, L] with respect to the family S := Sk,m of all m-adic intervals of the level at most k. We let MS denote the corresponding maximal operator for the family S:

1 MS g(t) := sup |I |

g(u) du: I t, I ∈ S .

(2.28)

I

Thus (2.27) is equivalent to the inequality MS g(t) Cg g(t),

t ∈ [0, L].

Put q :=

log m log(m − (m − 1)/Cg )

and q ∗ := (1 + q )/2. Clearly, 1 Cg < ∞ so that q , q ∗ > 1. We will be needed the following corollary of a general result proved in [25]. Theorem 2.8. For any A1 -weight g (with respect to S) and any q ∈ [1, q ∗ ] the following inequality

1 L

1

L

q

q

(MS g) dt

L 1 C g dt L

0

(2.29)

0

is a constant depending only on m and Cg . holds. Here C depending on m, Cg and q, Remark 2.9. Actually the theorem is true for q ∈ [1, q ) but with C see [25]. Corollary 2.10. For any A1 -weight g (with respect to S), any family A of non-overlapping madic intervals of the level at most k and any q, 1 q q ∗ , we have

L

q 1 q 1 1 1 g dt |I | C g dt . L |I | L I ∈A

I

0

(2.30)

2224


Proof. In fact, by definition (2.28), for every I ∈ A and every t ∈ I 1 |I |

g ds MS g(t) I

so that

1 |I |

q

g dt

|I |

I

(MS g)q dt. I

Therefore the left-hand side of (2.30) does not exceed

q 1 q (MS g) dt L 1

I ∈A I

1 L

1

L

q

(MS g)q dt 0

which together with (2.29) implies the required inequality (2.30).

2

We turn to construction of a family A ⊂ Sk,m of non-overlapping m-adic intervals of the level at most k such that for each I ∈ A sup g C inf g I

I

and I : I ∈ A < ε. [0, L] \ Here C is a constant depending only on n, α, and Cα,Ω . Let I = [t1 , t2 ] ∈ Sk−1,m be an m-adic interval of the level at most k − 1 and let u := Γ (t1 ), v := Γ (t2 ). By (2.24) and part (i) of Lemma 2.2, there exists tI ∈ I such the point zI = Γ (tI ) ∈ Γuv satisfies the following inequality: length(Γuv ) C dist(zI , ∂Ω) 1

(2.31)

with C := (2Cα,Ω ) 1−α . Let us split the interval I into m equal subintervals I (1) , . . . , I (m) . Then tI ∈ I (j ) for some j ∈ {1, . . . , m}. By Γ (j ) := Γ |I (j ) we denote the arc corresponding to the interval I (j ) . Thus I (j ) ∈ Sk,m is an m-adic interval of the level at most k and length Γ (j ) = I (j ) = |I |/m = length(Γuv )/m. Since dist(·, ∂Ω) is a Lipschitz function, for every t ∈ I (j ) we have dist(zI , ∂Ω) − dist z(t), ∂Ω zI − z(t) length Γ (j ) = length(Γuv )/m.


2225

Combining this inequality with (2.31) we obtain dist(zI , ∂Ω) − dist z(t), ∂Ω C dist(zI , ∂Ω). m

But m := [2C ] + 1, see (2.18), so that dist(zI , ∂Ω) − dist z(t), ∂Ω 1 dist(zI , ∂Ω). 2 Hence 3 1 dist(zI , ∂Ω) dist z(t), ∂Ω dist(zI , ∂Ω), 2 2

t ∈ I (j ) .

We let I denote the interval I (j ) . Thus we have proved that for each I ∈ Sk−1,m there exists a subinterval I ∈ Sk,m , I ⊂ I , such that max dist z(t), ∂Ω 3 min dist z(t), ∂Ω . t∈I

t∈I

Recall that g(t) := dist(z(t), ∂Ω)α−1 so that by this inequality max g(t) 31−α min g(t). t∈I

t∈I

Now we construct the family A ⊂ Sk,m as follows. At the first stage for the interval I0 := [0, L] we determine an m-adic interval I 0 ∈ I1,m of the first level and put A1 := {I 0 } and U1 := I 0 . Let us consider the set [0, L] \ U1 which consists of m − 1 m-adic intervals of the first level. We let B1 denote the family of these intervals. For every I ∈ B1 we construct the interval I ∈ I2,m and put A2 := {I ∈ I2,m : I ∈ B1 }. By U2 we denote the set U2 := U1 ∪

{I : I ∈ A2 } .

Now the set [0, L] \ U2 consists of (m − 1)2 m-adic intervals of the second level. We denote the family of these intervals by B2 and finish the second stage of the procedure. After the k-th stages of this procedure we obtain the families Aj ⊂ Ij,m , j = 1, 2, . . . , k, of m-adic intervals. We put A=

{Aj : j = 1, . . . , k}.

Thus A ⊂ Sk,m is a family of m-adic intervals of the level at most k. We know that for every interval I ∈ A the following inequality

2226


max g(t) 31−α min g(t)

(2.32)

t∈I

t∈I

holds. We also know that the set U = Uk :=

{I : I ∈ A}

has the following property: the set E := [0, L] \ U = [0, L] \

{I : I ∈ A}

consists of (m − 1)k m-adic intervals of the k-th level. Since |I | = m−k L for each I ∈ Ik,m , we obtain |E| =

(m − 1)k L. mk

But, by (2.22), L = length(Γ ) C x − y where C := 2e2Cα,Ω . Hence, |E| C x − y(1 − 1/m)k . Combining this inequality with (2.19), we obtain the required estimate |E| = [0, L] \ U < ε.

(2.33)

Now for the family A constructed above let us estimate from below the quantity

T :=

q 1 q 1 1 g dt |I | L |I | I ∈A

I

which appears in the left-hand side of inequality (2.30). By (2.32), for each I ∈ A we have g dt |I | max g (t) 3 q

q

t∈I

I

q(1−α)

q q 1 q(1−α) |I | min g(t) 3 |I | g dt t∈I |I | I

so that T 3 q

q(α−1)

1 q q(α−1) 1 g dt = 3 g q dt. L L I ∈A I

(Recall that U =

{I : I ∈ A}.)

U


2227

By Corollary 2.10, L 1 g dt T C L 0

so that 1 L

g dt 3 q

q(1−α)

T C1 q

1 L

U

q

L g dt 0

On the other hand, by inequality (2.21), with C1 := 3q(1−α) C.

L g dt =

dist(z, ∂Ω)α−1 ds(z) 2Cα,Ω x − yα . Γ

0

Hence 1 L

g dt C1 q

1 L

U

q

L g dt

C2 x − yqα /Lq

(2.34)

0

with C2 := (2Cα,Ω )q C1 . Recall that this inequality holds for every q ∈ [1, q ∗ ], see Corollary 2.10. We put q˜ := ∗ ˜ − α). Since q ∗ > 1 and 0 < α < 1, we have 1 < q˜ q ∗ min{q ∗ , 1−α/2 1−α } and α := 1 − q(1 ∗ and 0 < α < α. 1−τ . Then q ∈ [1, q ∗ ] so that, by (2.34), Let τ ∈ [α ∗ , α] and let q := 1−α

τ −1 dist Γ (t), ∂Ω dt =

U

g

τ −1 α−1

(t) dt =

U

L g (t) dt q

U

g q (t) dt 0

C2 x − yqα /Lq−1 . Since x − y L, we obtain τ −1 C2 dist Γ (t), ∂Ω dt q−1 x − yqα C2 x − yqα−q+1 = C2 x − yτ . L U

Finally, we put Γ := Γ |U . Then the last inequality, (2.21) and (2.33) show that inequalities (1.8), (1.11) and (1.10) of Theorem 1.5 are satisfied.

2228


It remains to prove inequality (1.9). We observe that the set U = Uk is obtained by the standard Cantor procedure (for m-adic intervals; recall that in the classical case m = 3). The reader can easily see that this Cantor set possess the following property: for each interval I centered in U with |I | 2L we have |I | 4m|I ∩ U |.

(2.35)

Let B = B(c, r) be a ball of radius r centered at a point c ∈ Γ . We may assume that r x − y/4. Then either x or y does not belong to B. Suppose that y ∈ / B. Then there exists a point v ∈ ∂B ∩ Γcy such that the arc Γcv ⊂ B. Let us define a positive number as follows. If Γxc ⊂ B, we put := length(Γcv ).

(2.36)

Assume that Γxc B. Then there exists a point u ∈ ∂B ∩ Γxc such that the arc Γuc ⊂ B. In this case we put := min length(Γuc ), length(Γcv ) . (2.37) Clearly, in the both cases r. Recall that c ∈ Γ so that c = Γ (a) for some a ∈ U . By I we denote the interval I := [a − , a + ]. Then, by definitions (2.36) and (2.37), the arc Γ (I ) ⊂ B. Since I is centered in U and |I | = 2 2L, by (2.35), |I | 4m|I ∩ U |. Since Γ (I ) ⊂ B, we have |I ∩ U | length(B ∩ Γ ) so that diam B = 2r 2 = |I | 4m|I ∩ U | 4m length(B ∩ Γ ) proving (1.9). Theorem 1.5 is completely proved.

2

3. Sobolev functions on subhyperbolic domains Let us fix some additional notation. In what follows, the terminology “cube” will mean a closed cube in Rn whose sides are parallel to the coordinate axes. We let Q(x, r) denote the cube in Rn centered at x with side length 2r. Given λ > 0 and a cube Q we let λQ denote the dilation of Q with respect to its center by a factor of λ. (Thus λQ(x, r) = Q(x, λr).) It will be convenient for us to measure distances in Rn in the uniform norm x := max |xi |: i = 1, . . . , n ,

x = (x1 , . . . , xn ) ∈ Rn .

Thus every cube Q = Q(x, r) := y ∈ Rn : y − x r is a “ball” in · -norm of “radius” r centered at x.


2229

Let A = {Q} be a family of cubes in Rn . By M(A) we denote its covering multiplicity, i.e., the minimal positive integer M such that every point x ∈ Rn is covered by at most M cubes from A. Recall that Pk denotes the space of polynomials of degree at most k defined on Rn . Also recall that, given a k-times differentiable function F and a point x ∈ Rn , we let Txk (F ) denote the Taylor polynomial of F at x of degree at most k: Txk (F )(y) :=

1 D β F (x)(y − x)β , β!

y ∈ Rn .

|β|k

Let Ω be a domain in Rn and let k ∈ N and p ∈ [1, ∞]. We let Lkp (Ω) denote the (homogeneous) Sobolev space of all functions f ∈ L1,loc (Ω) whose distributional partial derivatives on Ω of order k belong to Lp (Ω). Lkp (Ω) is normed by

f Lkp (Ω) :=

k p ∇ f dx

1

p

Ω

where ∇ k f denotes the vector with components D β f , |β| = k, and k ∇ f (x) :=

1 2 β D f (x) 2 ,

x ∈ Ω.

|β|=k

By the Sobolev imbedding theorem, see e.g., [23, p. 60], every f ∈ Lkp (Ω), p > n, can be redefined, if necessary, in a set of Lebesgue measure zero so that it belongs to the space C k−1 (Ω). Moreover, for every cube Q ⊂ Ω, every x, y ∈ Q and every multiindex β, |β| k − 1, the following inequality β k−1 k−|β|− pn D T (f ) − T k−1 (f ) (x) Cx − y x

y

k p ∇ f dx

1

p

(3.1)

Q

holds. Here C = C(n, p). In particular, the partial derivatives of order k − 1 satisfy a (local) Hölder condition of order α := 1 − pn : β D f (x) − D β f (y) C(n, p)f

Lkp (Ω) x

− y

1− pn

,

|β| = k − 1,

provided Q is a cube in Ω and x, y ∈ Q. Thus, for p > n, we can identify each element f ∈ Lkp (Ω) with its unique C k−1 -representative on Ω. This will allow us to restrict our attention to the case of Sobolev C k−1 -functions.

2230


The main result of this section is the following Theorem 3.1. Let n 0 depending only on n, p, in Rn . There exist a constant p, k, Cα,Ω and θα,Ω , such that the following is true: Let f ∈ Lkp (Ω), x, y ∈ Ω, x − y θ , and let Qxy := Q(x, x − y). Then for every multiindex β, |β| k − 1, the following inequality β k−1 k−|β|− pn˜ D T (f )(x) − D β T k−1 (f )(x) Cx − y x

y

k p˜ ∇ f dx

1

p˜

(λQxy )∩Ω

holds. Proof. We will be needed the following Lemma 3.2. Let x, y ∈ Ω and let γ ⊂ Ω be a continuous curve joining x to y. There is a finite family of cubes Q = {Q0 , . . . , Qm } such that: (i) Q0 x, Qm y, Qi = Qj , i = j , 0 i, j m, and Qi ∩ Qi+1 = ∅,

i = 0, . . . , m − 1.

(ii) For every cube Q = Q(z, r) ∈ Q we have z ∈ γ and r = 18 dist(z, ∂Ω). (iii) For each Q ∈ Q the cube 2Q ⊂ Ω. Moreover, the covering multiplicity of the family of cubes 2Q := {2Q: Q ∈ Q} is bounded by a constant C = C(n). Proof. For every z ∈ Γ we let Q(z) denote the cube (z)

Q

1 := Q z, dist(z, ∂Ω) . 8

We put A := {Q(z) : z ∈ Γ }. By the Besicovitch covering theorem, see e.g. [13], there exists a finite subcollection B ⊂ A such that B still covers Γ but no point which lies in more than C(n) of the cubes of B. (Thus the covering multiplicity M(B) C(n).) Given Q , Q ∈ B we write Q ∼ Q if there exists a family of cubes {K0 , . . . , K } ⊂ B such that K0 = Q ,

K = Q ,

Ki = Kj

for every i = j, 0 i, j ,

∈ B such that x ∈ Q, and put and Ki ∩ Ki+1 = ∅ for every i = 0, . . . , − 1. Fix a cube Q B := {Q ∈ B: Q ∼ Q} and E :=

{Q: Q ∈ B }.

(3.2)


2231

x, we have x ∈ E so that Prove that y ∈ E. Assume that this is not true, i.e., y ∈ / E. Since Q E ∩ γ = ∅. Put F :=

Q: Q ∈ B \ B .

Observe that every cube Q ∈ B such that Q ∩ E = ∅ belongs to B so that E ∩ F = ∅. On the other hand, since γ ⊂ E ∪ F and y ∈ / E, we have y ∈ F proving that F ∩ γ = ∅. Thus the sets E ∩ γ and F ∩ γ are a partition of the continuous curve γ into two closed disjoint sets; a contradiction. We have proved that y ∈ Q∗ for some Q∗ ∈ B so that, by definition (3.2), there exists a family of cubes Q = {Q0 , . . . , Qm } satisfying conditions (i) and (ii) of the lemma. Prove (iii). Let Q = Q(z, r) ∈ Q. Then z ∈ γ and r = 18 dist(z, ∂Ω) so that for every u ∈ 2Q = Q(z, 2r) we have dist(z, ∂Ω) dist(u, ∂Ω) + u − z dist(u, ∂Ω) + 2r = dist(u, ∂Ω) +

1 dist(z, ∂Ω). 4

Hence, 0
0 such that for every ε > 0 there exist a rectifiable curve Γ ⊂ Ω and a finite family of arcs Γ ⊂ Γ satisfying conditions (i), (ii) of the theorem. Observe that, by inequality (1.8) (with τ = α) and by (1.11), dist(z, ∂Ω)α−1 ds(z) Cx − yα . (3.4) Γ

Also, by part (ii) of Theorem 1.5, − y length(Γ ) Cx

(3.5)

= 2e2Cα,Ω , see (2.22). with C By Lemma 3.2, there exists a collection of cubes Q = {Q0 , . . . , Qm } satisfying conditions (i)–(iii) of the lemma. In particular, by (i), Q0 x,

Qm y,

Qi = Qj

for every i = j,

and Qi ∩ Qi+1 = ∅,

i = 0, . . . , m − 1.

Let Qi = Q(zi , ri ), i = 0, . . . , m. (Recall that by (ii) we have zi ∈ Γ and ri = 18 dist(zi , ∂Ω).) Let ai ∈ Qi−1 ∩ Qi , i = 1, . . . , m. Put a0 := x and am+1 := y. We may assume that for every Q ∈ Q either x ∈ / Q or y ∈ / Q. In fact, otherwise x, y ∈ Q. But, by condition (iii) of Lemma 3.2, 2Q ⊂ Ω. Then the cube Q(x, x − y) ⊂ 2Q ⊂ Ω as well. It remains to apply inequality (3.1) to x and y (replacing in this inequality p by an arbitrary p˜ ∈ (n, p)), and the theorem’s inequality follows. Thus we may assume that for every cube Qi = Q(zi , ri ) ∈ Q either x ∈ / Qi

or

y∈ / Qi .

Since x, y, zi ∈ Γ , we have ∂Qi ∩ Γ = ∅ so that there exists a point ai ∈ ∂Qi ∩ Γ . Hence ri = zi − ai length(Γ ∩ Qi )

(3.6)

so that, by (3.5), − y, ri length(Γ ) Cx

i = 0, . . . , m.

We fix a multiindex β, |β| k − 1, and put A := D β Txk−1 (f )(x) − D β Tyk−1 (f )(x) .

(3.7)


2233

Then A = D β Tak−1 (f )(a0 ) − D β Tak−1 (f )(a0 ) 0 m+1 m β k−1 β k−1 D T ai (f )(a0 ) − D Tai+1 (f )(a0 ) .

i=0

Put Pi (z) := Tak−1 (f )(z) − Tak−1 (f )(z), i i+1

i = 0, . . . , m.

The polynomial Pi ∈ Pk−1 so that

D β Pi (z) =

|η|k−1−|β|

1 η+β D Pi (ai )(z − ai )η , η!

z ∈ Rn .

Hence

β D Pi (a0 ) C

η+β D Pi (ai ) a0 − ai |η| .

|η|k−1−|β|

We put Q1 := {Q ∈ Q: Q ∩ Γ = ∅},

I1 := i ∈ {0, . . . , m}: Qi ∈ Q1 ,

(3.8)

and I2 := i ∈ {0, . . . , m}: Qi ∈ Q2 .

Q2 := Q \ Q1 , Then A

m m β D Pi (a0 ) C i=0

=C

η+β D Pi (ai ) a0 − ai |η|

i=0 |η|k−1−|β|

η+β |η| D Pi (ai ) a0 − ai .

m

|η|k−1−|β|

i=0

Let Aη :=

D η+β Pi (ai ) a0 − ai |η| i∈I1

and Aη :=

D η+β Pi (ai ) a0 − ai |η| . i∈I2

(3.9)

2234


We have proved that

AC

Aη + Aη .

(3.10)

|η|k−1−|β|

Fix a multiindex η, |η| k − 1 − |β|. Our aim is to show that Aη

Cx − y

k−|β|− pn˜

k p˜ ∇ f dx

1

p˜

(3.11)

(λQxy )∩Ω

where p˜ :=

n − α∗ 1 − α∗

see (3.5). (Recall that Qxy := Q(x, x − y).) Also we will prove that and λ := 2C, Aη

Cx − y

k−|β|− pn

k p ∇ f dx

1

p

(3.12)

E

where E is a subset of Ω of the Lebesgue measure |E| Cε n . Since 0 < α∗ =

p−n p˜ − n 0

Recall that a function f ∈ Wpk (Rn ), 1 < p ∞, if and only if f and fk are both in Lp (Rn ), see Calderón [10]. Moreover, up to constants depending only on n, k and p the following equivalence, f Wpk (Rn ) ∼ f Lp (Rn ) + fk L

p (R

n)

(4.1)

,

holds. This characterization motivates the following definition. Let S be a measurable subset of Rn . Given a function f ∈ Lq,loc (S), and a cube Q whose center is in S, we let Ek (f ; Q)Lq (S) denote the normalized best approximation of f on Q in Lq (S)-norm: − q1

Ek (f ; Q)Lq (S) := |Q|

inf f − P Lq (Q∩S) =

P ∈ Pk

inf

P ∈Pk−1

1 |Q|

|f − P | dx q

Q∩S

By fk,S , we denote the sharp maximal function of f on S,

fk,S (x) := sup r −k Ek f ; Q(x, r) L r>0

(Thus, fk = fα,Rn .)

1 (S)

,

1

x ∈ S.

q

.


2241

Let Ω ⊂ Rn be a subhyperbolic domain. The following two corollaries of Theorem 3.1 present estimates of the local best approximations and the sharp maximal function of a function f ∈ Wpk (Ω) via the local Lp -norms and the maximal function of ∇ k f . Corollary 4.1. Let p ∈ (n, ∞), α = (p − n)/(p − 1), and let Ω be an α-subhyperbolic domain in Rn . There exist a constant p˜ ∈ (n, p) and constants θ, λ, C > 0 depending only on n, p, k, Cα,Ω and θα,Ω , such that the following is true: Let f ∈ Lkp (Ω). Then for every cube Q = Q(x, r) with x ∈ Ω and 0 < r θ the following inequality

r

−k

1 Ek (f ; Q)L∞ (Ω) C |λQ|

k p˜ ∇ f dx

1

p˜

(λQ)∩Ω

holds. Proof. Let p˜ and θ be the constants from Theorem 3.1. Let y ∈ Q(x, r) so that y − x r θ . Applying Theorem 3.1 to the points y, x (with β = 0), we obtain n f (y) − T k−1 (f )(y) Cx − yk− p˜

k p˜ ∇ f dx

x

1

p˜

.

(λQxy )∩Ω

Recall that Qxy := Q(x, x − y). Since n < p, ˜ we have k − pn˜ > 0 so that n f (y) − T k−1 (f )(y) Cr k− p˜

k p˜ ∇ f dx

x

1

p˜

(λQxy )∩Ω

Cr

k

1 |λQ|

k p˜ ∇ f dx

1

p˜

.

(λQ)∩Ω

Hence, Ek (f ; Q)L∞ (Ω) :=

inf

Cr k

proving the corollary.

sup f (y) − P (y) sup f (y) − Txk−1 (f )(y)

P ∈Pk−1 y∈Q∩Ω

1 |λQ|

y∈Q∩Ω

k p˜ ∇ f dx

1

p˜

(λQ)∩Ω

2

Given a function g defined on Ω we let g denote its extension by zero to all of Rn . Thus x ∈ Ω, and g (x) := 0, x ∈ / Ω.

g (x) := g(x),

2242


As usual, given a function f ∈ L1,loc (Rn ) by M[f ] we denote the Hardy–Littlewood maximal function 1 f (y) dy. M[f ](x) := sup t>0 |Q(x, t)| Q(x,t)

Corollary 4.2. Let n < p < ∞, α = (p − n)/(p − 1), and let Ω be an α-subhyperbolic domain in Rn . There exists a constant p, ˜ n < p˜ < p, such that for every function f ∈ Lkp (Ω) and every x ∈ Ω the following inequality p˜ 1

fk,Ω (x) C M ∇ k f (x) p˜ + M f (x) holds. The constants p˜ and C depend only on n, p, k, Cα,Ω and θα,Ω . Proof. Let p, ˜ λ and θ be the constants from Corollary 4.1. By this corollary, sup r −k Ek f ; Q(x, r) L

1 (Ω)

0θ

1 (Ω)

1 |Q(x, r)|

θ −k M f (x) proving the corollary.

|f | dx Q(x,r)∩Ω

2

In [28] we show that the restrictions of Sobolev functions to regular subsets of Rn can be described in a way similar to the Calderón’s criterion (4.1), i.e., via Lp -norms of a function and its sharp maximal function on a set. We recall that a measurable set S ⊂ Rn is said to be regular if there are constants σS 1 and δS > 0 such that, for every cube Q with center in S and with diameter diam Q δS , |Q| σS |Q ∩ S|. Theorem 4.3. (See [28].) Let S be a regular subset of Rn . Then a function f ∈ Lp (S), 1 < p ∞, can be extended to a function F ∈ Wpk (Rn ) if and only if its sharp maximal function

fk,S ∈ Lp (S). In addition,


2243

f Wpk (Rn )|S ∼ f Lp (S) + fk,S L

p (S)

with constants of equivalence depending only on n, k, p, σS and δS . Moreover, there exists a continuous linear extension operator from the trace space Wpk (Rn )|S into Wpk (Rn ). Its operator norm is bounded by a constant depending only on n, k, p, σS and δS . Recall that the existence of a continuous linear extension operator for a regular subset of Rn has been earlier proved by Rychkov [27]. Proof of Theorem 1.1. By inequality (1.1), Ω is an α-subhyperbolic domain with α = p−n p−1 , so n that, by Lemma 2.3, Ω is a regular subset of R . Let p˜ ∈ (n, p) be the constant from Corollary 4.2. Let q > p˜ and let f ∈ Wqk (Ω). We have to prove that f can be extended to a function F ∈ Wqk (Rn ). Since Ω is regular, by Theorem 4.3 it

suffices to show that the sharp maximal function fk,Ω belongs to Lq (Ω). By Corollary 4.2, p˜ 1

(x) p˜ + M f (x) , fk,Ω (x) C M ∇ k f

x ∈ Ω,

so that f

k,Ω Lq (Ω)

p˜ 1 p˜ C M ∇ k f L

q (Ω)

p˜ 1 p˜ C M ∇ k f L

q (R

n)

+ M f L

q (Ω)

+ M f L

q (R

n)

.

By the Hardy–Littlewood maximal theorem M f

Lq (Rn )

C f L

q (R

n)

= Cf Lq (Ω) .

(Recall that f denotes the extension of f by zero to all of Rn .) Applying this theorem to the function g := (∇ k f )p˜ in the space Ls (Rn ) with s := q/p˜ > 1, we obtain p˜ 1 p˜ A := M ∇ k f L

n q (R )

1 C gLs (Rn ) p˜ = C

= M[g]L

1

n s (R )

k p˜ q p˜ dx ∇ f

p˜

p˜ 1 q

p˜

Rn

so that

AC Rn

k q ∇ f dx

1 q

=C Ω

k q ∇ f dx

1 q

= Cf Lkq (Ω) .

2244


Hence f

k,Ω Lq (Ω)

C f Lkq (Ω) + f Lq (Ω) Cf Wqk (Ω) .

The proof of Theorem 1.1 is complete.

2

Acknowledgments The author is greatly indebted to Michael Cwikel, Charles Fefferman, Vladimir Gol’dstein, Vladimir Maz’ya and Naum Zobin for interesting discussions and helpful remarks. The author also very grateful to the referee for useful suggestions and comments. References [1] K. Astala, K. Hag, P. Hag, V. Lappalainen, Lipschitz classes and the Hardy–Littlewood property, Monatsh. Math. 115 (1993) 267–279. [2] Yu.A. Brudnyi, Spaces that are definable by means of local approximations, Tr. Mosk. Mat. Obs. 24 (1971) 69–132; English transl.: Trans. Moscow Math. Soc. 24 (1974) 73–139. [3] Yu.A. Brudnyi, B.D. Kotljar, A certain problem of combinatorial geometry, Sibirsk. Mat. Zh. 11 (1970) 1171–1173; English transl.: Siberian Math. J. 11 (1970) 870–871. [4] S. Buckley, P. Koskela, Criteria for imbeddings of Sobolev–Poincaré type, Int. Math. Res. Not. 18 (1996) 881–902. [5] S. Buckley, A. Stanoyevitch, Weak slice conditions and Hölder imbeddings, J. London Math. Soc. 66 (2001) 690– 706. [6] S. Buckley, A. Stanoyevitch, Weak slice conditions, product domains, and quasiconformal mappings, Rev. Mat. Iberoamericana 17 (2001) 1–37. [7] S. Buckley, A. Stanoyevitch, Distinguishing properties of weak slice conditions, Conformal geometry and dynamics, Electron. J. Amer. Math. Soc. 7 (2003) 49–75. [8] Yu.D. Burago, V.G. Maz’ya, Certain questions of potential theory and function theory for regions with irregular boundaries, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) vol. 3, 1967, 152 pp.; English transl.: Potential Theory and Function Theory for Irregular Regions, Seminars in Math., V.A. Steklov Math. Inst., Leningrad, vol. 3, Consultants Bureau, New York, 1969, vii+68 pp. [9] A.P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math. IV (1961) 33–49. [10] A.P. Calderón, Estimates for singular integral operators in terms of maximal functions, Studia Math. 44 (1972) 563–582. [11] R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974) 241–250. [12] J. Garcia-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland Math. Stud., vol. 116, North-Holland, Amsterdam, 1985. [13] F.W. Gehring, The Lp integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973) 265–277. [14] F.W. Gehring, O. Martio, Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. AI Math. 10 (1985) 203–219. [15] V.M. Gol’dshtein, T.G. Latfullin, S.K. Vodop’janov, Criteria for extension of functions of the class L12 from unbounded plain domains, Siberian Math. J. 20 (1979) 298–301. [16] V.M. Gol’dshtein, S.K. Vodop’janov, Prolongement des fonctions de classe L1p et applications quasi conformes, C. R. Acad. Sci. Paris Ser. A–B 290 (10) (1980) A453–A456. [17] V.M. Gol’dshtein, S.K. Vodop’janov, Prolongement des fonctions differentiables hors de domaines plans, C. R. Acad. Sci. Paris Ser. I Math. 293 (12) (1981) 581–584. [18] P. Hajlasz, P. Koskela, H. Tuominen, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254 (2008) 1217–1234. [19] P. Hajlasz, P. Koskela, H. Tuominen, Measure density and extendability of Sobolev functions, Rev. Mat. Iberoamericana 24 (2) (2008) 645–669.


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The L2 -cutoff for reversible Markov processes Guan-Yu Chen a,∗,1 , Laurent Saloff-Coste b,2 a Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan b Malott Hall, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201, United States

Received 21 May 2009; accepted 27 August 2009 Available online 31 October 2009 Communicated by L. Gross

Abstract We consider the problem of proving the existence of an L2 -cutoff for families of ergodic Markov processes started from given initial distributions and associated with reversible (more, generally, normal) Markov semigroups. This includes classical examples such as families of finite reversible Markov chains and Brownian motion on compact Riemannian manifolds. We give conditions that are equivalent to the existence of an L2 -cutoff and describe the L2 -cutoff time in terms of the spectral decomposition. This is illustrated by several examples including the Ehrenfest process and the biased (p, q)-random walk on the non-negative integers, both started from an arbitrary point. © 2009 Elsevier Inc. All rights reserved. Keywords: L2 -cutoff; Markov semigroups; Normal operators

1. Introduction Consider a (time-homogeneous) Markov chain on a finite set Ω with one-step transition kernel K. Let K l (x, ·) denote the probability distribution of this chain at time l starting from the state x. Assuming irreducibility and aperiodicity, it is known that lim K l (x, ·) = π

l→∞


E-mail addresses: [email protected] (G.-Y. Chen), [email protected] (L. Saloff-Coste). 1 Partially supported by NSC grant NSC 96-2115-M-009-015 and NCTS, Taiwan. 2 Partially supported by NSF DMS 0603886.


G.-Y. Chen, L. Saloff-Coste / Journal of Functional Analysis 258 (2010) 2246–2315

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where π is the unique invariant probability of K on Ω. Set kxl = K l (x, ·)/π , the relative density of K l (x, ·) w.r.t. π . For 1 p ∞, set l maxy {|k l (y) − 1|} if p = ∞, Dp (x, l) = kx − 1p (π) = l x ( y |kx (y) − 1|p π(y))1/p if 1 p < ∞. For p = 1, this is exactly twice the total variation distance between K l (x, ·) and π and, for p = 2, it is the so-called chi-square distance. For any > 0, set Tp (x, ) = min l 1: Dp (x, l) .

(1.1)

The concept of cutoff was introduced by Aldous and Diaconis in [1–3] to capture the fact that many ergodic Markov processes appear to converge abruptly to their stationary measure. We refer the reader to [5,6,15,19] for detailed discussions and the description of various examples. In its simplest form, the cutoff phenomenon in L2 for a family of finite ergodic Markov chains (with given starting points) (Ωn , xn , Kn , πn ) is defined as follows. There is an L2 -cutoff with cutoff sequence tn if lim Dn,2 (xn , atn ) =

n→∞

0 if a ∈ (1, ∞), ∞ if a ∈ (0, 1).

Here Dn,2 denotes the chi-square distance on Ωn relative to πn . In [5], the authors discussed a number of variants of this definition and produced, in the reversible case, a necessary and sufficient condition for the existence of a max-L2 -cutoff, that is, a cutoff for maxx∈Ω D2 (x, ·) (some of the results in [5] holds for Lp , 1 < p < ∞). The aim of the present paper is twofold. Our first goal is to establish a criterion for the existence of an L2 -cutoff for families of Markov processes starting from specific initial distributions when the associated semigroup is normal (i.e., commutes with its adjoint on a proper Hilbert space). Our second goal is to derive formulas for the L2 -cutoff time sequence using spectral information. To attain these two goals, we will take advantage of the very specific structure of the chi-square distance and its direct relationship with spectral decomposition. This is in contrast to the techniques and results of [5] which do not involve much spectral theory and treats Lp distances, 1 < p < ∞ as well as L2 . The following theorem illustrates the goals outled above. Theorem 1.1. Let Ω = {0, 1, 2, . . .} and K be the Markov kernel of the birth and death chain on Ω with uniform birth rate p ∈ (0, 1/2), uniform death rate 1 − p and K(0, 0) = 1 − p. Let xn be a sequence of states in Ω. Then, the discrete time family of birth and death chains with respective starting states x1 , x2 , . . . presents an L2 -cutoff if and only if xn tends to infinity. Moreover, if there is a cutoff then tn =

log(1 − p) − log p xn − log(4p(1 − p))

is a cutoff time sequence as n → ∞. We will also obtain variants of this result that involve finite state spaces Ωn = {0, 1, . . . , n} and birth and death rates (pn , qn ) that are allowed to depend on n. Our second main example is

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the Ehrenfest chain, treated in Theorem 6.3. The treatment of these examples occupies the bulk of the paper and illustrates very well the delicacies of the cutoff phenomenon in the context of specified starting distributions. This paper is organized as follows. In Section 2, we recall various notions of cutoffs and quote useful results from [5]. In Section 3, we give criteria for the existence of a cutoff as well as formulas for the cutoff times in the case of families of Laplace transforms. The main result of Section 3 is Theorem 3.5 which is the technical cornerstone of this work. In Section 4, we observe that the chi-square distance between the distribution of a normal Markov process and its invariant probability measure can be expressed as a Laplace transform. This gives criteria and formula for cutoffs of families of ergodic normal Markov processes (normal here means that the generator is a normal operator, i.e., commutes with its adjoint). In Section 5, we spell out the results in the case of families of finite Markov chains (in discrete and continuous time). See Theorems 5.1, 5.3 and Theorems 5.4, 5.5. In Section 6, we apply these results to study the cutoff phenomenon for the Ehrenfest chain started at an arbitrary point. See Theorems 6.3, 6.5. In Section 7, we prove Theorem 1.1 and a number of related results. In Section 8, we study a family of birth and death chains on {−n, . . . , n} containing examples whose stationary measure has either a unique maximum or a unique minimum at 0. 2. Cutoff terminology In this section, we recall various notions of cutoffs and a series of related results from [5]. The notion of cutoff can be developed for any family of non-increasing functions taking values in [0, ∞]. The following definition treats functions defined on the natural integers. We refer the reader to [5] for additional details and examples. Definition 2.1. Let N be the set of non-negative integers. For n 1, let fn : N → [0, ∞] be a non-increasing function vanishing at infinity. Assume that M = lim sup fn (0) > 0. n→∞

Then the family F = {fn : n = 1, 2, . . .} is said to present: (i) A precutoff if there exist a sequence of positive numbers tn and constants b > a > 0 such that lim fn (kn ) = 0,

n→∞

lim inf fn (ln ) > 0, n→∞

where kn = min{j 0: j > btn } and ln = max{j 0: j < atn }. (ii) A cutoff if there exists a sequence of positive numbers tn such that, for all ∈ (0, 1), lim fn kn () = 0,

n→∞

lim fn kn (−) = M,

n→∞

where kn () = min{j 0: j > (1 + )tn } and kn () = max{j 0: j < (1 + )tn }. (iii) A (tn , bn )-cutoff if tn > 0, bn 0, bn = o(tn ) and lim F (c) = 0,

c→∞

lim F (c) = M,

c→−∞


2249

where, for c ∈ R, k(n, c) = min{j ∈ N: j > tn + cbn }, k(n, c) = max{j ∈ N: j < tn + cbn } and F (c) = lim sup fn k(n, c) , F (c) = lim inf fn k(n, c) . (2.1) n→∞

n→∞

This definition agrees with the one used in [5]. In (ii) and (iii), (tn )∞ 1 is referred to as a cutoff sequence and bn as a window with respect to tn . Obviously, (iii) ⇒ (ii) ⇒ (i). Remark 2.1. When the fn ’s are functions on [0, ∞), cutoffs are defined in a similar way. For a precutoff, we set ln = atn and kn = btn . For a cutoff, kn and kn are replaced respectively by (1+)tn and (1−)tn . These notions of precutoff, cutoff and their cutoff sequences coincide with the notion in [5]. For a (tn , bn )-cutoff in continuous time, we require bn > 0 and use k(n, c) = k(n, c) = tn + cbn . Assuming bn > 0, this agrees with the (tn , bn )-cutoff of [5]. In Definition 2.1(iii), the window of a cutoff captures explicitly how sharp the cutoff is. It is quite sensitive to the choice of the cutoff sequence tn . Window optimality is addressed in the following definition (when bn > 0 this definition is equivalent to the one in [5]). Definition 2.2. Let F and M be as in Definition 2.1. Assume that F presents a (tn , bn )-cutoff. Then, the cutoff is called: (i) weakly optimal if, for any (tn , dn )-cutoff for F , one has bn = O(dn ), (ii) optimal if, for any (sn , dn )-cutoff for F , we have bn = O(dn ). In this case, bn is called an optimal window for the cutoff, (iii) strongly optimal if 0 < F (c) F (−c) < M

∀c > 0.

Remark 2.2. An optimal window is a minimal window (in the sense of order) over all cutoff sequences and, hence, an optimal cutoff is also a weakly optimal cutoff, i.e. (ii) ⇒(i). If a (tn , bn )cutoff is strongly optimal, it is easy to see that bn > 0 for all n and that infn bn > 0 if the domain of the functions in F is N (bn may tend to 0 when the domain is [0, ∞)). Hence, a strongly optimal (tn , bn )-cutoff implies that for any −∞ < c1 < c2 < ∞ we have 0 < fn (tn + c2 bn ) fn (tn + c1 bn ) < M for n large enough. This implies that if (sn , dn ) is another cutoff sequence for F , then the window dn has order at least bn , and thus (iii) ⇒ (ii). Remark 2.3. Let F be a family of functions defined on N. If F has a (tn , bn )-cutoff with bn → 0, instead of looking for the optimal window, it is better to determine the limsup and liminf of the sequences fn ([tn ]), where [t] is any integer in [t − 1, t + 1]. Remark 2.4. For any family {fn : T → [0, ∞], n = 1, 2, . . .} with T = [0, ∞), a necessary condition for a strongly optimal (tn , bn )-cutoff is that 0 < lim inf fn (tn ) lim sup fn (tn ) < M. n→∞

n→∞

2250


When T = N, we need instead 0 < lim inf fn tn lim sup fn tn < M. The cutoff phenomenon and its optimality are closely related to the way the functions in F converge to 0. This is captured by the following simple concept. Definition 2.3. Let f be an extended real-valued non-negative function defined on T , which is either N or [0, ∞). For > 0, set T (f, ) = inf t ∈ T : f (t) if the right-hand side above is non-empty and let T (f, ) = ∞ otherwise. In the context of ergodic Markov chains, T (f, ) is interpreted as the -mixing time. The simplest relationship with the notions of cutoff discussed above is as follows. Let F = {fn : T → [0, ∞]: n = 1, 2, . . .} be a family of non-increasing functions vanishing at infinity. Assume that M = lim supn→∞ fn (0) > 0. Then F has a cutoff if and only if ∀, η ∈ (0, M),

lim T (fn , )/T (fn , η) = 1.

n→∞

See [5, Propositions 2.3–2.4] for further relationships and details. We end this section with a technical result concerning cutoffs and optimality which is useful in either proving or disproving a cutoff and its optimality when the cutoff or window sequences contain both bounded and unbounded subsequences. The proof is straightforward and is omitted. Proposition 2.1. Let T be [0, ∞) or N. Consider F = {fn : T → [0, ∞], n = 1, 2, . . .} as a family of non-increasing functions vanishing at infinity. For any subsequence ξ = (ξi ) of positive integers, denote by Fξ the subfamily {fξi , i = 1, 2, . . .}. Assume that M = lim supn→∞ fn (0) > 0. For T = [0, ∞), the following are equivalent. (i-1) F has a cutoff (resp. (tn , bn )-cutoff ). (i-2) For any subsequence ξ = (ξi ), the family Fξ has a cutoff (resp. (tξn , bξn )-cutoff ). (i-3) For any subsequence ξ = (ξi ), we may choose a further subsequence ξ = (ξi ) such that the family Fξ has a cutoff (resp. (tξn , bξn )-cutoff ). Moreover, assume that F has a (tn , bn )-cutoff. Then the following are equivalent. (ii-1) F has an optimal (resp. weakly, strongly optimal ) (tn , bn )-cutoff. (ii-2) For any subsequence ξ = (ξi ), the family Fξ has an optimal (resp. weakly, strongly optimal ) (tξn , bξn )-cutoff. (ii-3) For any subsequence ξ = (ξi ), we may choose a further subsequence ξ = (ξi ) such that the family Fξ has an optimal (resp. weakly, strongly optimal ) (tξn , bξn )-cutoff. For T = N, all equivalences remain true if tn → ∞, lim infn→∞ bn > 0 and, for some δ ∈ (0, M), T (fn , δ) → ∞.


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3. Cutoffs for Laplace transforms In this section, we deal with cutoffs for family of functions which are Lebesgue–Stieltjes integral of exponential functions, that is, generalized Laplace transforms. Such functions appear naturally in the context of chi-square distance for reversible Markov process. More precisely, if the infinitesimal generator is self-adjoint and the initial distribution has an L2 Radon–Nikodym derivative w.r.t. the invariant probability measure, then the square of the chi-square distance to stationarity is such an integral. This will be discussed in details in Section 4. 3.1. Laplace transforms For n 1, let Vn : (0, ∞) → (0, ∞) be a non-decreasing and right-continuous function. Consider fn as a Lebesgue–Stieltjes integral defined by

e−tλ dVn (λ)

fn (t) =

∀t 0.

(3.1)

(0,∞)

It is easy to see that fn is non-increasing. Observe that sums of exponential functions are of this special type. For example, let fn (t) =

an,i e−tλn,i

∀t 0,

i1

where an,i 0 and λn,i+1 λn,i > 0 for n 1, i 1. Then fn is of the form in (3.1) with an,0 = λn,0 = 0, λn = λn,1 and Vn (t) =

j −1

an,i ,

for

i=0

j −1

λn,i < t

i=0

j

λn,i

and j 1.

i=0

Lemma 3.1. Let V : (0, ∞) → (0, ∞) be a non-decreasing and right-continuous function and let f be a function on [0, ∞) defined by

f (t) =

e−tλ dV (λ).

(0,∞)

Assume that V is bounded. Then f is analytic on (0, ∞). Proof. See [21, Theorem 5, p. 57].

2

The following is an application of the above lemma which is helpful when examining cutoffs and their optimality. Lemma 3.2. For n 1, let fn be a function on [0, ∞) defined by (3.1). Assume that supn fn (0) < ∞. Then, for any sequence of positive numbers (tn )∞ 1 , there exists a subsequence

2252


(tnk )k1 such that the sequence gk : a → fnk (atnk ) converges uniformly on any compact subset of (0, ∞) to an analytic function on (0, ∞). Moreover, if cnk is such that |cnk | = o(tnk ), then ∀a > 0, Proof. See Appendix A.

lim fnk (atnk + cnk ) = lim fnk (atnk ).

k→∞

k→∞

2

Remark 3.1. Let fn be the function as in Lemma 3.2 and let tn be any sequence of positive numbers. If tn tends to infinity, we may choose a subsequence (nk )∞ 1 such that a → fnk ([atnk ]) converges to a function analytic on (0, ∞), where [t] is any integer in [t − 1, t + 1]. The following two corollaries are simple applications of Lemma 3.2. Corollary 3.3. Let (fn )∞ 1 be as in Lemma 3.2 such that fn (0) is bounded. For any sequence of positive numbers tn , set F1 (a) = lim sup fn (atn ), n→∞

F2 (a) = lim inf fn (atn ), n→∞

∀a > 0.

Then F1 and F2 are continuous on (0, ∞). Furthermore, either F1 > 0 (resp. F2 > 0) or F1 ≡ 0 (resp. F2 ≡ 0). Proof. See Appendix A.

2

Remark 3.2. By Remark 3.1, if tn → ∞, Corollary 3.3 also holds with the following replacements F1 (a) = lim sup fn [atn ] , n→∞

F2 (a) = lim inf fn [atn ] , n→∞

where [t] is any integer in [t − 1, t + 1]. Corollary 3.4. Let F = {fn : [0, ∞) → [0, ∞]: n = 1, 2, . . .} be a family of functions defined by (3.1). Assume that F has a (tn , bn )-cutoff with bn > 0 and let F , F be functions in (2.1). (i) If F (0) < ∞, then, on the set (0, ∞), either F > 0 (resp. F > 0) or F ≡ 0 (resp. F ≡ 0). (ii) Assume that F (0) > 0. If F (c) = 0 (resp. F (c) = 0) for some c > 0, then F (c) = ∞ (resp. F (c) = ∞) for all c < 0. (iii) If tn = T (fn , δ), then the conclusions in (i) and (ii) hold true without the assumptions on F (0) and F (0). That is, for (i), either F > 0 (resp. F > 0) or F ≡ 0 (resp. F ≡ 0). For (ii), if F (c) = 0 (resp. F (c) = 0) for some c > 0, then F (c) = ∞ (resp. F (c) = ∞) for all c < 0. Proof. See Appendix A.

2

Remark 3.3. By Remarks 3.1–3.2, Corollary 3.4 also holds for families of functions defined on N if one assumes that bn → ∞.


2253

3.2. Cutoffs for Laplace transforms The following theorem is one of the main technical results of this paper. It provides a simple criterion to inspect cutoffs. If V is a non-decreasing and right-continuous function on (0, ∞), we also denote by V the measure on (0, ∞) such that V ((a, b]) = V (b) − V (a). Theorem 3.5. For n 1, let fn : [0, ∞) → [0, ∞] be a function defined by (3.1) and set M = lim inf fn (0). Assume that fn (t) vanishes at infinity for n 1. n→∞

(i) If M < ∞, then the family has no precutoff. (ii) If M = ∞, let tn = T (fn , δ) and λn = λn (C) = inf λ: Vn (0, λ] > C .

(3.2)

Then the family has a cutoff if and only if there exist constants δ, , C ∈ (0, ∞) such that (a) tn λn → ∞. (b) (0,λn ) e−λtn dVn (λ) → 0. Furthermore, if (a) and (b) hold, then the family has a (tn , λ−1 n )-cutoff. Proof. See Appendix A.

2

Remark 3.4. If there is a cutoff for (fn )∞ 1 , then (a) and (b) hold for any positive triple (δ, , C). Remark 3.5. It follows from the proof of this result that if fn is an extended real-valued function defined on N, then Theorem 3.5(i) can fail. The next theorem is a discrete time version of Theorem 3.5. Theorem 3.6. For n 1, let fn : N → [0, ∞] be a function defined by (3.1) and set M = lim inf fn (0) and tn = T (fn , δ). Assume that fn (t) vanishes at infinity for all n 1 and tn → ∞ n→∞ for some δ > 0. (i) If M < ∞, then the family has no precutoff. (ii) If M = ∞, then the family has a cutoff if and only if there exists C > 0 and > 0 such that Theorem 3.5 (a)–(b) hold true. Furthermore, if Theorem 3.5 (a)–(b) are satisfied, then the family has a (tn , γn−1 )-cutoff, where γn = min{λn , 1}. Proof. See Appendix A.

2

Remark 3.6. As in Remark 3.4, if, in Theorem 3.6, there is a cutoff for the family (fn )∞ 1 with cutoff time tending to infinity, then (a) and (b) are true for any positive constants C, δ, . Remark 3.7. It has been implicitly proved in the appendix that, for the family of functions fn defined on [0, ∞), Theorem 3.6 applied to the family of restricted functions {fn |N : n = 1, 2, . . .}

2254


still holds true if tn = T (fn |N , δ) is replaced by T (fn , δ) defined in Theorem 3.5. Furthermore, Remark 3.6 is also true under this replacement. The next proposition is concerned with the optimality of cutoffs for Laplace transforms. Proposition 3.7. Let F = {fn : T → [0, ∞] | n = 1, 2, . . .} be a family of functions of the form (3.1). Assume that F has a (tn , bn )-cutoff where tn = T (fn , δ) for all n 1 with δ ∈ (0, ∞) and bn > 0. Let F and F be functions in (2.1). For T = [0, ∞), the (tn , bn )-cutoff is (i) weakly optimal iff it is optimal iff F (c) > 0 for some c > 0, (ii) strongly optimal iff F (c) < ∞ for all c < 0. For T = N, the above remains true if bn → ∞. Proof. See Appendix A.

2

3.3. The cutoff time of Laplace transforms Theorems 3.5 and 3.6 can be used to examine the existence of a cutoff by checking whether the product of T (fn , δ) and λn tends to infinity or not. By Remarks 3.4 and 3.6, the constant C appearing in the definition of λn can be taken to be any positive number and, hence, the only unknown term that needs to be studied is the δ-mixing time T (fn , δ). Understanding this quantity with any precision is a difficult task. In this section, we describe potential cutoff time sequences in different terms. Theorem 3.8. Let F = {fn : [0, ∞) → [0, ∞]: n = 1, 2, . . .} be a family of functions defined by (3.1) which vanish at infinity. For n 1 and C > 0, let λn = λn (C) be the constant defined in (3.2) and set

log(1 + Vn ((0, λ])) . (3.3) τn = τn (C) = sup λ λλn Then F has a cutoff if and only if, for some C > 0 and > 0, (a) τ n λn → ∞, (b) (0,λn ) e−λτn dVn (λ) → 0. Moreover, if (a) and (b) hold true, then F has a (tn , bn )-cutoff with tn = τ n ,

bn = λ−1 n w(τn λn ),

where w : (0, ∞) → (0, ∞) is any function satisfying lim

t→∞

w(t) = 0, t

2 lim inf ew(t) 1 − e−w (t)/t > 0. t→∞

In particular, w(t) = log t is a function qualified for (3.4) and bn = λ−1 n log(τn λn ).

(3.4)


Proof. See Appendix A.

2255

2

C Remark 3.8. It is shown in the proof of Theorem 3.8 that τn (C) T (fn , C+1 ), C > 0.

Remark 3.9. In contrast with Theorem 3.5, Theorem 3.8 does not require explicitly that fn (0) → ∞ (i.e., M = ∞) for a cutoff. However, this is in fact contained implicitly in Theorem 3.8(a). Remark 3.10. What is proved in the appendix is that, if a family has a cutoff then the conditions (a)–(b) hold for any C > 0, > 0. This means that either (a)–(b) hold for all positive constants C, or one of (a) or (b) must fail for all C, . Hence, when using Theorem 3.8 to inspect the existence of a cutoff, one needs to check (a) and (b) only for one arbitrary pair (C, ). In practice, this is a very important remark. Remark 3.11. The second condition in (3.4) implies limt→∞ w(t) = ∞. It follows that the window given by Theorem 3.8 has order strictly larger than that the one in Theorem 3.5 and, hence, cannot be optimal. The following is a discrete time version of Theorem 3.8. Theorem 3.9. For n 1, let fn : N → [0, ∞] be the function defined by (3.1). For C > 0, let λn = λn (C) and τn = τn (C) be the quantities defined by (3.2) and (3.3). Assume that either, for some C > 0, τn → ∞ or, for some δ > 0, T (fn , δ) → ∞. Then F has a cutoff if and only if Theorem 3.8 (a) and (b) hold for some C > 0 and > 0. Moreover, if (a) and (b) hold, then F has a (tn , bn )-cutoff with tn = τ n , bn = max λ−1 n w(τn λn ), 1 , where w is a function satisfying (3.4). Proof. See Appendix A.

2

Remark 3.12. Remark 3.8 holds true in discrete time cases. Remark 3.13. Concerning functions defined on [0, ∞) and their restriction to the natural integers, the proof of Theorem 3.9 shows that under the assumption of τn → ∞, the existence of cutoff in Theorems 3.8 and 3.9 are equivalent and both families share the same cutoff type if the window is at least 1. Thus, by Remark 3.10, if the family in Theorem 3.9 has a cutoff, then Theorem 3.8(a), (b) (in both discrete time and continuous time setting) hold true for all C > 0 and > 0. 4. The main results Spectral theory is a standard tool to study the L2 -convergence of Markov processes to their stationarity. In particular, in the general context of reversible Markov processes, the square of the chi-square distance can be expressed in terms of the spectral decomposition of the infinitesimal generator and written in the form of (3.1). In the following, we start by recalling the definition of ergodic Markov processes discussed in [5].

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4.1. Markov processes and transition functions In what follows, let the time to be either N = {0, 1, 2, . . .} or [0, ∞). A Markov transition function on a space Ω equipped with a σ -algebra B, is a family of probability measures p(t, x, ·) indexed by t ∈ T (T = [0, ∞) or N) and x ∈ Ω such that p(0, x, Ω \ {x}) = 0 and, for each t ∈ T and A ∈ B, p(t, x, A) is B-measurable and satisfies

p(t + s, x, A) =

p(s, y, A)p(t, x, dy). Ω

We say that a Markov process X = (Xt , t ∈ T ) with filtration Ft = σ (Xs : s t) ⊂ B admits p(t, x, ·), t ∈ T , x ∈ Ω, as transition function if

E(f ◦ Xs | Ft ) =

f (y)p(s − t, Xt , dy) Ω

for all 0 < t < s < ∞ and all bounded measurable f . The measure μ0 (A) = P (X0 ∈ A) is called the initial distribution of the process X. All finite dimensional marginals of X can be expressed in terms of μ0 and the transition function. In particular,

μt (A) = P (Xt ∈ A) =

p(t, x, A) μ0 (dx).

Given a Markov transition function p(t, x, ·), t ∈ T , x ∈ Ω, for any bounded measurable function f , set

Pt f (x) =

f (y)p(t, x, dy).

(4.1)

For any measure ν on (Ω, B) with finite total mass, set

νPt (A) =

p(t, x, A) ν(dx).

We say that a probability measure π is invariant if πPt = π for all t ∈ T . In the general setting, invariant measures are not necessarily unique. 4.2. L2 -distances, mixing time and cutoffs The Markov processes of interest in this paper are ergodic in the sense that, for some initial measure μ of interest, the sequence μt converges (in some sense) to a probability measure as t tends to infinity. A simple argument shows that this limit must be an invariant probability measure. We now introduce the chi-square distance measuring the convergence to stationarity. Let pt be a Markov transition function on Ω with invariant probability measure π . Let μ be another


2257

probability measure on (Ω, B). For t 0, if μPt is absolutely continuous with respect to π with density h(t, μ, ·), we set

D2 (μ, t) =

1/2 h(t, μ, x) − 12 π(dx) .

(4.2)

Ω

In the case that such a density does not exist, D2 (μ, t) is set to be infinity. It is an exercise to show that the absolute continuity of μPt w.r.t π implies that of μPt+s for all s > 0. Moreover, the map t → D2 (μ, t) is non-increasing (see., e.g., [5]). Definition 4.1 (Mixing time). For any > 0, set T2 (μ, ) = T D2 (μ, t), = inf t ∈ T : D2 (μ, t) . If the infimum is taken over an empty set, T2 (μ, ) = ∞. This quantity, the so-called L2 -mixing time of a Markov transition function with initial distribution μ, plays an important role in the quantitative analysis of ergodic Markov processes. For any Markov transition function p(t, x, ·), t ∈ T and x ∈ Ω, let Pt be the operator defined in (4.1). Extend Pt as a bounded operator on the Hilbert space L2 (Ω, π). Definition 4.2. The spectral gap λ of p(t, x, ·), t ∈ T , is the supremum of all constants c such that ∀f ∈ L2 (Ω, π), ∀t ∈ T , (Pt − π)f 2 e−ct f 2 . Remark 4.1. If T = [0, ∞) and Pt is a strongly continuous semigroup of contractions on L2 (Ω, π), then λ can be characterized using the infinitesimal generator A of Pt = etA . That is, λ = inf −Af, f : f ∈ Dom(A), real-valued, π(f ) = 0, π f 2 = 1 . In general, λ is not in the spectrum of A but, assuming that Dom(A) = Dom(A∗ ) = D then λ is in the spectrum of any self-adjoint extension of the symmetric operator ( 12 (A + A∗ ), D). Remark 4.2. If T = N, then e−λ is the second largest singular value of the operator P1 on L2 (Ω, π), namely, λ = − log P1 − πL2 (Ω,π)→L2 (Ω,π) . Consider a family of measurable spaces (Ωn , Bn ) indexed by n = 1, 2, . . . . For each, n, let pn (t, x, ·) with t ∈ T , x ∈ Ωn , be a Markov transition function with invariant measure πn and spectral gap λn . Fix a sequence of probability measures μn on Ωn and let fn (t) = Dn,2 (μn , t) be the L2 -distance defined in (4.2). Then, the family {pn (t, μn , ·): n 1} is said to have an L2 -cutoff (resp. L2 -precutoff and (tn , bn )–L2 -cutoff) if {fn : n 1} has a cutoff (resp. precutoff and (tn , bn )-cutoff) in the sense of Definition 2.1. The following proposition gives a sufficient condition for the L2 -cutoff.

2258


Proposition 4.1. (See [5, Theorem 3.3].) Referring to the setting and notation introduced above, assume that D2 (μn , t) vanishes as t tends to infinity and set tn = Tn,2 (μn , ) = inf{t ∈ T : Dn,2 (μn , t) }. (i) Assume that T = [0, ∞) and tn λn → ∞. Then the family {pn (t, μn , ·): n 1} has a 2 (tn , λ−1 n )-L -cutoff. (ii) Assume that T = N and tn γn → ∞ where γn = max{λn , 1}. Then the family {pn (t, μn , ·): n 1} has a (tn , γn−1 )-L2 -cutoff. We refer the reader to [4,5] for further results in this direction. The goal of the present work is to provide a necessary and sufficient condition for an L2 -cutoff and to describe the cutoff time using spectral theory. 4.3. The L2 -distance for normal Markov kernels Let T = [0, ∞) or N. A Markov transition function p(t, ·,·), t ∈ T , with invariant probability measure π is called normal if, for t ∈ T ∩ [0, 1], the operator Pt : L2 (Ω, π) → L2 (Ω, π) defined by (4.1) is normal, that is, Pt Pt∗ = Pt∗ Pt on L2 (Ω, π). In the case that Pt is a strongly continuous semigroup with infinitesimal generator A, the normality of p(t, ·,·) is equivalent to that of A. When Pt is normal, Pt − πL2 (Ω,π)→L2 (Ω,π) = e−λt ,

∀t > 0.

Our next goal is to obtain a spectral formula for the chi-square distance. See Theorems 4.4–4.5 below. Lemma 4.2. Let {Pt : t > 0} be a strongly continuous semigroup of contractions associated to a transition function p(t, x, ·), x ∈ Ω, t 0, by (4.1). Let A be its infinitesimal generator. Assume that A is normal and let {EB : B ∈ B(C)} be a resolution of the identity corresponding to −A, where B(C) is the Borel algebra over C. Set C0 = {bi: b ∈ R},

C1 = {a + bi: a > 0, b ∈ R}.

Then, for g ∈ L2 (Ω, π), lim Pt g2 = EC0 g2 .

t→∞

In particular, if Pt g − π(g)2 → 0 as t tends to infinity, then EC0 g = π(g) and

Pt g − π(g)2 = e−2Re(γ )t dEγ g, gπ . 2 C1

Proof. Let C = C0 ∪ C1 . Since (Pt )t>0 are contractions, the spectrum of −A is contained in C. By the spectral theorem, for all g ∈ L2 (Ω, π),

Pt g22 = e−2 Re(γ )t dEγ g, gπ = EC0 g22 + e−2 Re(γ )t dEγ g, gπ . C

C1

For a reference on the resolution of the identity for normal operators, see [18].

2


2259

Lemma 4.3. Let {Pt : t > 0} be as in Lemma 4.2 with infinitesimal generator A and spectral gap λ. Let σ (−A) be the spectrum of −A and λ = inf{Re(c): Re(c) > 0, c ∈ σ (−A)}, where Re(c) denotes the real part of c. Then, (i) λ λ. (ii) Assume that V is a dense subspace of L2 (Ω, π) and the following limit holds lim Pt g − π(g)2 = 0, ∀g ∈ V . t→∞

Then λ = λ. In particular, if λ > 0, then λ = λ. Remark 4.3. The converse of Lemma 4.3(i) is not always true and a typical example is to consider reducible finite Markov chains. Proof of Lemma 4.3. For the spectrum of −A, since Pt is a contraction for t > 0, σ (−A) is a subset of the half plane {a + bi: a 0, b ∈ R}. In the case that Pt is normal, we may choose, by the spectral decomposition, a resolution of the identity {EB : B ∈ B(C)} corresponding to −A such that

γ dEγ g, gπ , ∀g ∈ D(A), (4.3) −Ag, g = σ (−A)

where B(C) is the Borel algebra over C and D(A) is the domain of A. By Remark 4.1, λ can be obtained by the formula inf −Ag, gπ : g ∈ D(A), π(g) = 0, π g 2 = 1 . For (i), note that if λ = 0, then obviously λ λ. We now assume that λ > 0. Fix δ ∈ (0, λ) and let Bδ = {c ∈ C: 0 < Re(c) < δ} and T = EBδ . Using (4.3), one may easily compute that, for g ∈ D(A) with π(g) = 0,

T (−A − δ)g, g

π

= (−A − δ)(T g), T g π (λ − δ)T g22 0

and

T (−A − δ)g, g

π

=

(γ − δ) dEγ g, gπ . Bδ

Since T (−A − δ)g, gπ is real, the above identity can be rewritten as

T (−A − δ)g, g π = Re(γ ) − δ dEγ g, gπ 0. Bδ

Combining the above three inequalities, we obtain that EBδ g = 0 for g ∈ D(A) satisfying / Bδ and 1 is contained in the range of E{0} ). π(g) = 0. It is also clear that EBδ 1 = 0 (since 0 ∈ Thus, using the fact D(A) = L2 (Ω, π), we have EBδ = 0 on L2 (Ω, π). Finally, because σ (−A)

2260


is exactly the essential range of the function Ψ (t) = t w.r.t. {EB : B ∈ B(C)}, Bδ and σ (−A) are mutually disjoint, which implies δ λ for δ ∈ (0, λ). This proves the first part. For (ii), it remains to show that λ λ. Obviously, this inequality holds for λ = 0. For the case λ > 0, we set C1 = {a + bi: a λ, b ∈ R}. By Lemma 4.2,

2 ∀g ∈ V , Pt g − π(g)2 = e−2 Re(γ )t dEγ g, gπ e−2λt g22 . C1

λ λ. Since V is dense in L2 (Ω, π), the above holds true on L2 (Ω, π). Thus,

2

We are now ready to compute the L2 -distance, D(μ, t), using the spectral information of the infinitesimal generator A of Pt . Theorem 4.4. Let {Pt : t > 0} be as in Lemma 4.2 with infinitesimal generator A and spectral gap λ > 0. Assume that A is normal and {EB : B ∈ B(C)} is a resolution of identity for −A. If μ is a probability measure with an L2 -density f w.r.t. π , then

2 e−2 Re(γ )t dEγ f, f π , D2 (μ, t) = C(λ)

where C(λ) = {c ∈ C: Re(c) λ}. Proof. Let d(μPt ) = ft dπ . Then for g ∈ L2 (Ω, π), g, ft π = (μPt )(g) = μ(Pt g) = Pt g, f π = g, Pt∗ f π , where Pt∗ denotes the adjoint operator of Pt . This implies that ft = Pt∗ f . Since Pt is normal, it is obvious that Pt g2 = Pt∗ g2 for all g ∈ L2 (Ω, π) and, hence,

∗ 2 2 2 D2 (μ, t) = Pt (f − 1) 2 = Pt f − π(f ) 2 = e−2 Re(γ )t dEγ f, f π C(λ)

where the last equality uses Lemmas 4.2 and 4.3.

2

The discrete time version for Theorem 4.4 is as follows. The proof is similar. Theorem 4.5. Let {Pt : t ∈ N} be the family of contractions on L2 (Ω, π) defined in (4.1) with spectral gap λ > 0. Assume that P1 is a normal operator whose corresponding resolution of the identity is {EB : B ∈ B(C)}. If μ is a probability measure with an L2 -density f w.r.t. π , then for t ∈ N,

|γ |2t dEγ f, f π , D2 (μ, t)2 = C(λ)

= {c ∈ C: |c| λ}. where C(λ)


2261

4.4. The L2 -cutoff time for families of normal Markov kernels Now, we shall assume that the initial probability has an L2 -density (with respect to the invariant probability). In this case, Theorems 4.4 and 4.5 imply that the L2 -distance between a normal Markov transition function and its stationary measure is a Laplace transform. Thus, the results in Section 3 are applicable. In detail, let (Ωn , Bn ) be a measurable space and pn (t, x, ·), t ∈ T (T = [0, ∞) or N) and x ∈ Ωn , be a Markov transition function on Ωn with invariant probability measure πn . Let Pn,t be the operator defined by

Pn,t g(x) =

g(y)pn (t, x, dy),

∀g ∈ L2 (Ωn , πn ), t ∈ T ,

(4.4)

Ωn

and μn be a probability on (Ωn , Bn ) with L2 -density fn w.r.t. πn . • For T = [0, ∞), assume that Pn,t is normal, strongly continuous, with positive spectral gap λn , and that the infinitesimal generator of Pn,t has resolution of the identity {En,B : B ∈ B(C)}, where B(C) is the Borel algebra over C. For λ > 0, let Sλ be the strip {c ∈ C: Re(c) ∈ (0, λ]} and set Vn (λ) = En,Sλ fn , fn πn .

(4.5)

• For T = N, assume that Pn,1 is normal with positive spectral gap λn and resolution of the identity {En,B : B ∈ B(C)}. For λ > 0, let Aλ be the annulus {c ∈ C: |c| ∈ [e−λ , 1)} and set Vn (λ) = En,Aλ fn , fn πn .

(4.6)

As a consequence of Theorems 4.4 and 4.5, the L2 -distance is given by

Dn,2 (μn , t) = 2

e−2λt dVn (λ).

[λn ,∞)

To state the main results of this paper, for δ > 0 and C > 0, set ⎧ tn (δ) = Tn,2 (μn , δ) = inf t ∈ T : Dn,2 (μn , t) δ , ⎪ ⎪ ⎨ λ (C) = inf λ: V [λ , λ] > C , n n n

⎪ log(1 + Vn ([λn , λ])) ⎪ ⎩ τn (C) = sup : λ λn (C) . 2λ

(4.7)

Further, set γn = λn (C)−1 , bn = λn (C)−1 log λn (C)τn (C) if T = [0, ∞), γn = max 1, λn (C)−1 , bn = max 1, λn (C)−1 log λn (C)τn (C) if T = N. If T = N, we assume in addition that either τn (C) or tn (δ) tends to infinity, for some C or some δ.

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Theorem 4.6. Referring to the setup and notation described in (4.4)–(4.7), (i) If lim inf πn (fn2 ) < ∞, then {pn (t, μn , ·): n 1} has no L2 -precutoff. n→∞

(ii) If πn (fn2 ) → ∞, then the following are equivalent. (a) {pn (t, μn , ·): t ∈ [0, ∞)} has an L2 -cutoff. (b) For some positive constants C, δ, ,

lim tn (δ)λn (C) = ∞,

n→∞

lim

n→∞ [λn ,λn (C))

e−λtn (δ) dVn (λ) = 0.

(c) For some positive constants C, ,

lim τn (C)λn (C) = ∞,

n→∞

lim

n→∞ [λn ,λn (C))

e−λτn (C) dVn (λ) = 0.

Further, – If (b) holds, then {pn (t, μn , ·): n 1} has a (tn (δ), γn )-L2 -cutoff. – If (c) holds, then {pn (t, μn , ·): n 1} has a (τn (C), bn )–L2 -cutoff. Proof. Immediate from Theorems 3.5 and 3.8

2

Remark 4.4. By Remark 3.4, if the family {pn (t, μn , ·): t ∈ [0, ∞)} has an L2 -cutoff, then Theorem 4.6 (b) and (c) hold for any positive C, δ, . Similarly, by Remark 3.6, if the family {pn (t, μn , ·): t ∈ N} has an L2 -cutoff with the L2 -mixing time T2 (μn , δ) tending to infinity, then Theorem 4.6 (b) and (c) are true for any positive C, δ, . 5. Applications to finite Markov chains In this section, we spell out how our main results apply to normal Markov chains on finite state spaces. Let Ω be a finite set and K be an irreducible Markov kernel on Ω with invariant probability measure π . Denote by p d (t, ·,·) the associated discrete time Markov transition function, that is, p d (t, x, y) = K t (x, y), t ∈ N. Let p c (t, ·,·) be the associated continuous time Markov transition function defined by p c (t, x, y) = e−t (I −K) (x, y) = e−t

∞ n t n=0

n!

K n (x, y),

t 0.

(5.1)

To facilitate applications, we discuss continuous and discrete time separately. For n 1, let Ωn be a finite set and Kn be an irreducible Markov kernel on Ωn with invariant probability πn . Let μn be some given initial distribution with density fn with respect to πn . We assume that Kn is normal. Its eigenvalues and eigenfunctions will be ordered in different ways in the discrete and continuous time cases. We let pnc (t, x, y), pnd (t, x, y) be the corresponding continuous and discrete time transition functions.


2263

5.1. Continuous time Let βn,0 = 1, βn,1 , . . . , βn,|Ωn |−1 be the eigenvalues of Kn with orthonormal eigenvectors ψn,0 ≡ 1, ψn,1 , . . . , ψn,|Ωn |−1 on L2 (Ωn , πn ), ordered in such a way that Re βn,i Re βn,i+1 ,

∀1 i |Ωn | − 2.

Let λn,i = 1 − Re βn,i and set, for C > 0,

j μn (ψn,i )2 > C jn = jn (C) = min j 1:

(5.2)

i=1

and j log i=0 |μn (ψn,i )|2 . τn = τn (C) = max 2λn,j j jn

(5.3)

Note that c Dn,2 (μn , t) =

1/2 μ(ψn,i )2 e−2λn,i t

(5.4)

i1

and set c c tn = tn (δ) = Tn,2 (μn , δ) = inf t 0: Dn,2 (μn , t) δ .

(5.5)

Theorem 4.6 yields the following result. Theorem 5.1. Referring to the above setting and notation, (i) If lim infn→∞ πn (fn2 ) < ∞, then {pnc (t, μn , ·): n 1} has no L2 -precutoff. (ii) If πn (fn2 ) → ∞, then the following are equivalent. (a) {pnc (t, μn , ·): n 1} has an L2 -cutoff. (b) For some positive constants C, , δ, lim tn λn,jn = ∞,

n→∞

lim

n→∞

j n −1

μn (ψn,i )2 e−tn λn,i = 0.

i=1

(c) For some positive constants C, , lim τn λn,jn = ∞,

n→∞

lim

n→∞

j n −1

μn (ψn,i )2 e−τn λn,i = 0.

i=1

2 Furthermore, in case (ii), if (b)/ (c) holds, then {pnc (t, μn , ·): n 1} has a (tn , λ−1 n,jn )–L -

cutoff and a (τn , bn )–L2 -cutoff with bn = λ−1 n,jn log(τn λn,jn ).

2264


C Remark 5.1. By Remark 3.8, τn (C) T2c (μn , C+1 ) for C > 0.

Remark 5.2. Theorem 5.1 is useful in proving an L2 -cutoff if there is indeed one but, as stated, in order to disprove the existence of an L2 -cutoff, one has to show that Theorem 5.1 (b) and (c) fail for all C, δ, . In fact, as stated in Remark 4.4, a stronger version of Theorem 5.1 says that if pnc has an L2 -cutoff, then (b) holds for any triple (C, , δ) and (c) holds for any pair (C, ). Hence, to disprove the existence of an L2 -cutoff, we only need to check that (b) or (c) fails for some constants C, , δ. The following is a simple application of Theorem 5.1 which deals with the L2 -cutoff for a specific class of chains, those whose spectral gap is bounded away from 0 as in the case of expander graphs. The notation is as above. c Corollary 5.2. Assume that Kn is normal and λ−1 n is bounded. Then the family {pn (t, μn , ·): 2 2 n 1} has an L -cutoff if and only if πn (fn ) → ∞. Furthermore, if πn (fn2 ) → ∞, then the family {pnc (t, μn , ·): n 1} presents a strongly optimal (tn , 1)-L2 -cutoff, where tn is the constant in (5.5) and δ is any positive constant.

Proof. The first part of this corollary is obvious from Theorem 5.1 whereas the second part follows from c δe−2c Dn,2 (μn , tn + c) δe−cλn ,

∀ − tn < c < 0

and c δe−cλn Dn,2 μn , tn + cλ−1 δe−2c , n

∀c > 0.

2

5.2. Discrete time To treat the discrete time case, order the eigenvalues βn,0 = 1, βn,1 , . . . , βn,|Ωn |−1 and orthonormal eigenvectors ψn,0 ≡ 1, ψn,1 , . . . , ψn,|Ωn |−1 in such a way that |βn,i | |βn,i+1 |,

∀1 i |Ωn | − 2.

Set λn,i = − log |βn,i | and define jn = jn (C) and τn = τn (C) by (5.2). The L2 -distance takes the form d Dn,2 (μn , t) =

1/2 μ(ψn,i )2 |βn,i |2t .

(5.6)

i1

For δ > 0, set d d (μn , δ) = inf t 0: Dn,2 (μn , t) δ . tn = tn (δ) = Tn,2

(5.7)

Theorem 5.3. Referring to the above setting and notation and assuming that either tn (δ) → ∞ for some δ > 0 or τn (C) → ∞ for some C > 0, we have:


2265

(i) If lim infn→∞ πn (fn2 ) < ∞, then {pnd (t, μn , ·): n 1} has no L2 -precutoff. (ii) If πn (fn2 ) → ∞, then the following are equivalent. (a) {pnd (t, μn , ·): n 1} has an L2 -cutoff. (b) For some positive constants C, , δ, lim tn λn,jn = ∞,

n→∞

lim

n→∞

j n −1

μn (ψn,i )2 |βn,i |tn = 0.

i=1

(c) For some positive constants C, , lim τn λn,jn = ∞,

n→∞

lim

n→∞

j n −1

μn (ψn,i )2 |βn,i |τn = 0.

i=1

In case (ii), if (b)/ (c) holds, then {pnd (t, μn , ·): n 1} has a (tn , γn−1 )–L2 -cutoff with γn = min{λn,jn , 1} and a (τn , bn )-L2 -cutoff with bn = max{λ−1 n,jn log(τn λn,jn ), 1}. Remark 5.3. Remarks 5.1 and 5.2 remain true in discrete time cases. One also easily obtains a discrete version of Corollary 5.2 under the assumption that the eigenvalues βn,i , 0 i |Ωn |−1, βn,0 = 1 of the normal operator Kn satisfy inf{1 − |βn,i |, |βn,i |: 1 i |Ωn | − 1, n 1} > 0. 5.3. Invariant kernels Next, we specialize Theorems 5.1–5.3 to the case when the kernels Kn are invariant under some transitive group action, i.e., for each n, there is a group Gn acting transitively on Ωn and such that Kn (gx, gy) = Kn (x, y)

∀g ∈ Gn , x, y ∈ Ωn .

(5.8)

If |Ωn | ∞, then the families {pnc (t, xn , ·): n 1} and pnd (t, xn , ·) have no L2 -precutoff so we assume that |Ωn | → ∞. The notable contribution of these results is in the explicit spectral description of the cutoff time τn . Theorem 5.4 (Continuous time). Assume that |Ωn | → ∞ and that Kn satisfies (5.8) and is irreducible normal with eigenvalues βn,0 = 1, βn,1 , . . . , βn,|Ωn |−1 , Re βn,i Re βn,i+1 , ∀1 i |Ωn | − 2. Let λn,i = 1 − Re βn,i , λn = λn,1 and set

log(j + 1) , τn = sup 2λn,j j 1 The following properties are equivalent. (a) {pnc (t, xn , ·): n 1} has a L2 -cutoff. (b) tn λn → ∞ for some δ > 0. (c) τn λn → ∞.

c tn = Tn,2 (xn , δ).

(5.9)

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2 Furthermore, if (b)/ (c) holds, then {pnc (t, xn , ·): n 1} has a (tn , λ−1 n )-L -cutoff and a (τn , bn )L2 -cutoff with bn = λ−1 log(τ λ ). n n n

Theorem 5.5 (Discrete time). Assume that |Ωn | → ∞ and that Kn satisfies (5.8) and is irreducible normal with eigenvalues |βn,i | |βn,i+1 |, 0 i |Ωn | − 2. Set λn,i = − log |βn,i |, d (x , δ). λn = λn,1 , and let τn be defined in terms of these λn,i as in (5.9). Set also tn = Tn,2 n Assume that either tn → ∞ for some δ > 0 or τn → ∞. Then the following are equivalent. (a) {pnd (t, xn , ·): n 1} has an L2 -cutoff. (b) tn λn → ∞ for some δ > 0. (c) τn λn → ∞. Furthermore, if (b)/ (c) holds, then {pnd (t, xn , ·): n 1} has a (tn , γn−1 )-L2 -cutoff with γn = min{λn , 1} and a (τn , bn )-L2 -cutoff with bn = max{λ−1 n log(τn λn ), 1}. 6. The Ehrenfest chain The Ehrenfest chain is one of the most celebrated example of finite Markov chain. Its state space is Ωn = {0, . . . , n} and its kernel is given by i Kn (i, i + 1) = 1 − , n

Kn (i + 1, i) =

i+1 , n

∀0 i < n.

(6.1)

It is clear that Kn is irreducible with stationary distribution πn (i) = ni 2−n , i ∈ {0, 1, . . . , n}. Note that Kn is periodic. The L2 -distance of the Ehrenfest chain to its stationary measure has been studied by many authors. By lifting the chain to a walk on the hypercube, the representation theory of (Z2 )n can be used to identify the eigenvalues and eigenvectors of the Ehrenfest chain and to compute the L2 -distance. The following well-known result gives a description on the eigenvalues and eigenvectors of Kn . Theorem 6.1. The matrix Kn defined in (6.1) has eigenvalues βn,i = 1 −

2i , n

0 i n,

with L2 (πn )-normalized right eigenvectors −1/2 i n−x n k x ψn,i (x) = , (−1) k i−k i

0 i, x n.

(6.2)

k=0

Proof. See, e.g., [7]. The vectors ψn,i are in fact the Krawtchouk polynomials (up to a constant multiple) and the desired properties are the orthogonality and recurrence relation of Krawtchouk polynomials. See [14,16]. 2


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To apply our main result using the above spectral information, we need to study ψn,i . Recall the classical notation ∞ (a1 )k · · · (ar )k zk a1 , . . . , ar z = r Fs b 1 , . . . , bs (b1 )k · · · (bs )k k! k=0

where, for a ∈ R and n 0, (a)n is the Pochhammer symbol defined by (a)0 = 1,

(a)n = a(a + 1) · · · (a + n − 1),

∀n 1.

Using this notation, Krawtchouk polynomials are defined by −i, −x 1 Pi (x, p, n) = 2 F1 −n p for i = 0, 1, . . . , n. Then, the eigenvector ψn,i of Kn can be rewritten as ψn,i (x) =

1/2 n Pi (x, 1/2, n). i

(6.3)

The recurrence relation for Pi (j, 1/2, n) is (n − 2x)Pi (x, 1/2, n) = (n − i)Pi+1 (x, 1/2, n) + iPi−1 (x, 1/2, n).

(6.4)

Note that this is exactly saying that βn,i and ψn,i are eigenvalues and eigenvectors for Kn . Using the above identity, we are able to apply the results from Section 5 to the Ehrenfest chain. 6.1. The continuous time Ehrenfest process The transition function of the continuous time Ehrenfest process is given by pnc (t, ·,·) = e−t (I −Kn ) . Theorem 6.2. Given starting states xn , the family Fc of the continuous time Ehrenfest chains {pnc (t, xn , ·), n = 1, 2, . . .} has an L2 -cutoff if and only if lim

n→∞

|n − 2xn | = ∞. √ n

Our second result concerns the L2 -cutoff time and the optimality of window sequences. Theorem 6.3. Referring to the Ehrenfest family Fc , Assume that (6.5) holds and let tn =

|n − 2xn | n log √ . 2 n

Then, there exist universal positive constants A, N such that for all n N e−2c Dn,2 (xn , tn + cn) Ae−2c ,

(6.5)

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where the first inequality holds true for all real c with tn + cn 0 and the second inequality is true for c > 0. Remark 6.1. By Proposition 3.7, this result says that there is an optimal (tn , n)–L2 -cutoff. In fact, the (tn , n)–L2 -cutoff is strongly optimal. Using the relation between the first and the second eigenvectors of Kn , we also obtain a result concerning the total variation cutoff or equivalently, the L1 -cutoff. The details of the proof are omitted. Theorem 6.4. Referring to the Ehrenfest family Fc , let Dn,TV (xn , t) be the total variation distance between the distribution of the nth chain at time t starting from xn and πn . Then, for all n 1 and all c such that tn + cn 0, we have Dn,TV (xn , tn + cn) 1 − 8e4c . Remark 6.2. By Theorems 6.3–6.4, if (6.5) holds, then there is a (tn , n)-total variation cutoff. Before proving these results, we make some analysis on the eigenvectors ψn,i . Let xn = + yn ) with |yn | n. Using (6.3), the recurrence relation of Krawtchouk polynomials in (6.4) yields the following identity 1 2 (n

−yn an,i − an,i+1 = √ (i + 1)(n − i)

i(n − i + 1) an,i−1 (i + 1)(n − i)

= −An,i bn an,i − Bn,i an,i−1

(6.6)

√ where an,i = ψn,i (xn ), bn = yn / n and An,i =

n , (i + 1)(n − i)

Bn,i =

i(n − i + 1) . (i + 1)(n − i)

(6.7)

To compute an,i using the above iterative formula, one needs the boundary conditions an,1 and an,2 , which can be easily determined using the formula given in Theorem 6.1. They are an,1 = −bn ,

an,2 =

2 n bn − 1 . 2(n − 1)

(6.8)

Concerning the L2 -distance, the symmetry of the chain implies that there is no loss of generality in assuming xn n/2, that is, yn 0. (Otherwise, one only needs to replace xn with n − xn without any change on the Lp -distance.) From now on, we assume that xn n/2. Before starting the proofs, let λn,i , jn (C) and τn (C) be as in Theorem 5.1. It can be easily seen from Theorem 6.1 that λn,i = 2i/n. Also, note that (6.5) is equivalent to bn → ∞.


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Proof of Theorem 6.2. We first prove that (6.5) implies an L2 -cutoff. Since |ψn,1 (xn )| = bn → ∞, we have jn (1) = 1 for n large enough. This implies that τn (1)

log |an,1 | λn,1

and, hence, λn,1 τn log |an,1 | → ∞ as n → ∞. By Theorem 5.1, Fc has an L2 -cutoff. Suppose now that bn ∞. By Proposition 2.1, we can assume that bn is bounded from above, say by B. Observe that, by (6.8), 2 2 an,1 + an,2 = bn2 +

2 2 n bn − 1 1/2. 2(n − 1)

This implies that jn (1/2) 2 for all n. Also, it is obvious from (6.7) that An,i 1,

Bn,i 1,

∀0 i < n.

By setting γ = supn bn ∨ 1 using these inequalities and (6.6), one derives that |an,i+1 | γ |an,i | + |an,i−1 |,

∀1 i < n, n 1.

Then, an inductive argument along with the fact |an,1 | B and |an,2 | B 2 for n > 1 implies that |an,i | B 2 (γ + 1)i ,

∀1 i n,

which gives j i=0

|an,i |2 B 4

j (γ + 1)2i B 4 (γ + 1)2j +1 ,

∀j n.

i=0

Hence, 2j + 1 < ∞. j 1j n

τn (1/2)λn,jn (1/2) log B 4 + log(γ + 1) sup By Theorem 5.1, this shows there is no L2 -cutoff.

2

Proof of Theorem 6.3. Assume that bn → ∞ and set fn (c) = Dn,2 (xn , tn + cbn ) where tn is the sequence defined in Theorem 6.3, that is, tn = 12 n log bn . To prove the desired result, we need to investigate an,i or instead An,i and Bn,i . The following claim is the only fact we need. Claim: There exists N > 0 such that i +1 i +1 An,i + Bn,i bn−2 1, i i −1

∀2 i n − 3, n N.

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To prove this claim, set θ ∈ (1, 4/3). Then, for 2 i (1 − 1/θ )n, √ √ i +1 3θ θ (i + 1) n An,i = × n/2, since (i + 1)/i is decreasing in i and An,i = An,n−i−1 , the same bound holds. The claim is then proved by choosing N large enough so that i+1 Bn,i bn−2 3bn−2 < 1 − α, i−1

∀n N.

Returning to the proof of Theorem 6.3, we apply the triangle inequality in (6.6) to get |an,i+1 | An,i bn an,i + Bn,i an,i−1 .

(6.9)

Let N be the integer chosen in the claim above. Then, this iterative inequality and induction yields 2 |an,i | bni , i

1 i n − 2, n N.

Using (6.9) for i = n − 2 and n − 1 implies that |an,i |

β i b , i n

1 i n, n N,

for some β bigger than 2. To finish the proof, recall that Dn,2 (x, t)2 =

n ψn,i (xn )2 e−2tλn,i . i=1

Hence, for c > 0 and n N ,

∞ 1 −4c Dn,2 (xn , tn + cn) β e , i2 2

2

i=1

and for c ∈ R and n 1, 2 Dn,2 (xn , tn + cn)2 ψn,1 (xn ) e−2(tn +cn)λn,1 = e−4c .

2


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Proof of Remark 6.1. (The cutoff is strongly optimal.) The proof is similar to that of Theorem 6.3. Fix c < 0 and choose N = N ( c ) > 0 such that e

−2 c

2 + e−4c bn−2 1 ∀n N. N +1

Note that, for N i n − N − 1 and n 2N , n 2 n . (i + 1)(n − i) (N + 1)(n − N ) N + 1 Combining the above two inequalities then gives e−2c An,i + e−4c Bn,i bn−2 1 ∀N i < n − N, n 2N. As before, the iterative inequality (6.9) implies that |an,i | βe2ci bni for all 1 i n and n 2N , where β is a universal constant only depending on c . Hence, for c > c, Dn,2 (xn , tn + cn) β 2

2

∞

e4(c−c)i < ∞.

i=1

By Proposition 3.7, the family has a strongly optimal (tn , n)–L2 -cutoff.

2

6.2. Discrete time Ehrenfest chains Let Kn be the Markov kernel obtained by Kn =

1 n In+1 + Kn n+1 n+1

(6.10)

where Kn is the Ehrenfest kernel and In is the n × n identity matrix. As a consequence of = 1 − 2i with corresponding eigenvectors ψ Theorem 6.1, Kn has eigenvalues βn,i n,i given n+1 by (6.2). To apply Theorem 5.3 to this chain, we need to reorder the eigenvalues. For 1 i n/2, let λ n,2i−1

= λ n,2i

2i , = − log 1 − n+1

ψn,2i−1 = ψn,n−i+1 ,

ψn,2i = ψn,i .

(6.11)

in the discrete time case is given by Then, the L2 -distance Dn,2 (x, t) = Dn,2

n ψ (x)2 e−2tλ n,i . n,i

(6.12)

i=1

Write (6.3) in the form n − 2x ψn,i+1 (x) = − √ An,i ψn,i (x) − Bn,i ψn,i−1 (x), n

1 i < n,

2272


and ψn,1 (x) =

n − 2x √ ψn,0 (x), n

ψn,n−1 (x) =

n − 2x √ ψn,n (x), n

where ψn,0 ≡ 1, ψn,n (x) = (−1)x with An,i =

n , (i + 1)(n − i)

Bn,i =

i(n − i + 1) . (i + 1)(n − i)

Note that for 1 i n/2, An,n−i = An,i /Bn,i ,

Bn,n−i = 1/Bn,i .

This implies n − 2x ψn,n−i−1 (x) = − √ An,i ψn,n−i (x) − Bn,i ψn,n−i+1 (x). n Since ψn,i+1 and ψn,n−i−1 are derived by the same iterative formulae with respective initial values 1 and (−1)x , they are related as follows. ψn,n−i (x) = (−1)x ψn,i (x) This identity implies ψ (x) = 1, n,1

ψ

∀x, i ∈ {0, 1, . . . , n}.

= ψ

n,2i (x)

n,2i+1 (x)

∀1 i n/2.

(6.13)

This discussion will be used for the proof of the following theorem. Theorem 6.5. Let Ωn = {0, 1, . . . , n} and F = {(Ωn , Kn , πn ): n = 1, 2, . . .} be the family of Markov chains given by (6.10) with starting states (xn )∞ 1 . Then, the following are equivalent. √ (i) |n − 2xn |/ n → ∞; (ii) The family F has an L2 -cutoff. Furthermore, if (i) holds true, then F has a strongly optimal (tn , n)–L2 -cutoff and a (tn , n)-total variation cutoff, where tn =

|n − 2xn | n log √ . 2 n

Proof. The standard way to prove the above result would be to apply Theorem 5.3. Here, instead, using the L2 -distance, D , of the Ehrenfest process discussed in Theorem 6.3. we bound Dn,2 n,2 In detail, by (6.11), (6.12) and (6.13), we have

(x, t)2 e−2λn,1 t + 2 Dn,2

[n/2]

ψ

i=1

2 −2λ t e n,2i e−2λ n,1 t + Dn,2 (x, t)2

n,2i (x)


2273

where the last inequality uses the fact log(1 − t) −t for all t ∈ (0, 1), which implies that λ n,2i 2i/n = 1 − βn,i and βn,i is the term defined in Theorem 6.1. For the lower bound, we (x, t)2 , that is, use the second term in the series of Dn,i 2 (n − 2x)2 −2λ t e n,2 . Dn,2 (x, t)2 ψn,2 (x) e−2λn,2 t = n By writing λ n,2 = 2/n(1 + cn ), it can be easily shown that cn = O(1/n). √ Recall that xn = (n + yn )/2 with 0 yn n. By Theorem 6.3, if yn / n is bounded, then d (x , ) of D is of order at most n. (In fact, there exists > 0 such that the -mixing time Tn,2 n n,2 it is of order n using the lower bound obtained above.) Let jn (C) be the integer in Theorem 5.3. It is clear that jn (1) = 1 and, hence, λ n,1 = λ n,2 and 2 Tn,2 (xn , )λ n,1 = O(n) × (1 + cn ) = O(1). n √ By Theorem 5.3, F has no L2 -cutoff. In the case yn / n → ∞, Theorem 6.3 and Remark 6.1 imply that lim sup Dn,2 (xn , tn + cn) n→∞

0,

√ established above, where A is a constant and tn = (n/2) log(yn / n). Using the bounds for Dn,2 we get lim sup Dn,2 (xn , tn n→∞

+ cn)

0

and lim inf Dn,2 (xn , tn + cn) e−2c n→∞

∀c ∈ R.

This implies that F has a strongly optimal (tn , n)–L2 -cutoff. The proof of the total variation cutoff is as in the continuous time case. 2 7. Constant rate birth and death chains This section applies the main results of this paper to the study of constant rate birth and death chains. Finding the L2 -cutoff of family of Markov chains from arbitrary starting points is a difficult task that requires a great deal of spectral information. The following examples illustrate this very well. First, we treat families of finite constant rate birth and death chains on {0, . . . , n} with n tending to infinity and arbitrary constant rates pn , qn . Second, we discuss the case when the state space is the countable set {0, 1, . . .}.

2274


7.1. Chains of finite length Karlin and McGregor [12,13] observed that the spectral analysis of any given birth and death chain can be treated as an orthogonal polynomial problem. This sometimes leads to the exact computation of the spectrum. See, e.g., [10,12,13,20] and also [17] for a somewhat different approach based on continued fractions. The families of interest here are of the following simple type. For n 1, let Ωn = {0, 1, . . . , n} and let Kn be the Markov kernel of a birth and death chain on Ωn with constant rates pn (x) = pn ,

qn (x) = qn = 1 − pn ,

∀0 x n,

(7.1)

where pn (x) and qn (x) denote respectively the birth rate and the death rate with the usual convention that qn (0) = rn (0), pn (n) = rn (n) are holding probabilities. This has stationary (reversible) distribution πn given by πn (x) = cn

pn qn

x ,

n+1 pn pn 1− with cn = 1 − qn qn

−1

.

(7.2)

Set jπ √ , βn,j = 2 pn qn cos n+1

βn,0 = 1,

∀1 j n,

(7.3)

and let ψn,j be a vector on Ωn defined by ψn,0 ≡ 1 and, for 1 j n and x ∈ Ωn , ψn,j (x) = Cn,j

qn pn

(x+1)/2

(x+2)/2

qn j (x + 1)π j xπ − , sin sin n+1 pn n+1

(7.4)

−2 where Cn,j = cn (n + 1)qn λn,j /(2pn2 ) and λn,j = 1 − βn,j . Then, βn,j is an eigenvalue of Kn with corresponding normalized eigenvector ψn,j . See [9, Chapter XVI.3]. γ Let xn ∈ Ωn , n 1, be a sequence of initial states and set as before Dn,2 (xn , t), γ ∈ {c, d}, to be the L2 -distance for the nth chain starting from xn (c denotes the continuous time case and d stands for the discrete time case). Then, by Theorem 5.1(i) and Theorem 5.3(i), a necessary γ condition for the family {Dn,2 (xn , t), n = 1, 2, . . .}, γ ∈ {c, d}, to have a cutoff is πn (xn ) → 0 as n → ∞. The following lemma gives an equivalent condition for such a limit using pn and xn .

Lemma 7.1. For n 1, let πn (·) be the probability defined in (7.2) with pn ∈ (0, 1/2). Then, for xn ∈ {0, 1, . . . , n}, πn (xn ) → 0 if and only if lim

n→∞

1 xn + 1 − 2pn pn

= ∞.

Proof. Set, for n 1, bn = (pn /qn )xn . Then, πn (xn ) = bn cn . Assume that πn (xn ) → 0. Using the fact log(1 + t) t, we have log cn = − log 1 + pn /qn + · · · + (pn /qn )n −(qn − pn )−1


2275

and xn q n − pn − . log bn = −xn log 1 + pn pn Thus, bn cn → 0 implies xn /pn + 1/(qn − pn ) → ∞ as desired. For the other direction, assume that lim supn→∞ πn (xn ) > 0. Since bn 1 and cn 1, we may choose a subsequence (nk )k1 such that infk bnk > 0 and infk cnk > 0. Consider the following identity. 1 − cn = (pn /qn ) 1 − cn (pn /qn )n . This implies that cn → 0

⇔

pn → 1/2

and, hence, lim sup pnk = p < 1/2. k→∞

Using the last observation and the fact infk bnk > 0, it is clear that xnk has to be bounded. Concerning the value of xnk , let A = {nk : xnk = 0} = {n k : k 1} and B = {nk : k = 1, 2, . . .} \ A = {n k : k 1}. Observe that |A| = ∞ or |B| = ∞ must hold. In the former case, it is easy to see that 1 1 xn 1 lim sup < ∞. + lim inf n→∞ 1 − 2pn pn 1 − 2p 1 − 2p k→∞ nk In the latter case, since xn k 1 and infk bn k infk bnk > 0, it must be true that infk pn k > 0. Hence, we obtain lim inf n→∞

1 xn + 1 − 2pn pn

lim sup k→∞

xn 1 + lim sup k 1 − 2pn k k→∞ pn k

supk xn k 1 + < ∞. 1 − 2p infk pn k

2

The next theorem concerns the L2 -cutoff for these birth and death chains and the associated cutoff time. γ

Theorem 7.2. Referring to the setting introduced above, for n 1 and γ ∈ {c, d}, let pn (t, ·,·) be the (continuous/discrete) associated Markov transition function. Fix a sequence of states γ xn ∈ Ωn . Assume that 0 < pn < 1/2. Then, for γ ∈ {c, d}, the family {pn (t, xn , ·): n 1} has an L2 -cutoff if and only if lim xn

n→∞

qn − 1 = ∞. pn

(7.5)

2276

G.-Y. Chen, L. Saloff-Coste / Journal of Functional Analysis 258 (2010) 2246–2315 γ

γ

γ

Moreover, if the above condition holds, then, for γ ∈ {c, d}, the family pn (t, xn , ·) has a (tn , bn )– L2 -cutoff where tnc

" xn (log qn − log pn ) , = − log(4pn qn ) !

xn (log qn − log pn ) , = √ 2(1 − 2 pn qn )

tnd

and bnc =

log log(qn /pn )xn , √ 1 − 2 pn q n

bnd =

log log(qn /pn )xn . − log(4pn qn )

Remark 7.1. Note that the cutoff time (if there is an L2 -cutoff) is independent of the size of the state space. Remark 7.2. Concerning the case pn ≡ p ∈ (0, 1/2), by Theorem 7.2, the existence of the L2 γ cutoff for pn (t, xn , ·), γ ∈ {c, d}, is equivalent to the condition xn → ∞. As a consequence of γ γ γ Theorem 7.2, if xn → ∞, the family pn (t, xn , ·), τ ∈ {c, d}, has a (tn , bn )–L2 -cutoff with tnc =

(log q − log p)xn , √ 2(1 − 2 pq)

tnd =

(log q − log p)xn , − log(4pq)

bnc = bnd = log xn .

Diaconis and Saloff-Coste proved in [8] that both families pnc (t, n, ·) and pnd (t, n, ·) have a sepn aration cutoff at time q−p . One can check that log q − log p log q − log p 1 > > √ 2(1 − 2 pq) − log(4pq) q −p

∀p ∈ (0, 1/2).

(7.6)

Thus, tnc tnd and the L2 -cutoff occurs later than the separation cutoff (this is not always true). Note that the window given here is not optimal. For example, in continuous time case, it can be proved directly using the expression in (5.4) and the formulas (7.3), (7.4) that pnc (t, n, ·) has a strongly optimal (tnc , 1)–L2 -cutoff, where the strong optimality uses Corollary 5.2. Similarly, for any integer m, the L2 -distance between pnd (tnd + m, n, ·) and πn always converges to 0 as n → ∞. Remark 7.3. In the case pn → 0, the equivalent condition for the existence of the L2 -cutoff is γ γ γ xn → ∞. If this holds true, then the family pn (t, xn , ·) has a (tn , bn )–L2 -cutoff with 1 tnc = xn log(1/pn ), 2

bnc = log xn

and tnd = xn , Note that tnc and tnd are of different order.

bnd =

log xn . log(1/pn )


2277

Remark 7.4. In the case pn → 1/2, write pn = 12 − δnn , where δn = o(n). By Theorem 7.2, for γ γ ∈ {c, d}, pn (t, xn , ·) has an L2 -cutoff if and only if xn δn /n → ∞. Moreover, if xn δn /n → ∞, then both families in continuous time and discrete time cases have a (tn , bn )–L2 -cutoff with tn =

nxn , δn

bn = xn +

n2 x n δn . log 2 n δn

Theorem 7.2 also holds in the case pn = qn = 1/2 where there is no cutoff. This is well known and we omit the details. Proof of Theorem 7.2. The proof of Theorem 7.2 involves considering several cases. We shall use the convention that, for any two sequences of positive numbers sn , tn , ⎧ ⎨ sn ∼ tn if limn→∞ sn /tn = 1; sn tn if lim supn→∞ {sn /tn } < ∞; (7.7) ⎩ sn tn if sn tn , tn sn . Set pn = 12 − δnn and let xn ∈ {0, 1, . . . , n}. Then, for any sequence of pairs (xn , pn ), there exists a subsequence nk such that the conjunction of one A(i) and one B(j ) holds, where ⎧ A(1): δnk = o(1); B(1): xnk ≡ 0; ⎪ ⎪ ⎪ ⎪ ⎪ B(2): xnk 1; ⎨ A(2): δnk 1; A(3): δnk → ∞, δnk = o(nk ); B(3): xnk → ∞, xnk δnk = o(nk ); ⎪ ⎪ ⎪ A(4): δnk /nk → δ ∈ (0, 1/2); B(4): xnk → ∞, xnk δnk nk ; ⎪ ⎪ ⎩ A(5): δnk /nk → 1/2; B(5): xnk → ∞, nk = o(xnk δnk ). Let R(i, j ) denote the case when A(i) and B(j ) hold. Clearly, R(1, 4), R(1, 5), R(2, 5), R(4, 3), R(4, 4), R(5, 3) and R(5, 4) can not happen. By Lemma 7.1, it is easy to see that πn (xn ) 1 is equivalent to cases R(4, 1), R(4, 2) and R(5, 1). Thus, by Theorem 5.1, the family of continuous time chains has no L2 -cutoff in those cases. For the family of discrete time chains, we can show that 2 d p (0, 0, ·) − 1 = 0 in R(5, 1) lim πn n n→∞ πn (·) and ∀t > 0

2 d p (t, 0, ·) − 1 > 0 lim inf πn n n→∞ π (·)

in R(4, 1) and R(4, 2).

n

This implies that no L2 -cutoff exists, where the case R(5, 1) uses the first equality and cases R(4, 1) and R(4, 2) use the second inequality and Corollary 3.3. Let ξ = {nk : k 1} and Fξ be the subfamily of F indexed by ξ . By Proposition 2.1, to prove Theorem 7.2, in cases R(3, 5), R(4, 5), R(5, 2) and R(5, 5), it suffices to show that Fξ has an L2 -cutoff. In cases R(i, j ) with i, j ∈ {1, 2, 3} and R(2, 4), R(3, 4), it suffices to show that Fξ c = ψ d ≡ 1 and set has no L2 -cutoff. To simplify the notations, we write F for Fξ . Let ψn,0 n,0 λcn,i = λn,i = 1 − βn,i ,

c ψn,i = ψn,i ,

∀1 i n,

(7.8)

2278


and, for 1 i [(n + 1)/2], λdn,2i−1 = λdn,2i = − log βn,i ,

d ψn,2i−1 = ψn,i ,

d ψn,2i = ψn,n+1−i .

(7.9)

As before, c and d represent the continuous time and discrete time cases, and Fc and Fd are the corresponding families. Necessity of (7.5). Consider the cases R(1, j ) with 1 j 3 and R(i, j ) with i ∈ {2, 3} and j ∈ {1, 2, 3, 4}. Rewrite (7.4) as follows. 2 ψn,i (xn ) =

where n,i = s < t,

22n,i (n + 1)πn (xn )λn,i

,

∀1 i n,

(7.10)

√ √ n +1)π nπ pn sin i(xn+1 − qn sin ix n+1 . Note that, for 0 s π/2 and 0 t π with 1 2 1 (t − s 2 ) cos s − cos t t 2 − s 2 . 8 2

This implies that, for 1 i n, λn,i

αn,i /5, iπ (qn − pn )2 √ = √ √ 2 + 2 pn qn 1 − cos 5αn,i n+1 ( q n + pn )

(7.11)

where αn,i

1 4δ 2 = 2 δn2 + i 2 1 − 2n . n n

Note also that, for j ∈ {1, 2, 3, 4}, πn (xn )

1/n δn /n

for R(1, j ), R(2, j ), for R(3, j ).

(7.12)

Now, we are going to disprove the existence of L2 -cutoff using Theorems 5.1–5.3. We first treat the continuous time cases. For C > 0, let jn (C), τn (C) be as defined in (5.2), (5.3). Step 1 and Step 2 below treat the cases R(1, j ) with j ∈ {1, 2, 3} and R(2, j ) with j ∈ {1, 2, 3, 4}, whereas Step 3 and Step 4 discuss the cases R(3, j ) with 1 j 4. Step 1: There exists C0 > 0 such that jn (C0 ) 1. Note that A(1) or A(2) implies pn → 1/2 and δn = O(1). When A(1) holds, by writing √ i(xn + 1)π ixn π ixn π √ √ − sin + pn − qn sin , (7.13) n,i = pn sin n+1 n+1 n+1 we have |n,1 | 1/n and |n,2 | 1/n. In a detailed computation, one can get

|n,1 | 1/n if xn /n ∈ [0, 3/8] ∪ [5/8, 1], |n,2 | 1/n if xn /n ∈ [3/8, 5/8].


2279

Thus, 2n,1 + 2n,2 n−2 . When A(2) holds, we may choose a constant ∈ (0, 1/2) such that |n,1 | 1/n,

∀xn ∈ [0, n] ∪ [n/2, n].

(7.14)

Clearly, for all k 1, |n,k | 1/n using (7.13). To get a similar bound as in A(1), observe that for fixed k 1, if xn ∈ [n/(2k), n/k], then √ kxn π kxn π k(xn + 1)π √ √ + qn − pn sin |n,k | = pn sin − sin n+1 n+1 n+1 √ kxn π (kxn + 1)π kxn π √ √ pn sin − sin + qn − pn sin 1/n, n+1 n+1 n+1 where the last asymptotic inequality is given by (7.14). Consequently, by setting K = 1/(2) , we have 2n,1 + · · · + 2n,K n−2 . Hence, in either case of A(1) and A(2), into (7.10) then gives K i=1

K

2 ψn,i (xn )

2 i=1 n,i

n−2 . Plugging this result, (7.11) and (7.12)

2(2n,1 + · · · + 2n,K ) (n + 1)πn (xn )λn,K

1.

This proves Step 1. Step 2: Let C0 be as in Step 1. Then, τn (C0 ) n2 . In order to prove this fact, we need the following computations. ixn π ixn π √ √ i(xn + 1)π √ + qn − pn sin − sin |n,i | pn sin n+1 n + 1 n + 1 iπ 4xn δn (1 + 4δn )πi 1+ . n n n Using the last inequality and (7.10)–(7.12), we obtain 2 2 2 2 2 10π 2 (1 + 4δn )2 ψ (xn ) 10π (1 + 4δn ) i /n # 1, n,i (n + 1)πn (xn )αn,i (n + 1)πn (xn ) 1 − 4δn2 /n2

where the last asymptotic relation is uniform for 1 i n. This implies that 2 sup ψn,i (xn ): 1 i n, n 1 = M < ∞ and, hence, n2 This proves Step 2.

log(1 + Mi) log(1 + C0 ) τn (C0 ) max n2 . 2λn,jn (C0 ) 2λn,i 1in

(7.15)

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It is immediate from Step 1 and Step 2 that λn,jn (C0 ) τn (C0 ) 1 and then, by Theorem 5.1, the family {pnc (t, xn , ·): n = 1, 2, . . .} has no L2 -cutoff. In Step 3 and Step 4, we treat the cases R(3, j ) with 1 j 4. Step 3: There exists C1 > 0 such that jn (C1 ) δn . To see the detail, recall those identities introduced in (7.10)–(7.12). It is an immediate result of (7.11) that if A(3) holds, then λn,i δn2 + i 2 n−2 ,

uniformly for 1 i n.

(7.16)

We first consider R(3, j ) with j ∈ {1, 2, 3}. In these cases, it is obvious that xn δn = o(n). Using this fact, one can easily compute i(xn + 1)π ixn π i sin , − sin n+1 n + 1 n

√ ixn π ixn δn i √ 2 =o , qn − pn sin n + 1 n n

where and o(·) are uniform for 1 i (n + 1)/(4(xn + 1)). This implies 2n,i i 2 /n2 ,

uniformly for 1 i δn .

(7.17)

By replacing corresponding terms in (7.10) with (7.12), (7.16) and (7.17), we obtain 2 (xn ) ψn,i

i2 , δn3

uniformly for 1 i δn .

Thus, for C small enough, jn (C) δn . We now consider the case R(3, 4), that is, δn → ∞, xn → ∞ and δn xn n. As before, applying (7.12), (7.15) and (7.16) to (7.10) gives 2 (xn ) ψn,i

i2 + i2)

δn (δn2


This implies jn (C) δn for all C > 0. To see the inverse direction, observe that for n+1 xn , |n,i | =

√

i(xn + 1)π ixn π ixn π √ √ + . sin − sin pn sin q − p n n n+1 n + 1 n + 1

(7.18) n+1 2xn

i

(7.19)

This can be easily seen from (7.13). To analyze the right side summation, we compute that ∀

n+1 3(n + 1) i , 2xn 4xn

sin

1 2xn i ixn π , n + 1 2 3(n + 1)

and 3(n + 1) n+1 ∀ i , 4xn xn

i(xn + 1)π iπ ixn π sin − sin . n+1 n+1 2(n + 1)


2281

Putting these two inequalities back to (7.19) gives |n,i | i/n,

uniformly for

n+1 n+1 i . 2xn xn

Hence, by applying this result with (7.10), (7.12) and (7.16), we get

n+1 xn

i=1

n+1 xn

2 ψn,i (xn )

n+1 xn 2 ψn,i (xn )

i= n+1 2xn

i= n+1 2xn

i2 1. δn (δn2 + i 2 )

This implies jn (C) n/xn δn for C small enough. Consequently, in R(3, 4), jn (C) δn for C small enough. Step 4: Let C1 be as in Step 3. Then, τn (C1 ) n2 /δn2 . Note that, in cases B(1)–B(4), xn δn /n 1. By the second inequality of (7.15), this implies |n,i | i/n


As before, applying this result with (7.10), (7.12) and (7.16) gives 2 sup ψn,i (xn )δn : 1 i n, n 1 = N < ∞. Thus, we have log(1 + iN/δn ) n2 n2 log(1 + C1 ) τn (C1 ) max 2. 2 λn,jn (C1 ) λn,i δn jn (C1 )in δn As a consequence of Step 3 and Step 4, we have λn,jn (C1 ) τn (C1 ) 1. By Theorem 5.1, this implies that the family {pnc (t, xn , ·): n = 1, 2, . . .} has no L2 -cutoff. The proof for discrete time cases goes in a similar way. Recall in the following the spectral information displayed in (7.9) using the setting given by (7.3) and (7.4). For 1 i n/2, d = ψn,i , ψn,2i−1

d ψn,2i = ψn,n+1−i ,

(7.20)

and λdn,2i−1 = λdn,2i = − log βn,i . Note that ∀1 i n,

− log βn,i = − log(1 − λn,i ) λn,i

and, for all L > 2, − log βn,i λn,i

uniformly for 1 i n/L.

Using the above comparison relationship, it is easy to show from the definition of jn (C) and τn (C) given in (5.2) and (5.3) that Step 1 and Step 2 remain true in cases R(1, j ) with j =

2282


1, 2, 3 and R(2, j ) with j = 1, 2, 3, 4. As a consequence of Theorem 5.3, the family {pnd (t, xn , ·): n = 1, 2, . . .} has no L2 -cutoff. For cases R(3, j ) with j ∈ {1, 2, 3, 4}, let C1 be the constant for families of continuous time chains selected in Step 3. Using (7.20), one can easily show that, for discrete time chains, jn (C1 ) δn , whereas (7.18) gives jn (C) δn for all C > 0. This implies jn (C1 ) δn . A similar proof as that for Step 4 implies τn (C) n2 /δn2 . By Theorem 5.3, the family {pnd (t, xn , ·): n = 1, 2, . . .} has no L2 -cutoff. Sufficiency of (7.5). First of all, recall the notations defined in (7.2)–(7.4) and (7.7)–(7.9), and rewrite (7.10) and λn,i in the following way. ∀1 i n,

2 (xn ) = ψn,i

ixn + θn,i )π) 2 sin2 (( n+1

(n + 1)πn (xn )

(7.21)

,

where θn,i ∈ (1/2, 1) is such that sin(θn,i π) =

√ iπ pn sin n+1 # , λn,i

cos(θn,i π) =

√ √ iπ pn cos n+1 − qn # , λn,i

(7.22)

and ∀1 i n,

iπ √ 2 √ qn + 2 pn qn 1 − cos n+1 2 2 2 i 4δn /n 1+O 2 , = √ 1 + 2 pn q n δn

λn,i =

√

pn −

(7.23)

where O is uniform for 1 i n. According to the discussion in the beginning of the proof, only cases R(3, 5), R(4, 5), R(5, 2) and R(5, 5) are needed to be considered. Obviously, either of them implies lim δn = ∞,

n→∞

lim

n→∞

x n δn =∞ n

and further that πn (x) ∼

2δn pn x . nqn qn

(7.24) γ

We will prove the sufficiency of (7.5) using Theorems 5.1–5.3. For C > 0, let jn (C) and γ ∈ {c, d}, be as defined in (5.2) and (5.3). In what follows, Steps 5, 6 and 7 deal with cases R(3, 5), R(4, 5) and R(5, 5), whereas Step 8 consider R(5, 2). γ τn (C),

Step 5: For C > 0, jnd (C) 2jnc (C) − 1 and jnc (C) δn (pn /qn )xn /3 . d c . To see the second one, = ψn,i Clearly, the first inequality follows from the setting ψn,2i−1 observe that √ p n + pn q n i uniformly for 1 i n/xn . 1 − θn,i ∼ × 2 δn


2283

This can be proved without difficulty using (7.23). By this fact, one can show that ixn ixn + θn,i − 1 ∼ n+1 n

uniformly for 1 i

n . xn

Hence, we have sin

2

ixn ixn 2 + θn,i π n+1 n

uniformly for 1 i n/(2xn ).

(7.25)

As the above result can hold only for xn n/2, we consider two subcases. Case 1: x n n/2. In this case, one may use (7.25) to show that for 1 j n/(2xn ),

xn 3 2 j c 2 j xn qn ψn,i (xn ) = log + O(1). log pn δn n 2

(7.26)

i=1

Using the inequality log(1 + t) 12 (t ∧ 1) for t 0, we obtain qn δn 1 δn ∧ . pn n 2 2n This implies δn

pn qn

xn /3

n x n δn n =o . = δn × o δn exp − 6n x n δn xn

(7.27)

Hence, for C > 0, $ jn (C) c

pn qn

xn /3

δn n2 xn2

1/3 %

$ xn /3 % pn δn . qn

Case 2: x n > n/2. In this case, we go back to (7.10). Note that for xn > n/2, √ xn π (xn + 1)π xn π √ √ − sin + qn − pn sin , |n,1 | = pn sin n+1 n+1 n+1 where the right side is a sum of positive terms. In a few computations, one can show that for pn < 1/4 or xn < 3n/4, √ xn π 1 √ qn − pn sin , n+1 n and for pn 1/4 and xn 3n/4, (xn + 1)π xn π 1 √ − sin pn sin . n+1 n+1 n

2284


Thus, |n,1 | n−1 . Applying this result with (7.10), (7.23) and (7.24) gives 2 ψn,1 (xn )

1 qn xn . δn3 pn

Moreover, using the fact log(1 + t) 12 (t ∧ 1), we have 2 log ψn,1 (xn ) xn log

qn + O(log δn ) n (δn /n) ∧ 1 + O(log δn ), pn

(7.28)

where the most right summation tends to infinity. This implies that for any C > 0, jnc (C) = 1 as n large enough. Then, Step 5 is an immediate result of (7.27). Step 6: For C > 0 and γ ∈ {c, d}, γ

τn (C)

xn log(qn /pn ) + O(log log(qn /pn )xn ) . γ 2λn,1

To prove this inequality, we set, for n 1, n =

n qn −1/2 xn log . xn pn

Using the first inequality of (7.27), one can show that ∀C > 0,

δn

pn qn

xn /3

= o(n ),

n = o

n . xn

(7.29)

As the proof for Step 5, we consider the following two cases. Case 1: x n n/2. An immediate result of (7.29) is that for any C > 0, jnc (C) n

n , 2xn

for n large enough.

(7.30)

Putting this fact with (7.23), (7.26) and (7.27) together gives τnc (C) =

log

n

c 2 i=0 |ψn,i (xn )| 2λcn,n

xn log(qn /pn ) + O(log log(qn /pn )xn ) 2λcn,1 (1 + O((n − 1)2 /δn2 ))

xn log(qn /pn ) + O(log log(qn /pn )xn ) xn log(qn /pn ) ∼ . 2λcn,1 2λcn,1

(7.31)

For discrete time chains, one can compute without difficulty that λdn,2i−1

= λdn,2i

(i − 1)2 iπ √ d = λn,1 1 + O = − log 2 pn qn cos n+1 δn2

(7.32)

where O is uniformly for 1 i n/xn . Applying this fact with (7.20) and (7.31), we have


2n i=0

τnd (C) =

d (x )|2 |ψn,i n

2λdn,2n

n

i=0

2285

c (x )|2 |ψn,i n

2λdn,1 (1 + O((n − 1)2 /δn2 ))

xn log(qn /pn ) + O(log log(qn /pn )xn ) xn log(qn /pn ) ∼ . 2λdn,1 2λdn,1

(7.33)

γ

Case 2: x n > n/2. It has been shown in Case 2 of Step 5 that for any C > 0, jn (C) = 1 for n large enough. Then, by (7.28), we have γ

τn (C)

2 (x )) log(ψn,1 n γ

2λn,1

=

xn log(qn /pn ) + O(log(xn δn /n)) . γ 2λn,1

This proves Step 6 using the first inequality of (7.27). To determine the existence of the L2 -cutoff using Theorems 5.1 and 5.3, we have to compute γ λn,j γ (C) . Using (7.23) and (7.32), one can show that n

γ

γ

λn,i ∼ λn,1

uniform for 1 i Cn/xn , γ ∈ {c, d} γ

where C is any positive constant. By Step 5 and (7.29), it is easy to see that jn (C) n/xn . Putting these two results together gives γ

γ

λn,j γ (C) ∼ λn,1 , n

∀C > 0.

Then, by Step 6, we obtain that for γ ∈ {c, d},

γ γ τn (C)λn,j γ (C) n

γ γ ∼ τn (C)λn,1

qn xn log pn

→∞

as n → ∞

and γ

jn (C)−1

γ γ γ γ γ ψ (xn )2 e−2λn,i τn (C) Ce−2λn,1 τn (C) → 0 as n → ∞. n,i

i=1 γ

γ

γ

As a consequently of Theorems 5.1 and 5.3, the family {pn (t, xn , ·): n 1} has a (τn (C), ln )– L2 -cutoff with −1 c lnc = λcn,1 log τn (C)λcn,1

(7.34)

−1 d lnd = max 1, λdn,1 log τn (C)λdn,1 .

(7.35)

and

This proves the sufficiency of the L2 -cutoff for cases R(3, 5), R(4, 5) and R(5, 5). In the next step, we make a detailed computation on the L2 -cutoff times and cutoff windows yielded above.

2286


Step 7: For C > 0 and γ ∈ {c, d}, γ

τn (C) =

xn log(qn /pn ) + O(log log(qn /pn )xn ) . γ 2λn,1 γ

By Step 6, it remains to give an adequate upper bound for τn (C). Obviously, by (7.2) and (7.21), we have 2 ψn,i (xn )

1 qn xn 2 (n + 1)πn (xn ) δn pn

∀1 i n.

This implies for γ ∈ {c, d}, γ τn (C)

γ

max

jn (C)in

xn log(qn /pn ) + log((i + 1)/δn ) . γ 2λn,i

(7.36)

We consider two subcases concerning the value of pn . In the case pn < 1/4, or equivalently δn n/4, it is obvious from (7.36) that ∀C > 0, γ ∈ {c, d},

γ

τn (C)

xn log(qn /pn ) + log 4 . γ 2λn,1

In the case pn 1/4, one can show that there is a constant N > 0 such that, for n large enough, γ λn,i

γ λn,1

i2 − 1 1+ ∀1 i n, γ ∈ {c, d}. N δn2

Using this fact, we may prove that for

xn δn2 n

< i n,

N log((i + 1)/δn ) xn log(qn /pn ) + log((i + 1)/δn ) xn log(qn /pn ) × max γ γ (i 2 − 1)/δn2 2λn,i 2λn,1 xn δn2 /n 0 and x 0 such that m x + t, d 2 2 2 Dm (x, t) + 1 = D d (x, t) + 1 × 1 − (p/q)m+1 , 2 d 2 2 d (x, t) = D d (x, t) + 1 (p/q)m+1 2 − (p/q)m+1 . D (x, t) − Dm Moreover, for m x, tj c D (x, t) 2 − D c (x, t) 2 6 D d (x, 0) 2 + 1 (p/q)m+1 + e−t . m j! j >m−x

Proof. Let π, πm be the invariant probabilities associated with p(t, ·,·), pm (t, ·,·). Then, the first and second identities follow immediately from the fact π(y) = πm (y)[1 − (p/q)m+1 ] for 0 y m and x+t d 2 2 d p (t, x, y) /π(y) − 1, D (x, t) = y=0 x+t d d 2 2 pm (t, x, y) /πm (y) − 1. Dm (x, t) = y=0


2291

To see the last inequality, set A = {0, 1, . . . , m − x} and Ac = Ω \ A. A simple computation shows that c 2 p (t, x, y) =

e−t

2 2 ti d tj e−t p d (j, x, y) p (i, x, y) + i! j! c j ∈A

i∈A

+ 2e−2t

i∈A,j ∈Ac

t i+j d p (i, x, y)p d (j, x, y). i!j !

For the second and third terms on the right side, we may prove using Jensen’s and Cauchy inequality that

e

−t

i∈Ac

2 i 2 ti d tj d −t t p (i, x, y) p (j, x, y) e e−t i! i! j! c c j ∈A

i∈A

and y∈Ω

p d (i, x, y)p d (j, x, y) π(y)

2

(p d (i, x, y))2 π(y)

y∈Ω

y∈Ω

(p d (j, x, y))2 . π(y)

This implies 2 i c 1 −t t d D (x, t) 2 + 1 − p e (i, x, y) i! π(y) y∈Ω

d 2 D (x, 0) + 1

i∈A

e−t

i∈Ac

ti . i!

c (t, ·,·) and p d (t, ·,·), we have Similarly, for the transition functions pm m

2 i c 1 −t t d D (x, t) 2 + 1 − p (i, x, y) e m i! m πm (y) y∈Ωm

i∈A

d 2 −t t i 3 Dm . (x, 0) + 1 e i! c i∈A

Note that for m x,

1 1 d D d (x, 0) = Dm √ = D d (x, 0) (x, 0) = √ πm (x) π(x)

and p (i, x, y) = d

d (i, x, y) pm 0

for i ∈ A, y m, for i ∈ A, y > m.

2292


Putting all above together and applying the triangle inequality gives 2 i c 1 −t t d D (x, t) 2 − D c (x, t) 2 (p/q)m+1 p (i, x, y) e m i! π(y) y∈Ω

i∈A

2 −t t j + 6 D d (x, 0) + 1 e j! c 2 6 D (x, 0) + 1

d

j ∈A

m+1

(p/q)

+

e

−t

j ∈Ac

where the last inequality uses Jensen’s inequality on the summation w.r.t. i.

tj , j!

2

The next theorem concerns birth and death chains on non-negative integers and contains Theorem 7.2. Theorem 7.4. Let Ω be the set of non-negative integers. For n 1, let pn ∈ (0, 1/2), qn = 1 − pn γ and let pn (t, ·,·), γ ∈ {c, d}, be the continuous or discrete time Markov transition function on Ω γ associated with (7.39) for p = pn . Then, the family pn (t, xn , ·) with xn ∈ Ω has an L2 -cutoff if γ γ γ and only if (7.5) holds. Moreover, if (7.5) holds, then for γ ∈ {c, d}, pn (t, xn , ·) has a (tn , bn )– γ γ L2 -cutoff, where tn and bn are as defined in Theorem 7.2. γ

Proof of Theorem 7.4. For n 1 and γ ∈ {c, d}, let Dn (t) be the L2 -distance between γ γ pn (t, xn , ·) and its stationary distribution. Let tn be as in Theorem 7.2 and set γ sn = inf t > 0: Dn (t) 1, γ ∈ {c, d} + tnc . Note that, for n 1, we may choose mn max{xn , mn−1 + 1} with m0 = 0 such that d 2 lim Dn (xn , 0) + 1 (pn /qn )mn + e−2sn

n→∞

∞ i=mn −xn

(2sn )i i!

= 0.

Let p nd (t, ·,·) be the transition function on Ωmn satisfying ∀x, y ∈ Ωmn ,

p nd (1, x, y) =

pnd (1, x, y) pn

if (x, y) = (mn , mn ), if (x, y) = (mn , mn ),

nγ (t) be the L2 -distance between and let p nc (t, ·,·) be the associated continuous time chain. Let D γ p n (t, xn , ·) and its stationary distribution. In the above setting, one may prove using Lemma 7.3 that for τ ∈ {c, d}, γ nγ (t) = o(1). sup Dn (t) − D 0t2sn

(7.40)


2293

Fig. 1. This is the graph associated with K4 in (8.2). For undefined arrows, those away from 0 and the loops at 4, −4 have weight q4 , whereas those into 0 and the loop at 0 have weight p4 .

nγ (t, xn , ·) has a (tnγ , bnγ )–L2 -cutoff. Since If (7.5) holds true, then by Theorem 7.2, the family P γ γ γ tnd tnc sn , (7.40) implies that pn (t, xn , ·) also has a (tn , bn )–L2 -cutoff. Conversely, assume γ γ that the family pn (t, xn , ·) has an L2 -cutoff with cutoff time s¯n . In this case, it is clear that γ

lim sup n→∞

s¯n 1, sn

for γ ∈ {c, d}.

γ

By (7.5), this implies that p n (t, xn , ·) also has an L2 -cutoff. As a consequence of Theorem 7.2, we obtain (7.5). This proves Theorem 7.4. 2 8. A peak/valley example Recall that, by Proposition 4.1, a family of ergodic Markov processes has an L2 -cutoff if lim Tn,2 (μn , )λn = ∞.

(8.1)

n→∞

In the normal case, this sufficient condition for an L2 -cutoff can be regarded as a special case of Theorems 5.1 and 5.3 with jn (C) = 1. However, it is possible that an L2 -cutoff exists but (8.1) fails. That is, (8.1) is not a necessary condition. This is illustrated by the examples in this section. Consider the following birth and death chain. Let n be a positive integer and Kn be the Markov kernel on Ωn = {−n, . . . , −1, 0, 1, . . . , n} defined by Kn (−i, −j ) = Kn (i, j ),

∀i 0, j 0,

Kn (i, i + 1) = Kn (n, n) = qn ,

∀0 < i < n, Kn (0, 1) = qn /2,

Kn (i + 1, i) = Kn (0, 0) = pn ,

∀0 i < n,

(8.2)

where pn + qn = 1. See Fig. 1 for an example of n = 4. Obviously, Kn has invariant probability πn (0) = cn ,

cn qn |x| πn (x) = , 2 pn

∀x = 0

(8.3)

where cn =

(1 − qn /pn )[1 − (qn /pn )n+1 ]−1 1/(n + 1)

if pn = qn , if pn = qn .

Using the method in [9, Chapter XVI], Kn has eigenvalues βn,0 = 1 and √ pn qn (an + an−1 ) βn,1 = √ 2 pn qn cos θn,1

if pn /qn n2 /(n + 1)2 , if pn /qn > n2 /(n + 1)2

(8.4)

2294


and √ 2 pn qn cos θn,j βn,l = √ jπ 2 pn qn cos n+1

if l = 2j − 1 and 2 j n, if l = 2j and 1 j n

(8.5)

where, for 1 j n, θn,j is a solution to sin nθ = sin(n + 1)θ and an solves fn (t) =

pn , qn

θ∈

(j − 1)π j π , n n+1

√ pn /qn with ⎧ t n −t −n ⎪ ⎪ ⎪ t n+1 −t −n−1 ⎨ 0 fn (t) = n ⎪ ⎪ n+1 ⎪ ⎩ −n n+1

if t ∈ / {0, ±1}, if t = 0, if t = 1,

(8.6)

if t = −1.

Let ψn,i be a normalized (in L2 (πn )) eigenvector for Kn associated with βn,i . Then, ψn,0 = 1 is the constant function with value 1 and ψn,1 (x) = Cn,1

⎧ |x|/2 ⎨ anx − an−x pn x ⎩ qn sin xθn,1

if pn /qn < n2 /(n + 1)2 , if pn /qn = n2 /(n + 1)2 , if pn /qn > n2 /(n + 1)2

and ψn,2j −1 (x) = Cn,2j −1

pn qn

|x|/2 sin xθn,j ,

2 j n,

and ψn,2j (x) = Cn,2j

pn qn

|x|/2

j (|x| + 1)π √ j |x|π √ qn sin − pn sin n+1 n+1

where ⎧ a 2n+1 −a −2n−1 n n ⎪ ⎪ ⎨ [ an −an−1 − (2n + 1)] −2 Cn,1 = cn n(n + 1)(2n + 1)/6 ⎪ ⎪ ⎩1 sin nθn,1 cos(n+1)θn,1 ] 2 [n − sin θn,1

if pn /qn < n2 /(n + 1)2 , if pn /qn = n2 /(n + 1)2 , if pn /qn > n2 /(n + 1)2

and −2 Cn,2j −1

sin nθn,j cos(n + 1)θn,j cn , n− = 2 sin θn,j

−2 Cn,2j =

cn (n + 1)(1 − βn,2j ) . 2


2295

Clearly, βn,i βn,i+1 for all 1 i < 2n and max |βn,1 |, |βn,2n | = |βn,1 | ∀n. Remark 8.1. Note that 1, 1 and βn,2j −1 (x), ψn,2j −1 (x) with 1 j n and x ∈ {0, 1, . . . , n} are the eigenvalues and eigenfunctions of the transition matrix ∀1 x < n,

K(x, x + 1) = K(n, n) = qn ,

K(x, x − 1) = pn ,

K(0, 0) = 1,

whereas 1, 1 and βn,2j , ψn,2j (n − x) with 1 j n and x ∈ {0, 1, . . . , n} are the eigenvalues and eigenvectors for the transition matrix in (7.1). With work, the above spectral information leads to the following result. Theorem 8.1. Let {(Ωn , Kn , πn ): n = 1, 2, . . .} be the family introduced in (8.2) and let xn be the initial state of the nth chain. Then, in the continuous and the discrete time cases, if pn qn , the family has an L2 -cutoff if and only if pn |xn | − 1 → ∞. qn

(8.7)

If pn < qn , then the family has an L2 -cutoff if and only if n(qn − pn ) → ∞

and |xn |(qn − pn ) → 0. γ

(8.8) γ

Moreover, if there is an L2 -cutoff, then the cutoff time is tn (|xn |) if pn > qn and tn (n − |xn |) if pn < qn , where c and d represent for continuous time and discrete time cases and tnc (x) =

x| log pn − log qn | , √ 2(1 − pn qn )

! tnd (x) =

" x| log pn − log qn | . − log(4pn qn )

Remark 8.2. A (non-optimal) window size can be obtained by arguments similar to those in Theorem 7.2. It is not included because it involves additional long computations. Remark 8.3. As illustrated in (8.5) and (8.15), except perhaps for the second largest one, the eigenvalues of Kn are distributed in way that is very similar to those of the chains treated in Theorem 7.2. In the case pn > qn , this is true even for the second largest eigenvalue. When pn < qn , however, the spectral gap 1 − βn,1 is of much smaller order than for the chain in Theorem 7.2 and βn,1 is separated from the rest of the eigenvalues. In the latter case, it is easy to see from Theorem 8.1 and (8.15) that if there is an L2 -cutoff, then the cutoff time is of order smaller than the inverse of the spectral gap. This means the optimal window size is not directly related to the spectral gap but depends on the rest of the eigenvalues. Remark 8.4. Theorem 8.1 covers two very different cases depending on whether pn >> qn or pn qn , the stationary distribution has a sharp peak at 0 and this case is not much different from the one treated in Theorem 7.2. The spectral gap is relatively large in this case

2296


(bounded away from 0 when pn /qn > 1 stays bounded away from 1). To reach stationarity, the walk must have a chance to visit the peak. To present a cutoff, the walk must start far enough from the peak, just as in Theorem 7.2. When pn < qn , the stationary measure has a unique valley bottom at 0. In this case, to reach stationarity, the walk must have a chance to cross the bottom. The bottom creates a bottle neck which implies that the spectral gap 1 − βn,1 is very close to 0 if pn /qn < 1 stays bounded away from 1. However, the rest of the spectrum (in the continuous time case, say, i.e., 1 − βn,j , j > 1) is bounded away from 0. In this case, there is no cutoff, except if one starts very close to 0 where the eigenvector associated with the spectral gap takes very small values. This illustrates one of the main feature of the central results of this paper: in order to understand the cutoff and the cutoff time from specified starting points, one may have to drop those eigenvalues (including possibly the spectral gap) whose eigenvectors take very small values at the specified starting points. Before proving Theorem 8.1, we make some analysis on 1 − βn,1 , where βn,1 is defined in (8.4). Set pn = 1/2 − δn /n and assume first that |δn | = o(n). In the case pn /qn > n2 /(n + 1)2 , the fact θn,1 ∈ (0, π/(n + 1)) yields √ √ 1 − βn,1 = 1 − 2 pn qn + 2 pn qn (1 − cos θn,1 ) 2 δ 2 /n2 + θn,1 if |δn | = O(1), n2 2 δn /n if |δn | → ∞.

(8.9)

In the subcase |δn | = O(1), one may use the following identity sin nθn,1 = sin(n + 1)θn,1

pn , qn

to derive sin nθn,1

qn − 1 = sin(n + 1)θn,1 − sin nθn,1 = pn

(n+1)θ

n,1

cos t dt.

(8.10)

nθn,1

This implies θn,1 ∈

(0, π/(2n + 1) if δn > 0, (π/(2n + 1), π/(n + 1)) if δn < 0.

(8.11)

Thus, by (8.9), if |δn | = O(1) and δn < 0, then 1 − βn,1 1/n2 . For the further subcase |δn | = O(1) and δn > 0, consider the following computations. sin nθn,1 δn θn,1

qn 1 −1 = pn θn,1

(n+1)θ

n,1

cos t dt = cos θn ,

nθn,1


2297

where θn ∈ (nθn,1 , (n + 1)θn,1 ). Note that the first asymptote uses the fact θn,1 ∈ (0, π/(2n + 1)) whereas the first equality applies (8.10). Hence, if δn → 0 and δn > 0, then θn → π2 or equivalently nθn,1 → π/2. As a consequence of (8.9), if |δn | = O(1) and δn > 0, then 1 − βn,1

2 δn2 + n2 θn,1

n2

1 . n2

In the case pn /qn = n2 /(n + 1)2 , it is obvious that δn ∼ 2 2δn /n2 ∼ 1/(2n2 ). In the case pn /qn < n2 /(n + 1)2 , let √ pn /qn , where fn (t) is the function in (8.6). That is,

√ 1/2 and 1 − βn,1 = 1 − 2 pn qn ∼ an ∈ (0, 1) be such that fn (an ) =

pn a n − an−n (1 − an2 )an2n+1 = n+1n = a − . n qn an − an−n−1 1 − an2n+2

Then, we have 0 < 1 − an < 1 − Write an = 1 − and

ζn n

pn 2δn ∼ qn n

as n → ∞.

(8.12)

with ζn 0. If |δn | = O(1), then the last asymptote implies that ζn = O(1)

ann = exp ζn 1 + o(1) 1,

1 − an2n = 1 − exp 2ζn 1 + o(1) ζn ,

and 1 − an2n+2 = 1 − an2 + an2 1 − an2n ζn . Thus, we have 1 − βn,1 =

√ pn qn (1 − an2 )2 an2n−1 1 2 (1 − a 2n )(1 − a 2n+2 ) n

as n → ∞.

If |δn | → ∞ or equivalently δn → ∞, one can compute 1 − an ∼

2δn , n

1 an

pn qn

n

(1 − an2 )an2n n = 1− ∼ 1 as n → ∞, 1 − an2n+2

(8.13)

which yields √ pn qn (1 − an2 )2 an2n−1 8δn2 pn n 1 − βn,1 = ∼ 2 . qn (1 − a 2n )(1 − a 2n+2 ) n For the case |δn | n, it is clear that if δn < 0, then 1 − βn,1 1 and if δn > 0, then lim sup pn < 1/2. n→∞

(8.14)

2298


Note that the function fn in (8.6) converges uniformly to the identity map on [0, 1]. Thus, in the case δn n, we have lim sup an < 1. n→∞

This implies that the second part of (8.13) holds true and (8.14) becomes 1 − βn,1

pn qn

n+1/2 .

Summarizing from the above discussions, we achieve ⎧1 ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ δn /n 1 − βn,1 1/n2 ⎪ ⎪ 2 2 n ⎪ ⎪ ⎩ (δn /n )(pn /qn ) (pn /qn )n+1/2

if δn → −∞ and |δn | n, if δn → −∞ and |δn | = o(n), if |δn | = O(1), if δn → ∞ and |δn | = o(n), if δn n.

(8.15)

Proof of Theorem 8.1. Recall those notations introduced in (7.7). Write pn = 1/2 − δn /n. In this setting, (8.7) is equivalent to |xn δn | →∞ nqn

(8.16)

and (8.8) becomes δn → ∞ and

|xn |δn → 0. n

(8.17)

Due to the symmetry of the transition probabilities about 0, we can assume that xn 0. In the case xn = 0, by binding states i and −i together, the origin chain in (8.2) collapses to the chain in (7.1) with the exchange of pn and qn . Then, the results in Theorem 7.2 and Remark 7.4 yield the equivalent conditions in (8.16) and (8.17) and the desired cutoff time. We assume in the following that xn 1 and prove this theorem by considering all possible cases of δn and xn . γ γ Throughout this proof, we let jn (C) and τn (C) be those defined in (5.2) and (5.3), where c and d denote respectively continuous time cases and discrete time cases. Let λcn,j and λdn,j be the rearrangements of 1 − βn,j and − log |βn,j | in the way that γ

γ

λn,j λn,j +1 ,

∀1 j < 2n, γ ∈ {c, d}.

γ

γ

Similarly, let ψn,i be the rearrangement of ψn,i according to λn,i . In this setting, one can see that c = ψd = ψ ψn,1 n,1 and n,1 λcn,j = 1 − βn,j

∀1 j 2n,

(8.18)

and λdn,2j −1 = − log |βn,j |,

λdn,2j = − log |βn,2n−j +1 |

∀1 j n.

(8.19)


2299

Case 1: |δ n | = O(1). In this case, it is easy to see that none of (8.16) and (8.17) are satisfied and we shall prove that there is no L2 -cutoff. To achieve this conclusion, one needs to compute γ γ 2 (x) should be determined. In the assumption τn (C) and jn (C) and, first of all, the order of ψn,i |δn | = O(1), it is clear that (pn /qn )x 1 uniformly for |x| n, and then the normalizing constant for the stationary distribution πn satisfies cn =

1 − qn /pn 1 1 = . 1 − (qn /pn )n+1 1 + qn /pn + · · · + (qn /pn )n n

In the case pn /qn < n2 /(n + 1)2 , one may apply (8.12) to get t x 1 uniformly for |x| n + 1, t ∈ an , an−1 and −1

an−x

an − anx

=

xt x−1 dt x an−1 − an uniformly for 1 |x| n.

an

The last asymptote leads to the following estimations, −2 = cn Cn,1

n −x 2 2 an − anx n2 an−1 − an x=1

and 2 (x) ψn,1

x2 n2

uniformly for 1 |x| n.

Such a conclusion is obviously true for the case pn /qn = n2 /(n + 1)2 . When pn /qn > n2 / (n + 1)2 , observe that n 1 sin nθ cos(n + 1)θ n− = sin2 xθ, 2 sin θ x=1

1 sin xθ θ

xθ

2

sin2 t dt (x−1)θ

where the second asymptote holds true uniformly for θ ∈ (0, π/(n + 1)), x ∈ {1, 2, . . . , n} and n 1. Using these observations, we have for θ ∈ (0, π/(n + 1)), n x=1

1 sin xθ θ

nθ sin2 t dt n3 θ 2 .

2

0

2300


−2 2 and Hence, Cn,1 n2 θn,1 2 (x) ψn,1

x2 n2

uniformly for 1 |x| n.

For ψn,2j −1 , the fact θn,j ∈ [(j − 1)π/n, j π/(n + 1)] implies sin nθn,j cos(n + 1)θn,j 1 1 n+1 sin θ sin π 2 sin θn,j n,j n+1

(8.20)

−2 and, hence, Cn,2j −1 1 uniformly for 1 < j n and 2 2 ψn,2j −1 (x) sin xθn,j 1 uniformly for 1 |x| n, 1 < j n.

To estimate ψn,2j , note that −2 Cn,2j =

cn (n + 1)(1 − βn,2j ) 1 − βn,2j 2

uniformly for 1 j n.

By setting √ jπ qn sin n+1 sin θn,j = # , 1 − βn,2j

cos θn,j =

√ √ jπ qn cos n+1 − pn # , 1 − βn,2j

(8.21)

the last asymptote yields 2 ψn,2j (x) sin2

j |x|π + θn,j n+1

1 uniformly for 1 |x| n, 1 j n.

As a consequence of the above discussions, we have 2 ψn,j (x) 1 uniformly for 1 |x| n, 1 j n. γ

γ

(8.22)

Now, it is ready to estimate jn (C) and τn (C). By (8.5), (8.15), (8.18) and (8.19), one can compute γ

j2 n2

uniformly for 1 j 2n, γ ∈ {c, d}

γ

j2 n2

uniformly for 1 j n, γ ∈ {c, d}.

λn,j and λn,j

These two facts and (8.22) then lead to

log(j + 1) γ n2 τn (C) sup 2 /n2 γ j j jn (C)

∀C > 0, γ ∈ {c, d}


2301

regardless of the initial states (xn )∞ n=1 . For jn (C), we may choose, by Remark 7.4, Remark 8.1 and Step 1 in the proof of Theorem 7.2, two constants C0 and N such that γ

N γ ψ

2 C0

∀0 x n, n 1, γ ∈ {c, d}.

n,2j (x)

j =1 γ

This implies for γ ∈ {c, d}, jn (C0 ) 1 and 1

γ

τn (C0 )

γ λn,j γ (C ) 0 n

n2 ,

γ

γ

λn,j γ (C ) τn (C0 ) 1. n

0

Hence, by Theorems 5.1 and 5.3, both families in discrete time and continuous time cases have no L2 -cutoffs. Case 2: δ n → −∞ and |δ n | = o(n). In this case, we will prove that the L2 -cutoff exists if and only if |δn |xn /n → ∞. By (8.5) and the conclusion in (8.15), it is easy to see that γ

δn2 + j 2 n2

uniformly for 1 j 2n, γ ∈ {c, d}

γ

δn2 + j 2 n2

uniformly for 1 j n, γ ∈ {c, d}.

λn,j and λn,j

γ

To estimate the order of |ψn,j (xn )|2 , we have to determine the constants Cn,2j −1 . First, the normalizing constant cn in (8.3) satisfies cn =

1 − qn /pn 2|δn | . ∼ n 1 − (qn /pn )n+1

For Cn,1 , note that the fact δn < 0 implies θn,1 ∈ [π/(2n + 1), π/(n + 1)] and sin2 xθn,1

x2 n2

uniformly for 1 x n/2.

This yields n

n

−2 Cn,1 =

sin2 xθn,1 n,

x=0

n cn 2 sin xθn,1 ncn |δn |. 2 x=1

For Cn,2j −1 , observe that the conclusion developed in (8.20) is also valid here. Thus, we have −2 Cn,2j −1 ncn |δn | uniformly for 1 < j n.

Consequently, the above discussion gives ψn,2j −1 (x)2

pn qn

|x|

sin2 xθn,j |δn |

uniformly for 1 j, |x| n.

(8.23)

2302


Consider two subcases, |δn |xn = O(n) and |δn |xn /n → ∞. In the former situation, it is easy to check that ψn,2j −1 (xn )2 1 |δn |


This implies sup

j c (xn )|2 log 1 + i=1 |ψn,2i−1 λcn,2j −1

1j n

n2 log(1 + j/|δn |) n2 2 sup 2 2 2 δn 1j n 1 + j /δn δn

(8.24)

and similarly, sup

j d d (x )|2 ) (xn )|2 + |ψn,4i log 1 + i=1 (|ψn,4i−3 n λdn,4j −3

1j n

n2 . δn2

(8.25)

Recall the conclusions of Step 3 and Step 4 in the proof of Theorem 7.2: There exist M and C1 such that M|δ n |

ψn,2i (xn )2 C1

i=1

and sup

j c (x )|2 log 1 + i=1 |ψn,2i n λcn,2j

M|δn |j n

n2 δn2

(8.26)

and sup

j d d (xn )|2 + |ψ2,4i−1 (xn )|2 log 1 + i=1 |ψn,4i−2 λdn,4j −2

M|δn |j n

n2 . δn2

(8.27)

γ

The first inequality implies jn (C1 ) |δn | and ∀γ ∈ {c, d},

γ

τn (C1 )

1 γ λn,jn (C1 )

n2 . δn2

Using the fact log(1 + a + b) < log(1 + a) + log(1 + b) for a, b > 0, one may conclude from γ γ (8.24)–(8.27) that τn (C1 ) n2 /δn2 for γ ∈ {c, d}, which yields τn (C1 ) n2 /δn2 for γ ∈ {c, d}. γ γ Consequently, τn (C1 )λn,j γ (C ) 1 and, by Theorems 5.1 and 5.3, both families in discrete time n

1

and continuous time cases have no L2 -cutoff. γ For the subcase |δn |xn /n → ∞, let Dn,2 (xn , t) be the L2 -distance for the nth Markov chain. Then, for γ ∈ {c, d}, 2 γ γ γ Dn,2 (xn , t) = Ln,1 (t) + Ln,2 (t)

(8.28)


2303

where Lcn,1 (t) =

n ψn,2j −1 (xn )2 e−2t (1−βn,2j −1 ) ,

Ldn,1 (t) =

j =1

n ψn,2j −1 (xn )2 |βn,2j −1 |2t j =1

and Lcn,2 (t) =

n ψn,2j (xn )2 e−2t (1−βn,2j ) ,

Ldn,2 (t) =

j =1

n ψn,2j (xn )2 |βn,2j |2t . j =1

γ

Note that Ln,2 (t) is exactly the square of the L2 -distance for the chain in (7.1) starting from xn . In the assumption of |δn |xn /n → ∞, Theorem 7.2 implies that, for γ ∈ {c, d}, the family γ {Ln,2 (t): n = 1, 2, . . .} presents an L2 -cutoff with cutoff time nxn /|δn |. Using this fact, it reγ mains to show that {Ln,1 (t): n = 1, 2, . . .} also possesses an L2 -cutoff with the same cutoff time. In detail, write λcn,j = 1 − βn,2j −1

∀1 j n

and λdn,2j −1 = − log |βn,2j −1 |,

λdn,2j = − log |βn,2n−2j +1 |

∀1 j n/2,

be the rearrangement of ψn,2j −1 associated with λn,j . In this setting, it is clear that and let ψ n,j γ γ λn,j λn,j +1 for 1 j < n and γ ∈ {c, d} and γ

γ

γ

Ln,1 (t) =

n γ 2 γ ψ exp −2t λn,j . n,j j =1

γ γ γ γ . Then, by (8.11) Let jn (C) and τn (C) be those in (5.2) and (5.3) associated with λn,j and ψ n,j and (8.23), we have

xn 3 2 j γ j xn pn ψ (xn )2 n,i qn |δn |n2

$ uniformly for 1 j

i=1

% n . 2xn

Using this, one can compute log 1 +

n/2x n i=1

2 γ 4x |δ | ψ (xn ) n n 1 + o(1) → ∞ n,i n

as n → ∞,

which gives jn (C) n/xn for n large enough and γ

γ

τn (C)

4xn |δn |/n(1 + o(1)) 4xn |δn |/n nxn ∼ ∼ γ γ |δn | 2 λn,n/xn 2 λn,1

∀C > 0.

(8.29)

2304


γ γ Hence, τn (C) λn, γ j (C) n

lim

xn |δn | n

→ ∞ and

γ jn (C)−1

n→∞

γ γ γ γ γ τn (C) λn,j τn (C) λn,1 ψ (xn )2 e−2 lim Ce−2 = 0. n,j

n→∞

j =1

By Theorems 5.1 and 5.3, both families have an L2 -cutoff. To see nxn /|δn | is a cutoff time, we need the following facts. For some universal constant N > 0, j2 − 1 γ γ ∀1 j n, n 1, γ ∈ {c, d}, λn,j λn,1 1 + N δn2 and γ ψ (xn )2 n,j

pn qn

xn

1 |δn |

uniformly for 1 j n. γ

The former comes immediate from the definition of λn,2j −1 whereas the latter is a simple corollary of (8.23). Using these two inequalities, one can prove that j γ (xn )|2 log 1 + i=1 |ψ xn |δn |/n + [log(j + 1)/|δn |] n,i γ γ λn,1 (j 2 − 1)/δn2 2λn,i nxn =o uniformly for δn2 xn /n j n. |δn | and j γ (xn )|2 log 1 + i=1 |ψ xn log(pn /qn ) + log(xn |δn |/n) + O(1) n,i γ γ 2 λ 2 λ n,i

n,1

nxn ∼ |δn |

uniformly for 1 j xn δn2 /n.

As a consequence of the above computations and (8.29), the L2 -cutoff time for both families is nxn /|δn |. Case 3: δn → ∞ and δn = o(n). In this case, one can use (8.13) to get an−2n−1 − an2n+1 n(an−1

− an )

∼

exp{4δn (1 + o(1))} → ∞. 4δn

This implies Cn,1 ∼ 1 and ψn,1 (xn )2 ∼

pn qn

xn

−x 2 an n − anxn

2 1 if xn δn /n 1, ∼ 1 − an2x = o(1) if xn δn /n = o(1).

(8.30)


2305

We discuss the L2 -cutoff by considering these two subcase. In the assumption xn δn /n 1, one may choose C2 such that γ

jn (C2 ) = 1, γ

γ

γ

which implies τn (C2 ) 1/λn,1 . To find τn (C2 ), note that (8.20) implies −2 C2j −1

ncn δn

pn qn

n .

Thus, we have n ψn,2j −1 (xn )2 1 qn δn p n


(8.31)

−2 ∼ 2δn (1 − βn,2j )(pn /qn )n uniformly for 1 j n. Using the notations inObserve that Cn,2j troduced at (8.21), one can derive

n ψn,2j (xn )2 1 qn δn p n


(8.32)

By the fact γ

λn,j

δn2 + j 2 n2

uniformly for 1 < j 2n,

(8.31) and (8.32) yield

n log(qn /pn ) + log(j/δn ) 1 n2 γ τn (C2 ) max γ , 2 sup 1 + j 2 /δn2 λn,1 δn 2j 2n

1 n2 1 max γ , γ , λn,1 δn λn,1 γ

γ

where the last asymptotic is a result of (8.15). Consequently, τn (C2 )λn,1 1 for γ ∈ {c, d} and, by Theorems 5.1 and 5.3, there is no L2 -cutoff in either case. γ In the case xn δn /n = o(1), recall (8.28). By Theorem 7.2, the family {Ln,2 : n = 1, 2, . . .} has an L2 -cutoff with cutoff time (n − xn ) log(qn /pn ) n2 ∼ γ δn 2λn,2

∀γ ∈ {c, d}.

be those defined in Case 3. Note that one may choose a universal For Ln,1 (t), let λn,j and ψ n,j constant N > 0 such that j 2 N 2δn2 γ λn,j 2 1 + 2 ∀2 j n, γ ∈ {c, d}. n δn γ

γ

γ

2306


As before, (8.20) implies 1 qn n e4δn (1+o(1)) ψ n,j (xn )2 = δn p n δn

uniformly for 1 < j n.

By setting tn = n2 /δn , we may compute using (8.30) that, for any > 0 and γ ∈ {c, d},

n γ γ γ ψ (xn )2 exp −2(1 + )tn λ Ln,1 (1 + )tn = n,j n,j + o(1) j =2

∞ 2 1 −4Nj + o(1) exp δn δn j =2

∞ j 1 −4N 1 √ exp √ 1+ √ + o(1) δn δn √ δn j δn

∞ 1 z −4N dz + o(1) = o(1). √ 1+ e δn 0

Consequently, for ∈ (0, 1) and γ ∈ {c, d}, γ lim Dn,2 xn , (1 + )n2 /δn

n→∞

γ γ = lim Ln,1 (1 + )n2 /δn + lim Ln,2 (1 + )n2 /δn = 0 n→∞

n→∞

and γ γ lim Dn,2 xn , (1 − )n2 /δn lim Ln,2 (1 − )n2 /δn = ∞.

n→∞

n→∞

This means that both families have an L2 -cutoff with cutoff time n2 /δn as desired. Case 4: |δ n | n. We first deal with the case δn > 0. Recall that

qn /pn an2

n ∼ 1.

Using this fact, it is easy to check ψn,1 (xn )2 1 − a 2xn 2 1, n

γ

where the last asymptote uses the assumption xn 1. Thus, by (8.15), we have λn,1 (pn /qn )n+1/2 and

G.-Y. Chen, L. Saloff-Coste / Journal of Functional Analysis 258 (2010) 2246–2315 2n γ ψ (xn )2 exp −2 λγ −1 λγ n,j n,j n,1 j =2

2307

1 γ γ exp −2λn,2 /λn,1 πn (xn )

exp n log(qn /pn ) − C(qn /pn )n+1/2 + O(1) = o(1)

where C is a universal positive constant. This yields γ γ Dn,2 xn , /λn,1 e−

∀ > 0

which means that both families have no L2 -cutoff. In the case δn < 0, (8.7) becomes xn /qn → ∞. First, assume that xn /qn = O(1) or equivalently xn = O(1) and 1/qn = O(1). For continuous time cases, since 1/πn (xn ) is bounded, Corollary 5.2 implies that no L2 -cutoff exists. For discrete time cases, using the notation in (8.28), one can show without difficulty that ∀t > 0,

d (xn , t) lim inf Ldn,2 (t) > 0. lim inf Dn,2 n→∞

n→∞

Hence, by Corollary 3.3, these is no L2 -cutoff. To see the sufficiency of xn /qn → ∞, set snc

xn (log pn − log qn ) = , √ 2(1 − 2 pn qn )

" xn (log pn − log qn ) . = − log(4pn qn ) !

snd

γ

γ

By Theorem 7.2, the family {Ln,2 : n = 1, 2, . . .} has an L2 -cutoff with cutoff time sn . This implies for ∈ (0, 1) and γ ∈ {c, d}, γ γ γ γ lim inf Dn,2 xn , (1 − )sn Ln,2 (1 − )sn = ∞. n→∞

To get an upper bound on the L2 -distance, note that √ λcn,1 ∼ 1 − 2 pn qn ,

λdn,1 ∼ − log(4pn qn ).

This implies, for > 0, 1 γ γ exp −2(1 + )sn λn,1 πn (xn ) γ γ = exp xn (log pn − log qn ) − 2(1 + )sn λn,1 + O(1) .

γ γ 2 Dn,2 xn , (1 + )sn

Hence, in the assumption xn /qn → ∞, we have snd ∼

xn (log pn − log qn ) , − log(4pn qn )

xn (log pn − log qn ) → ∞,

and, for γ ∈ {c, d} and > 0, γ γ 2 Dn,2 xn , (1 + )sn exp −2 1 + o(1) xn (log pn − log qn ) = o(1).

2

2308


Acknowledgment We thank the anonymous referee for his/her very careful reading of the manuscript. Appendix A. Techniques and proofs Proof of Lemma 3.2. There is no loss of generality in assuming that tn = 1 for all n since one may always consider the following sequence of functions.

gn (t) = fn (ttn ) =

e−tλ dVn (λ)

(0,∞)

where Vn (λ) = Vn (λ/tn ). By letting Vn (0) = lim Vn (λ) and λ↓0

hn (s) = sup e−λ : Vn (λ) − Vn (0) > sfn (0) ,

∀s ∈ (0, 1), we may express fn as follows.

fn (t) = fn (0)

htn (s) ds,

∀t > 0.

(A.1)

(0,1)

It is clear that hn is a non-increasing non-negative function bounded from above by 1. Using the sequential compactness of monotonic functions, we may choose a subsequence nk such that hnk converges almost surely to a non-increasing function h and fnk (0) converges to C 0. Consequently, one can show without difficulty that

lim fnk (a) = C

k→∞

∀a > 0.

ha (s) ds,

(0,1)

Using a similar argument as before, one may show that the right-hand side above is in fact a Laplace transform and then, by Lemma 3.1, is analytic on (0, ∞). It remains to prove that such a convergence is uniform on any compact subset of (0, ∞). Note that b x − y b b x a − y a , a

∀x, y ∈ (0, 1), b > a > 0.

Using this fact, one can show that sup fnk (b) − fnl (b) fnk (0) − fnl (0) + fnk (0)

b∈[2a,3a]

sup b∈[2a,3a] (0,1)

fnk (0) − fnl (0) + 3fnk (0)

(0,1)

b h (s) − hb (s) ds nk nl

a h (s) − ha (s) ds nk nl


2309

which converges to 0 as k, l tends to infinity. This proves that fnk converges uniformly on [2a, 3a] for all a > 0 as desired. The last part of this lemma is easy to show using the locally uniform convergence of fnk and the continuity of the limiting function. 2 Proof of Corollary 3.3. By scaling the time t up to a constant, one only needs to prove the continuity of F1 , F2 at t = 1. We give the proof for F1 but omit the similar proof for F2 . By Lemma 3.2, we may choose a subsequence fnk such that lim fnk (atnk ) = f (a)

k→∞

∀a > 0

and f (1) = F1 (1), where f is continuous on (0, ∞). Clearly, F1 and f are non-increasing and satisfy f F1 . This implies F1 (1) = f (1) = lim f (a) lim inf F1 (a) lim sup F1 (a) F1 (1) a↓1

a↓1

a↓1

which proves the right-continuity of F1 at 1. Concerning the left-continuity, set L = lim F1 (a). a↑1

Let m0 = 1. For k 1, we may choose xk ∈ (1 − 2−k , 1) and mk mk−1 such that fmk (xk tmk ) ∈ (L − 1/k, L + 1/k). Referring to the subsequence sequence mk , we may choose by Lemma 3.2 a further subsequence m k such that the function a → fm k (atm k ) converges uniformly to a continuous function g on any compact subset of (0, ∞). This implies L = lim fm k (xk tm k ) = lim g(xk ) = g(1). k→∞

k→∞

Again, since F1 is non-increasing and g F1 , we get F1 (1) L = g(1) F1 (1), that is, F1 is left-continuous. For the second part of this corollary, assume that F1 (c) > 0 for some c > 0. As before, we may choose, by Lemma 3.2, a subsequence nk such that fnk converges to an analytic function f and f (c) = F1 (c) > 0. Clearly, F1 f and then, by the analyticity of f on (0, ∞), F1 > 0. 2 Proof of Corollary 3.4. Set gn (s) = fn (tn + s). It is clear that

n (λ) for s > 0, gn (s) = fn (tn ) e−sλ d V (0,∞)

is a probability distribution defined by where V e−tn λ dVn (λ) n (γ ) = (0,γ ] V −tn λ dV (λ) n (0,∞) e

∀γ > 0.

2310


Part (i) is then obtained by applying Corollary 3.3 to gn and bn . In the first case of (ii), assume gn (s) = fn (tn + c0 bn + s). Since F the inverse that F (c0 ) < ∞ for some c0 < 0. For n 1, let has a (tn , bn )-cutoff, hn is well-defined on [0, ∞) for n large enough. For c > 0, let G(c) = lim inf gn (cbn ). n→∞

Obviously, G(c) = F (c + c0 ),

G(−c0 ) = F (0) > 0,

H (0) F (c0 ) < ∞.

As a consequence of Corollary 3.3, the analyticity of H implies that G > 0 on (0, ∞) or equivalently F > 0 on (c0 , ∞). This contradicts the assumption that F (c) = 0 for some c > 0. Thus, F = ∞ on (−∞, 0). In the second case of (ii), we prove as before by contradiction. Assume the inverse that F (c1 ) < ∞ for some c1 < 0. This is equivalent to the existence of a subsequence nk such that fnk (c1 ) is bounded. By considering the subsequence fnk , a similar proof as that of the first case will derive a confliction. Hence, F = ∞ on (−∞, 0). For (iii), let tn = T (fn , δ) with δ > 0 and set F (c) = F (c + ),

F (c) = F (c + ),

∀ ∈ R.

According to the definition of the δ-mixing time, it can be easily shown that F (0) = F () δ < ∞,

∀ > 0,

F (0) = F () δ > 0,

∀ < 0.

and

By [5, Corollary 2.4], the family F also presents a (tn + bn , bn )-cutoff for all ∈ R. Using the former inequality in the above, we may conclude from (i) that, for > 0, either F > 0 or F ≡ 0 on (0, ∞). This is equivalent to say that either F > 0 or F ≡ 0 on (0, ∞). The proof for (ii) in this case is similar to that of (i) using the latter inequality. 2 Proof of Theorem 3.5. Part (i) is an immediate result of Corollary 3.3. For (ii), we assume that there is a cutoff for F = {fn : n = 1, 2, . . .}. By [5, Corollary 2.5(i)], the cutoff time sequence can be chosen to be tn = T (fn , δ) for any δ > 0. Let C be any positive number and λn = λn (C) be the constant defined in (3.2). Note that, for n 1,

−2λtn −4λn tn −2λtn e dVn (λ) max Ce , e dVn (λ) . fn (2tn ) (0,2λn ]

(0,λn )

Then, the existence of the cutoff for F implies that fn (2tn ) → 0 as n → ∞. This proves (a) and (b) with arbitrary C > 0, δ > 0 and = 2. In fact, (c) is true for all > 0. To see this, let Vn be a function defined by Vn (λ) =

Vn (λ) limt↑λn Vn (t)

if λ ∈ (0, λn ), if λ ∈ [λn , ∞)


2311

and set

gn (t) =

e

−λt

dVn (λ) =

(0,∞)

e−λt dVn (λ).

(0,λn )

Clearly, gn (0) = Vn ((0, λn )) C for all n 1 and lim sup gn (2tn ) = 0. n→∞

By Corollary 3.3, we obtain lim sup gn (tn ) = 0 ∀ > 0. n→∞

For the other direction, assume that C, δ, are positive constants such that (a) and (b) hold and let tn = T (fn , δ),

bn = 1/λn = 1/λn (C).

In this setting, one can show that for c > 0 and n N = N (c),

e−λtn /2 dVn (λ) + δe−c/2 fn (tn + cbn ) (0,λn )

and fn (tn − cbn ) e

c/2

δ−

e

−λtn /2

dVn (λ) .

(0,λn )

By Corollary 3.3, (b) implies a (tn , bn )-cutoff as desired. 2

(0,λn ) e

−λtn /2 dV (λ) n

→ 0 as n → ∞. Consequently, F has

Proof of Theorem 3.6. Part (i) is immediate from Remark 3.2. For (ii), let T d (fn , δ) and T c (fn , δ) be respectively the mixing time for fn with domain N and [0, ∞). By Definition 2.3, |T d (fn , δ) − T c (fn , δ)| 1 and using the assumption T d (fn , δ) → ∞, we know that T d (fn , δ) ∼ T c (fn , δ) for all δ > 0. This implies that Theorem 3.5 (a)–(b) hold for T c (fn , δ), λn (C) if and only if they are true for T d (fn , δ), λn (C). Also [5, Propositions 2.3–2.4], {fn : [0, ∞) → [0, ∞] | n = 1, 2, . . .} has a cutoff if and only if {fn : N → [0, ∞] | n = 1, 2, . . .} has a cutoff. Consequently, Theorem 3.6 is then a corollary of Theorem 3.5. To see a cutoff window, note that, by Theorem 3.5, {fn : [0, ∞) → [0, ∞] | n = 1, 2, . . .} d c has a (T c (fn , δ), λ−1 n )-cutoff. Recall the fact |T (fn , δ) − T (fn , δ)| 1. Then, by [5, Proposid tions 2.3–2.4], {fn : N → [0, ∞] | n = 1, 2, . . .} has a (T (fn , δ), γn−1 )-cutoff. 2 Proof of Proposition 3.7. We only consider the case where the domain of fn is [0, ∞). To see why the assumption bn → ∞ arises in the case of discrete domain, confer [5, Remark 2.9]. Since F has a cutoff, Theorem 3.5 implies that M = lim supn fn (0) = ∞. Let F , F be functions in (2.1). Part (ii) is an immediate result of Corollary 3.4. For (i), we first assume that

2312


F (c) > 0 for some c > 0. From the definition of mixing time and the monotonicity of fn , it is clear that F (c/2) δ. Then, by [5, Proposition 2.2], the (tn , bn )-cutoff is optimal. Since an optimal cutoff must be a weakly optimal cutoff, it remains to show that if there is a weakly optimal cutoff, then F (c) > 0 for some c > 0, which is equivalent to F (c) > 0 for all c > 0 using Corollary 3.4. Assume the inverse that F (c0 ) = 0 for some c0 > 0 and let nk be a subsequence such that fnk (tnk + c0 bnk ) → 0 as k → ∞. Consider the subfamily G = {fnk : k 1} and let G(c) = lim inf fnk (tnk + cbnk ), k→∞

G(c) = lim sup fnk (tnk + cbnk ). k→∞

Obviously, G(c0 ) = G(c0 ) = 0 and, by Corollary 3.4, this implies G(c) = G(c) = 0 ∀c > 0,

G(c) = G(c) = ∞ ∀c < 0.

Then, by [5, Proposition 2.2], the (tnk , bnk )-cutoff for G can not be weakly optimal and this contradicts Proposition 2.1. 2 Proof of Theorem 3.8. We first assume that (a) and (b) hold for some positive constants C, . Note that (a) restricts us to case (ii) of Theorem 3.5 because one may choose a sequence λ n > λn such that log(1 + Vn ((0, λ n ])) τn /2. λ n This implies τn λn τn λ n 2 log 1 + Vn 0, λ n 2 log 1 + Vn (0, ∞) → ∞, as n → ∞. By Corollary 3.3, (b) is true for all > 0. Note that, for n 1, we may choose a non-decreasing sequence (λn,k )∞ k=1 such that λn,k λn

∀k 1,

rn,k =

log(1 + Vn ((0, λn,k ])) → τn λn,k

as k → ∞.

In this setting, it is easy to see that, for k 1,

fn (rn,k )

e−λrn,k dVn (λ) e−λn,k rn,k Vn (0, λn,k ]

(0,λn,k ]

=

Vn ((0, λn,k ]) Vn ((0, λn ]) C . 1 + Vn ((0, λn,k ]) 1 + Vn ((0, λn ]) 1 + C

By letting C = C/(1 + C), we obtain from the above computations that τn T (fn , C). Consequently, lim T (fn , C)λn lim τn λn = ∞

n→∞

n→∞


2313

and

e−λT (fn ,C) dVn (λ)

(0,λn )

e−λτn dVn (λ) = 0.

(0,λn )

By Theorem 3.5, F presents a cutoff. For the inverse direction, assume the existence of the cutoff for F . By Theorem 3.5 and Remark 3.4, the following are true for any positive constants C, δ, .

e−λT (fn ,δ) dVn (λ) → 0, λn T (fn , δ) → ∞, (0,λn )

where λn = λn (C) is the constant defined in (3.2). Using these facts, it remains to show that, for some δ > 0, T (fn , δ) = O(τn ). Let C > 0 and τn = τn (C) be the quantity defined in (3.3). For η > 0 and n 1, we let An,j = [λn (1 + η)j , λn (1 + η)j +1 ) for j 0. Consider the following computations.

fn (t) = (0,∞)

C+

e−λt dVn (λ) C +

e−λt dVn (λ)

j 0A

j 0

C+

n,j

e−λn

(1+η)j t

Vn 0, λn (1 + η)j +1

exp −λn (1 + η)j +1 t/(1 + η) − τn .

j 0

By letting t = (1 + η)2 τn , we have fn (1 + η)2 τn C +

exp{−ητn λn } . 1 − exp{−η2 τn λn }

(A.2)

Let v : (0, ∞) → (0, ∞) be any function satisfying sup v(t)/t: t log(1 + C) < ∞,

2 inf ev(t) 1 − e−v (t)/t = L > 0.

t>0

If one puts η = v(τn λn )/(τn λn ) in (A.2), then there exists some positive constant N such that ∀n N,

fn (τn + dn ) C,

where = C + 2/L, C

dn = 2bn (1 + bn /τn ),

bn = λ−1 n v(τn λn ).

τn + dn for n N . To derive the desired identity T (fn , C) = O(τn ), it suffices Thus, T (fn , C) to show that bn = O(τn ), which can be easily computed out using the fact τn λn log(1+C) > 0. For a realization of v, one may choose v(t) = t 1/2 .

2314


To see a cutoff sequence, assume that F has a cutoff or, equivalently, Theorem 3.8 (a)–(b) hold. In this case, one may go through all arguments in the above to choose positive constants such that C, C − dn τn T (fn , C) T (fn , C)

for n large enough,

(A.3)

where dn = 2bn (1 + bn /τn ), bn = λ−1 n w(τn λn ) and w : (0, ∞) → (0, ∞) is a function satisfying lim sup t→∞

2 lim inf ew(t) 1 − e−w (t)/t > 0.

w(t) < ∞, t

t→∞

(A.4)

Thus, τn is a cutoff sequence for F if and only if there exists a function w satisfying (A.4) such that dn = o(τn ). This is equivalent to bn = o(τn ) or w(t) = o(t) as t → ∞. As one can see that w(t) = t 1/2 is qualified for (3.4), τn is a cutoff sequence. To get a window sequence corresponding to τn , assume that w is a function satisfying (3.4). By Theorem 3.5, F has a (T (fn , δ), λ−1 n )-cutoff for any δ > 0 and, by [5, Proposition 2.3], there exists C1 > 0 such that + C1 λ−1 T (fn , C) T (fn , C) n

for n large enough.

Putting this inequality and (A.3) together gives τn − T (fn , C) = O bn + λ−1 = O λ−1 n n w(τn λn ) + 1 . Note that the second condition of (3.4) implies that w(t) → ∞ as t → ∞. This implies = O(bn ) and then, by [5, Corollary 2.5(v)], F has a (τn , bn )-cutoff. 2 |τn − T (fn , C)| Proof of Theorem 3.9. Let Fc and Fd be families in Theorems 3.8 and 3.9 and, for δ > 0, let T c (fn , δ) and T d (fn , δ) be respectively their mixing time sequences. In this setting, it is clear that T c (fn , δ) T d (fn , δ) T c (fn , δ) + 1. Recall in the proof of Theorem 3.8 that τn (C) T c fn , C/(C + 1)

(A.5)

∀C > 0, n 1.

This implies τn (C) → ∞

⇒

T d fn , C/(C + 1) → ∞.

Thus, in Theorem 3.9, we always have T d (fn , δ) → ∞ for some δ > 0. Consequently, by [5, Propositions 2.3–2.4], the above fact and (A.5) imply Fc has a cutoff

⇔

Fd has a cutoff

⇔

Fd has a (tn , bn )-cutoff.

and, for bn such that infn bn > 0, Fc has a (tn , bn )-cutoff

Hence, Theorem 3.9 is an immediate result of Theorem 3.8.

2


2315

References [1] David Aldous, Random walks on finite groups and rapidly mixing Markov chains, in: Seminar on Probability, XVII, in: Lecture Notes in Math., vol. 986, Springer, Berlin, 1983, pp. 243–297. [2] David Aldous, Persi Diaconis, Shuffling cards and stopping times, Amer. Math. Monthly 93 (5) (1986) 333–348. [3] David Aldous, Persi Diaconis, Strong uniform times and finite random walks, Adv. Appl. Math. 8 (1) (1987) 69–97. [4] Guan-Yu Chen, The cut-off phenomenon for finite Markov chains, PhD thesis, Cornell University, 2006. [5] Guan-Yu Chen, Laurent Saloff-Coste, The cutoff phenomenon for ergodic Markov processes, Electron. J. Probab. 13 (2008) 26–78. [6] Persi Diaconis, The cutoff phenomenon in finite Markov chains, Proc. Natl. Acad. Sci. USA 93 (4) (1996) 1659– 1664. [7] Persi Diaconis, Phil Hanlon, Eigen-analysis for some examples of the Metropolis algorithm, in: Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, Tampa, FL, 1991, in: Contemp. Math., vol. 138, Amer. Math. Soc., Providence, RI, 1992, pp. 99–117. [8] Persi Diaconis, Laurent Saloff-Coste, Separation cut-offs for birth and death chains, Ann. Appl. Probab. 16 (4) (2006) 2098–2122. [9] William Feller, An Introduction to Probability Theory and Its Applications, vol. I, third ed., John Wiley & Sons Inc., New York, 1968. [10] Mourad E.H. Ismail, David R. Masson, Jean Letessier, Galliano Valent, Birth and death processes and orthogonal polynomials, in: Orthogonal Polynomials, Columbus, OH, 1989, in: NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 229–255. [11] Samuel Karlin, James McGregor, Random walks, Illinois J. Math. 3 (1959) 66–81. [12] Samuel Karlin, James McGregor, Ehrenfest urn models, J. Appl. Probab. 2 (1965) 352–376. [13] Samuel Karlin, James L. McGregor, The Hahn polynomials, formulas and an application, Scripta Math. 26 (1961) 33–46. [14] R. Koekoek, R. Swarttouw, The askey-scheme of hypergeometric orthogonal polynomials and its q-analog, http://math.nist.gov/opsf/projects/koekoek.html, 1998. [15] David A. Levin, Yuval Peres, Elizabeth L. Wilmer, Markov Chains and Mixing Times, Amer. Math. Soc., Providence, RI, 2009, with a chapter by James G. Propp and David B. Wilson. [16] F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-correcting Codes. I, North-Holland Math. Library, vol. 16, North-Holland Publishing Co., Amsterdam, 1977. [17] P.R. Parthasarathy, R.B. Lenin, On the exact transient solution of finite birth and death processes with specific quadratic rates, Math. Sci. 22 (2) (1997) 92–105. [18] Walter Rudin, Functional Analysis, second ed., Int. Ser. Pure Appl. Math., McGraw–Hill Inc., New York, 1991. [19] Laurent Saloff-Coste, Lectures on finite Markov chains, in: Lectures on Probability Theory and Statistics, SaintFlour, 1996, in: Lecture Notes in Math., vol. 1665, Springer, Berlin, 1997, pp. 301–413. [20] W. Van Assche, P.R. Parthasarathy, R.B. Lenin, Spectral representation of four finite birth and death processes, Math. Sci. 24 (2) (1999) 105–112. [21] David Vernon Widder, The Laplace Transform, Princeton Math. Ser., vol. 6, Princeton University Press, Princeton, NJ, 1941.


Reduced limits for nonlinear equations with measures ✩ Moshe Marcus a,∗ , Augusto C. Ponce b a Technion, Department of Mathematics, Haifa 32000, Israel b Université catholique de Louvain, Département de mathématique, Chemin du Cyclotron 2, 1348 Louvain-la-Neuve,

Belgium Received 26 May 2009; accepted 11 September 2009 Available online 27 November 2009 Communicated by H. Brezis

Abstract We consider equations (E) −u + g(u) = μ in smooth bounded domains Ω ⊂ RN , where g is a continuous nondecreasing function and μ is a finite measure in Ω. Given a bounded sequence of measures (μk ), assume that for each k 1 there exists a solution uk of (E) with datum μk and zero boundary data. We show that if uk → u# in L1 (Ω), then u# is a solution of (E) relative to some finite measure μ# . We call μ# the reduced limit of (μk ). This reduced limit has the remarkable property that it does not depend on the boundary data, but only on (μk ) and on g. For power nonlinearities g(t) = |t|q−1 t, ∀t ∈ R, we show that if (μk ) is nonnegative and bounded in W −2,q (Ω), then μ and μ# are absolutely continuous with respect to each other; we then produce an example where μ# = μ. © 2009 Elsevier Inc. All rights reserved. Keywords: Semilinear elliptic equations; Outer measure; Equidiffuse sequence of measures; Diffuse limit; Biting lemma; Inverse maximum principle; Kato’s inequality

Contents 1. 2. 3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2317 Diffuse and concentrated limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2321 The diffuse limit of (g(uk )) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323

✩

The first author (M.M.) wishes to acknowledge the support of the Israeli Science Foundation through grant No. 145-05. The second author (A.C.P.) was supported by the Fonds spécial de Recherche—FNRS. * Corresponding author. E-mail address: [email protected] (M. Marcus). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.09.007

M. Marcus, A.C. Ponce / Journal of Functional Analysis 258 (2010) 2316–2372

4. The Inverse Maximum Principle for sequences . . 5. Supersolutions always converge to supersolutions 6. Proofs of Theorems 1.1 and 1.2 . . . . . . . . . . . . 7. Some properties of μ# . . . . . . . . . . . . . . . . . . 8. Absolute continuity between μ and μ# . . . . . . . 9. Reduced limits and W −2,q -weak convergence . . 10. Reduced limits for g(t) = |t|q−1 t . . . . . . . . . . . 11. Sufficient conditions for the equality μ# = μ . . . 12. Characterization of sequences for which μ# = μ . 13. Absolute continuity between μ# and ν # . . . . . . . 14. Reduced limit of max {μk , νk } . . . . . . . . . . . . . 15. Open problems . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. G = G0 . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2326 2329 2332 2336 2339 2342 2350 2355 2358 2364 2367 2369 2370 2370 2372

1. Introduction In this paper we investigate the convergence of solutions of the equation −u + g(u) = μ

in Ω,

(1.1)

where Ω ⊂ RN , N 2, is a smooth bounded domain, g : R → R is a nondecreasing continuous function with g(0) = 0, and μ is a finite measure in Ω. By a solution of (1.1) we mean a function u ∈ L1loc (Ω) such that g(u) ∈ L1loc (Ω) and (1.1) holds in the sense of distributions. In general, Eq. (1.1) is not solvable for every finite measure μ. We shall denote by G(g) the set of finite measures for which a solution exists. When there is no risk of confusion we shall simply write G, even though this set depends on the nonlinearity g. Questions related to the convergence and stability of solutions of

−u + g(u) = μ

in Ω,

u=0

on ∂Ω,

(1.2)

have been addressed in various contexts. We recall that a function u is a solution of (1.2) if u ∈ L1 (Ω), g(u) ∈ L1 (Ω) and − uζ + g(u)ζ = ζ dμ, Ω

Ω

Ω

for every ζ ∈ C02 (Ω) (= space of functions in C 2 (Ω) vanishing on ∂Ω). Let us denote by G0 (g) the set of finite measures for which (1.2) has a solution. Clearly, G0 (g) ⊂ G(g). We prove in Appendix A below that G0 (g) = G(g). The space of finite measures in Ω is denoted by M(Ω). If (μk ) is a sequence in this space, the notation ∗

μk μ

(1.3)

2318


means that (μk ) converges weakly∗ in [C0 (Ω)]∗ , where C0 (Ω) denotes the space of continuous functions in Ω vanishing on the boundary. For brevity, we shall refer to this convergence as weak∗ convergence in Ω. It is known that if (μk ) is a bounded sequence of measures in Ω converging strongly to μ, then the solutions uk of (1.2) with data μk always converge strongly in L1 (Ω) to the solution of (1.2) (see [6, Appendix 4B]). Similarly, if g(t) = |t|q−1 t where 1 < q < NN−2 , then (1.2) has a solution for every finite measure and if (μk ) is a sequence converging weakly∗ to μ, then the solutions uk also converge strongly in L1 (Ω) to the solution u associated to μ. However, for ∗ q NN−2 , this conclusion fails; see [6, Example 1]. In fact, it may even happen that μk 1 weakly∗ but uk → 0 in L1 (Ω), even though the function identically equal to 0 is not the solution of (1.2) with datum μ = 1! A natural question that comes up in this connection is the following: assuming that q NN−2 ∗ and μk μ, what additional ‘minimal’ assumptions would guarantee that solutions of (1.2) with data μk converge to the solution of (1.2) with datum μ? When this is not the case, what can we still say about the limit of the solutions? These are the types of problems that we address in this paper. Our first result shows that if the sequence of solutions converges strongly in L1 then the limit is a solution of (1.2) with some measure μ# , in general different from the weak∗ limit μ. ∗

Theorem 1.1. Let (μk ) ⊂ G be a bounded sequence such that μk μ. For each k 1, denote by uk the unique solution of (1.2) with datum μk . If uk → u#

in L1 (Ω),

(1.4)

then g(u# ) ∈ L1 (Ω) and there exists a finite measure μ# in Ω such that −u# + g u# = μ# in Ω, u# = 0

(1.5)

on ∂Ω.

Surprisingly, the measure μ# does not depend on the Dirichlet boundary condition. In fact, the sequence (uk ) may be replaced by any sequence of solutions of equation (1.1) with μ = μk , which may not even possess a boundary trace. This is the content of our next result: ∗

Theorem 1.2. Let (μk ) ⊂ G be a bounded sequence such that μk μ. For every k 1, assume that vk ∈ L1 (Ω) satisfies −vk + g(vk ) = μk

in Ω.

(1.6)

If vk → v #

in L1 (Ω),

(1.7)

then −v # + g v # = μ# where μ# is the measure given by Theorem 1.1.

in Ω,

(1.8)


2319

We say that a sequence (μk ) in G(g) has a reduced limit if it converges weakly∗ in M(Ω) and if there exists a sequence (vk ) ⊂ L1 (Ω) satisfying (1.6)–(1.7); the reduced limit μ# is defined by (1.8). We use this notation because of its simplicity, but we emphasize that the reduced limit μ# depends on (μk ) and not just on its weak∗ limit. Indeed it is possible that different sequences converging weakly∗ to the same measure μ lead to different limits with respect to the same nonlinearity g. However, μ# does not depend on the domain: for any domain ω Ω, the reduced limit of (μk ) in ω is simply the restriction of μ# to ω. Further we note that every bounded sequence (μk ) in G possesses a subsequence which satisfies the conditions of Theorem 1.2 and consequently has a reduced limit (see Section 6). Following these results, we investigate some properties of μ# ; in particular, to what extent # μ inherits properties of the sequence (μk ). Our next result illustrates the kind of properties that we are interested in. Theorem 1.3. Assume that (μk ) ⊂ G has reduced limit μ# . If μk 0 ∀k 1,

(1.9)

μ# 0.

(1.10)

then

Observe that (1.10) does not follow from Fatou’s lemma, which only implies in this case that μ# μ, where μ is the weak∗ limit of the sequence (μk ). Remark 1.1. The notion of reduced limit is reminiscent of the notion of reduced measure introduced by Brezis, Marcus and Ponce [6]. We recall that if g(t) = 0, ∀t 0, the reduced measure μ∗ is the largest measure less than or equal to μ for which problem (1.2) has a solution. Our main concern in [6] was to study the approximation mechanism behind (1.2), for example via truncation of the nonlinearity g for a fixed measure μ, or via some special approximations of the datum μ for a fixed g. For instance, given a sequence of mollifiers (ρk ) we have shown that, if g is convex, then solutions uk of (1.2) with data μk = ρk ∗ μ converge to the largest subsolution u∗ associated to μ. Since this function satisfies (1.2) with measure μ∗ , one deduces in this case that μ# = μ∗ . We now focus on the case of equations with power nonlinearities, namely −u + |u|q−1 u = μ in Ω, in the supercritical range q if and only if

(1.11)

N N −2 . We recall that for a finite measure μ, Eq. (1.11) has a solution

μ ∈ L1 (Ω) + W −2,q (Ω). In [6], we have showed that if (μk ) is a bounded sequence of measures converging strongly to μ in W −2,q (Ω), then μ# = μ. One might ask what happens if (μk ) is just bounded in W −2,q (Ω). In Theorem 1.3 the reduced limit μ# can be identically zero even if the sequence (μk ) has a

2320


nonzero weak∗ limit. However, if g(t) = |t|q−1 t then, boundedness in W −2,q guarantees that this cannot happen: Theorem 1.4. Assume that (μk ) ⊂ G is a nonnegative sequence with weak∗ limit μ and reduced limit μ# . If (μk ) is bounded in W −2,q (Ω), then μ# = 0 if and only if μ = 0.

(1.12)

For the proof see Section 8 below. Under the assumptions of this theorem, Eq. (1.11) has a solution with datum μ. Therefore, in view of (1.12) one may expect that the reduced limit μ# coincides with μ. Surprisingly, this conclusion does not hold in general; a counterexample is provided by Theorem 9.2 below. Following is a description of some basic concepts and tools employed in this paper. (i) The notion of equidiffuse sequence of measures (μk ) relative to an outer measure T . This means that (μk ) is uniformly absolutely continuous with respect to T ; more precisely, for every ε > 0 there exists δ > 0 such that E ⊂ Ω Borel

and T (E) < δ

⇒

|μk |(E) < ε

∀k 1.

(ii) The notion of concentrating sequence of measures (μk ) relative to an outer measure T . This means that there exists a sequence of Borel sets (Ek ) of Ω such that T (Ek ) → 0 and |μk |(Ω \ Ek ) → 0. Let us consider for example the special case where T is a measure and μ1 = μ2 = · · · = μ for some fixed measure μ. Then the sequence (μk ) is equidiffuse if and only if μ is absolutely continuous with respect to T (denoted μ T ) and (μk ) is concentrating if and only if μ is singular with respect to T (denoted μ ⊥ T ). Two important ingredients, related to the above concepts, are: (iii) The Biting lemma of R. Chacon and H. Rosenthal according to which every bounded sequence of measures (μk ) can be decomposed as a sum of an equidiffuse and a concentrating sequences; see Theorem 2.1 below. (iv) The Inverse Maximum Principle for sequences, extending a previous result of Dupaigne and Ponce [14]. Using the Biting lemma we introduce the notions of diffuse limit and concentrated limit of a bounded sequence of measures (see Definition 2.1 below) and study some of the properties of these limits. In particular we identify the diffuse limit of a sequence (g(uk )) where (uk ) converges in L1 (Ω) and (g(uk )) is bounded in this space. These results, together with the counterpart of the Inverse Maximum Principle for sequences, play a crucial role in the proofs of Theorems 1.2 and 1.3.


2321

2. Diffuse and concentrated limits We denote by T a nonnegative outer measure defined on the class of Borel subsets of Ω. The space of finite Borel measures in Ω is denoted by M(Ω) and is equipped with the norm μM =

|μ|; Ω

by the Riesz representation theorem, M(Ω) = [C0 (Ω)]∗ . The following result, independently proved by R. Chacon and H. Rosenthal (see Brooks and Chacon [11]), plays a central role in this section. Theorem 2.1 (Biting lemma). For every bounded sequence (μk ) ⊂ M(Ω), there exist bounded sequences (αk ), (σk ) ⊂ M(Ω) such that (B1 ) μk = αk + σk , ∀k 1; (B2 ) (αk ) is equidiffuse and (σk ) is concentrating with respect to T . It is not difficult to see that the sequences (αk ) and (σk ) can be chosen so that (B3 ) αk ⊥ σk , ∀k 1. ∗

∗

∗

Lemma 2.1. Using the notation of the Biting lemma, assume that μk μ, αk α and σk σ . ∗ ∗ If (αk ) and (σk ) is another pair of sequences satisfying (B1 )–(B2 ), then αk α and σk σ . Proof. From the definition of equidiffuse sequences, one shows that α T . Therefore, if μ = 0 then α = σ = 0. Let (αk j ) and (σk j ) be subsequences converging weakly∗ to α and σ respectively. The previ∗

∗

ous statement implies that α = α and σ = σ . This further implies that αk α and σk σ .

2

In order to analyze in more detail the weak∗ limit of (μk ) we shall study the weak∗ limits of the sequences (αk ) and (σk ). Definition 2.1. Let (μk ) be a bounded sequence in M(Ω) and let (αk ) and (σk ) be sequences satisfying conditions (B1 )–(B2 ) of the Biting lemma. Assume that (μk ) converges weakly∗ . ∗

(a) If αk α, we call α the diffuse limit of (μk ). ∗ (b) If σk σ , we call σ the concentrated limit of (μk ). If a sequence of measures (μk ) is bounded (but not necessarily weakly∗ convergent) and if every weak∗ convergent subsequence of (μk ) possesses a diffuse limit α independent of the subsequence, we shall still say that this common limit α is the diffuse limit of (μk ). Note that if (μk ) is merely bounded, then it may possess a diffuse limit in this sense, but not a concentrated limit.

2322


In view of Lemma 2.1, if (μk ) possesses a diffuse limit and a concentrated limit then these limits are independent of the decomposition given by (B1 )–(B2 ). The diffuse and concentrated limits of (μk ) depend on T . For instance, if (ρk ) ⊂ C0∞ (−1, 1) is a sequence of mollifiers, ∗

ρk δ0

weakly∗ in M(−1, 1)

and one verifies that (a) if T is the Lebesgue measure in R, then (ρk ) has diffuse limit 0 and concentrated limit δ0 ; (b) if T is the Newtonian capacity capH 1 , then (ρk ) has diffuse limit δ0 and concentrated limit 0, since every nonempty set in R has positive capacity. ∗

We recall that if μk μ weakly∗ in M(Ω), then μM lim inf μk M . k→∞

It is worth noting the following improved version of this estimate. Corollary 2.1. Let (μk ) ⊂ M(Ω) be a bounded sequence possessing diffuse and concentrated limits α and σ , respectively. Then, αM + σ M lim inf μk M . k→∞

(2.1)

Proof. Take sequences (αk ), (σk ) ⊂ M(Ω) satisfying (B1 )–(B3 ). Then, ∗

∗

and σk σ

αk α

weakly∗ in M(Ω).

Hence, αM lim inf αk M k→∞

and σ M lim inf σk M . k→∞

(2.2)

On the other hand, since μk = αk + σk and αk ⊥ σk , we have μk M = αk M + σk M Combining (2.2)–(2.3) we obtain (2.1).

∀k 1.

(2.3)

2

Corollary 2.2. Let (μk ) ⊂ M(Ω) be a bounded sequence of nonnegative measures with weak∗ limit μ. If (μk ) has diffuse and concentrated limits α and σ , respectively, then 0 α μ and 0 σ μ.

(2.4)


2323

Proof. Take sequences (αk ), (σk ) ⊂ M(Ω) satisfying (B1 )–(B2 ) and such that αk ⊥ σk , ∀k 1. Since αk + σk = μk 0 and αk ⊥ σk , we must have αk , σk 0, ∀k 1; hence, α, σ 0. The corollary now follows from the equality μ = α + σ. 2 As a final remark, we point out that if (μk ) ⊂ M(Ω) has diffuse and concentrated limits equal to α and σ , respectively, then α T , but σ need not be a measure concentrated with respect to T or with respect to α. For instance, if T is the Lebesgue measure in RN , f ∈ L1 (Ω) and (λk ) is a convex combination of Dirac masses such that ∗

λk 1

weakly∗ in M(Ω),

then the sequence (μk ) given by μ k = f + λk

∀k 1,

has f as diffuse limit and 1 as concentrated limit. 3. The diffuse limit of (g(uk )) In this section we study the diffuse limit of the nonlinear term in Eq. (1.2) with data μk . We start with a basic result which is independent of the PDE. Proposition 3.1. Let (uk ) ⊂ L1 (Ω) be a sequence such that (g(uk )) is bounded in L1 (Ω). If uk → u#

in L1 (Ω),

(3.1)

then g(u# ) is the diffuse limit of (g(uk )) with respect to Lebesgue measure in RN . Given a > 0, we denote by Ta : R → R the truncation at ±a, defined as ⎧ ⎨t Ta (t) = a ⎩ −a

if |t| a, if t > a, if t < −a.

(3.2)

We first prove the following Lemma 3.1. Assume that (uk ) ⊂ L1 (Ω) satisfies the assumptions of Proposition 3.1. Then, there exists a subsequence (ukj ) such that g(ukj )χ[|ukj |j ] → g u# in L1 (Ω).

(3.3)

2324


Proof. For every j ∈ N, we have by dominated convergence, g Tj (uk ) → g Tj u# in L1 (Ω). On the other hand, if follows from Fatou’s lemma that g(u# ) ∈ L1 (Ω). Thus, by monotone convergence, g Tj u# → g u#

in L1 (Ω).

Using a diagonalization argument, one then finds an increasing sequence of integers (kj ) such that g Tj (ukj ) → g u# in L1 (Ω). Since for every j 1, 0 g(ukj ) χ[|ukj |j ] g Tj (ukj ) the conclusion follows by dominated convergence.

a.e.,

2

Proof of Proposition 3.1. Passing to a subsequence if necessary, we may assume that (g(uk )) has diffuse and concentrated limits α and σ , respectively. Let (ukj ) be the subsequence given by Lemma 3.1. Set αj = g(ukj )χ[|ukj |j ]

and σj = g(ukj )χ[|ukj |>j ] .

(3.4)

We claim that (αj ) and (σj ) satisfy conditions (B1 )–(B2 ). Indeed, since (αj ) strongly converges in L1 (Ω), the sequence (αj ) is equidiffuse (or, equivalently in this case, equiintegrable). On the other hand, by the Chebyshev inequality,

|uk | > j 1 uk 1 C j j L j j

∀j 1.

Thus, the sequence (σj ) is concentrating. Therefore, α = g(u# ). Since α is independent of the subsequence, we conclude that g(u# ) is the diffuse limit of (g(uk )). 2 We now examine the weak∗ limit of the sequence (g(uk )) when uk is a solution of (1.1) with datum μk . In this case, the conclusion can be improved by replacing the Lebesgue measure with the Newtonian capacity capH 1 as the outer measure T . Proposition 3.2. Let (μk ) ⊂ M(Ω) be a bounded sequence. Assume that, for each k 1, there exists uk ∈ L1 (Ω) such that −uk + g(uk ) = μk If (g(uk )) is bounded in L1 (Ω) and

in Ω.

(3.5)


uk → u#

in L1 (Ω),

2325

(3.6)

then g(u# ) is the diffuse limit of (g(uk )) with respect to capH 1 . For the proof of the proposition we need the following lemma. Lemma 3.2. Let u ∈ L1 (Ω) be such that u ∈ M(Ω). Then, 1 Ta (u) ∈ Hloc (Ω)

∀a > 0.

(3.7)

Moreover, for every ω Ω there exists Cω > 0 such that for every a > 0,

∇Ta (u) 2 Cω a u

L1 (Ω)

+ uM(Ω)

(3.8)

ω

and

Cω uL1 (Ω) + uM(Ω) . capH 1 |u| > a ∩ ω a

(3.9)

Proof. Let ϕ ∈ C0∞ (Ω) be such that 0 ϕ 1 in Ω and ϕ = 1 on ω. Set v = uϕ. For every a > 0, we have ∇Ta (v) 2 = ∇Ta (v) · ∇v = − Ta (v)v a |v|. (3.10) Ω

Ω

Ω

Ω

Since v = ϕu + 2∇ϕ · ∇u + uϕ

in Ω,

we have

|v| uM(Ω) + 2Cϕ

|∇u| + Cϕ uL1 (Ω) .

(3.11)

supp ϕ

Ω

We recall that

|∇u| Cϕ uL1 (Ω) + uM(Ω) .

supp ϕ

Combining (3.10)–(3.12), we get Ω

This implies (3.8). Since

∇Ta (v) 2 Cϕ a u

L1 (Ω)

+ uM(Ω) .

(3.12)

2326


1 capH 1 |u| > a ∩ ω capH 1 |v| > a 2 a

∇Ta (v) 2 ,

Ω

the conclusion follows.

2

Proof of Proposition 3.2. Passing to a subsequence if necessary, we may assume that (g(uk )) has diffuse and concentrated limits α and σ , respectively. Take (αj ) and (σj ) as in (3.4). Since (αj ) converges strongly in L1 (Ω), it is in particular equidiffuse with respect to capH 1 . We show that the sequence (σk ) is concentrating with respect to capH 1 in every subdomain ω Ω. For this purpose, let

Ej = |ukj | > j ∩ ω. By Lemma 3.2, given ω Ω we have capH 1 (Ej )

C ukj L1 (Ω) + μkj M(Ω) + g(ukj ) L1 (Ω) . j

Thus, capH 1 (Ej ) Cj and so (σj ) is concentrating in ω with respect to capH 1 . Therefore, α = g(u# ) in ω for every ω Ω, whence g(u# ) is the diffuse limit of (g(uk )) relative to capH 1 . 2 4. The Inverse Maximum Principle for sequences An important tool in the present work is an extension to sequences of the Inverse Maximum Principle of Dupaigne and Ponce [14]. We first recall their result. Theorem 4.1 (Inverse Maximum Principle). Let u ∈ L1 (Ω) be such that u ∈ M(Ω). If u 0 a.e., then (u)c 0.

(4.1)

Here, “c” denotes the concentrated part of the measure with respect to capH 1 . In fact, every finite measure μ can be uniquely decomposed in terms of a diffuse part μd and a concentrated part μc with respect to an outer measure T , so that μ = μd + μc , μd T and μc ⊥ T ; see e.g. [6, Lemma 4.A.1]. We prove the following extension of this result. Theorem 4.2. Let (uk ) ⊂ L1 (Ω) be a bounded sequence such that uk ∈ M(Ω), ∀k 1. Assume that (uk ) is bounded in M(Ω) and has concentrated limit σ ∈ M(Ω) with respect to capH 1 . If uk 0 a.e., ∀k 1, then σ 0. For the proof we use an extension of Kato’s inequality (see [8]).

(4.2)


2327

Lemma 4.1. Let u ∈ L1 (Ω) be such that u ∈ M(Ω). Then, u+ χ[u0] (u)d − |u|c

in Ω.

(4.3)

We recall that if u ∈ L1 (Ω) and u ∈ M(Ω), then u is quasicontinuous with respect to capH 1 ; see e.g. [1,7]. More precisely, there exists a quasicontinuous function u˜ : Ω → R, unique up to sets of zero H 1 -capacity, such that u = u˜ a.e. We shall henceforth identify u with u˜ (u)d . pointwise in Ω. In particular, the term χ[u0] (u)d is well defined, meaning χ[u0] ˜ Proof of Theorem 4.2. For every k 1, let μk := uk . We denote by (αk ), (σk ) ⊂ M(Ω) two sequences satisfying (B1 )–(B2 ). Passing to a subsequence if necessary, we may assume that uk → u a.e. for some function u ∈ L1 (Ω) and also ∗

αk α

∗

and σk σ

weakly∗ in M(Ω).

In particular, σ is the concentrated limit of the original sequence (μk ). Given a > 0, let Ta be as in (3.2). Since uk 0 a.e., Ta (uk ) = a − (a − u)+ . Thus, by Lemma 4.1, Ta (uk ) χ[uk a] (uk )d + |uk |c .

(4.4)

On the other hand, since each measure αk is diffuse, one verifies that (uk )d = (αk )d + (σk )d = αk + (σk )d , |uk |c = |σk |c . Thus, Ta (uk ) αk χ[uk a] + |σk | = αk − αk χ[uk >a] + |σk |.

(4.5)

Let ε > 0. Since (αk ) is equidiffuse with respect to capH 1 , there exists δ > 0 such that E ⊂ Ω Borel and

capH 1 (E) < δ

⇒

|αk |(E) < ε

∀k 1.

(4.6)

On the other hand, given a subdomain ω Ω, by Lemma 3.2 we have Cω capH 1 [uk > a] ∩ ω a

∀a > 0.

(4.7)

Keeping ω fixed, by (4.6)–(4.7) there exists a0 > 0 such that if a a0 , then |αk | [uk > a] ∩ ω ε

∀k 1.

Since (σk ) is concentrating, there exists a sequence of Borel sets Ek ⊂ Ω such that

(4.8)

2328


capH 1 (Ek ) → 0 and |σk |(Ω \ Ek ) → 0. By inner regularity of σk , one can then find compact subsets Kk ⊂ Ek such that capH 1 (Kk ) → 0 and |σk |(Ω \ Kk ) → 0.

(4.9)

For each k 1, let ζk ∈ C0∞ (Ω) be such that 0 ζk 1 in Ω, ζk = 1 on Kk , and

1 |∇ζk |2 capH 1 (Kk ) + . k

Ω

Given ψ ∈ C0∞ (Ω) with ψ 0 in Ω and supp ψ ⊂ ω, set ϕk = ψ(1 − ζk ) in Ω. Then, the sequence (ϕk ) satisfies 0 ϕk ψ

in Ω,

ϕk = 0 on Kk , ϕk → ψ

in H01 (Ω).

Passing to a subsequence if necessary, we may also assume that ϕk → ψ

q.e.,

(4.10)

where q.e. (= quasi-everywhere) means: outside some set of zero H 1 -capacity. By (4.5), for every k 1 and a > 0, we have

∇Ta (uk ) · ∇ϕk

− Ω

ϕk dαk −

ϕk dαk +

[uk >a]

Ω

ϕk d|σk |.

(4.11)

Ω

It follows from Lemma 3.2 that the sequence (Ta (uk )) is bounded in H 1 (ω). Since supp ϕk ⊂ ω and ϕk → ψ in H01 (Ω), we then have

∇Ta (uk ) · ∇ϕk → Ω

∇Ta (u) · ∇ψ

as k → ∞.

(4.12)

Ω

Since ϕk → ψ q.e. and (αk ) is equidiffuse (see e.g. [9, Lemma 1])

ϕk dαk → Ω

ψ dα

as k → ∞.

(4.13)

Ω

By (4.8), [uk >a]

ϕk dαk εϕk L∞ εψL∞

∀a a0 .

(4.14)


2329

Using (4.9), we also get

ϕk d|σk | = Ω

ϕk d|σk | ψL∞ |σk |(Ω \ Kk ) → 0 as k → ∞.

(4.15)

Ω\Kk

As k → ∞ in (4.11), we then obtain −

∇Ta (u) · ∇ψ

Ω

ψ dα + εψL∞

∀a a0 .

Ω

Thus,

Ta (u)ψ

Ω

ψ dα + εψL∞

∀a a0 .

Ω

Letting a → ∞ and ε → 0, we get

uψ

ψ dα.

Ω

Ω

Since uψ = Ω

ψu = Ω

ψ dα +

Ω

ψ dσ, Ω

we conclude that

ψ dσ 0 ∀ψ ∈ C0∞ (Ω), ψ 0 in Ω.

Ω

Therefore, σ 0. The proof of Theorem 4.2 is complete.

2

5. Supersolutions always converge to supersolutions In this section we prove a result about convergence of supersolutions of Eq. (1.1) which appears to be stronger than Theorem 1.3 but is, in fact, equivalent to it. Theorem 5.1. Let (uk ) ⊂ L1 (Ω) be a sequence such that −uk + g(uk ) 0

in Ω.

(5.1)

If (g(uk )) is bounded in L1 (Ω) and uk → u in L1 (Ω), then −u + g(u) 0 in Ω.

(5.2)

2330


In the proof we need a variant of Kato’s inequality up to the boundary (see [6, Proposition 4.B.5]). Lemma 5.1. Let u ∈ L1 (Ω) be such that

uζ

Ω

∀ζ ∈ C02 (Ω), ζ 0 in Ω,

fζ

(5.3)

Ω

where f ∈ L1 (Ω). Then,

u+ ζ

Ω

fζ

∀ζ ∈ C02 (Ω), ζ 0 in Ω.

(5.4)

Ω [u0]

Here, we use the notation

C02 (Ω) = ζ ∈ C 2 (Ω); ζ = 0 on ∂Ω . Proof of Theorem 5.1. Let μk = −uk + g(uk )

in Ω.

Since the right-hand side is a nonnegative distribution in Ω, μk is a locally finite (nonnegative) measure. We first show that for every ω Ω the sequence (μk ) is bounded in M(ω). In fact, take ϕω ∈ C0∞ (Ω) such that 0 ϕω 1 in Ω and ϕω = 1 on ω. Then,

ϕω dμk = −

Ω

uk ϕω +

Ω

g(uk )ϕω Cω uk L1 (Ω) + g(uk ) L1 (Ω) .

Ω

Since μk 0 and the sequences (uk ) and (g(uk )) are bounded in L1 (Ω), we then have μk M(ω) Cω uk L1 (Ω) + g(uk ) L1 (Ω) C˜ ω

∀k 1.

Thus, (μk ) is bounded in M(ω). By Fatou’s lemma, g(u) ∈ L1 (Ω). Passing to a subsequence if necessary, we may assume that ∗

∗

μk μ and g(uk ) g(u) + τ

weakly∗ in M(ω),

for some μ, τ ∈ M(ω). Thus, u satisfies −u + g(u) = μ − τ

in ω.

(5.5)

From Proposition 3.2 we know that g(u) is the diffuse limit of (g(uk )) with respect to capH 1 and, consequently, τ must be its concentrated limit. In view of (5.5), our goal is to show that


μ − τ 0 in ω.

2331

(5.6)

We may assume that (μk ) has a concentrated limit in M(ω), which we denote by λ. By Corollary 2.2, μk 0, ∀k 1, implies that λ μ. Since uk = g(uk ) − μk

∀k 1,

the concentrated limit of (uk ) in ω is then given by τ − λ. Note that τ −μτ −λ

in ω.

(5.7)

uk 0 a.e., ∀k 1.

(5.8)

Let us assume temporarily that

In this case, it follows from Theorem 4.2 that the concentrated limit of (uk ) is nonpositive. In other words, τ −λ0

in ω.

(5.9)

Combining (5.7) and (5.9), we obtain (5.6) under the additional assumption (5.8). In the general case where the functions uk need not be nonnegative we proceed as follows. 1,1 (Ω), we have uk ∈ L1 (∂ω). Let vk be the harmonic function in ω with boundary Since uk ∈ Wloc value −|uk | on ∂ω. We claim that uk vk

a.e.

Indeed, for every ζ ∈ C02 (ω), ζ 0 in ω, we have

(vk − uk )ζ =

ω

∂ζ + (vk − uk ) ∂n

0 on ∂ω; thus,

μk − g(uk ) ζ −

ω

∂ω

Applying Lemma 5.1 we get (vk − uk )+ ζ − ω

∂ζ ∂n

(5.10)

g(uk )ζ. ω

g(uk )ζ 0 ∀ζ ∈ C02 (ω), ζ 0 in ω, ω [vk uk ]

since vk 0 in ω and g(t) 0, ∀t 0. This gives (5.10). Because (uk − vk ) = uk = g(uk ) − μk

∀k 1,

we can apply Theorem 4.2 to the sequence (uk − vk ) and deduce (5.9). Hence, u satisfies −u + g(u) 0 in ω. Since ω Ω is arbitrary, (5.2) holds.

2

(5.11)

2332


6. Proofs of Theorems 1.1 and 1.2 Proof of Theorem 1.1. By standard estimates (see [6, Appendix 4B]), g(uk )

L1

μk M

∀k 1.

In particular, the sequence (g(uk )) is bounded in L1 (Ω) and, by Fatou’s lemma, g(u# ) ∈ L1 (Ω), with # g u

L1

lim inf μk M . k→∞

Moreover, passing to a subsequence if necessary, there exists λ ∈ M(Ω) such that ∗

g(uk ) λ weakly∗ in M(Ω). Hence, the function u# satisfies

−u# + g u# = μ#

in Ω,

u# = 0

on ∂Ω,

where μ# = μ + g(u# ) − λ. Since μ, λ ∈ M(Ω) and g(u# ) ∈ L1 (Ω), the conclusion follows.

2

In order to prove Theorem 1.2 we need a few lemmas. We first prove a local estimate for solutions of (1.1). Lemma 6.1. Let u ∈ L1 (Ω) and μ ∈ M(Ω) be such that −u + g(u) = μ in Ω.

(6.1)

1,1 Then, u ∈ Wloc (Ω) and for every ω Ω,

∇uL1 (ω) + g(u) L1 (ω) Cω uL1 (Ω) + μM(Ω) .

(6.2)

Proof. Given δ > 0, let

Ωδ = x ∈ Ω; d(x, ∂Ω) > δ .

(6.3)

1,1 Let δ0 > 0 be such that ω Ω2δ0 . By standard elliptic linear estimates (see [17]), u ∈ Wloc (Ω) and

∇uL1 (ω) Cδ0 uL1 (Ωδ ) + μM(Ωδ0 ) + g(u) L1 (Ω ) δ0 0 Cδ0 uL1 (Ω) + μM(Ω) + g(u) L1 (Ω ) . δ0

(6.4)

Therefore, for every smooth subdomain ω Ω, u possesses a boundary trace in L1 (∂ω). Consequently, using a Fubini-type argument, one can find δ1 ∈ (0, δ0 /2) such that


uL1 (∂Ωδ ) 1

2333

C uL1 (Ω) . δ0

On the other hand (see [15])

g u(x) ρδ (x) dx C u 1 L (∂Ωδ ) + μM(Ωδ1 ) , 1 1

Ωδ 1

where ρδ (x) = d(x, ∂Ωδ )

∀x ∈ Ωδ .

Therefore, g(u)

L1 (Ω

δ0

)

2 δ0

g u(x) ρδ (x) dx 1

Ωδ 1

Cδ0 uL1 (∂Ωδ ) + μM(Ωδ1 ) 1 Cδ0 uL1 (Ω) + μM(Ω) . Combining (6.4)–(6.5), the conclusion follows.

(6.5)

2

We recall a result concerning the existence of solutions of (1.2) with L1 -boundary data (see [10]). Lemma 6.2. Let μ ∈ M(Ω). If the problem

−u + g(u) = μ u=f

in Ω, on ∂Ω,

(6.6)

has a solution for some f ∈ L1 (∂Ω), in the sense that for every ζ ∈ C02 (Ω), g(u)ζ ∈ L1 (Ω) and

uζ +

− Ω

g(u)ζ = −

Ω

∂ζ f+ ∂n

∂Ω

ζ dμ,

(6.7)

Ω

then it has a solution for every f ∈ L1 (∂Ω). In the next lemma, given two solutions u and v of (1.1), we show the existence of a solution above the subsolution max {u, v}. Lemma 6.3. Let μ ∈ M(Ω). Assume that u, v ∈ L1 (Ω) satisfy −z + g(z) = μ in Ω. Then, for every ω Ω there exists w ∈ L1 (ω) such that

(6.8)

2334


−w + g(w) = μ in ω, w max {u, v} a.e., wL1 (ω) Cω uL1 (Ω) + vL1 (Ω) + μM(Ω) . Proof. Using a Fubini-type argument, one can find δ > 0 such that ω Ωδ and zL1 (∂Ωδ ) Cδ zL1 (Ω)

for z = u, v.

Let f = max {u, v} on ∂Ωδ . By Lemma 6.2, there exists w ∈ L1 (Ωδ ) such that

−w + g(w) = μ in Ωδ , w=f on ∂Ωδ .

By elliptic estimates, wL1 (Ωδ ) C f L1 (∂Ωδ ) + μM(Ωδ ) . Since f L1 (∂Ωδ ) uL1 (∂Ωδ ) + vL1 (∂Ωδ ) Cδ uL1 (Ωδ ) + vL1 (Ωδ ) , we deduce that wL1 (ω) wL1 (Ωδ ) C uL1 (Ωδ ) + vL1 (Ωδ ) + μM(Ωδ ) . We now show for instance that wu

a.e.

(6.9)

For every ζ ∈ C02 (Ω), ζ 0 in Ω, we have

(u − w)ζ = Ω

(u − w)

∂Ω

Thus, by Lemma 5.1, (u − w)+ ζ Ω

∂ζ + ∂n

Ω

g(u) − g(w) ζ

g(u) − g(w) ζ.

Ω

g(u) − g(w) ζ 0 ∀ζ ∈ C02 (Ω), ζ 0 in Ω.

Ω [uw]

Therefore, (u − w)+ = 0 a.e. In other words, (6.9) holds. A similar argument shows that w v a.e. 2


2335

Proof of Theorem 1.2. For every k 1, we denote by uk the solution of (1.2) with datum μk . We split the proof in two steps: Step 1. Conclusion holds if uk vk a.e., ∀k 1. Let ω Ω. By Lemma 6.1, both sequences (g(uk )) and (g(vk )) are bounded in L1 (ω). Passing to a subsequence if necessary, one can find τ1 , τ2 ∈ M(ω) such that ∗ g(uk ) g u# + τ1

∗ and g(vk ) g v # + τ2

weakly∗ in M(ω).

Thus, −u# + g u# = μ − τ1

and

− v # + g v # = μ − τ2 .

Our goal is to show that τ1 = τ2 . Since uk vk a.e. and g is nondecreasing, g(vk ) − g(uk ) 0 a.e. Moreover, ∗ g(vk ) − g(uk ) g v # − g u# + (τ2 − τ1 )

weakly∗ in M(ω).

By Proposition 3.1, g(v # ) − g(u# ) is the diffuse limit of (g(vk ) − g(uk )) with respect to Lebesgue measure; hence, τ2 − τ1 is its concentrated limit. Thus, by Corollary 2.2, τ2 − τ1 0.

(6.10)

On the other hand, (vk − uk ) = g(vk ) − g(uk )

in ω.

Since τ2 − τ1 is also the concentrated limit of (g(vk ) − g(uk )) with respect to capH 1 (see Proposition 3.2), it follows from Theorem 4.2 that τ2 − τ1 0. Combining (6.10)–(6.11), we deduce that τ1 = τ2 . In other words, −u# + g u# = −v # + g v # in ω. Since ω Ω is arbitrary, the conclusion follows. Step 2. Proof of Theorem 1.2 completed.

(6.11)

2336


Take ω ω˜ Ω. By Lemma 6.3, there exists a bounded sequence (wk ) ⊂ L1 (ω) ˜ such that −wk + g(wk ) = μk wk max {uk , vk }

in ω, ˜ a.e.

1,1 By Lemma 6.1, (wk ) is bounded in Wloc (ω). ˜ Passing to a subsequence if necessary, we may assume that

wk → w #

in L1 (ω).

By the previous step, −u# + g u# = −w # + g w # −v # + g v # = −w # + g w #

in ω, in ω.

Hence, −u# + g u# = −v # + g v # This concludes the proof.

in ω.

2

7. Some properties of μ# In this section we present comparison results for reduced limits in terms of the sequences (μk ) or in terms of the nonlinearities g with which they are associated. We prove in particular a stronger version of Theorem 1.3. Proposition 7.1. Let (μk ), (νk ) ⊂ G be two bounded sequences with weak∗ limits μ, ν and reduced limits μ# , ν # , respectively. Then, # μ − ν #

M

μ − νM + lim inf μk − νk M . k→∞

(7.1)

In particular, if μ = ν, then # μ − ν #

M

lim inf μk − νk M . k→∞

(7.2)

Proof. Let uk and vk be the solutions of

−z + g(z) = γ

in Ω,

z=0

on ∂Ω,

(7.3)

associated to the measures μk and νk , respectively. By standard estimates (see [6, Corollary 4.B.1]), we have


g(uk ) − g(vk ) μk − νk M

2337

∀k 1.

Ω

On the other hand, we know from Proposition 3.1 that (μ − μ# ) − (ν − ν # ) is the concentrated limit of the sequence (g(uk ) − g(vk )) with respect to Lebesgue measure. Letting k → ∞, we deduce from Corollary 2.1 that μ − μ# − ν − ν #

M lim inf k→∞

g(uk ) − g(vk ) lim inf μk − νk M . k→∞

Ω

The conclusion follows using the triangle inequality.

2

If we know in addition that νk μk , ∀k 1, then one can deduce a stronger statement which implies Theorem 1.3 by taking νk = 0, ∀k 1. Theorem 7.1. Let (μk ), (νk ) ⊂ G be two bounded sequences with weak∗ limits μ, ν and reduced limits μ# , ν # , respectively. If νk μk

∀k 1,

(7.4)

then 0 μ# − ν # μ − ν.

(7.5)

Proof. Let uk , vk ∈ L1 (Ω) be the solutions of (1.2) with data μk and νk , respectively. Then, both sequences (uk ), (vk ) ⊂ L1 (Ω) are bounded in L1 (Ω) and uk vk a.e. Thus, g(uk ) − g(vk ) 0 a.e. Since (μ − μ# ) − (ν − ν # ) is the concentrated limit of (g(uk ) − g(vk )), we deduce from Corollary 2.2 that μ − μ# − ν − ν # 0.

(7.6)

It remains to show that μ# ν # . For this purpose, write (uk − vk ) = g(uk ) − g(vk ) − (μk − νk ). Passing to a subsequence, we may assume that (μk − νk ) has a concentrated limit with respect to capH 1 , which we will denote by σ . By Corollary 2.2, 0 σ μ − ν. On the other hand, it follows from Proposition 3.2 that (μ−μ# )−(ν −ν # )−σ is the concentrated limit of (g(uk ) − g(vk ) − (μk − νk )) with respect to capH 1 . Therefore, since uk vk a.e., ∀k 1, we deduce from Theorem 4.2 that

2338


μ − μ# − ν − ν # − σ 0. Hence, μ# − ν # μ − ν − σ 0. This establishes the proposition.

(7.7)

2

We now compare reduced limits associated to different nonlinearities. Proposition 7.2. Let (μk ) ⊂ G(g1 ) ∩ G(g2 ) be a bounded sequence with reduced limits μ#1 and μ#2 associated to g1 and g2 , respectively. If g1 g2 , then μ#1 μ#2 .

(7.8)

Proof. Let uk , vk ∈ L1 (Ω) be the solutions associated to (1.2) with datum μk and nonlinearities g1 and g2 , respectively. Since g1 g2 , by comparison we have uk vk

a.e., ∀k 1.

On the other hand, (uk − vk ) = g(uk ) − g(vk ). Since the concentrated limit of (g(uk ) − g(vk )) with respect to capH 1 is μ − μ#1 − μ − μ#2 = μ#2 − μ#1 , it follows from Theorem 4.2 that μ#2 − μ#1 0.

2

The next result gives the main tool for studying reduced limits of sequences signed measures. Proposition 7.3. Let (μk ) ⊂ G be a bounded sequence with weak∗ limit μ. Assume that ∗

+ μ+ k μ

∗

− and μ− k μ

weakly∗ in M(Ω).

(7.9)

− # # Then, (μk ) has a reduced limit μ# if and only if (μ+ k ) and (−μk ) have reduced limits μ1 and μ2 , respectively. In this case,

+ μ#1 = μ#

− and μ#2 = − μ# .

(7.10)

In particular, μ# = μ#1 + μ#2

(7.11)


2339

and μ# = μ

and μ#2 = −μ− .

if and only if μ#1 = μ+

(7.12)

Proof. Passing to a subsequence if necessary, we may assume that μ# , μ#1 and μ#2 exist. From Theorem 7.1, we have 0 μ#1 − μ# μ+ − μ = μ− .

(7.13)

Applying the Hahn decomposition with respect to μ, we can write Ω in terms of two disjoint sets E1 , E2 ⊂ Ω, Ω = E1 ∪ E2 such that μ 0 in E1

and μ 0

in E2 .

On the other hand, by Theorem 1.3, 0 μ#1 μ+

and

− μ− μ#2 0.

(7.14)

In particular, μ#1 is concentrated on E1 . It then follows from (7.13) that # μ E = μ#1 E = μ#1 . 1

1

Similarly, μ#2 is concentrated on E2 and # μ E = μ#2 . 2

In particular, μ#1 and μ#2 are singular with respect to each other. Moreover, μ# = μ# E + μ# E = μ#1 + μ#2 . 1

2

Since, by (7.14), μ#1 0 and μ#2 0, (7.10) follows.

2

8. Absolute continuity between μ and μ# We showed in Theorem 7.1 that if (μk ) ⊂ G is a bounded nonnegative sequence, then 0 μ# μ, and thus μ# μ. Our next result provides a sufficient condition on the sequence (μk ) so that μ μ# . This implies in particular that μ# = 0 if and only if μ = 0. Theorem 8.1. Assume that g : R → R is a continuous nondecreasing function such that g(0) = 0 and g(at) = +∞. a,t→+∞ ag(t) lim

(8.1)

2340


Let (μk ) ⊂ G be a bounded nonnegative sequence with weak∗ limit μ and reduced limit μ# . Suppose that there exists (Uk ) ⊂ L1 (Ω) such that for every k 1, −Uk = μk

and g(Uk ) ∈ L1 (Ω).

in Ω

(8.2)

If g(Uk ) is bounded in L1 (Ω),

(8.3)

then μ and μ# are absolutely continuous with respect to each other. Remark 8.1. If g is given by g(t) = |t|q−1 t, ∀t ∈ R, where q > 1, then (8.1) holds and assumption (8.2)–(8.3) on (μk ) is satisfied whenever (μk ) is bounded in W −2,q (Ω). In the next section, we shall study this nonlinearity in more detail in the supercritical case q NN−2 . Proof. Replacing Ω by a smaller domain if necessary, we may assume that (Uk |∂Ω ) is bounded in L1 (∂Ω). Replacing g by g + if necessary, we may assume that g(t) = 0

∀t 0.

Given α ∈ (0, 1), we then have 0 g(αUk ) g(Uk )

a.e.

Thus, there exists C0 > 0, independent of α, such that g(αUk )

L1

C0

∀k 1.

Let (g(αUkj )) be a subsequence having diffuse and concentrated limits with respect to Lebesgue measure; denote by σα its concentrated limit. The proof of the theorem is based on the following assertions: Claim 1. For every α ∈ (0, 1), αμ σα + μ# .

(8.4)

Indeed, let vj be such that

−vj + g(vj ) = αμkj

in Ω,

vj = αUkj

on ∂Ω.

(8.5)

Then, (vj ) is bounded in L1 (Ω) and, by comparison, vj αUkj a.e. Thus, g(vj ) g(αUkj )

a.e.

Passing to a further subsequence, we may assume that (αμkj ) has a reduced limit μ#α . It follows from Proposition 3.1 that the sequence (g(vj )) has concentrated limit αμ − μ#α . Thus,

M. Marcus, A.C. Ponce / Journal of Functional Analysis 258 (2010) 2316–2372 ∗

2341

weakly∗ in M(Ω),

g(vj ) g(vα ) + αμ − μ#α

where vα is the solution of (8.5) associated to μ#α . Applying Corollary 2.2 to the nonnegative sequence (g(αUkj ) − g(vj )), we deduce that its concentrated limit is nonnegative, σα − αμ + μ#α 0.

(8.6)

On the other hand, since αμ μ, it follows from Theorem 7.1 that μ#α μ# .

(8.7)

σα M = 0. α

(8.8)

Combining (8.6)–(8.7), we obtain (8.4). Claim 2. lim

α→0

Given ε > 0, take a0 , t0 > 1 such that g(at) 1 ag(t) ε

∀a a0 , ∀t t0 .

(8.9)

For every α ∈ (0, 1/a0 ), we write g(αUkj ) = g(αUkj )χ[αUkj 0 is arbitrary, the claim follows.

∀α ∈ (0, 1/a0 ).

(8.10)

2342


We now complete the proof of Theorem 8.1. Since 0 μ# μ, we only need to show that μ μ# . For this purpose, take a Borel set E ⊂ Ω such that μ# (E) = 0. By Claim 1, αμ(E) σα (E)

∀α ∈ (0, 1).

Thus, σα (E) σα M α α

μ(E)

∀α ∈ (0, 1).

Letting α → 0, by Claim 2 we deduce that μ(E) = 0. The proof is complete.

2

9. Reduced limits and W −2,q -weak convergence In this section we assume that N 3 and we focus on the case of power nonlinearities g(t) = |t|q−1 t in the supercritical range q equation

N N −2 .

∀t ∈ R,

(9.1)

Denote by G q the set of finite measures in Ω for which the

−u + |u|q−1 u = μ in Ω,

(9.2)

q

has a solution and we denote by G0 the set of finite measures in Ω for which the Dirichlet problem

−u + |u|q−1 u = μ u=0

in Ω, on ∂Ω,

(9.3)

has a solution. For every μ ∈ M(Ω), q

μ ∈ G0

if and only if μ ∈ L1 (Ω) + W −2,q (Ω), q

and Baras and Pierre [2] proved that μ ∈ G0 if and only if the measure μ is diffuse relative to q the capacity capW 2,q . Since, by Theorem A.1 in Appendix A, G q = G0 , we have in this way a q complete characterization of measures in G . Concerning sequences, if (μk ) ⊂ G q is a bounded sequence strongly converging in W −2,q (Ω), then its reduced limit and its weak∗ limit coincide; see [6, Theorem 4.13]. The goal of this section is to investigate what happens if (μk ) is bounded in W −2,q (Ω) but does not necessarily converge strongly in this space. We start by proving a more precise version of Theorem 1.4. Theorem 9.1. Given q NN−2 , let (μk ) ⊂ G q be a bounded sequence of nonnegative measures with weak∗ limit μ and reduced limit μ# . If in addition (μk ) is bounded in W −2,q (Ω), then μ and μ# are absolutely continuous with respect to each other. Moreover, there exists Cq > 0 such that for every Borel set E ⊂ Ω,


q Cq

μ(E) q−1 μ# (E) μ(E), 1 Γ0q−1

2343

(9.4)

q

where Γ0 = supk1 {μk M + μk W −2,q }. Proof. We use the same notation as in the proof of Theorem 8.1. This theorem applies in the present case. In addition, by Theorem 7.1, μ# μ. Therefore we only have to prove the first inequality in (9.4). Recall that, by (8.4), αμ − σα μ#

∀α ∈ (0, 1).

On the other hand, by (8.10), σα M α q C0 α q Γ0 . Therefore, given a Borel set E ⊂ Ω, αμ(E) − α q Γ0 αμ(E) − σα (E) μ# (E)

∀α ∈ (0, 1).

Since μ(E) Γ0 , the left-hand side achieves a positive maximum in the interval (0, 1). Computing this maximum we obtain

This completes the proof.

q q − 1 [μ(E)] q−1 μ# (E). 1 q q−1 q−1 Γ0

(9.5)

2

For every bounded sequence of nonnegative measures (μk ) ⊂ G q converging weakly∗ to μ, 0 μ# μ. We have just showed that if in addition (μk ) is bounded in W −2,q (Ω), then μ μ# . Since μ ∈ W −2,q (Ω) and this space is contained in G q , one might expect that μ# = μ. We now present a striking example showing that this need not be the case. Theorem 9.2. For every q NN−2 there exists a sequence of nonnegative functions (fk ) ⊂ C ∞ (Ω), bounded in L1 (Ω) and in W −2,q (Ω), such that its weak∗ limit f and its reduced limit f # associated to the equation −u + |u|q−1 u = h

in Ω,

(9.6)

are different. In other words, if uk is a solution of (9.6) with datum fk and if uk → u# in L1 (Ω), then u# is not a solution of (9.6) with datum f . We first recall some known estimates. In what follows, we say that A ∼ B if there exist constants C1 , C2 > 0 such that A C1 B and B C2 A.

2344


Lemma 9.1. Let a > 0. For every R > a we have BR

dx ∼ (|x| + a)p

a N −p

if p > N,

1 + log Ra

if p = N.

(9.7)

The proof is straightforward and will be omitted. Given f ∈ L1 (RN ), consider the Newtonian potential associated to f :

f (y) dy |x − y|N −2

Gf (x) = RN

∀x ∈ RN .

(9.8)

It is well known that −(Gf ) = γN f

in RN ,

where γN = N(N − 2)|B1 | and |B1 | denotes the Lebesgue measure of the unit ball in RN . Lemma 9.2. Given p N and a > 0, let hp (x) =

1 (|x| + a)p

∀x ∈ RN .

(9.9)

Then, for every R > a and every x ∈ BR , G[hp χBR ](x) ∼

⎧ ⎨

a N−p (|x|+a)N−2

if p > N,

⎩ 1+log+ (|x|/a) (|x|+a)N−2

if p = N.

Proof. Clearly, G[hp χBR ] is radial and G[hp χBR ](x) → 0 as |x| → ∞. Denote v(r) := G[hp χBR ](x), where r = |x|. We then have 1 v (r) = |∂Br |

CN = N −1 r

∂Br

∂ G[hp χBR ] ∂n C˜ N G[hp χBR ] = − N −1 hp χBR . r

Br

Br

Assume that p > N . In this case, a straightforward computation shows that

hp χBR ∼ Br

rN ap a N −p

if r a, if r > a.

(9.10)


2345

Thus,

v (r) ∼

if r a,

− arp a N−p

− r N−1

if r > a.

Since ∞ G[hp χBR ](x) = v(r) = −

v (t) dt,

r

estimate (9.10) for p > N follows. The case p = N can be deduced in a similar way using ⎧ N r ⎪ ⎪ ⎨ aN ∼ 1 + log ar ⎪ ⎪ ⎩ 1 + log R

hp χBR Br

This establishes the lemma.

a

if r a, if a < r < R, if r R.

2

Given k 1, we write the unit cube [0, 1]N as a union of k N cubes of sides k1 such that their interiors, Q1 , . . . , Qk N , are disjoint. If we denote by xi the center of the open cube Qi , then Qi = Q0 + xi , where 1 1 N . Q0 = − , 2k 2k Lemma 9.3. Given a radially nonincreasing function h ∈ C ∞ (RN ) with h 0, let N

k

H (x) =

h(x − xi )χQi (x)

∀x ∈ (0, 1)N .

(9.11)

i=1

Then, for every i ∈ {1, . . . , k N }, GH (x) ∼ G[hχQ0 ](x − xi ) + k

N

h

on Qi .

Q0

Proof. Given i ∈ {1, . . . , k N }, let J1 = {j ; Qj ∩ Qi = ∅} and J2 = {j ; Qj ∩ Qi = ∅}. Denote hi (x) := h(x − xi )χQi (x). Using this notation, Ghi (x) = G[hχQ0 ](x − xi ).

(9.12)

2346


Since h is radially nonincreasing, for every x ∈ Qi and j ∈ {1, . . . , k N } we have Ghi (x) = G[hχQ0 ](x − xi ) G[hχQ0 ](x − xj ) = Ghj (x). In particular,

Ghj (x) ∼ Ghi (x)

on Qi .

(9.13)

j ∈J1

On the other hand, for every x ∈ Qi and j ∈ J2 , Ghj (x) ∼

1 [d(Qj , Qi )]N −2

h. Q0

Since the number of cubes Qt at distance ∼ /k from Qi is of the order of N −1 , then for every x ∈ Qi we have

Ghj (x) ∼

j ∈J2

∼

k

t =1 d(Qt ,Qi )∼ k k =1

N −1 (/k)N −2

1 [d(Qt , Qi )]N −2

h Q0

h ∼ kN

Q0

Combining (9.13)–(9.14), we obtain (9.12).

(9.14)

h. Q0

2

Proof of Theorem 9.2. Without loss of generality, we may assume that Ω = (0, 1)N . We split the proof in two parts: Case 1. q >

N N −2 .

Let ϕ ∈ C0∞ (B1 ) be a radially nonincreasing function with ϕ 0 in Ω and α > 0, we take ak > 0 so that

B1

ϕ = 1. Given

N −(N −2)q

ak k N (q−1)

=α

∀k 1,

(9.15)

and define k 1 x − xi ∀x ∈ (0, 1)N , ϕ Hk (x) = ak k N akN i=1 N

N

(9.16)

N

where (xi )ki=1 are the centers of the open cubes (Qi )ki=1 . Let fk = γN Hk + (GHk )q .

(9.17)


2347

We show that for α > 0 sufficiently large the weak∗ limit and the reduced limit of (fk ) are different. For this end, let ϕk (x) =

1 x − xi ∀x ∈ RN . ϕ ak akN

Since Gϕ(x) ∼

1 (|x| + 1)N −2

Gϕk (x) ∼

1 (|x| + ak )N −2

∀x ∈ RN ,

one obtains, by scaling, ∀x ∈ RN .

It thus follows from Lemma 9.3 that for every x ∈ Qi , i = 1, . . . , k N , GHk (x) ∼

1 1 1 Gϕk (x − xi ) + 1 ∼ N + 1. N k k (|x − xi | + ak )N −2

(9.18)

Thus, by Lemma 9.1, (GHk )q ∼ (0,1)N

kN kN q

N −(N −2)q

Q0

a dx + 1 ∼ k N (q−1) (N −2)q (|x| + ak ) k

+ 1 = α + 1.

(9.19)

In particular, fk ∼ α + 1

∀k 1.

(9.20)

(0,1)N

Let Aδ = (0, 1)N \ (δ, 1 − δ)N . A similar computation shows that given ε > 0 there exists δ > 0 such that fk < ε

∀k 1.

(9.21)

Aδ

By (9.18), (GHk )q (x) ∼

1 1 +1 k N q (|x − xi | + ak )(N −2)q

Applying Lemmas 9.2–9.3, for every x ∈ Qi we have

in Qi .

2348

M. Marcus, A.C. Ponce / Journal of Functional Analysis 258 (2010) 2316–2372 N −(N −2)q

ak 1 1 G (GHk )q (x) ∼ N q + +α+1 N −2 k (|x − xi | + ak ) (|x − xi | + 1)N −2

∼

N −(N −2)q

1 kN q

ak + α + 1. (|x − xi | + ak )N −2

Thus, by Lemma 9.1,

N −(N −2)q q

N −(N −2)q q q N ak G (GHk ) ∼k ak + αq + 1 kN q

(0,1)N

=

N −(N −2)q q+1

ak k N (q−1)

+ α q + 1 = α q+1 + α q + 1 ∼ α q+1 + 1.

Let vk be such that

−vk = fk vk = 0

in (0, 1)N , on ∂(0, 1)N .

Since 0 vk Gfk , we have

q vk

(0,1)N

(Gfk )q α q+1 + 1 ∀k 1.

(0,1)N

In particular, the sequence (fk ) is bounded in W −2,q (Ω) and fk W −2,q α

q+1 q

+ 1 ∀k 1.

Let uk = GHk

in (0, 1)N .

Then, uk satisfies the equation q

−uk + uk = fk

in (0, 1)N ,

and uk → u

in L1 (0, 1)N ,

where u satisfies −u = 1 in (0, 1)N . In other words, f # = 1 + uq is the reduced limit of the sequence (fk ); hence,

(9.22)


2349

f # ∼ 1, (0,1)N

independently of α. On the other hand, passing to a subsequence if necessary, we have weakly∗ in M (0, 1)N .

∗

fk f In view of (9.20)–(9.21),

f ∼ α + 1. (0,1)N

Thus, by taking α > 0 sufficiently large, we must have f # = f . This establishes the result when q > NN−2 . Case 2. q =

N N −2 .

Let Hk and fk be given by (9.16) and (9.17), respectively, where ak > 0 is now given by 1 k

1 =α kak

log

2N N−2

(9.15 )

∀k 1.

Note that (9.18) still holds. Hence, by Lemma 9.1,

1

N

(GHk ) N−2 ∼ k

(0,1)N

2N N−2

1 + log

1 kak

(9.19 )

+ 1 ∼ α + 1,

from which (9.20) follows. By Lemmas 9.2–9.3, estimate (9.22) now becomes

N G (GHk ) N−2 (x) ∼

i| 1 + log+ ( |x−x ak )

1 k

N2 N−2

(|x − xi | + ak )N −2

+α+1

(9.22 )

in Qi .

Therefore,

N N G (GHk ) N−2 N−2 ∼

kN N3

k (N−2)2

(0,1)N

1

∼ k

2N N−2

2(N−1) N−2 N 1 1 + log + α N−2 + 1 kak 1 log kak

2(N−1) N−2

N

+ α N−2 + 1 ∼ α

2(N−1) N−2

+ 1.

Proceeding as in the previous case, we deduce that the weak∗ limit and the reduced limit of the sequence (fk ) are different for α > 0 sufficiently large. The proof is complete. 2

2350


10. Reduced limits for g(t) = |t|q−1 t Given a bounded sequence (μk ) ⊂ G q , consider a splitting (αk ) and (σk ) into an equidiffuse and a concentrating parts relative to capW 2,q . In this section, we show that the reduced limits of (μk ) and (αk ) associated to the nonlinearity g(t) = |t|q−1 t coincide. We first study the case where the sequence (μk ) is concentrating. Proposition 10.1. Given q NN−2 , let (μk ) ⊂ G q be a bounded sequence with reduced limit μ# . If (μk ) is concentrating with respect to capW 2,q , then μ# = 0.

(10.1)

Proof. In view of Proposition 7.3, it suffices to prove the result when the sequence (μk ) is nonnegative. For each k 1, assume that uk satisfies

−uk + |uk |q−1 uk = μk uk = 0

in Ω, on ∂Ω.

(10.2)

Passing to a subsequence if necessary, we may assume that uk → u# in L1 (Ω) and a.e. By a comparison principle, uk 0 a.e. Let (Ek ) be a sequence of Borel subset of Ω such that capW 2,q (Ek ) → 0 and |μk |(Ω \ Ek ) → 0.

(10.3)

From the regularity of capW 2,q and μk , we may assume that each Ek is compact. Moreover, there exists a sequence (ϕk ) ⊂ C0∞ (Ω) such that 0 ϕk 1 in Ω,

ϕk = 1 on Ek

and

2 p D ϕk C cap

W 2,q

(Ek ).

(10.4)

Ω

Let

Fk = x ∈ Ω; ϕk (x) 1/2 . Then, capW

2,q

(Fk ) 2

q

2 q D ϕk → 0.

Ω q

We claim that the sequence (uk ) is concentrating with respect to capW 2,q . In order to prove this, it suffices to show that q uk → 0. (10.5) Ω\Fk

Using ϕk as a test function in (10.2), we get


q uk ϕk

=

Ω

2351

ϕk dμk +

Ω

uk ϕk

∀k 1.

(10.6)

Ω

In view of (10.2), uk Lq μk M . Therefore, by (10.6), 1 2

q

q

uk

uk (1 − ϕk ) Ω

Ω\Fk

(1 − ϕk ) dμk −

Ω

uk ϕk .

(10.7)

Ω

We show that both terms in the right-hand side of this estimate converge to 0 as k → ∞. By (10.3), (1 − ϕk ) d|μk | |μk |(Ω \ Ek ) → 0.

(10.8)

Ω

Furthermore, by (10.4), uk ϕk uk Lq ϕk q C D 2 ϕk q → 0. L L

(10.9)

Ω

Combining (10.7)–(10.9), we get

q

uk → 0. Ω\Fk q

Thus, the sequence (uk ) is concentrating. Since uk → u# a.e., this implies that u# = 0 a.e. We deduce that uk → 0 in L1 (Ω) and μ# = 0. 2 Remark 10.1. Let q NN−2 . Then, for every μ ∈ M(Ω) there exists a bounded sequence (μk ) ⊂ G q converging weakly∗ to μ but having reduced limit zero with respect to g(t) = |t|q−1 t. In fact, let (τk ) be a sequence consisting of linear combinations of Dirac masses such that ∗

τk μ

weakly∗ in M(Ω),

and let (ρk ) be a sequence of smooth mollifiers. For every j 1, the reduced limit of the sequence (ρk ∗ τj )k1 equals the reduced measure τj∗ , which is zero. Hence, there exists kj j such that the solution of −uj + |uj |q−1 uj = ρkj ∗ τj in Ω, uj = 0 on ∂Ω, satisfies uj L1 j1 . Therefore, the sequence (ρkj ∗ τj ) has weak∗ limit μ but its reduced limit is zero. We now present the main result of this section.

2352


Theorem 10.1. Given q NN−2 , let (μk ) ⊂ G q be a bounded sequence, and let (αk ), (σk ) ⊂ M(Ω) be a decomposition of (μk ) satisfying (B1 )–(B2 ) with respect to capW 2,q . If (μk ) has a reduced limit μ# , then μ# is also the reduced limit of (αk ). By Theorem 9.2, μ# need not coincide with the diffuse limit of (μk ) with respect to capW 2,q , which is by definition the weak∗ limit of the sequence (αk ). However, we show that the reduced limits of the two sequences coincide. For the proof of Theorem 10.1, we need two lemmas. Lemma 10.1. Let (μk ) ⊂ G q be a bounded sequence. For each k 1, let uk be the solution of

−uk + |uk |q−1 uk = μk uk = 0

in Ω, on ∂Ω.

(10.10)

If (μk ) is equidiffuse with respect to capW 2,q , then so is the sequence (|uk |q ). Proof. Assume by contradiction that (|uk |q ) is not equidiffuse. Then, passing to a subsequence if necessary, one can find ε > 0 and a sequence of Borel subsets (Ek ) of Ω such that capW 2,q (Ek ) → 0 and

|uk |q ε

∀k 1.

Ek

By regularity of capW 2,q and of the Lebesgue measure, we may assume that each set Ek is compact. Moreover, there exists a sequence (ϕk ) ⊂ C0∞ (Ω) satisfying (10.4). In particular, ϕk → 0 in W 2,q (Ω). Passing to a subsequence if necessary, we may assume that ϕk → 0 q.e. with respect to capW 2,q . Let vk be the solution of

−vk + |vk |q−1 vk = |μk | vk = 0

in Ω, on ∂Ω.

(10.11)

Since |μk | 0, we have vk 0 a.e. Using ϕk as a test function, we get Ω

q vk ϕk

=

ϕk d|μk | +

Ω

vk ϕk

∀k 1.

(10.12)

Ω

Since (ϕk ) is uniformly bounded, ϕk → 0 q.e. with respect to capW 2,q , and (μk ) is equidiffuse, ϕk d|μk | → 0.

(10.13)

Ω

Moreover, as in the proof of Proposition 10.1, vk ϕk → 0. Ω

(10.14)


2353

Combining (10.12)–(10.14), we deduce that

q

vk ϕk → 0. Ω

Since |uk | vk a.e., this contradicts the assumption |uk |q ϕk ε

∀k 1.

Ek

Therefore, the sequence (|uk |q ) must be equidiffuse.

2

The following estimate will be used in the proof of Theorem 10.1. Lemma 10.2. Given v, w ∈ Lq (Ω), let h = |v + w|q−1 (v + w) − |v|q−1 v − |w|q−1 w.

(10.15)

Then, there exists a constant C > 0 such that for every Borel set F ⊂ Ω, q−1 q−1 hL1 (Ω) C vLq (Ω) + wLq (Ω) vLq (F ) + wLq (Ω\F ) .

(10.16)

Proof. We first write hL1 (Ω) =

|h| +

F

|h|.

(10.17)

Ω\F

We show that

q−1 q−1 |h| C vLq (Ω) + wLq (Ω) vLq (F ) .

(10.18)

F

By the triangle inequality,

|h|

F

F

|v + w|q−1 (v + w) − |w|q−1 w +

|v|q .

(10.19)

F

Denote by I the first integral in the right-hand side of this inequality. In order to estimate I we use the following elementary estimate, |a + b|q−1 (a + b) − |b|q−1 b q |a + b|q−1 + |b|q−1 |a| ∀a, b ∈ R. In fact, applying this estimate with a = v(x) and b = w(x), and integrating it over F , one gets

2354


I q

|v + w|q−1 |v| +

F

|w|q−1 |v| .

F

Thus, by Hölder’s inequality, q−1 q−1 q−1 q−1 I q v + wLq (F ) + wLq (F ) vLq (F ) C vLq (Ω) + wLq (Ω) vLq (F ) . Inserting this estimate into (10.19), we get

q−1 q−1 q−1 |h| C vLq (Ω) + wLq (Ω) vLq (F ) + vLq (Ω) vLq (F ) .

F

This gives (10.18). Interchanging the roles of v and w, and replacing F by Ω \ F , one gets a similar estimate for the last integral in (10.17). Combining these estimates, one deduces (10.16). 2 Proof of Theorem 10.1. For every k 1, let vk and wk be the solutions of

−z + |z|q−1 z = γ z=0

in Ω, on ∂Ω,

(10.20)

with data αk and σk , respectively. Adding both equations, we observe that vk + wk also satisfies problem (9.4) with datum λk = μk + hk ,

(10.21)

where hk ∈ L1 (Ω) is given by hk = |vk + wk |q−1 (vk + wk ) − |vk |q−1 vk − |wk |q−1 wk . We claim that hk → 0 in L1 (Ω).

(10.22)

Since the sequence (σk ) is concentrating, it follows from the proof of Proposition 10.1 that the sequence (|wk |q ) is concentrating with respect to the capacity capW 2,q . Let (Fk ) be a sequence of Borel subsets of Ω such that capW 2,q (Fk ) → 0 and |wk |q → 0. Ω\Fk

Applying Lemma 10.2 with functions vk and wk , and Borel set Fk , we have q−1 q−1 hk L1 (Ω) C vk Lq (Ω) + wk Lq (Ω) vk Lq (Fk ) + wk Lq (Ω\Fk ) . Since (αk ) and (σk ) are bounded in M(Ω), the sequences (vk ) and (wk ) are bounded in Lq (Ω). Thus,


2355

hk L1 (Ω) C˜ vk Lq (Fk ) + wk Lq (Ω\Fk ) ∀k 1. By the choice of the sets Fk , wk Lq (Ω\Fk ) → 0. On the other hand, since the sequence (αk ) is equidiffuse with respect to capW 2,q , (|vk |q ) is also equidiffuse by Lemma 10.1. Thus, vk Lq (Fk ) → 0. This implies (10.22). We have thus showed that λk − μk M = hk L1 → 0. In particular, the sequences (λk ) and (μk ) have the same weak∗ limit μ. In order to identify their reduced limit, we note that if vk → v #

in L1 (Ω),

then, since wk → 0 in L1 (Ω), uk + vk → v #

in L1 (Ω).

Thus, the reduced limit of (λk ) coincides with the reduced limit of (αk ), namely α # . But since by Proposition 7.1 the sequences (μk ) and (λk ) have the same reduced limits, we conclude that μ# = α # . This concludes the proof of the theorem. 2 11. Sufficient conditions for the equality μ# = μ We present in this section some cases where the weak∗ limit and the reduced limit μ# of a given sequence (μk ) are equal. The first result should be compared with Theorems 9.1 and 9.2. Proposition 11.1. Let (μk ) ⊂ G be a bounded sequence with weak∗ limit μ and reduced limit μ# . If (μk ) is bounded in H −1 (Ω), then μ# = μ. Proof. For each k 1, let uk be such that

−uk + g(uk ) = μk uk = 0

in Ω, on ∂Ω.

(11.1)

Passing to a subsequence if necessary, we may assume that uk → u# in L1 (Ω) and a.e. Since μk ∈ H −1 (Ω), uk ∈ H 1 (Ω) and (see [4,6])

|∇uk | +

Ω

g(uk )uk =

2

Ω

(11.2)

uk dμk . Ω

In particular, from the boundedness of (μk ) in H −1 (Ω), we deduce that the sequence (uk ) is bounded in H 1 (Ω). Thus,

g(uk )uk Ω

uk dμk uk H 1 μk M C Ω

∀k 1.

2356


Since g(t)t 0, ∀t ∈ R, this implies that (g(uk )) is an equiintegrable sequence in L1 (Ω). As g(uk ) → g(u# ) a.e., it follows from Egorov’s lemma that g(uk ) → g(u# ) in L1 (Ω). Therefore, μ# = μ. 2 Proposition 11.2. Let (μk ) ⊂ G be a bounded sequence with weak∗ limit μ and reduced limit μ# . Assume that there exists ν ∈ M(Ω) such that |μk | ν

∀k 1.

(11.3)

Then, μ# = μ.

(11.4)

Proof. We split the proof in two steps: Step 1. (11.4) holds if, in addition, λ1 μ k λ2

∀k 1,

(11.5)

where λ1 , λ2 ∈ G. For each k 1, let uk be such that

−uk + g(uk ) = μk uk = 0

in Ω, on ∂Ω.

(11.6)

Denote by v1 and v2 the solutions of (11.6) with data λ1 and λ2 , respectively. By the comparison principle, we have v1 uk v2

a.e., ∀k 1.

Hence, since g is nondecreasing, g(v1 ) g(uk ) g(v2 )

a.e., ∀k 1.

On the other hand, passing to a subsequence if necessary, we may assume that uk → u in L1 (Ω) and a.e. Since g(v1 ), g(v2 ) ∈ L1 (Ω), we conclude that g(uk ) → g(u)

in L1 (Ω).

Therefore, u satisfies (11.6) with right-hand side μ, whence μ is the reduced limit of the (μk ). Step 2. Proof completed.


2357

In view of the previous step, it suffices to find λ1 , λ2 ∈ G satisfying (11.5). For this purpose, note that by (11.3) we have −ν − μk ν +

∀k 1.

We recall (see [6, Section 6]) that the reduced measure (ν + )∗ is the largest measure in G which + + is dominated by ν + . Since μ+ k ∈ G and μk ν , + ∗ μ+ k ν

∀k 1.

Similarly, (−ν − )∗ is the smallest measure in G which dominates −ν − . Since −μ− k ∈ G and , −ν − −μ− k − ∗ −ν (−μk )−

∀k 1.

Thus, (11.5) holds with λ1 = (−ν − )∗ and λ2 = (ν + )∗ . By the previous step, (11.4) follows.

2

We now show that the reduced limit and the weak∗ limit always coincide under weak-L1 convergence. Proposition 11.3. Given ν ∈ M(Ω), let (hk ) ⊂ G ∩ L1 (Ω; ν). If hk h

weakly in L1 (Ω; ν),

(11.7)

then hν is the reduced limit of the sequence (hk ν). Proof. By a diagonalization procedure, one can find an increasing sequence of integers (kj ) such that, for every integer n 1, the sequence (Tn (hkj ))j 1 converges weakly in L1 (Ω; ν) to some function h˜ n , where Tn is given by (3.2). We may also assume that the reduced limit μ# of (hkj ν) exists. Since Tn (hk )ν nν j

∀j 1,

it follows from Proposition 11.2 that h˜ n ν is the reduced limit of the sequence (Tn (hkj )ν). On the other hand, by the Dunford–Pettis theorem (see [13]), the sequence (hk ) converges weakly in L1 (Ω; ν) if and only if (hk ) is bounded in L1 (Ω; ν) and for every ε > 0 there exists δ > 0 such that E ⊂ Ω Borel and ν(E) < δ ⇒ |hk | dν < ε ∀k 1. (11.8) E

Let C0 > 0 be such that |hk | dν C0 Ω

∀k 1.

(11.9)

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Let Aj,n = [|hkj | > n]; by the Chebyshev inequality, ν(Aj,n )

1 n

|hkj | dν

C0 n

∀j, n 1.

Ω

Take n 1 sufficiently large so that C0 /n < δ. Then, by (11.8) we have hk ν − Tn (hk )ν = j j M

hk − Tn (hk ) dν j j

Ω

|hkj | dν < ε.

(11.10)

Aj,n

By lower semicontinuity of the norm in M(Ω), as we let j → ∞ we get hν − h˜ n νM ε.

(11.11)

Denote by μ# the reduced limit of the sequence (hkj ν). By Proposition 7.1 applied to (hkj ν) and (Tn (hkj )ν), # μ − h˜ n ν

M

hν − h˜ n νM + lim inf hkj ν − Tn (hkj )ν M 2ε. j →∞

(11.12)

Combining (11.11)–(11.12) we deduce that # μ − hν

M

3ε.

Since ε > 0 is arbitrary, we must have μ# = hν. In particular, the reduced limit μ# does not depend on the sequence (kj ). Therefore, the reduced limit of the whole sequence (hk ν) exists and equals hν. 2 12. Characterization of sequences for which μ# = μ In the previous section, we presented some sufficient conditions in order that the weak∗ limit and the reduced limit of a given sequence (μk ) coincide. Our goal in this section is to provide necessary and sufficient conditions for this property to hold. Before we present our next result, we observe that every μ ∈ G has a decomposition of the form μ = f − v

in Ω,

(12.1)

where f ∈ L1 (Ω), v ∈ L1 (Ω) and g(v) ∈ L1 (Ω). For instance, we can take f = g(u) and v = u, where u is the solution of problem (1.2). But the decomposition (12.1) of μ is not unique. Theorem 12.1. Let (μk ) ⊂ G be a bounded nonnegative sequence with weak∗ limit μ and reduced limit μ# . Then, μ# = μ if and only if for every k 1 there exist fk ∈ L1loc (Ω) and vk ∈ L1loc (Ω) such that

(12.2)


μk = fk − vk

g(vk ) ∈ L1loc (Ω),

in Ω,

2359

(12.3)

where both sequences (fk ) and (g(vk )) converge strongly in L1 (ω) for every subdomain ω Ω. For the proof of Theorem 12.1 we need the following auxiliary results. Lemma 12.1. Let (μk ) ⊂ G be a bounded nonnegative sequence with weak∗ limit μ and reduced limit μ# . Let uk ∈ L1 (Ω) be the solution of

−uk + g(uk ) = μk uk = 0

in Ω, on ∂Ω,

(12.4)

and assume that (uk ) converges in L1 (Ω). Then, the following assertions are equivalent: (i) μ = μ# ; (ii) (g(uk )) converges in L1 (ω) for every subdomain ω Ω; (iii) (g(uk )) is equidiffuse with respect to capH 1 in every subdomain ω Ω. Proof. (i) ⇒ (ii). Since μk 0, we have uk 0 a.e., ∀k 1. Let u# ∈ L1 (Ω) be such that uk → u#

in L1 (Ω).

Passing to a subsequence if necessary, we may also assume that uk → u# a.e. By assumption, μ = μ# . Thus,

g(uk )ζ →

Ω

g u# ζ

∀ζ ∈ C02 (Ω).

Ω

By a density argument, we get

g(uk )ρ0 →

Ω

g u# ρ0 ,

Ω

where ρ0 (x) = d(x, ∂Ω)

∀x ∈ Ω.

(12.5)

Since g(uk ) 0 a.e., ∀k 1, and g(uk )ρ0 → g(u# )ρ0 a.e., it follows from the Brezis–Lieb lemma (see [5]) that g(uk )ρ0 → g u# ρ0

in L1 (Ω).

(ii) ⇒ (iii). By the Poincaré inequality, |K|1/2 C capH 1 (K),

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for every compact set K ⊂ Ω. By regularity, this inequality holds for every Borel subset of Ω. Thus, if (g(uk )) converges strongly in L1 (ω), then it is equidiffuse with respect to capH 1 in ω. (iii) ⇒ (i). By Proposition 3.2, μ − μ# is the concentrated limit of (g(uk )) with respect to capH 1 . In particular, if (g(uk )) is equidiffuse in ω for every ω Ω, then we must have μ − μ# = 0. 2 Lemma 12.2. Let (μk ) ⊂ G be a bounded nonnegative sequence with weak∗ limit μ and reduced limit μ# . If μ# = μ, then for every sequence (hk ) ⊂ L1 (Ω) such that hk → h strongly in L1 (Ω), the sequence (λk ) given by λk = μk + hk

∀k 1,

(12.6)

has reduced limit λ# = μ + h. Proof. For every k 1, let uk be the solution of the problem

−z + g(z) = γ

in Ω,

z=0

on ∂Ω,

(12.7)

with datum γ = μk . Given a ∈ (0, 1), let vk be the solution of the linear problem

−v = f

in Ω,

v=0

on ∂Ω,

(12.8)

with datum f = T1/a (hk ). Since vk ∈ L∞ (Ω) and a ∈ (0, 1), it follows that g(auk +vk ) ∈ L1 (Ω) and, consequently, νk := aμk + T1/a (hk ) + g(auk + vk ) − ag(uk ) ∈ M(Ω). We observe that auk + vk is the solution of (12.7) with datum γ = νk . If uk → u in L1 (Ω) then, by Lemma 12.1, g(uk ) → g(u) in L1 (ω) for every ω Ω. By dominated convergence, it follows that g(auk + vk ) → g(au + v)

in L1 (ω),

where v is the solution of (12.8) with f = T1/a (h). Let wk and w˜ k denote the solutions of (12.7) with data βk = g(auk ) − ag(uk )

and τk = aμk + T1/a (hk ) − ag(uk ) + g(auk ),

respectively. Passing to a subsequence if necessary we may assume that wk → w and w˜ k → w˜ in L1 (Ω) and a.e. For every ω Ω, g(auk ) − ag(uk ) → g(au) − ag(u) Therefore, by Lemma 12.1,

in L1 (ω).


g(wk ) → g(w),

2361

in L1 (ω).

Since βk τk νk we have wk w˜ k auk + vk

a.e.,

which implies that g(wk ) g(w˜ k ) g(auk + vk )

a.e.

Since (g(w ˜ k )) converges a.e. to g(w), ˜ by dominated convergence, g(w˜ k ) → g(w) ˜

in L1 (ω),

for every subdomain ω Ω. This implies that w˜ is the solution of (12.7) with datum τa where τa is the weak∗ limit of (τk ), τa = aμ + T1/a (h) − ag(u) + g(au). Thus, w˜ does not depend on the subsequence and τa is the reduced limit of the whole sequence (τk ). By Proposition 7.1, # λ − τa

M(ω)

(μ + h) − τa M(ω) + lim inf λk − τk M(ω) k→∞ (1 − a)μM(ω) + 2 h − T1/a (h) L1 (ω) + 2 ag(u) − g(au) L1 (ω) + (1 − a) lim sup μk M(ω) . k→∞

As a → 1, the right-hand side of this inequality tends to 0, while τa → μ + h

strongly in M(ω).

Therefore, λ# = μ + h in every subdomain ω Ω, whence in Ω.

2

Proof of Theorem 12.1. (⇒) Assume that μ# = μ. For each k 1, let uk be such that

−uk + g(uk ) = μk uk = 0

in Ω, on ∂Ω.

(12.9)

Then, uk → u in L1 (Ω), where u is the solution of (12.9) with datum μ. Since by Lemma 12.1, g(uk ) → g(u) in L1 (ω) for every ω Ω, we have the conclusion with fk = g(uk ) and vk = uk .

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(⇐) We fix a subdomain ω˜ Ω. By Lemma 6.1, the sequence (vk ) is relatively compact in L1 (Ω). Thus, passing to a subsequence if necessary, vk → v in L1 (Ω). By assumption, for every k 1, −vk + g(vk ) = μk − fk + g(vk )

in ω. ˜

Since g(vk ) → g(v) strongly in L1 (ω), ˜ the reduced limit ν # of (μk − fk + g(vk )) coincides with ∗ its weak limit. Thus, ˜ ν # = μ − f + g(v) in ω. Since fk − g(vk ) → f − g(v) in L1 (ω), ˜ it follows from the previous lemma applied to the sequences (μk − fk + g(vk )) and (fk − g(vk )) that ˜ μ# = μ − f + g(v) + f − g(v) = μ in ω. Since μ# = μ in every subdomain ω˜ Ω, the conclusion follows.

2

In [6, Theorem 4.5], we prove that μ ∈ G(g) for every nonlinearity g if and only if the measure μ is diffuse with respect to capH 1 . Using this result we characterize the sequences of measures (μk ) for which the weak∗ limit and the reduced limit coincide for every g. Theorem 12.2. Let (μk ) ⊂ M(Ω) be a bounded sequence of nonnegative measures with weak∗ limit μ. Assume that every measure μk is diffuse with respect to capH 1 . Then, μ# = μ for every nonlinearity g

(12.10)

if and only if (μk ) is equidiffuse with respect to capH 1 in every subdomain ω Ω. Proof. First we observe that, since μk is diffuse, μk ∈ G(g) for every nonlinearity g. (⇐) Without loss of generality, we may assume that the sequence (μk ) is equidiffuse in Ω. Let uk be such that

−uk + g(uk ) = μk uk = 0

in Ω, on ∂Ω.

(12.11)

Passing to a subsequence if necessary, we may assume that uk → u#

in L1 (Ω).

Since (μk ) is equidiffuse, it follows from [9, Lemma 3] that (g(uk )) is also equidiffuse. By Lemma 12.1, μ is the reduced limit of (μk ) with respect to g. (⇒) Assume that μ = μ# . We closely follow the proof of [6, Theorem 4.5]. Suppose by contradiction that (μk ) is not equidiffuse in some subdomain ω Ω. Passing to a subsequence if necessary, one finds ε > 0 and a sequence of compact sets (Kk ) in ω such that μk (Kk ) ε

and

capH 1 (Kk ) → 0.


2363

By [6, Lemma 4E.1], for every k 1 there exists ϕk ∈ C0∞ (Ω) such that 0 ϕk 1 in Ω, ϕk = 1 on Kk and 1 (12.12) |ϕk | 2 capH 1 (Kk ) + → 0. k Ω

We may assume that supp ϕk ⊂ ω, ˜ ∀k 1, where ω ω˜ Ω. Up to a subsequence we also have ϕk → 0 a.e., ϕk → 0 a.e. and there exists F1 ∈ L1 (Ω) such that |ϕk | F1

a.e., ∀k 1.

According to a result of de La Vallée Poussin [12, Remarque 23], there exists a convex function h : [0, ∞) → [0, ∞) such that h(0) = 0, h(s) > 0 for s > 0, lim

t→∞

h(t) = +∞ and h(F ) ∈ L1 (Ω). t

Let g(t) =

h∗ (t) 0

if t 0, if t < 0,

where h∗ is the convex conjugate (or Fenchel transform) of h. For each k 1, let uk be the solution of (12.11) for this nonlinearity g. Since μ coincides with the reduced limit of (μk ), by Lemma 12.1 above we have g(uk ) → g(u)

in L1 (ω). ˜

˜ with Passing to a subsequence if necessary, one finds F2 ∈ L1 (ω), 0 g(uk ) F2

a.e., ∀k 1.

On the other hand, for every k 1, ε μk (Kk )

ϕk dμk =

Ω

g(uk )ϕk − uk ϕk .

(12.13)

Ω

Note that g(uk )ϕk − uk ϕk → 0 a.e. and g(uk )ϕk − uk ϕk 2g(uk )χω˜ + h |ϕk | 2F2 χω˜ + F1

∀k 1.

By dominated convergence, the right-hand side of (12.13) converges to 0 as k → ∞. This is a contradiction. Therefore, the sequence (μk ) is equidiffuse in ω with respect to capH 1 . 2

2364


13. Absolute continuity between μ# and ν # In addition to our standard assumptions on the nonlinearity g (continuity and monotonicity), throughout this section we assume that g is convex.

(13.1)

The goal of this section is to prove that if a sequence (νk ) is uniformly absolutely continuous with respect to another sequence (μk ), then the reduced limit ν # is absolutely continuous with respect to μ# . More precisely, Theorem 13.1. Let (μk ), (νk ) ⊂ G be bounded sequences of nonnegative measures with reduced limits μ# and ν # , respectively. If for every ε > 0 there exists δ > 0 such that E ⊂ Ω Borel and νk (E) < δ

⇒

μk (E) < ε

∀k 1,

(13.2)

then μ# ν # .

(13.3)

We first establish the following Lemma 13.1. Given nonnegative measures μ, ν ∈ G, let u and v be the solutions of

−z + g(z) = γ z=0

in Ω, on ∂Ω,

(13.4)

with data μ and ν, respectively. If μ aν for some a 1, then u av

a.e.

(13.5)

Proof. Since μ aν, subtracting the equations satisfied by u and v we get

(u − av)ζ =

Ω

g(u) − μ − ag(v) + aν ζ

Ω

g(u) − ag(v) ζ,

Ω

for every ζ ∈ C02 (Ω), ζ 0 in Ω. Thus, by Lemma 5.1,

+

(u − av) ζ Ω

g(u) − ag(v) ζ.

(13.6)

Ω [uav]

On the other hand, since g is convex and g(0) = 0, the function g(t)/t is nondecreasing on (0, ∞). Hence, for a 1 we have g(at) ag(t)

∀t 0.


2365

In particular, g(u) − ag(v) 0 a.e. on [u av].

(13.7)

It follows from (13.6)–(13.7) that

(u − av)+ ζ 0 ∀ζ ∈ C02 (Ω), ζ 0 in Ω.

(13.8)

Ω

This immediately gives (13.5).

2

Proposition 13.1. Let (μk ), (νk ) ⊂ G be bounded sequences of nonnegative measures with reduced limits μ# and ν # , respectively. Assume that there exists a 1 such that μk aνk

∀k 1.

(13.9)

Then, μ# aν # .

(13.10)

Proof. Denote by uk , vk ∈ L1 (Ω) the solutions of (13.4) with data μk and νk , respectively. In particular, for every k 1 we have (avk − uk ) = ag(vk ) − g(uk ) − aνk + μk

in Ω.

Passing to a subsequence if necessary, we may assume that (μk ) and (νk ) have concentrated limits σ and τ , respectively. On the other hand, the sequences (g(uk )) and (g(vk )) have concentrated limits μ − μ# and ν − ν # . Since avk − uk 0 a.e. for every k 1, it follows from Theorem 4.2 that a ν − ν # − μ − μ# − aτ + σ 0.

(13.11)

Note that (aνk − μk ) is a sequence of nonnegative measures with weak∗ limit aν − μ and concentrated limit aτ − σ . Hence, by Corollary 2.2, aτ − σ aν − μ.

(13.12)

Combining (13.11)–(13.12), we deduce that −aν # + μ# 0, which is precisely (13.10).

2

Proof of Theorem 13.1. Given a 1, we apply the Hahn decomposition to μk − aνk . We may thus write Ω = Ek ∪ Fk as a disjoint union of measurable sets such that

2366


μk aνk

on Ek

and μk aνk

on Fk

(for simplicity of notation we omit the dependence of Ek and Fk on a). In particular, 1 1 C0 νk (Ek ) μk (Ek ) μk M a a a

∀k 1,

since the sequence (μk ) is bounded in M(Ω). Thus, for a 1 sufficiently large, we have C0 /a < δ. By (13.2) we deduce that μk (Ek ) < ε

∀k 1.

(13.13)

Consider the sequences λ k = μ k Fk

and τk = νk Fk

∀k 1.

Then, λk aτk

∀k 1.

Passing to a subsequence if necessary, we may assume that (λk ) and (τk ) have reduced limits λ# and τ # , respectively. Thus, by Proposition 13.1, λ# aτ # .

(13.14)

Let E ⊂ Ω be a Borel set such that ν # (E) = 0. Since 0 τk νk , ∀k 1, by Theorem 7.1 we have τ # (E) = ν # (E) = 0. It follows from (13.14) and λ# 0 that λ# (E) = 0.

(13.15)

On the other hand, applying Proposition 7.1 to the sequences (μk ) and (λk ), we get # μ − λ #

M

μ − λM lim inf μk − λk M = lim inf μk (Ek ) ε. k→∞

k→∞

Thus, in view of (13.15), μ# (E) = μ# (E) − λ# (E) μ# − λ# M ε. Since ε > 0 is arbitrary we conclude that μ# (E) = 0. Therefore, μ# ν # .

2


2367

14. Reduced limit of max {μk , νk } Throughout this section, we assume in addition to our usual assumptions on g that g is convex. Given bounded sequences (μk ), (νk ) ⊂ M(Ω) converging weakly∗ to μ and ν, if μ ⊥ ν, then the measures λk = max {μk , νk } converge weakly∗ to max {μ, ν}. In this section we prove the counterpart of this statement for reduced limits. In order to do so we need the following result proved in [6, Corollary 4.4]: if μ, ν ∈ G, then max {μ, ν} ∈ G. Theorem 14.1. Let (μk ), (νk ) ⊂ G be bounded sequences of nonnegative measures with reduced limits μ# and ν # , respectively. If μ# ⊥ ν # , then the sequence (λk ) given by λk = max {μk , νk } ∀k 1,

(14.1)

has reduced limit λ# = max {μ# , ν # }. We first prove a variant of Lemma 13.1. Lemma 14.1. Given nonnegative measures λ, μ, ν ∈ G, let w, u, v ∈ L1 (Ω) be the solutions of

−z + g(z) = γ z=0

in Ω, on ∂Ω,

(14.2)

with data λ, μ and ν, respectively. If λ μ + ν, then wu+v

a.e.

(14.3)

Proof. Since λ μ + ν, we have

(w − u − v)ζ =

Ω

g(w) − λ − g(u) + μ − g(v) + ν ζ

Ω

g(w) − g(u) − g(v) ζ,

Ω

for every ζ ∈ C02 (Ω), ζ 0 in Ω. Thus, by Lemma 5.1,

(w − u − v)+ ζ

Ω

g(w) − g(u) − g(v) ζ 0,

Ω [wu+v]

where we used the property g(s + t) g(s) + g(t) From estimate (14.4) we deduce (14.3).

2

∀s, t 0.

(14.4)

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Proof of Theorem 14.1. Since μk , νk ∈ G, we have λk ∈ G. We observe that by Proposition 7.1, μ# λ# . Thus,

max μ# , ν # λ# .

(14.5)

λ# μ# + ν # .

(14.6)

We now prove that

For this purpose, let wk , uk , vk ∈ L1 (Ω) be the solutions of (14.2) with data μk , νk and λ˜ k , respectively, where λ˜ k = (μk + νk )∗ . In particular, since λk ∈ G and λk μk + νk , λk λ˜ k . Passing to a subsequence if necessary, we may assume that (λ˜ k ) has reduced limit λ˜ # . By Lemma 14.1, we have wk uk + vk

a.e., ∀k 1.

(14.7)

On the other hand, (uk + vk − wk ) = g(uk ) + g(vk ) − g(wk ) − μk − νk + λ˜ k

∀k 1.

Proceeding as in the proof of Proposition 13.1, one deduces that λ˜ # μ# + ν # .

(14.8)

On the other hand, since λk λ˜ k , ∀k 1, by Theorem 7.1 we also have λ# λ˜ # .

(14.9)

Combining (14.8)–(14.9) we deduce (14.6). Since μ# and ν # are nonnegative and, by assumption, μ# ⊥ ν # ,

μ# + ν # = max μ# , ν # . Thus,

λ# max μ# , ν # . The conclusion follows from (14.5) and (14.10).

2

(14.10)


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15. Open problems This section is devoted to questions related to the present work. The first open problem concerns a possible extension of Theorem 1.4. Open Problem 1. Given q NN−2 , let (μk ) ⊂ G q be a bounded nonnegative sequence with weak∗ limit μ. For every k 1, let uk be such that

−uk + |uk |q−1 uk = μk uk = 0

in Ω, on ∂Ω.

If (μk ) is equidiffuse with respect to capW 2,q and if uk → 0 in L1 (Ω), does μ = 0? In terms of reduced limits, this problem is equivalent to the question of whether μ# = 0 implies μ = 0. More generally, we would like to know whether the measure μ is absolutely continuous with respect to the reduced limit μ# . By Theorem 1.4, if one makes the stronger assumption that (μk ) is bounded in W −2,q (Ω), then indeed μ μ# . We recall that by a result of Boccardo, Gallouët and Orsina [3] (see also [6, Theorem 4.3]) every finite measure μ in Ω, diffuse relative to capacity capH 1 , can be written as μ = f + S, where f ∈ L1 (Ω) and S ∈ H −1 (Ω). In connection with this decomposition, it would be interesting to have the following counterpart for equidiffuse sequences. Open Problem 2. Let (μk ) ⊂ M(Ω) be a bounded sequence converging weakly∗ to μ. Assume that, for every k 1, μk is diffuse with respect to capH 1 . If (μk ) is equidiffuse with respect to capH 1 , is it possible to find sequences (fk ) ⊂ L1 (Ω) and (Sk ) ⊂ H −1 (Ω) such that, for every k 1, μk = fk + Sk

in Ω,

(15.1)

where (fk ) converges strongly in L1 (Ω) and (Sk ) is bounded in H −1 (Ω)? Let q NN−2 . By a result of Baras and Pierre [2], every finite measure μ in Ω, diffuse relative to capW 2,q can be written as μ = f + S, where f ∈ L1 (Ω) and S ∈ W −2,q (Ω). One can pose a similar question with respect to this capacity: Open Problem 3. Let q NN−2 . Let (μk ) ⊂ M(Ω) be a bounded sequence converging weakly∗ to μ. Assume that, for every k 1, μk is diffuse with respect to capW 2,q . If (μk ) is equidiffuse with respect to capW 2,q , is it possible to find sequences (fk ) ⊂ L1 (Ω) and (Sk ) ⊂ W −2,q (Ω) such that, for every k 1, μk = fk + Sk

in Ω,

(15.2)

where (fk ) converges strongly in L1 (Ω) and (Sk ) is bounded in W −2,q (Ω)? If one replaces the assumption of boundedness of (Sk ) in W −2,q (Ω) by the condition that (Sk ) converges strongly in this space, then the answer is negative. In fact, if such decomposition

2370


were true, then by Theorem 12.1 we would have μ# = μ for every equidiffuse sequence, but this is impossible by Theorem 9.2. In this paper we present some conditions that assure that the reduced limit and the weak∗ limit of a given sequence (μk ) ⊂ G coincide. It would be interesting to fully investigate what happens in other cases, for instance with the sequence of convolutions (ρn ∗ μ) for some given measure μ. Open Problem 4. Given μ ∈ G and a sequence of smooth mollifiers (ρk ), let μ# be the reduced limit associated to the sequence (ρn ∗ μ). Does μ# = μ? The answer is known to be yes if g + and g − are both convex (see [6]). If the answer to Open Problem 4 is negative for some nondecreasing nonlinearity g, then is it possible to find some sequence of smooth functions (ψk ) ⊂ C ∞ (Ω) such that ∗

ψk μ weakly∗ in M(Ω), and (ψk ) possesses a reduced limit μ# equal to μ? Acknowledgments Both authors wish to thank H. Brezis for interesting discussions. The second author would like to thank the Mathematics Department of the Technion for the invitation and the warm hospitality. Appendix A. G = G0 In this appendix we prove the following result: Theorem A.1. For each nonlinearity g, let G(g) and G0 (g) be defined as in the Introduction. Then, G(g) = G0 (g). The proof is based on two lemmas. Lemma A.1. If μ ∈ G0 (g), then μ+ ∈ G0 (g) and −μ− ∈ G0 (g). Proof. First we show that μ ∈ G0 (g + ). Since u ∈ G0 (g) problem (1.2) possesses a (unique) solution u. It follows that u is a supersolution of the problem

−v + g + (v) = μ in Ω, v=0 on ∂Ω.

(A.1)

Next w be such that

−w = −μ− w=0

in Ω, on ∂Ω.

(A.2)

Then, w 0, hence g + (w) = 0. Consequently, w is a subsolution of (A.1). By [16, Corollary 5.4], this implies the existence of a solution of (A.1).


2371

Let ν ∗ denote the reduced limit of a measure ν ∈ M(Ω) relative to the nonlinearity g + (for the definition of reduced limit see [6]). Since μ μ+ it follows that μ∗ (μ+ )∗ (see [6, Proposition 4.4]). As μ ∈ G0 (g + ), μ = μ∗ . On the other hand, for any finite measure ν, ν ∗ ν. In particular (μ+ )∗ μ+ . We thus have ∗ μ = μ∗ μ+ μ+ . Since the measure (μ+ )∗ is nonnegative (see [6, Corollary 4.1]), this implies that ∗ μ+ μ+ μ+ . Thus, μ+ = (μ+ )∗ ∈ G0 (g + ). But if v is a solution of (A.1) with μ replaced by μ+ , then v is positive and consequently satisfies

−u + g(u) = μ+

in Ω,

u=0

on ∂Ω.

(A.3)

Therefore, μ+ ∈ G0 (g). Observe that the function g˜ : R → R defined by g(t) ˜ = −g(−t) is a nonlinearity possessing ˜ Hence, by the first the same properties as g. Furthermore, μ ∈ G0 (g) if and only if −μ ∈ G0 (g). ˜ which in turn implies that −μ− ∈ G0 (g). 2 part of the proof, μ− ∈ G0 (g), Lemma A.2. G0 (g) + L1 (Ω) = G0 (g). Proof. Clearly, G0 (g) + L1 (Ω) ⊃ G0 (g). In order to prove the reverse inclusion, let ν ∈ G0 (g) and f ∈ L1 (Ω). We have to show that ν + f ∈ G0 (g). Let u and v denote the solutions of (1.2) with μ = ν and μ = f respectively. If both ν and f are nonnegative, then u and v are nonnegative functions. Therefore, u and v satisfy the problem

−v + g + (v) = μ in Ω, v=0 on ∂Ω,

(A.4)

with μ = ν and μ = f , respectively. By [6, Corollary 4.7], ν + f ∈ G0 (g + ) and therefore ν + f ∈ G0 (g) since ν + f is nonnegative. Similarly, one verifies that if ν and f are nonpositive then ν + f ∈ G0 (g). In the general case, we observe that by Lemma A.1, ν + and −ν − belong to G0 (g) and therefore, by the first part of the proof, ν + + f + and −ν − − f − belong to G0 (g). Since −ν − − f − ν + f ν + + f + the existence of a solution of (A.1) for μ = ν + f follows from the existence of a supersolution and a subsolution for the problem (see [16]). 2 Proof of Theorem A.1. We only need to establish the inclusion G(g) ⊂ G0 (g). We first prove that if μ ∈ G(g) and if ϕ ∈ C0∞ (Ω) is such that 0 ϕ 1, then ϕμ ∈ G0 (g).

2372


Indeed, let u be a solution of (1.1). We first observe that |g(ϕu)| |g(u)|. Since g(u) ∈ L1loc (Ω) and ϕ has compact support in Ω, g(ϕu) ∈ L1 (Ω). Next, −(ϕu) + g(ϕu) = ϕμ + h in Ω, where h = g(ϕu) − uϕ + 2∇ϕ · ∇u + ϕg(u) . Since ϕ has compact support, h ∈ L1 (Ω). Thus, ϕμ + h ∈ G0 (g) and consequently, by Lemma A.2, ϕμ ∈ G0 . Now let (ϕk ) be a sequence of nonnegative functions in C0∞ (Ω) such that 0 ϕk 1 and ϕk 1 locally uniformly in Ω. It follows by dominated convergence that ϕk μ → μ in M(Ω). Consequently, if uk is the solution of (1.2) with μ replaced by ϕμk , then (uk ) converges in L1 (Ω) to a solution u of (1.2). Thus, μ ∈ G0 (g). 2 References [1] A. Ancona, Une propriété d’invariance des ensembles absorbants par perturbation d’un opérateur elliptique, Comm. Partial Differential Equations 4 (1979) 321–337. [2] P. Baras, M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984) 185–206. [3] L. Boccardo, T. Gallouët, L. Orsina, Existence and uniqueness of entropy solutions for nonlinear elliptic equations with measure data, Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996) 539–551. [4] H. Brezis, F.E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 5 (1978) 587–603. [5] H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. [6] H. Brezis, M. Marcus, A.C. Ponce, Nonlinear elliptic equations with measures revisited, in: J. Bourgain, C. Kenig, S. Klainerman (Eds.), Mathematical Aspects of Nonlinear Dispersive Equations, in: Ann. of Math. Stud., vol. 163, Princeton University Press, Princeton, NJ, 2007, pp. 55–110. [7] H. Brezis, A.C. Ponce, Remarks on the strong maximum principle, Differential Integral Equations 16 (2003) 1–12. [8] H. Brezis, A.C. Ponce, Kato’s inequality when u is a measure, C. R. Math. Acad. Sci. Paris Ser. I 338 (2004) 599–604. [9] H. Brezis, A.C. Ponce, Reduced measures for obstacle problems, Adv. Differential Equations 10 (2005) 1201–1234. [10] H. Brezis, A.C. Ponce, Reduced measures on the boundary, J. Funct. Anal. 229 (2005) 95–120. [11] J.K. Brooks, R.V. Chacon, Continuity and compactness of measures, Adv. Math. 37 (1980) 16–26. [12] C. de La Vallée Poussin, Sur l’intégrale de Lebesgue, Trans. Amer. Math. Soc. 16 (1915) 435–501. [13] N. Dunford, J.T. Schwartz, Linear Operators. Part I, Wiley Classics Lib., John Wiley & Sons Inc., New York, 1988. [14] L. Dupaigne, A.C. Ponce, Singularities of positive supersolutions in elliptic PDEs, Selecta Math. (N.S.) 10 (2004) 341–358. [15] M. Marcus, L. Véron, The boundary trace of positive solutions of semilinear elliptic equations: The subcritical case, Arch. Ration. Mech. Anal. 144 (1998) 201–231. [16] M. Montenegro, A.C. Ponce, The sub-supersolution method for weak solutions, Proc. Amer. Math. Soc. 136 (2008) 2429–2438. [17] G. Stampacchia, Équations elliptiques du second ordre à coefficients discontinus, Sém. Math. Supér., vol. 16, Les Presses de l’Université de Montréal, Montréal, Québec, 1966 (été 1965).


Global existence for some radial, low regularity nonlinear Schrödinger equations Benjamin Dodson University of California – Riverside, Department of Mathematics, 900 University Ave., Riverside, CA 92521, United States Received 9 June 2009; accepted 3 December 2009 Available online 23 December 2009 Communicated by I. Rodnianski

Abstract We prove the nonlinear Schrödinger equation has a local solution for any energy – subcritical nonlinearity when u0 is the characteristic function of a ball in Rn . Additionally, we establish the existence of a global 2 < α < 4 and α 2. Finally, we establish the existence of a global solution solution for n 3 when n−1 n−2 when the initial function is radial, the nonlinear Schrödinger equation has an energy subcritical nonlinearity, and the initial function lies in H ρ+ (Rn ) ∩ H 1/2+ (Rn ) ∩ H 1/2+,1 (Rn ). © 2009 Elsevier Inc. All rights reserved. Keywords: Nonlinear Schrödinger equation; Partial differential equations; Gibbs phenomenon; Harmonic analysis

1. Introduction The nonlinear Schrödinger equation with power-type nonlinearity, iut + u = ±|u|α u, u(0) = u0 ,

(1.1)

is called defocusing if the sign is + and focusing if the sign is −. One solution to (1.1) gives, in fact, an entire class of solutions due to scaling. If u(t, x) is a solution on some interval [T− , T+ ] ⊂ R then E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.002

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B. Dodson / Journal of Functional Analysis 258 (2010) 2373–2421

λ2/α u λ2 t, λx 4 n−2ρ ,

0 ρ < n2 , the H˙ ρ norm is

= u(x)H˙ ρ (Rn ) .

(1.2)

is also a solution on the interval [λ−2 T− , λ−2 T+ ]. When α = invariant under the scaling n/2−ρ λ u(λx)

H˙ ρ (Rn )

Therefore (1.1) is called an H˙ ρ (Rn )-critical nonlinear Schrödinger equation. Remark. Throughout this paper, the spaces H˙ σ (Rn ), H σ (Rn ) refer to the L2 -based Sobolev spaces H˙ σ,2 (Rn ) and H σ,2 (Rn ). For Lp Sobolev spaces, p = 2, we will write H σ,p (Rn ). Lemma 1.1. If u0 ∈ H˙ ρ (Rn ) for some 0 ρ 1, n 3, then (1.1) has a solution for some interval [0, T ], where T (u0 ) > 0. Remark. This was first proved in [3]. Remark. (1.1) is H˙ ρ -critical, then (1.1) fails to be locally well-posed for u0 ∈ H s (Rn ), s < ρ. See [5,4]. Nevertheless, if u0 ∈ H s (Rn ) for s < ρ but u0 also satisfies certain other conditions, then a local solution will exist. Let Ω be a smoothly bounded, compact region in Rn , χΩ is the characteristic function of the region. Theorem 1.2. If u0 = χΩ , when n = 1, 2, there exists a T∗ > 0 such that iut + u = F (u), u(0, x) = u0 (x) = χB(0;1) ,

(1.3)

has a solution in the function space uX = sup 0tT∗

u(t, ·)

H σ (Rn )

+ u(t, ·)L∞ (Rn ) ,

(1.4)

when the nonlinearity F : C → C obeys F (0) = 0, DF (0) = 0, and F ∈ C ∞ . Proof. See [22].

2

In particular, for any nonlinearity of the form F (u) = |u|2k u there will be a local solution. This is despite the fact that when n = 2, iut + u = |u|2k u is H˙ 1−1/k (R2 )-critical, while χΩ ∈ H 1/2− (R2 ) only. In Section 3 we will partially extend this result to higher dimensions. Let u0 = χB(0;1) , B(0; 1) is a ball of radius 1 centered at the origin in Rn .


2375

Theorem 1.3. There exists T∗ (α, n) > 0 such that the nonlinear Schrödinger equation iut + u = |u|α u, u0 = χB(0;1) ,

(1.5)

p n L∞ t Lx [0, T∗ ] × R

(1.6)

has a local solution in

when p = 2 + α, 0 α
2,

ξ eix·ξ dξ, uˆ 0 (ξ )φ N

uh (x) = u0 (x) − ul (x). Then let v(t, x) be the solution to the equation ivt + v = |v|2 v, v(0, x) = ul (x).

(1.11)

The energy E(v(t)) N 2(1−s) u0 H s (Rn ) is conserved, E v(t) N 2(1−s) u0 2H s . Then, treating the remainder as a perturbation of the original equation, iwt + w = 2|v|2 w + v 2 w¯ + 2|w|2 v + w 2 v¯ + |w|2 w, w(0, x) = uh (x).

(1.12)

The solution of (1.12) on [0, T ], T ∼ N −2(1−s) , is of the form eit uh + z(t, x), where z(t, x)H 1 is small due to the fact that uh L2 (R3 ) N −s u0 H s (R2 ) . Adding z(T , x) to v(T , x) and repeating the process, [1] was able to control the H 1 (R2 ) norm of z(t, x) for s > 3/5. This inspired the introduction of the I-method in [8]. If u(t) solves (1.1), then I u(t) solves the equation iI ut + I u = I |u|α u .

(1.13)

Where I is a smooth Fourier multiplier, I u(ξ ) = mN |ξ | u(ξ ˆ ), 1, |ξ | N ; mN |ξ | = s−1 |ξ | , |ξ | 2N ,

(1.14) (1.15)

I u0 H 1 (Rn ) CN 1−s u0 H s (Rn ) , so E(I u0 ) < ∞. However, the new difficulty is that since I (|u|α u) = |I u|α (I u), d dt E(I u(t)) = 0. Therefore, to prove global existence one must endeavor to control the modified energy


1 E I u(t) = 2

∇I u(t, x)2 dx +

1 2+α

2377

I u(t, x)2+α dx.

(1.16)

At this point, the best that can be hoped for is some sort of bound of the form E(I u(t, x)) CT β(s) . Currently the best results are global well-posedness for s > 1/3 for iut + u = |u|2 u

(1.17)

in R2 (see [7]) and s > 4/5 in R3 (see [9]). The I-method was also modified in [11], making a resonant decomposition, but we will not discuss that here. For the L2 -critical nonlinear Schrödinger equation iut + u = |u|4/n u,

(1.18) √

√

−(n−2)+

(n−2)2 +8(n−2)

when n = 3, and s > for there is global well-posedness for s > 7−1 3 4 n 4 (see [12]). In this paper, a method similar to Fourier truncation is used. For certain initial data, the Duhamel term smooths the local solution, and it is of the form u(t, x) + v(t, x),

(1.19)

1 n where v(t, x) ∈ L∞ t ([0, T ], H (R )). By [2] (1.1) has a global solution when u0 = v(T , x). i(t−T ) If e u(T , x) has nice asymptotic behavior then (1.1) with u0 = u(T , x) + v(T , x) can be treated as a perturbation of (1.1) with u0 = v(T , x). There are two main advantages to this method. First, for certain initial data it is possible to obtain global existence of a solution for supercritical data. In Section 6 we will prove

Theorem 1.5. (1.1) has a global solution for u0 = χB(0;1) and Remark. The case when α =

4 n

4 n

0, ρ it |∇| e u0

2(n+2) Lt,x n ([0,T ]×Rn )

+ |∇|ρ eit u0

α(n+2) p 2 Lx ([0,T ]×Rn )

.

(2.9)

Lt

Then, by applying the continuity method to (2.8), we have ρ |∇| u

2(n+2) Lt,x n ([0,T ]×Rn )

+ |∇|ρ u

α(n+2) p 2 Lx ([0,T ]×Rn )

.

(2.10)

Lt

If u0 H˙ ρ (Rn ) is small, then (2.9) is true for (−∞, ∞).

2

Lemma 2.3. There exists a T (u0 H ρ+ (Rn ) ) > 0 such that (1.1) has a local solution for [0, T ]. Proof. By the Sobolev embedding theorem, ρ it |∇| e u0

2n/(n−2)

L2t Lx

([0,T ]×Rn )

for some admissible pair (p, q) with q1 = embedding theorem imply (2.11). 2

T δ(,n) |∇|ρ+ eit u0 Lp Lq ([0,T ]×Rn ) , t

n−2 2n

x

(2.11)

+ n . Then Hölder’s inequality and the Sobolev

The radial Sobolev embedding will also be needed later in the paper. This lemma is not new. We prove it here because we will make arguments similar to this proof elsewhere in the paper.

2380


Lemma 2.4. For u radial, (n−1)/2 |x| u

L∞ (Rn )

uH 1/2+,2 (Rn ) .

(2.12)

Proof. This is the standard Sobolev estimate when n = 1. Take n 2. By a change of variables, e

ix·ξ

fˆ |ξ | dξ =

|ξ |1/|x|

∞

π/2

fˆ(r)r n−1

n−2 e−2ir|x| sin(θ) cos(θ ) dθ dr

−π/2

1/|x|

(letting u = sin(θ )) ∞ =

fˆ(r)r n−1

1

(n−3)/2 e−2ir|x|u 1 − u2 du dr.

−1

1/|x|

If n−3 2 is an integer, then we integrate by parts f (u) ∈ C ∞ ([−1, 1]). 1 e

−2ir|x|u

k 1 − u2 f (u) du =

−1

1 −2ir|x|

−1 = −2ir|x|

n−3 2

1 −1

1 −1

times. Suppose k 1 is an integer and

k d −2ir|x|u e 1 − u2 f (u) du du −2ir|x|u d k 1 − u2 f (u) du. e du

The boundary term disappears when k 1 since (1 − u2 ) = 0 at ±1. Furthermore, k k−1 d 1 − u2 f (u) = 1 − u2 g(u), du where g(u) ∈ C ∞ ([−1, 1]). Therefore, by induction, if n is odd, n 3, 1 e

−2i|x|ru

−1

Integrating

1

−1 e

∞ 1/|x|

n−3 C( n−3 2 ) 1 − u2 2 du = n−3 (|x|r) 2

−2ir|x|u f (u) du

fˆ(r)r n−1

1 e

(2.13)

1 e

−2i|x|ru

−1

d du

n−3 2

n−3 1 − u2 2 du.

by parts one more time,

−2ir|x|u

(n−3)/2 1 − u2 du dr C

−1

Similarly when n is even, integrate by parts

∞

1/|x| n−2 2

r (n−1)/2 fˆ(r) (n−1)/2 dr. |x|

times, reducing to an integral of the form


C(n) |x|

n−2 2

r

n−2 2

1

−1/2 e−2ir|x|u 1 − u2 g(u) du.

−1

To control this term, we use the stationary phase-type estimate once again we have ∞ (2.13) C 1/|x|

∞ C() |x|

n−1 2

1 0

x −1/2 eiAx dx = O(A−1/2 ). So

r (n−1)/2 fˆ(r) (n−1)/2 dr |x| fˆ(r)r (n−1)/2 1 + r 1/2+ 2 dr

1/2

1/|x|

2381

−2 1 + r 1/2+ dr

1/2

1/|x|

f H 1/2+ (Rn ) .

By simple integration, n−1 f L2 (Rn ) = C(n)f (r)r 2 L2 (R) . To estimate the |ξ |

1 |x|

term,

1/|x| 1/2 1/2 2 n−1 n−1 n−1 ˆ ˆ f (r) r dr r dr f (r)r 0

1 |ξ | |x|

1/|x|

1 |x|(n−1)/2

f H 1/2 (Rn ) .

2

Remark. Lemmas 2.2 and 2.4 are not new. See [3,15] respectively. Lemma 2.5. For any fixed K > 0, A > 1, K

eiAx x −1/2 dx = O A−1/2 .

0

Proof. Make a change of variables. Let x 1/2 = u, x −1/2 dx = 2 du. K 0

Integrate via a contour integral.

1 eiAx x −1/2 dx = 2

K 2 2 eiAu du. 0

(2.14)

2382

B. Dodson / Journal of Functional Analysis 258 (2010) 2373–2421 K2 K 2 K 2 √ iA((1+i)u)2 2 iAu2 2 e du = e du + eiA(K +iu) du 0

0

0

√ 2 2 2 = 2 e−2Au du = eiAu du + eiAK e−Au du. K2

K2

K2

0

0

0

Therefore, by algebra, K 2 2 eiAu du = O A−1 + O A−1/2 . 0

2

See [18] for more details on the stationary phase method. 3. Proof of Theorem 1.3

In this paper, Ω will always refer to a compact region in Rn bounded by a smooth manifold. Proof of local existence in [22] for n = 1, 2 relied heavily on the bound it e χΩ

L∞ t,x ([0,∞))

C.

(3.1)

Indeed, the only restriction placed on the nonlinearity F (u) was that F ∈ C ∞ (C), F (0) = F (0) = 0. When n 3, this bound is not available due the Pinsky phenomenon or perfect focus caustic [22]. Take u0 = χB(0;1) .

e

it

χB(0;1) (0) = Ct

−n/2

1 r

n−1 ir 2 /4t

e

dr = Ct

−n/2

0

1 u(n−2)/2 eiu/t du, 0

after letting u = r 2 and du = 2r dr.

Ct

−n/2

1 u(n−2)/2 eiu/t du 0

1 = Ct 1−n/2 eiu/t u(n−2)/2 0

1 − C1 t

1−n/2

eiu/t u(n−4)/2 du. 0

If n 4, then integrating by parts,

(3.2)


1 t

1−n/2

e

iu/t (n−4)/2

0

u

2383

1 d (n−4)/2 t 2−n/2 iu/t (n−4)/2 1 t 2−n/2 u du du = − eiu/t e u i i du 0 0

= O t 2−n/2 .

If n = 3, then by Lemma 2.5,

t

−1/2

1

u−1/2 eiu/t du = O(1).

0

Therefore, (3.2) = Ct

1−n/2 i/t

e

+

O(1),

if n = 3;

O(t 2−n/2 ),

if n 4.

(3.3)

So for n > 2, (3.3) is not bounded as t 0. Remark. The Pinsky phenomenon can also be observed in pointwise Fourier inversion. See [17], for example. Instead we rely on the bound it e χΩ

n L∞ t Lx ([0,∞)×R ) p

C,

for some other p(n). Because p(n) < ∞ for n 3, it will be necessary to be more restrictive of the types of nonlinearities used, namely H˙ 1 -subcritical power-type nonlinearities. Lemma 3.1. p n eit χB(0;1) ∈ L∞ t Lx [0, ∞) × R for 2 p
t,

2n it e χB(0;1) (x) n−2 dx Ct 1/2

|x|t

1 r

1

n−1

2n n−2

r (n−1)/2

C .

t

For |x| > 1, eit χB(0;1) is bounded, so the bound for p = 2 lating between L∞ x and Lx on |x| > 1.

2n n−2

on |x| > 1 follows from interpo2n

2 it χ ∞ n−2 proves the lemma Interpolating between eit χB(0;1) ∈ L∞ B(0;1) ∈ Lt Lx t Lx and e when n is odd. When n is even,

it e χB(0;1) (x)p dx C

B(0;1)

B(0;1)

1

p dx,

|x|(n−2)/2

2n which is bounded for p < n−2 . Meanwhile, on |x| > 1, eit χB(0;1) is uniformly bounded in t, so ∞ 2 interpolating with the Lt Lx bound proves the lemma in the even case also. 2

Armed with this lemma, it is possible to prove a local existence theorem. Theorem 3.2. If α
0 such that iut + u = |u|α u, u0 = χB(0;1) ,

(3.17)

has a local solution in 2+α [0, T∗ ] × Rn . L∞ t Lx Proof. Set p = 2 + α. p 0. So if u0 is not continuous, then it is impossible for eit u0 to converge to u0 uniformly. This convergence must in fact fail in a neighborhood of a point x ∈ Rn , if u0 is discontinuous at x. The Gibbs phenomenon is the failure of this uniform convergence. Remark. Uniform convergence can also fail in a neighborhood of a point when u0 is C ∞ in a neighborhood of that point. See the Pinsky phenomenon (3.3) in the previous section. Theorem 4.1. The solution u : R3 → C to the nonlinear Schrödinger equation iut + u = |u|2 u, u0 = χB(0;1) ,

(4.1)

has a solution on some time interval [0, T ], T > 0 of the form eit u0 + w(t, x),

(4.2)

1 3 where w(t, x) ∈ L∞ t Hx ([0, T ] × R ).

Proof. If u(t, x) solves (4.1) then t u(t, x) = e

it

χB(0;1) − i

ei(t−s) F u(s) ds,

(4.3)

0

where F (u) = |u|2 u. Let w(t, x) = u(t, x) − eit χB(0;1) , t w(t, x) =

ei(t−s) F eis χB(0;1) + w(s) ds.

(4.4)

0

w is a fixed point of the map t Φw(t, x) =

ei(t−s) F eis χB(0;1) + w(s) ds.

(4.5)

0

We aim to show that for some T > 0, Φ : XT → XT is a contraction on a suitable space of

2390


functions 1 3 X T ⊂ L∞ t H [0, T ] × R .

(4.6)

We will take XT ⊂ S 1 ([0, T ] × R3 ), the inhomogeneous Strichartz space. uS 1 (I ×Rn ) = uS 0 (I ×R3 ) + ∇uS 0 (I ×R3 ) .

(4.7)

By the dual Strichartz estimates, t i(t−τ ) F u(τ ) dτ e 0

C F (u)

S 1 ([0,T ]×R3 )

N 0 ([0,T ]×R3 )

+ C ∇F (u)N 0 ([0,T ]×R3 ) .

(4.8)

Let An be the set of admissible pairs, (p, q) ∈ An

⇔

p 2,

1 1 2 =n − , p 2 q

(4.9)

and A n the set of Lebesgue duals to admissible pairs. Define a sequence of functions, w0 (t, x) = 0 and wm (t, x) = Φwm−1 (t, x). wm (t, x)

S 1 ([0,T ]×R3 )

t 2 = ei(t−s) eis u0 + wm−1 (s, x) eis u0 + wm−1 (s, x) ds

S 1 ([0,T ]×R3 )

0

t 2 = ei(t−s) ∇ eis u0 + wm−1 (s, x) eis u0 + wm−1 (s, x) ds 0

S 0 ([0,T ]×R3 )

t 2 is i(t−s) is e u0 + wm−1 (s, x) e u0 + wm−1 (s, x) ds + e 0

. S 0 ([0,T ]×R3 )

(4.10) The term with a gradient is more difficult, so that is the term that will be computed explicitly here. Split (4.10) into six terms. is ∇ e χΩ + wm−1 (s, x) eis χΩ + wm−1 (s, x)2 2 ∇eis χΩ eis χΩ + ∇eis χΩ eis χΩ wm−1 (s, x) 2 2 + ∇eis χΩ wm−1 (s, x) + eis χΩ ∇wm−1 (s, x) 2 + eis χΩ wm−1 (s, x)∇wm−1 (s, x) + wn−1 (s, x) ∇wm−1 (s, x),

(4.11)


(4.11) =

6

2391

(4.12)

vj (s),

j =1

where v1 (s) is the first term, v2 (s) is the second, and so forth. Additionally, let η be a C ∞ cutoff, η=

1,

|x| 2;

0,

|x| > 3,

(4.13)

vj (s) = vj 1 (s) + vj 2 (s) = ηvj (s) + (1 − η)vj (s).

(4.14)

XT = w ∈ S 1 [0, T ] × Rn : wS 1 ([0,T ]×Rn ) C ,

(4.15)

Define the set

for some C > 0 to be determined later. To prove Φ : XT → XT it suffices to prove that for some (pj k , qj k ) ∈ A n , vj k S 1 ([0,T ]×Rn ) Cj k T a ,

(4.16)

for some a > 0. From the previous section we know that in three dimensions p n eit χB(0;1) ∈ L∞ t Lx R ,

(4.17)

for 2 p 6. We also know that by (3.13),

1/2 it e χB(0;1) (x) C 1 + t , |x| so in fact p n 1 − η(x) eit χB(0;1) ∈ L∞ t Lx R ,

(4.18)

for 2 p ∞. The derivative of χB(0;1) is a finite measure supported on {|x| = 1}. d it 2 e χB(0;1) = Ct −3/2 ei|x| /4t ei/4t dr

π/2

e−2i|x|r sin(θ)/4t cos(θ ) dθ,

−π/2

after a change of variables u = sin θ , = Ct −3/2 ei|x|

2 /4t

1 ei/4t

e−2i|x|u/4t du

−1

= Ct −1/2 |x|−1 ei|x|

2 /4t

−2i|x|/4t e − e2i|x|/4t

1 . |x|t 1/2

2392


Term 1. Take q = 3/2 − , p =

12−8 9−11 .

v11 (s, x) p q Ls Lx ([0,T ]×R3 ) −1/2 (12−8)/(9−11) η|x|−1 3− 3 eis χB(0;1) 2 6 3 C()t L (R ) L (R ) L ([0,T ]) t

3−7 12−8

C ()T

.

Meanwhile (1 − η(x))|x|−1 ∈ Lp (R3 ) for 3 < p ∞, so v12 (s, x) 1 2 eis u0 2 ∞ 6 (1 − η) ∇eis u0 1 6 L L L L L L ([0,T ]×R3 ) t

t

x

T C

x

s

x

t −1/2 dt = CT 1/2 .

0

Term 2. In this term use the Sobolev embedding H 1 (R3 ) ⊂ L6 (Rn ), and perform the same estimates as in term one, only replace one of the eis χB(0;1) with wn−1 (s, x). Again take q = 3/2 − , p = 12−8 9−11 . v21 (s, x)

v22 (s, x)

p

q

Ls Lx ([0,T ]×R3 )

L1t L2x

C()s −1/2 L(12−8)/(9−11) ([0,T ]) η|x|−1 L3− (R3 ) s × eis χB(0;1) L6 (R3 ) wm−1 (s, x)L∞ H 1 (R3 ) t 3−7

C ()T 12−8 wm−1 (s, x) S 1 ([0,T ]×R3 ) ,

eis u0 L∞ L6 wm−1 (s, x)L∞ H 1,2 (R3 ) ∇eis u0 L1 L6 ([0,T ]×R3 ) t

T C

t

x

s

x

t −1/2 dt = CT 1/2 wm−1 (s, x)S 1 ([0,T ]×R3 ) .

0

Term 3. Once again take q = 3/2 − , p = v31 (s, x)

p

12−8 9−11 .

q

Ls Lx ([0,T ]×R3 )

2 C()s −1/2 L(12−8)/(9−11) ([0,T ]) η|x|−1 L3− (R3 ) wm−1 (s, x)L∞ H 1 ([0,T ]×R3 ) s t x 2 3−7

C ()T 12−8 wm−1 (s, x)S 1 ([0,T ]×R3 ) , is v32 (s, x) 1 2 wm−1 (s, x)2 ∞ 1 ∇e u0 1 6 L H ([0,T ]×R3 ) L L L L ([0,T ]×R3 ) t

t

x

T C

x

s

2 t −1/2 dt wm−1 (s, x) ∞

Lt Hx1 ([0,T ]×R3 )

0

2 CT 1/2 wm−1 (s, x)S 1 ([0,T ]×R3 ) .

x


2393

Term 4. Take q = 2 and p = 1. v4 (s, x)

p q Ls Lx ([0,T ]×R3 )

2 CT 1/2 ∇wm−1 L2s L6x ([0,T ]×R3 ) eis L∞ L6 s

CT

1/2

x

wm−1 S 1 ([0,T ]×R3 ) .


p

q

Ls Lx ([0,T ]×R3 )

CT 1/2 ∇wm−1 L2s L6x ([0,T ]×R3 ) eis L∞ L6 wm−1 L∞ 1 3 s Hx ([0,T ]×R ) s

CT

1/2

x

wm−1 2S 1 ([0,T ]×R3 ) .


L1s L2x ([0,T ]×R3 )

2 = ∇wm−1 (s, x)wm−1 (s, x) L1 L2 ([0,T ]×R3 ) s

CT

1/2

x

wm−1 3S 1 ([0,T ]×R3 ) .

Adding all this together gives the estimate, wm (t, x)

S 1 ([0,T ]×R3 )

1 + wm−1 S 1 ([0,T ]×R3 ) + wm−1 2S 1 ([0,T ]×R3 ) + CT 1/2 wm−1 S 1 ([0,T ]×R3 ) + wm−1 2S 1 ([0,T ]×R3 ) + wm−1 3S 1 ([0,T ]×R3 ) .

CT

3−7 9−11

(4.19)

Let X = w: wS 1 ([0,T ]×R3 ) 2CT

3−7 9−11

.

(4.20)

For T sufficiently small, Φ : X → X. Contraction. Using the same calculations suppose there are two sequences starting with w0 and w˜ 0 , wm (t, x) − w˜ m (t, x)

S 1 ([0,T ]×R3 )

2 3−7 CT 9−11 1 + wm−1 (t, x) + w˜ m−1 (t, x)S 1 ([0,T ]×R3 ) × wm−1 (t, x) − w˜ m−1 (t, x)S 1 ([0,T ]×R3 ) ,

(4.21)

which gives a contraction for T sufficiently small. This proves the existence of a solution of the form (4.2). 2

2394


The Gibbs phenomenon. Now we examine the Gibbs phenomenon for the solution to the nonlinear Schrödinger equation (4.1), that is, what is the behavior of the solution to (4.1), u(t, x) as t 0? The Gibbs phenomenon arising at the edges of B(0; 1) in the free solution eit u0 as t 0 is well-understood (see for example [21,23]). Now, take the cutoff χ=

0,

|x| 12 ;

1,

|x| > 3/4.

(4.22)

By Lemma 2.4, if u is a radial function, χ(x)u(x)

L∞ x

CuH 1 ,

(4.23)

because u is a radial function. Combining (2.12), and (4.19) gives a uniform estimate for the Gibbs phenomenon near the boundary. 3−7 χ(x)u(t, x) − χ(x)χB(0;1) = χ(x)eit χB(0;1) − χ(x)χB(0;1) + O t 9−11 .

(4.24)

Similarly, Lemma 4.2. iut + u = |u|α u, u(0, x) = χB(0;1) ,

(4.25)

has a local solution of the form eit χB(0;1) + w(t, x) on some time interval [0, T ], T > 0, where w(t, x)

2

S 1 ([0,T ]×R3 )

C(δ)T (1− 5−α −δ)(

5−α− 4 )

,

(4.26)

C(δ) ∞ as δ → 0 for 2 α < 3. 4 , so by Theorem 3.2, there is an interval [0, T ] such that u(t, x) ∈ Proof. When n = 3, α < n−2 p ∞ 3 Lt Lx ([0, T ] × R ) for 2 p < 6.

∇|u|α u

N 0 ([0,T ]×R3 )

Let

1 p

=

α 6−

+

C |∇u||u|α N 0 ([0,T ]×R3 ) .

1 3− .

η(x)∇eit u0 u(t, x)α

p

Lx (R3 )

Let

2 q

= 3( p1 − 12 ), 2 −

2 q

= 3( p1 − 12 ).

Ct −1/2 uαL∞ Lp ([0,T ]×R3 ) . t

x


T

2395

−1/2 q 2 t = C(δ, α)T 1− 5−α −δ ,

0

T

−1/2 q t

1/q 2

= C(δ, α)T (1− 5−α −δ)(

5−α− 4 )

.

0

As in the case of α = 2, the terms 1 − η(x) ∇eit u0 u(t, x)α

p

Lx (R3 )

and ∇w(t, x)u(t, x)α

p

Lx (R3 )

are better behaved.

2

Thus, for t ∈ [0, T ], |x| > 3/4, 2 −δ u(t, x) − u0 (x) = eit u0 (x) − u0 (x) + O t 1− 5−α .

(4.27)

Higher dimensions. Now consider u0 = χB(0;1) for higher dimensions. As the dimensions increase the blowup at the origin becomes worse and worse. d it 2 e u0 = t −n/2 ei|x| /4t dr

1

(n−3)/2 e−2i|x|u/4t 1 − u2 du,

−1

it ∇e u0 t −1/2 |x|−(n−1)/2 .

(4.28)

1 n By the Strichartz estimates, to place the remainder term in L∞ t Hx ([0, T∗ ] × R ), it suffices to have 2nα

3 eit χB(0;1) ∈ L∞ t Lx

To do this, we need Lemma 4.3. If

2 n

2nα 3

n−1 i(t+τ ) ∇e χB(0;1)

Lp (Rn )

n( p1 − 12 )

C(p, n)(t + τ )

(4.36)

.

Then, as in Theorem 4.1, ∇v(t, x)

S 0 ([0,T∗ ]×Rn )

2 vi (s, x) i=1

p

q

Lt i Lxi ([0,T∗ ]×Rn )

(4.37)

,

α v1 (s, x) = ∇ei(t+τ ) χB(0;1) ei(t+τ ) χB(0;1) + eit w(τ, x) + v(s) , α v2 (s, x) = ∇eit w(τ, x) + ∇v(s, x) ei(t+τ ) χB(0;1) + eit w(τ, x) + v(s) . From (1.9), (4.33) conserves the L2 norm of a solution. If v1 (s, x) 1 2 L L ([0,T t

x

∗ ]×R

n)

1 n

< α 1, take p =

(4.38)

2 1−α ,

T∗ ∇ei(t+τ ) χB(0;1) L∞ Lp χB(0;1) αL2 (Rn ) t

x

C(τ, α, n)T∗ . If 1 < α
2.

(6.12)

We now prove a local well-posedness result. Lemma 6.2. Suppose τ T∗ (α) and T∗ (α)

u0 = e

iτ

α ei(τ −s) χ u(s) u(s) ds + v0 ,

χB(0;1) +

(6.13)

0

u is the solution on [0, T ] found in Lemma 6.1, and v0 ∈ H 1 (Rn ). There exists T0 (v0 H 1 (Rn ) , T∗ (α)) such that iut + u = |u|α u, u(0, x) = u0 , has a solution on [0, T0 ] of the form

(6.14)

2404


eit u0 + v(t, x), v(t, x) ∞ 1 C v0 H 1 (Rn ) , T∗ (α) . L H ([0,T ]×Rn ) t

(6.15) (6.16)

0

x

Proof. We follow the by now familiar procedure. t v(t, x) =

α ei(t−s) eis u0 + v(s, x) eis u0 + v(s, x) ds.

0

Using Strichartz estimates, v(t, x) 1 S ([0,T0 ]×Rn ) α δ(α) T0 ∇v Lp L2+α ([0,T ]×Rn ) eis u0 + v(s, x)L∞ L2+α ([0,T ]×Rn ) 0 0 x t t x α is δ(α) it ∇e u0 p 2+α e u0 + v(s, x) ∞ 2+α +T n 0

Lt Lx

([0,T0 ]×R )

Lt Lx

([0,T0 ]×Rn )

.

This is implied by the following estimates: it iτ e e χB(0;1)

2+α n L∞ t Lx ([0,T0 ]×R )

T∗ (α) α it i(τ −s) e χ u(s) u(s) ds e 0 T∗ (α)

0

C,

2+α n L∞ t Lx ([0,T0 ]×R )

C ds C , (t + τ − s)r

since r < 1. v(t, x) + eit v0

vS 1 ([0,T ]×Rn ) + v0 H 1 (Rn ) ,

2+α n L∞ t Lx ([0,T0 ]×R )

by the Sobolev embedding theorem. Furthermore, ∇v(t, x) + ∇eit v0

p

n Lt L2+α x ([0,T0 ]×R )

T∗ (α) α ei(t+τ −s) u(s) u(s) ∇ 0

CvS 1 ([0,T ]×Rn ) + Cv0 H 1 (Rn ) ,

T∗ (α)

n L2+α x (R )

0

C uX ([0,T ]×Rn ) ds C , (t + τ − s)r

again since r < 1. Finally, it iτ ∇e e χB(0;1)

n L2+α x (R )

This completes the proof of the lemma.

2

C(τ ).


2405

Set up a system of equations ivt + v = 0, iwt + w = |v + w|α (v + w), T∗ (α)

v(0) = e

iT

α ei(T −s) χ u(s) u(s) ds,

χB(0;1) + 0

T e

w(0)

i(T −s)

α u(s) u(s) ds +

T∗

T∗ (α)

α ei(T −s) (1 − χ)u(s) u(s) ds.

(6.17)

0

Define the energy E(t) 2 2 v(t, ·) + w(t, ·)2+α E(t) = ∇w(t, ·)L2 (Rn ) + n , L2+α x (R ) x 2+α d E(t) C |∇v||v + w|α |∇v| + |∇w| dx. dt

(6.18)

Estimate the terms separately. Term 1.

∇v(x, t)2 v(x, t) + w(x, t)α dx 2 α ∇v(x, ·)L2+α (Rn ) v(x, t) + w(x, t)L2+α (Rn ) ,

(6.19)

i(t+T ) ∇e χB(0;1)

(6.20)

n L2+α x (R )

T∗ (α) α ei(t−s) χ u(s) u(s) ds ∇ 0

T∗ C 0

C , (t + T )1/2+(α)

n L2+α x (R )

1 ds ∇uX ([0,T∗ (α)]×Rn ) uL∞ L2+α ([0,T∗ ]×Rn ) x t (t + τ − s)r

C . (1 + t)r

(6.21)

Term 2.

∇v(x, t)∇w(x, t)v(x, t) + w(x, t)α dx α ∇v(·, t)Lp (Rn ) ∇w(·, t)L2 (Rn ) v(·, t) + w(·, t)L2+α (Rn ) ,

(6.22)

2406


where p = measure,

2−α 2(2+α) . Using the dispersive estimates, combined with the fact that ∇χB(0;1)

it iτ ∇e e χB(0;1)

n L∞ x (R )

C . (t + T )n/2

is a finite

(6.23)

Also, T∗ (α) α i(t+τ −s) u(s) u(s) ds e ∇ 0

n L∞ x (R )

C ∇u(s)χ u(s)α 1 Lt,x ([0,T∗ (α)]×Rn ) (t + τ − T∗ (α))n/2 C ∇uX ([0,T∗ (α)]×Rn ) uL∞ L2+α ([0,T∗ (α)]×Rn ) , x t (t + τ − T∗ (α))n/2

(6.24)

by Hölder’s inequality. Lemma 6.3. w(0) ∈ H 1 Rn . Proof. We have already proved T

α ei(T −s) u(s) u(s) ds ∈ H 1 Rn .

T∗ (α)

Now, we combine u(t) ∈ L∞ t Hx

1/2−

([0, T∗ (α)] × Rn ) with the radial Sobolev embedding

(n−1)/2 |x| u

n L∞ x (R )

CuH 1/2+ (Rn )

(6.25)

to prove (1 − χ)u

n L∞ t Lx ([0,T∗ (α)]×R ) p

C(p)uL∞ H 1/2+ ([0,T t

x

∗ (α)]×R

(6.26)

n)

for any p < ∞, if > 0 is sufficiently small. ∇(1 − χ)u(s)α u(s) 1 2 Lt Lx ([0,T∗ (α)]×Rn ) α CT∗ (α)δ(α) (1 − χ)uL∞ Lp ([0,T (α)]×Rn ) ∇uX ([0,T∗ (α)]×Rn ) . t

x

∗

2

Thus we have proved d E(t)α/2+1/2 E(t)α/2 E(t) C + s dt (T + t) 1 (T + t)s2

(6.27)


2407

and furthermore E(0) is bounded. s1 , s2 > 1 and by Gronwall’s inequality E(t) is bounded on [T∗ , ∞). This proves a global solution to (6.1) exists. 2 A digression to the heat equation. This method can be applied to a Schrödinger equation that has a damping term with ease. (a + ib)ut = u,

(6.28)

where a 0 and |a + ib| = 1. The solution is the Fourier multiplier 2 et/(a+ib) u0 = F −1 e−t|ξ | /(a+ib) uˆ 0 (ξ ) .

(6.29)

u(x, t) = K(x, y, t) ∗ u0 (y),

(6.30)

In other words

K(x, y, t) =

− ib)n/2

(−a (4πt)n/2

e

2 (a+ib) |x−y| 4t

(6.31)

.

This operator obeys the operator bounds et(a−ib) : t n/2 L1 → L∞ , et(a−ib) : L2 → L2 .

(6.32)

This equation obeys the same Strichartz estimates as the Schrödinger equation. Theorem 6.4. Suppose (p, q) are admissible pairs. e−isL F (s, ·) ds C(n, q, p)F 2

R

q

p

Lt Lx (R×Rn )

(6.33)

.

Definition 6.1. (p, q) is an admissible pair for n if p > 2, q > 2, and

1 1 2 =n − . q 2 p

(6.34)

Now consider the nonlinear equation with power-type nonlinearity. (a + ib)ut + u = |u|α u, 4 . When u0 ∈ H ρ+ (Rn ) for α = with |a + ib| = 1 is H˙ ρ (Rn )-critical for α = n−2ρ exists T (u0 H ρ+ (Rn ) > 0 such that a solution to (6.35) exists on [0, T ).

(6.35) 4 n−2ρ

there

Theorem 6.5. When a < 0, (6.35) has a global solution u(t). u(t)

H ρ+ (Rn )

F t, u0 H ρ+ (Rn ) , a .

(6.36)

2408


Proof. Apply the same expansion method as was used for the Schrödinger equation (a = 0). For some uniform constant C, a σ/2 t σ/2 |ξ |σ e−at|ξ | C(σ ). 2

(6.37)

Thus the Strichartz estimates hold ρ+ −t/(a+ib) ∇ e u0

2n/(n−2) Cu0 H˙ ρ+ , L2t Lx ρ++δ −t/(a+ib) ∇ e u0 2 |a|−δ/2 t −δ/2 u0 H˙ ρ+ .

(6.38)

Meanwhile by the Sobolev embedding −t/(a+ib) 4/(n−2ρ) e u0

L∞ t Lx

where

1 q

=

4 2n

−

4 n(n−2ρ) .

Let

1 p

=

2n n−2

+

4 n(n−2ρ) .

σ −t/(a+ib) ∇ e u0

q

1 q

=

1 2

−

2 n−2ρ ,

(6.39)

If

q

p

Lt Lx

for

u0 H ρ+ ,

u0 H˙ ρ+ ,

(6.40)

then the Duhamel term lies in H σ (Rn ). By interpolation, ρ++δ −t/(a+ib) ∇ e u0

q

p

Lt Lx

u0 H˙ ρ+ ,

(6.41)

for some δ() > 0, δ c for some c > 0. Thus expressing the solution in Duhamel’s equation u(t) = e

−t/(a+ib)

t u0 +

α e−(t−s)/(a+ib) u(s) u(s) ds,

(6.42)

0

the second term will be smoothed, belonging to H ρ++δ (Rn ) for some δ > c. But the first term also belongs to H ρ++δ (Rn ) by (6.38) on [T /2, T ). Thus u(T /2) ∈ H ρ++δ (Rn ). We can iterate this procedure, obtaining a smoother and smoother solution after each step. Eventually, the solution will lie in H 1 (Rn ), at which point we can use the conservation of H 1 norm. 2 Theorem 6.6. The energy E u(t) = ∇u, ∇u +

2 2+α

|u|α+2 dx

(6.43)

is decreasing. Proof. d dt

|∇u|2 dx = −ut , u = −

ut |u|α u¯ dx −

a|ut |2 dx.


2409

Therefore, d E u(t) 0. dt

(6.44)

So once the solution u(T ) ∈ H 1 (Rn ), the solution can be continued to a global solution. 7. Proof of Theorem 1.6 In the final section we prove Theorem 7.1. The initial value problem iut + u = |u|α u, u(0, x) = u0 (x), α=

4 , n − 2ρ

0 < ρ < 1,

(7.1)

with initial data radial and lying in the space H ρ+ Rn ∩ H 1/2+ Rn ∩ H 1/2+,1 Rn has a global solution. (7.1) is H˙ ρ -critical, and so Lemma 2.1 guarantees local well-posedness. Moreover, because u0 is radial and u0 ∈ H 1/2+,1 (Rn ), eit u0 and ∇eit u0 have long time asymptotics similar to eit χB(0;1) and ∇eit χB(0;1) respectively. Therefore, it is possible to apply the methods used in Sections 5 and 6. Remark. The cases with ρ = 0 and ρ = 1 have already been considered. See [14] for n = 2 and [15] for (n 3) when ρ = 0. See [10] for n = 3, [16] for n = 4, and [24] for n 5 when ρ = 1. The first order of business is to obtain some asymptotic estimates for eit u0 and ∇eit u0 . Lemma 7.2. Let u0 be a radial function. it ∇e u0 (x) Ct −3/2 |x|−(n−1)/2 + |x|−(n−3)/2 u0 2 n + u0 1 n , L (R ) L (R ) it ∇e u0 (x) C t −n/2 + t −n/2+1 |x|−1 u0 2 n + u0 1 n , L (R ) L (R ) it e u0 (x) Ct −n/2 u0 1 n , L (R ) it e u0 (x) Ct −1/2 |x|−(n−1)/2 u0 1 n + u0 2 n . L (R ) L (R )

(7.2) (7.3) (7.4) (7.5)

2410


Proof. ∂ 1 ∂xi t n/2

2 /4t

ei|x−y|

f (y) =

Rn

1 2it n/2+1

2 /4t

(xi − yi )ei|x−y|

f (y) dy.

Rn

It suffices to bound the two terms with stationary phase calculations. First term. t

−n/2−1

∞ |x|

f (r)r n−1 eir 0

=t

−n/2−1

2 /4t

e−2ix·ξ/4t dσr (ξ ) dr

Srn−1

∞ |x|

f (r)r

π/2

n−1 ir 2 /4t

e

n−2 e−2i|x|r sin(θ)/4t cos(θ ) dθ dr.

−π/2

0

Let u = sin θ =t

−n/2−1

∞ |x|

f (r)r

n−1 ir 2 /4t

1

e

(n−3)/2 e−2i|x|ru/4t 1 − u2 du dr.

(7.6)

−1

0

If n is odd, we apply the analysis used in the proof of Lemma 2.4 and integrate by parts in u times, yielding (7.6)

t −3/2 |x|(n−3)/2

t −3/2 = (n−3)/2 |x|

n−1 2

∞ f (r)r (n−1)/2 dr 0

1

f (r)r (n−1)/2 dr +

0

1 |x|(n−3)/2

∞ f (r)r (n−1)/2 dr. 1

Recall, f (r)r (n−1)/2 2 = f 2 n , L (R ) L (R) n−1 1 = f 1 n , f (r)r L (R ) L (R) (7.6)

t −3/2 f 1 (Rn ) + f L2 (Rn ) . L |x|(n−3)/2

Using the same arguments used in the proof of Lemma 2.4, when n is even, Ct −3/2 (7.6) (n−3)/2 |x|

∞ f (r)r (n−1)/2 dr 0

t −3/2 f L1 (Rn ) + f L2 (Rn ) . (n−3)/2 |x|

(7.7)


2411

To prove (7.3) in this case, we integrate by parts twice in u (we can do this since n 6), ∞ −n/2−1 n−1 ir 2 /4t −2ix·ξ/4t |x| f (r)r e e dσr (ξ ) dr t 0

Srn−1

t −n/2+1 |x|−1 f L2 (Rn ) + f L1 (Rn ) . Second term. The second term is t −n/2−1 ei|x|

2 /4t

∞ f (r)r n eir

2 /4t

0

e−2ix·ξ/4t dσr (ξ ) dr.

Srn−1

As in the case of the first term, after considering n even and n odd separately, Ct −3/2 (7.8) (n−1)/2 |x| 1

f (r)r (n+1)/2 dr

0

1

∞ f (r)r (n+1)/2 dr, 0

f (r)r (n−1)/2 dr f

L2 (Rn ) ,

0

∞

f (r)r (n+1)/2 dr

1

∞ f (r)r (n−1) dr f

L1 (Rn ) .

1

This takes care of the second term for (7.10). When proving (7.3) for the second term integrate by parts once.

t

−n/2−1

∞

n ir 2 /4t

f (r)r e 0

Ct −n/2 |x|−1

e−2ix·ξ/4t dσr (ξ ) dr

Srn−1

∞

f (r)r n−1 = Ct −n/2 |x|−1 f L1 (Rn ) .

0

This proves (7.2) and (7.3). (7.4) is just the dispersive estimate. Since we have already proved |x| it C e u0 |x|(n−3)/2 t 3/2 u0 L2 (Rn ) + u0 L1 (Rn ) t (the first term of (7.2)), (7.5) also follows.

2

(7.8)

2412


By Lemma 2.1, the local solution on [0, T ] has the form t eit u0 +

α ei(t−s) u(s) u(s) ds = eit u0 + w(t).

(7.9)

0

In Section 6, recall that for the local solution on [0, T ], we split the Duhamel term into a term in Hx1 (Rn ) and a term with “nice” asymptotics. We will do something similar here via the in-out decomposition. Remark. In general, when n > 4, the in-out decomposition will not split an L2 function into a sum of two L2 functions. Because of this, it will be necessary to exercise care in splitting the data. See [15] for more details on the in-out decomposition. Lemma 7.3. Suppose u0 ∈ L2 (Rd ) is supported on |y| ∼ 2k , k 10. Then, eit u0 = u1 (t) + u2 (t), ∇u2 (t) 2 d C(t) 2k u0 2 n , L (R ) L (R ) s n/2 u2 (t) 2 d Cu0 2 n , L (R ) L (R )

(7.10) (7.11) (7.12)

and for s > t, is ∇e u1 (t)(x) C(t) u0 2 n L (R ) 2k s n/2

(7.13)

on |x| 1. The constant C(t) is independent of k. Proof. Define a cutoff χ ∈ C0∞ (Rn ), χ(x) =

1, |x| 10 · 2k ;

0, |x| > 20 · 2k , i|x−y|2 Cχ(x) e 4t (x − y)i u0 (y) dy. χ(x)Dxi eit u0 (y) = n/2+1 t

(7.14)

Therefore, χ(x)Dx eit u0 (y) 2 n C2k u0 2 d . L (R ) i L (R ) x

Cu0

2

n

(7.15)

L (R ) Remark. [Dxi , χ(s)]eit u0 L2 (Rn ) , so swapping the order of Dxi and χ(x) will 2k it also obey (7.15). Let u2 (t) = χ(x)e u0 and u1 (t) = (1 − χ(x))eit u0 . Now compute

Dzj eis 1 − χ(x) eit u0 (x) (z), for |z| 1.


=

C n/2 s t n/2

(z − x)j ei

|z−x|2 4s

1 − χ(x)

ei

|y−x|2 4t

2413

u0 (y) dy dx,

ys zt |z − x|2 |y − x|2 |z − y|2 |x − ( s+t + s+t )|2 (s + t) + = + , 4s 4t 4(t + s) 4st

(t + s)n/2 s n/2 t n/2

(xj − zj )ei

(t + s)n/2 = n/2 n/2 s t =−

(t + s)n/2 s n/2 t n/2

zt + ys )|2 (t+s)|x−( t+s s+t 4st

1 − χ(x) dx

zt + 8st (1 − χ(x))(x − ( t+s

zt |x − ( t+s +

∇·

ys 2 t+s )| (t

ys t+s ))

+ s)

zt 8st (1 − χ(x))(x − ( t+s + zt |x − ( t+s +

zt when |z| 1, |y| ∼ 2k , |x − ( t+s + N times,

ys 2 t+s )| (t

ys |x| t+s )| > 2

(7.16) CN (t)

· ∇ei

zt + ys )|2 (t+s)|x−( t+s s+t 4st

dx

(t+s)|x−( zt + ys )|2 ys t+s s+t t+s )) i 4st

+ s)

e

dx,

(7.16)

on the support of 1 − χ(x). Integrating by parts

|x|>10·2k

1 dx 2k(n−N ) . |x|N

(7.17)

Now, u0 L1 (Rn ) C2kn/2 u0 L2 (Rn ) , so taking N independent of k, (7.13) is also satisfied.

2

Now we prove a global result. Theorem 7.4. Suppose the initial value problem iut + u = |u|α u, u(0, x) = u0 ∈ H ρ+ Rn ∩ H 1/2+ Rn , α=

4 , n − 2ρ

(7.18)

has a local solution on some interval [0, T ], uS ρ+ ([0,T ]×Rn ) + uS 1/2+ ([0,T ]×Rn ) < ∞.

(7.19)

Suppose also that u0 is radial and satisfies (7.2)–(7.5). Also suppose that there is an interval [T∗ , T ] such that the solution on [T∗ , T ] has the form ei(t−T∗ ) u(T∗ ) + w(t, x), 1 n w(t, x) ∈ L∞ t Hx [T∗ , T ] × R . Then the solution to (7.18) can be extended to a global solution of the form

(7.20) (7.21)

2414


ei(t−T∗ ) u(T∗ ) + w(t, x), 1 n w(t, x) ∈ L∞ t Hx [0, T0 ] × R ,

(7.22) (7.23)

for any T0 > T∗ . Proof. Let τ =

T +T∗ 2 .

T∗ u(T∗ ) = e

iT∗

u0 +

α ei(T∗ −t) u(t) u(t) dt.

0

Define the cutoffs, χ ∈ C0∞ , η ∈ C0∞ ,

1,

χ(x) =

1 2

|x| 2;

(7.24)

0, |x| < 14 or |x| > 4, 1, |x| 40; η(x) = 0, |x| > 80.

(7.25)

Let χj (x) = χ( 2xj ) and ηj (x) = η( 2xj ). Let u2 (τ ) =

∞

T∗ ηj (x)

j =10

α ei(τ −t) χj (y)u(t) u(t) dt.

(7.26)

0

By Lemma 7.3, ∞ ∇u2 (τ ) 2 d C χj (y)u(t)α u(t) 2 d L (R ) L (R ) x

x

j =10

∞

Cuα ∞

1/2+ Lt Hx ([0,T ]×Rn )

Cuα ∞

1/2+

Lt Hx

This is because α( n−1 2 )>

2(n−1) n

([0,T ]×Rn )

j =10

2j 2

j α( n−1 2 )

χj (y)u

2 n L∞ t Lx ([0,T ]×R )

uL∞ 2 n . t Lx ([0,T ]×R )

> 1 when n 6. Also by Lemma 7.3,

T∗ α i(s−τ ) i(s−τ ) i(τ −t) ∇e 1 − ηj (z) u1 (τ ) = ∇e e χj (y) u(t) u(t) dt 0

C(τ ) n/2 j u1+α , 1/2+ L∞ ([0,T ]×Rn ) s 2 t Hx on |x| 1. Finally,

(7.27)


χ10 =

2415

χj (y)

j −10

is compactly supported, so by Hölder’s inequality T∗ α C 1+α i(s−t) χ10 u(t) u(t) dt n/2+1 uL ∇ e ∞ L2 ([0,T ]×Rn ) . t x s

(7.28)

0

On |x| 1. Interpolating T∗ i(s−t) F (t) dt ∇ e

C ∇F (t)N 0 ([0,T

∗ ]×R

L2x (Rd )

0

d)

with T∗ 1 ∇ (s − t)ei(s−t) F (t) dt 1 + |x|

C 1 + |y| F (t, y)N 0 ([0,T

∗ ]×R

L2x (Rd )

0

d)

,

for |x| > 1, s > τ > T , s 1/2− T∗ α i(s−t) u(s) u(s) ds 1/2− ∇ e |x| 0

L2x (|x|>1)

uS1+α 1/2+ ([0,T ]×Rn ) ,

(7.29)

T∗ α i(s−t) u(s) u(s) ds ∇ e 0

T∗ i(s−t) α 1/2+ e 1/2− u(s) ∇ |x| u(s) ds (s − t)1/2− 0

T∗ α 1/2+ ei(s−t) 1/2− + u(s) ∇ |y| u(s) ds . (s − t)1/2− 0

Following the same arguments as in the proof of Lemma 2.1, u(s)α ∇ 1/2+ u(s) ∈ N 0 [0, T∗ ] × Rn , α |y|1/2− u(s) ∇ 1/2+ u(s) ∈ N 0 [0, T∗ ] × Rn .

(7.30) (7.31)

2416


The last embedding uses the radial Sobolev embedding n−1 |y| 2 u

n L∞ x (R )

CuH 1/2+ (Rn ) .

This gives (7.29). To simplify notation, shift time so that T = 0. Set up a system of equations ivt + v = 0, v(0, x) = ei(T −τ ) u1 (τ ) +

T∗

α ei(T −s) χ10 (y)u(s) u(s) ds + eiT u0 ,

(7.32)

0

iwt + w = |v + w|α (v + w), ˜ , x). w(0, x) = ei(T −τ ) u2 (τ ) + w(T

(7.33)

˜ x) in (7.22). Let w(t, ˜ x) is the w(t, x) for ei(t−T∗ ) u(T∗ ) + w(t, E(t) =

1 2

∇w(t, x)2 dx +

2 2+α

v(t, x) + w(t, x)2+α dx.

(7.34)

Combining the Sobolev embedding theorem with the dispersive estimates, E(0) is finite. d E(t) dt

|∇v| + |∇w| |∇v||v + w|α dx.

By Hölder’s inequality, (7.27), (7.28), and (7.3),

∇v(t, x)2 |v + w|α dx ∇v(t, x)2 n− Lx

(|x|1)

v(t, x) + w(t, x)α 2

Lx (Rn )

C.

|x|1

Similarly,

∇v(t, x)∇w(t, x)|v + w|α dx

|x|1

α ∇v(t, x)L∞ (|x|1) ∇w(t, x)L2 (Rn ) v(t, x) + w(t, x)L2 (Rn ) C. x

This is because α < 1 and n 6, so

x

α 2+α

1

2 T∗ α 1 i(t−τ ) i(T −s) 1/2− ∇ e u1 (τ ) + e χ10 u(s) u(s) 2 |x| 0

Lx (|x|>1)


2417

n−1 α × |x| 2 (v + w)L∞ (|x|>1) x i(t+T ) 2 + ∇e u0 2n − v + wαL2 (Rn ) CE(t)α/4+ . Lxn−2 (|x|>1)

x

This is by interpolating (7.2) and (7.5) in Lemma 7.2, and the radial Sobolev embedding. Finally, ∇v(t, x)∇w(t, x)|v + w|α dx |x|>1

T∗ α 1 i(t−τ ) i(T −s) u1 (τ ) + e χ10 u(s) u(s) 1/2− ∇ e |x| 0

L2x (|x|>1)

α n−1 × ∇w(t, x)L2 (Rn ) |x| 2 (v + w)L∞ (|x|>1) x x i(t+T ) ∇w(t, x)L2 (Rn ) v + wαL2 (Rn ) CE(t)1/2+α/4+ . + ∇e u0 2 Lx1−α (|x|>1)

x

x

This also follows from Lemma 7.2 and the radial Sobolev embedding. v + wH 1/2+ (Rn ) C vH 1/2+ (Rn ) + wH 1/2+ (Rn ) 1/2+ 1/2− C v(0) 1/2+ n + Cw 1 n w 2 n H

C v(0)

(R )

H (R )

L (R )

1/2− + CE(t)1/4+ wL2 (Rn ) . H 1/2+ (Rn )

Now, v(0)

H 1/2+ (Rn )

u0 H 1/2+ (Rn ) + u1 (τ )H 1/2+ (Rn ) T∗ α i(T −s) + e χ10 u(s) u(s) ds 0

. H 1/2+ (Rn )

u0 ∈ H 1/2+ (Rn ) and ∇1/2+ |u(s)|α u(s) ∈ N 0 ([0, T∗ ] × Rn ). Therefore, it remains to settle whether u1 (τ ) ∈ H 1/2+ (Rn ). u1 (τ )

H 1/2+ (Rn )

∞ χj (y)u(s)α u(s) j =10

T∗

∞

2−j (

j =10

(n−1)α ) 2

1/2+

L1s Hx

([0,T∗ ]×Rn )

uL∞ H 1/2+ ([0,T t

x

∗ ]×R

n)

< ∞.

Now if v(0) ∈ L2 (Rn ), then because v(t)L2 (Rn ) = v(0)L2 (Rn ) , then w(t)L2 (Rn ) is uniformly bounded by conservation of mass and the triangle inequality. We also have v(t)

H 1/2+ (Rn )

= v(0)H 1/2+ (Rn ) .

2418


All in all, d E(t) C + CE(t)1/2+α/4+ . dt Because E(0) is finite and α < 1, E(t) is bounded on any finite interval [0, T0 ]. This proves the existence of a global solution. 2 So it suffices to prove that the initial value problem (7.1) satisfies the conditions of Theorem 7.4. By Lemma 2.1, the initial value problem iut + u = |u|α u, u(0, x) = u0 , u0 ∈ H ρ+ Rn ∩ H 1/2+ Rn ,

(7.35)

is locally well-posed on some interval [0, T ]. Also, uS ρ+ ([0,T ]×Rn ) + uS 1/2+ ([0,T ]×Rn ) < ∞.

(7.36)

Therefore, all that remains is to show Lemma 7.5. There exists T∗ < T such that (7.35) has a solution on [T∗ , T ] of the form ei(t−T∗ ) u(T∗ ) + w(t, x), 1 n w(t, x) ∈ L∞ t Hx [T∗ , T ] × R , possibly after shrinking T . Proof. We prove this by an iteration argument. Suppose that the solution on [ak , bk ] has the form ei(t−ak ) u(ak ) + wk (t, x), σ n wk (t, x) ∈ L∞ t Hx [0, T ] × R . Then there exists bk+1 > bk , σ > 0 such that the solution on [ak+1 , bk+1 ] has the form ei(t−bk ) u(bk ) + wk+1 (t, x), σ +δ wk+1 (t, x) ∈ L∞ [ak+1 , bk+1 ] × Rn . t Hx To prove this, estimate the Duhamel term σ +δ |∇| u(t)α u(t) 0 N ([ak+1 ,bk+1 ]×Rn ) α σ +δ C |∇| wk+1 (t) u(t) 0

N ([ak+1 ,bk+1 ]×Rn )

α + C |∇|σ +δ ei(t−bk ) wk (bk ) u(t) α + C |∇|σ +δ ei(t−ak ) u(ak ) u(t)

N 0 ([ak+1 ,bk+1 ]×Rn )

N 0 ([ak+1 ,bk+1 ]×Rn )

,

(7.37)


2419

α σ +δ |∇| wk+1 (t) u(t)

N 0 ([ak ,bk ]×Rn )

C|bk+1 − ak+1 |η |∇|σ +δ wk+1 (t)

S 0 ([ak ,bk ]×Rn )

uαSρ+ ([ak ,bk ]×Rn ) ,

for some η() > 0. ak α ∇ ei(t−ak ) u(t) u(t) dt 0

ak ak ei(t−ak ) α α ei(t−ak ) |x| u(t) u(t) dt + |y|u(t) u(t) dt . t − ak t − ak 0

0

Using the radial Sobolev embedding theorem, 2 |y|u(t, y)α Cu n−1 1/2+

H

(Rn )

2 u(t, y)α− n−1 .

Therefore by interpolation, ak α α σ +δ i(t−ak ) u(s) u(s) ds u(t) e |∇|

N 0 ([ak+1 ,bk+1 ]×Rn )

0

1+α C|bk+1 − ak+1 |η uS 1/2+ ([ak+1 ,bk+1 ]×Rn ) + uS ρ+ ([ak+1 ,bk+1 ]×Rn ) . Also, for ak > 0, it ∇e u0 u(t)α

N 0 ([ak+1 ,bk+1 ]×Rn )

α C|bk+1 − ak+1 |η uS 1/2+ ([ak+1 ,bk+1 ]×Rn ) + uS ρ+ ([ak+1 ,bk+1 ]×Rn ) . Finally, for σ > ρ, we can use the bilinear estimate Lemma 7.6. Take an interval I = [0, T0 ]. (P>N u)(P<M v) 2 L

n t,x (I ×R )

M (n−1)/2 u>N S∗0 (I ×Rn ) v<M S∗0 (I ×Rn ) , N 1/2 . n + (iut + u) 2(d+2)

uS∗0 (I ×Rn ) = u0 L2 (R Proof. See [15].

2

)

Lt,xd+4 (I ×Rn )

(7.38) (7.39)

2420


Now let

1 p

=

1 2

−

ρ+ n .

α σ +ρ i(t−b ) k |∇| e wk (bk ) u(t)

∞

2n

L2t Lxn+2 ([ak+1 ,bk+1 ]×Rn )

1−η N σ +ρ ei(t−bk ) PN wk (bk ) 1 2n

L2t Lxn−2 ([ak+1 ,bk+1 ]×Rn )

N =1

× ei(t−bk ) PN wk (bk ) P

N

1 2(n−1)

η u(t) L12

t,x ([ak+1 ,bk+1 ]×R

n)

α−η

× uL∞ L1p ([a x

t

k+1 ,bk+1 ]×R

+ |bk+1 − ak+1 |η

∞

n)

1−η N σ +ρ ei(t−bk ) PN wk (bk ) 2 2n

L2t Lxn−2 ([ak+1 ,bk+1 ]×Rn )

N =1

η × ei(t−bk ) PN wk (bk ) 2 × P

>N

1 2(n−1)

η u(t) 2

2n n−2ρ

L∞ t Lx

α−η

× uL∞ L1p ([a t

x

k+1 ,bk+1 ]×R

2n

L2t Lxn−2 ([ak+1 ,bk+1 ]×Rn )

n)

([ak+1 ,bk+1 ]×Rn )

,

for some η1 , η2 , η depending on . Then take δ(η1 , η2 , η, n) > 0. Iterating over a finite number of intervals, we can find an interval [aK , bK ], for some finite K such that the Duhamel term lies in Hx1 (Rn ). Take aK = T∗ and bK = T . 2 Combining Lemma 7.5 with Theorem 7.4 proves Theorem 7.1. Acknowledgments This paper was part of my PhD thesis. I am grateful Michael Taylor, my advisor, for all his help with my thesis. I am also grateful to Jason Metcalfe and Mark Williams for their helpful comments. References [1] J. Bourgain, Refinements of Strichartz’ inequality and applications to 2D-NLS with critical nonlinearity, Int. Math. Res. Not. 5 (1998) 253–283. [2] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in H 1 , Manuscripta Math. 61 (1988) 477–494. [3] T. Cazenave, F. Weissler, The Cauchy problem for the nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990) 807–836. [4] M. Christ, Asymptotics, frequency modulation, and low regularity ill-posedness for canonical defocusing equations, Amer. J. Math. 125 (2003) 1235–1293. [5] M. Christ, J. Colliander, T. Tao, Ill-posedness for nonlinear Schrödinger and wave equations, Ann. Henry Poincaré, in press, arXiv:math/0311048v1. [6] M. Christ, A. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg–de Vries equations, J. Funct. Anal. 100 (1991) 87–109. [7] J. Colliander, T. Roy, Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on R2 , preprint, arXiv:0811.1803.


2421

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Almost conservation laws and global rough solutions to a nonlinear Schrödinger equation, Math. Res. Lett. 9 (5–6) (2002) 659–682. [9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global existence and scattering for rough solutions of a nonlinear Schrödinger equation on R3 , Comm. Pure Appl. Math. 21 (2008) 987–1014. [10] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in R3 , Ann. of Math. (2) 167 (3) (2008) 767–865. [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Resonant decompositions and the I-method for cubic nonlinear Schrödinger equation on R2 , Discrete Contin. Dyn. Syst. A 21 (2008) 665–686. [12] D. De Silva, N. Pavlović, G. Staffilani, N. Tzirakis, Global well-posedness for the L2 critical nonlinear Schrödinger equation in higher dimensions, Commun. Pure Appl. Anal. 6 (4) (2007) 1023–1041. [13] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980. [14] R. Killip, T. Tao, M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc., in press, arXiv:0707.3188v2. [15] R. Killip, M. Visan, X. Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. Partial Differential Equation 1 (2) (2008) 229–266. [16] A. Ruiz, L. Vega, On local regularity of Schrödinger equations, Int. Math. Res. Not. 1 (1993) 13–27. [17] M. Pinksy, M. Taylor, Pointwise Fourier inversion: A wave equation approach, J. Fourier Anal. Appl. 3 (1997) 647–703. [18] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Princeton University Press, Princeton, NJ, 1993, with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [19] T. Tao, Nonlinear Dispersive Equations, CBMS Reg. Conf. Ser. Math., vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2006, local and global analysis. [20] M. Taylor, Pseudodifferential Operators and Nonlinear PDE, Birkhäuser, 1991. [21] M. Taylor, Partial Differential Equations, Springer-Verlag, 1996. [22] M. Taylor, Short time behavior of solutions to nonlinear Schrödinger equations in one and two space dimensions, Comm. Partial Differential Equations 31 (2006) 945–957. [23] M. Taylor, Short time behavior of the nonlinear Schrödinger equation II, Canad. J. Math. 60 (2008) 1168–1200. [24] M. Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 (2007) 281–374.


Sampling and reconstruction of signals in a reproducing kernel subspace of Lp (Rd ) M. Zuhair Nashed, Qiyu Sun ∗ Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA Received 15 June 2009; accepted 22 December 2009 Available online 31 December 2009 Communicated by N. Kalton

Abstract In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of Lp (Rd ), 1 p ∞, associated with an idempotent integral operator whose kernel has certain offdiagonal decay and regularity. The space of p-integrable non-uniform splines and the shift-invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of Lp (Rd ). We show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. We also study the exponential convergence, consistency, and the asymptotic pointwise error estimate of the iterative approximation–projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap. © 2009 Elsevier Inc. All rights reserved. Keywords: Sampling; Iterative reconstruction algorithm; Reproducing kernel spaces; Idempotent operators; p-Frames

1. Introduction Sampling and reconstruction is a cornerstone of signal processing. The most common form of sampling is the uniform sampling of a bandlimited signal. In this case, perfect reconstruction of the signal from its uniform samples is possible when the samples are taken at a rate greater than twice the bandwidth [28,39]. Motivated by the intensive research activity taking place around wavelets, the paradigm for sampling and reconstructing band-limited signals has been extended * Corresponding author.

E-mail addresses: [email protected] (M.Z. Nashed), [email protected] (Q. Sun). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.012

M.Z. Nashed, Q. Sun / Journal of Functional Analysis 258 (2010) 2422–2452

2423

over the past decade to signals in shift-invariant spaces [4,46]. Recently, the above paradigm has been further extended to representing signals with finite rate of innovation, which are neither band-limited nor living in a shift-invariant space [17,31,43,44,47]. Here a signal is said to have finite rate of innovation if it has finite number of degrees of freedom per unit of time, that is, if it has requires only a finite number of samples per unit of time to specify the signal [47]. In this paper, we consider sampling and reconstruction of signals in a reproducing kernel subspace of Lp (Rd ), 1 p ∞. Here and henceforth Lp := Lp (Rd ) is the space of all pintegrable functions on the d-dimensional Euclidean space Rd with the standard norm · Lp (Rd ) , or · p for short. A reproducing kernel subspace of Lp (Rd ) [10] is a closed subspace V of Lp (Rd ) such that the evaluation functionals on V are continuous, i.e., for any x ∈ Rd there exists a positive constant Cx such that f (x) Cx f

for all f ∈ V .

Lp (Rd )

(1.1)

Let 1 p ∞. We say that a bounded linear operator T on Lp (Rd ) is an idempotent operator if it satisfies T 2 = T.

(1.2)

Denote by V the range space of the idempotent operator T on Lp (Rd ), i.e., V := Tf f ∈ Lp Rd .

(1.3)

We say that the range space V of the idempotent operator T on Lp (Rd ) is a reproducing kernel space V associated with the idempotent operator T on Lp (Rd ) if it is a reproducing kernel subspace of Lp (Rd ). A trivial example of idempotent linear operators is the identity operator. In this case, the range space is the whole space Lp (Rd ) on which the evaluation functional is not continuous. As pointed out in [34], the whole space L2 (Rd ) is too big to have stable sampling and reconstruction of signals belonging to this space. So it would be reasonable and necessary to have certain additional constraints on the idempotent operator T . In this paper, we further assume that the idempotent operator T is an integral operator Tf (x) =

K(x, y)f (y) dy,

f ∈ Lp R d ,

(1.4)

Rd

whose measurable kernel K has certain off-diagonal decay and regularity, namely, sup K(· + z, z) z∈Rd

L1 (Rd )

< ∞,

(1.5)

and lim sup ωδ (K)(· + z, z)

δ→0 z∈Rd

L1 (Rd )

=0

(1.6)

2424


[29,42]. Here the modulus of continuity ωδ (K) of a kernel function K on Rd × Rd is defined by ωδ (K)(x, y) =

sup x ,y ∈[−δ,δ]d

K x + x , y + y − K(x, y).

(1.7)

In this paper, we assume that signals to be sampled and represented live in a reproducing kernel space associated with an idempotent integral operator whose kernel satisfies (1.5) and (1.6). The reason for this setting is three-fold. First, the range space of an idempotent integral operator whose kernel satisfies (1.5) and (1.6) is a reproducing kernel subspace of Lp (Rd ), see Theorem A.1 in Appendix A. Secondly, signals in the range space of an idempotent integral operator whose kernel satisfies (1.5) and (1.6) have finite rate of innovation, see Theorem A.2 in Appendix A. Thirdly, the common model spaces in sampling theory such as the space of pintegrable non-uniform splines of order n satisfying n − 1 continuity conditions at each knot [38,48] and the finitely-generated shift-invariant space with its generators having certain regularity and decay at infinity [4,46], are the range space of some idempotent integral operators whose kernels satisfy (1.5) and (1.6), see Examples A.3 and A.4 in Appendix A. A discrete subset Γ of Rd is said to be relatively-separated if BΓ (δ) := sup

x∈Rd γ ∈Γ

χγ +[−δ/2,δ/2]d (x) < ∞

(1.8)

for some δ > 0, while a positive number δ is said to be a gap of a relatively-separated subset Γ of Rd if AΓ (δ) := inf

x∈Rd

γ ∈Γ

χγ +[−δ/2,δ/2]d (x) 1

(1.9)

[8]. Note that the set of all positive numbers δ with AΓ (δ) 1 is either an interval or an empty set because AΓ (δ) is an increasing function of δ > 0. Then for a relatively-separated subset Γ of Rd having positive gap, we define the smallest positive number δ with AΓ (δ) 1 as its maximal gap. One may verify that a bi-infinite increasing sequence Λ = {λk }k∈Z of real numbers is relativelyseparated if infk∈Z (λk+1 − λk ) > 0, and that it has maximal gap supk∈Z (λk+1 − λk ) if it is finite. In this paper, we assume that the sample Y := (f (γ ))γ ∈Γ of a signal f is taken on a relativelyseparated subset Γ of Rd with positive gap. The samplability is one of most important topics in sampling theory, see for instance [22,26,46] for band-limited signals, [4,43] for signals in a shift-invariant space, [16,20,21,24,25] for signals in a co-orbit space, and [27,33] for signals in reproducing kernel Hilbert and Banach spaces. In this paper, we study the samplability of signals in a reproducing kernel subspace of Lp (Rd ) associated with an idempotent operator. Particularly, in Section 2, we show that any signal in a reproducing kernel subspace V of Lp (Rd ) associated with an idempotent operator whose kernel satisfies (1.5) and (1.6) can be reconstructed in a stable way from its samples taken on a relatively-separated set Γ with sufficiently small gap δ, i.e., there exist positive constants A and B such that Af Lp (Rd ) f (γ ) γ ∈Γ p (Γ ) Bf Lp (Rd )

for all f ∈ V

(1.10)


2425

(see Theorem 2.1 for the precise statement). Here and henceforth, given a discrete set Γ , p := p (Γ ), 1 p ∞, is the space of all p-summable sequences on Γ with the standard norm · p (Γ ) , or · p for short. In this paper, we then study the linear reconstruction of a signal from its samples taken on a relatively-separated set with sufficiently small gap. The iterative approximation–projection reconstruction algorithm is an efficient algorithm to reconstruct a signal from its samples, which was introduced in [22] for reconstructing band-limited signals, and was later generalized to signals in shift-invariant spaces in [2]; see also [4,7,23] and the references therein for various generalizations and applications. In Section 3 of this paper, we introduce the iterative approximation–projection reconstruction algorithm for reconstructing a signal in a reproducing kernel subspace of Lp (Rd ) from its samples taken on a relatively-separated set with sufficiently small gap, and study its exponential convergence, consistency, and numerical implementation of the above iterative approximation–projection algorithm (see Theorem 3.1, Remark 3.1 and Remark 3.2 for details). Denote the standard action between functions f ∈ Lp (Rd ) and g ∈ Lp/(p−1) (Rd ) by f, g = f (x)g(x) dx. (1.11) Rd

Then the stability condition (1.10) can be interpreted as the p-frame property of {K(γ , ·)}γ ∈Γ on the space V . Here for a Banach subspace V of Lp (Rd ), we say that a family Φ = {ψγ }γ ∈Γ of functions in Lp/(p−1) (Rd ) is a p-frame for V [6] if there exist positive constants A and B such that (1.12) Af Lp (Rd ) f, ψγ γ ∈Γ p (Γ ) Bf Lp (Rd ) for all f ∈ V . Then a natural linear reconstruction algorithm is the frame reconstruction algorithm; see [11,49] for reconstructing band-limited signals, [4,9,15,30] for reconstructing signals in shift-invariant spaces, and [35] for reconstructing signals in some reproducing kernel Hilbert spaces. In Section 4, we introduce the preconditioned frame algorithm for reconstructing signals in a reproducing kernel space associated with an idempotent integral operator from its samples taken a relatively-separated set Γ with sufficiently small gap, and study its exponential convergence and consistency (see Theorem 4.1 for details). Reconstructing a function from data corrupted by noise and estimating the reconstruction error are leading problems in sampling theory, however they have not been given as much attention; see [18,36,40] for reconstructing bandlimited signals, [5,18] for reconstructing signals in shiftinvariant spaces, and [12,31,32] for reconstructing signals with finite rate of innovations. It is observed in [37] that reconstruction from noisy data may introduce spatially-dependent noise in the reconstructed signal (hence spatial dependent artifacts) that are undesirable for sub-pixel signal processing. Thus it is desirable to have an accurate error estimate of the reconstructed signal at each point. In this paper, we show that the reconstruction via the approximation–projection reconstruction algorithm and the frame reconstruction algorithm is unbiased, and we also provide an asymptotic estimate of the variance of the error between the reconstruction from noisy sample of a signal f via these algorithms and the signal f in a reproducing kernel space, see Theorem 5.1 and Remark 5.2. The range space V of an idempotent operator T on Lp (Rd ) has various properties. For instance, it is complementable and the null space N (T ) := {g ∈ Lp (Rd ) | T g = 0} is its algebraic

2426


and topological complement. In Appendix A, some properties of the range space of an idempotent integral operator on Lp (Rd ) whose kernel satisfies (1.5) and (1.6) are established, such as the reproducing kernel property in Theorem A.1 and the frame property in Theorem A.2. 2. Samplability of signals in a reproducing kernel space In this section, we consider the samplability of signals in a reproducing kernel subspace V of Lp (Rd ) associated with an idempotent integral operator whose kernel satisfies (1.5) and (1.6), by showing that any signal in V can be reconstructed in a stable way from its samples taken on a relatively-separated set with sufficiently small gap. Theorem 2.1. Let 1 p ∞, T be an idempotent integral operator whose kernel K satisfies (1.5) and (1.6), V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T , and δ0 > 0 be so chosen that r0 := sup ωδ0 /2 (K)(· + z, z) z∈Rd

L1 (Rd )

< 1.

(2.1)

Then any signal f in V can be reconstructed in a stable way from its samples f (γ ), γ ∈ Γ , taken on a relatively-separated subset Γ of Rd with gap δ0 . Moreover, 1/p (1 − r0 ) δ0−d AΓ (δ0 ) f Lp (Rd ) p f (γ ) γ ∈Γ (Γ )

1/p (1 + r0 ) δ0−d BΓ (δ0 ) f Lp (Rd )

for all f ∈ V .

(2.2)

Now we apply the above samplability result to signals in a shift-invariant space. Let

W := f f W :=

sup

f (x + k) < ∞

(2.3)

d k∈Zd x∈[−1/2,1/2]

be the Wiener amalgam space [4,19]. Let φ1 , . . . , φr ∈ W be continuous functions on Rd with the property that {φi (· − k): 1 i r, k ∈ Zd } is an orthonormal subset of L2 (Rd ). Then the integral operator T defined by

Tf (x) =

r Rd

φi (x − k)φi (y − k) f (y) dy

for all f ∈ L2 Rd

(2.4)

i=1 k∈Zd

is an idempotent operator whose kernel satisfies (1.5) and (1.6). This yields the samplability of signals in a finitely-generated shift-invariant space [2].


2427

Corollary 2.2. Let φ1 , . . . , φr ∈ W be continuous functions on Rd such that {φi (·−k) | 1 i r, k ∈ Zd } is an orthonormal subset of L2 (Rd ). Define the finitely-generated shift-invariant space V2 (φ1 , . . . , φr ) by V2 (φ1 , . . . , φr ) =

r

r 2 ci (k) < ∞ . ci (k)φi (· − k)

i=1 k∈Zd

i=1 k∈Zd

(2.5)

Then any signal f in V2 (φ1 , . . . , φr ) can be reconstructed in a stable way from its samples f (γ ), γ ∈ Γ , taken on a relatively-separated subset Γ of Rd with sufficiently small gap δ0 . The following theorem is a slight generalization of Theorem 2.1. Theorem 2.3. Let 1 p ∞, T be an idempotent integral operator whose kernel K is continuous and satisfies sup K(x, ·)L1 (Rd ) + sup K(·, y)L1 (Rd ) < ∞,

x∈Rd

(2.6)

y∈Rd

V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T , and δ0 > 0 be so chosen that r0 :=

sup sup K(x + t, ·) − K(x, ·)

x∈Rd |t|δ0 /2

×

1−1/p L1 (Rd )

sup sup K(· + t, y) − K(·, y)

y∈Rd |t|δ0 /2

1/p L1 (Rd )

< 1.

(2.7)

Then any signal f in V can be reconstructed in a stable way from its samples f (γ ), γ ∈ Γ , taken on a relatively-separated subset Γ of Rd with gap δ0 . Remark 2.1. The conclusion in Theorem 2.3 is established in [24, Section 7.5] when the kernel K of the idempotent operator T satisfies K(x, y) = K(y, x).

(2.8)

For p = 2, an idempotent operator T with kernel K satisfying (2.8) is a projection operator onto a closed subspace of L2 . Hence the idempotent operator T with its kernel satisfying (2.8) is uniquely determined by its range space V onto L2 . The above conclusion on the idempotent operator does not hold without the assumption (2.8) on its kernel. We leave the above option on the kernel of idempotent operators free for better estimate in the gap δ0 in Theorem 2.1, and also for our further study on local exact reconstruction (cf. [3,41,45] for signals in shift-invariant spaces). For instance, let us consider samplability of signals in the linear spline space V1 :=

k∈Z

c(k)h(x − k) sup c(k) < ∞ , k∈Z

2428


where h(x) := max(1 − |x|, 0) is the hat function. It is well known [3] that a signal f in the linear spline space V1 can be reconstructed in a stable way from its samples f (γk ), k ∈ Z, with maximal gap δ0 := supk∈Z (γk+1 − γk ) < 1. For any integer N 1, define |k−l| 3N 2 KN (x, y) = √ h(x − k)h N (y − l) 9N 2 − 6N − 3N + 1 , 9N 2 − 6N k,l∈Z and let TN be the integral operator with kernel KN . One may verify that TN , N 1, are idempotent operators with the same range space V1 and the kernel KN satisfies (2.8) only when N = 1. Recalling that KN (x − 1, y − 1) = KN (x, y) and KN (−x, −y) = KN (x, y), we have sup sup KN (x + t, ·) − KN (x, ·)

1

x∈R |t|δ0 /2

=

sup sup KN (x + t, ·) − KN (x, ·)

1

x∈[0,1/2] |t|δ0 /2

∞ |s| 3N 2 3N − 1 − 9N 2 − 6N √ 2 9N − 6N s=−∞ h(x − k) − h(x + t − k)h N (· − k − s) sup × sup x∈[0,1/2] |t|δ0 /2 k∈Z

1

9N δ0 . 6N − 4

This shows that the inequality (2.7) holds for K = KN and p = ∞ when δ0 < other hand, we have

2 3

−

4 9N .

On the

√ (9 − 3)δ0 , sup sup K1 (x + t, ·) − K1 (x, ·) K1 (δ0 /2, ·) − K1 (0, ·) 1 = 1 4 x∈R |t|δ0 /2 which implies that the inequality (2.7) does not hold for K = K1 and p = ∞ when δ0 0.5504 and so the theorem does not apply.

4√ (9− 3)

≈

We conclude this section by providing proofs of Theorems 2.1 and 2.3. To prove Theorem 2.1, we need a technical lemma. Lemma 2.4. Let 1 p ∞, δ0 ∈ (0, ∞), r ∈ (0, 1), and Γ be a discrete subset of Rd with the property that 1 AΓ (δ0 ) BΓ (δ0 ) < ∞.

(2.9)

Assume that f ∈ Lp (Rd ) satisfies ω δ

0 /2

(f )Lp (Rd ) rf Lp (Rd ) ,

(2.10)


2429

and U := {uγ }γ ∈Γ is a bounded uniform partition of unity (BUPU) associated with the covering {γ + [−δ0 /2, δ0 /2]d }γ ∈Γ of Rd , i.e., ⎧ 0 uγ (x) 1 for all x ∈ Rd , and γ ∈ Γ, ⎪ ⎪ ⎪ ⎨ uγ is supported in γ + [−δ0 /2, δ0 /2]d for each γ ∈ Γ, ⎪ ⎪ uγ (x) ≡ 1 for all x ∈ Rd . ⎪ ⎩

and

(2.11)

γ ∈Γ

Then 1/p (1 − r)f Lp (Rd ) f (γ )uγ L1 (Rd ) γ ∈Γ p (Γ ) (1 + r)f Lp (Rd ) .

(2.12)

Proof. By the definition of the modulus of continuity, f (x) − ωδ /2 (f )(x) f (γ ) f (x) + ωδ /2 (f )(x) 0 0

(2.13)

for all x ∈ γ + [−δ0 /2, δ0 /2]d and γ ∈ Γ . This together with (2.9) and (2.10) proves (2.12) with p = ∞. For 1 p < ∞, it follows from (2.10), (2.11), and (2.13) that f p =

γ ∈Γ

Rd

γ ∈Γ

1/p

f (x)p uγ (x) dx

f (γ )p uγ (x) dx

1/p +

γ ∈Γ

Rd

f (γ )p uγ 1

1/p

ωδ /2 (f )(x)p uγ (x) dx 0

1/p

Rd

+ rf p ,

γ ∈Γ

and

f (γ )p uγ 1

1/p

γ ∈Γ

γ ∈Γ

f (x) + ωδ

p uγ (x) dx

1/p

0 /2 (f )(x)

Rd

(1 + r)f p . Then (2.12) for 1 p < ∞ is proved.

2

Remark 2.2. Two popular examples of bounded uniform partitions of unity (BUPU) associated with the covering {γ + [−δ0 /2, δ0 /2]d }γ ∈Γ of Rd are given by uγ (x) =

χγ +[−δ0 /2,δ0 /2]d (x) γ ∈Γ

χγ +[−δ0 /2,δ0 /2]d (x)

,

γ ∈ Γ,

(2.14)

2430


and uγ (x) = χVγ (x),

γ ∈ Γ,

(2.15)

where Vγ is the Voronoi polygon whose interior consists of all points in Rd being closer to γ than any other point γ ∈ Γ . Given a continuously differentiable function f on the real line, its modulus of continuity ωδ (f )(x) is dominated by the integral of its derivative f on x + [−δ, δ], i.e., δ ωδ (f )(x)

f (x + t) dt

for all x ∈ R.

−δ

Then the following result (which is well known for band-limited signals [22]) follows easily from Lemma 2.4. Corollary 2.5. Let 1 p ∞, f be a time signal satisfying f p B0 f Lp (R) L (R)

(2.16)

for some positive constant B0 , and Γ = {γk }k∈Z be a relatively-separated subset of R with maximal gap δ0 < 1/B0 . Then there exists a positive constant C (that depends on B0 , BΓ (δ0 ) and AΓ (δ0 ) only) such that 1/p C −1 f Lp (R) f (γ )uγ L1 (Rd ) γ ∈Γ p (Γ ) Cf Lp (R) .

(2.17)

Now we prove Theorem 2.1. Proof of Theorem 2.1. For any f ∈ V , ω δ

0 /2

(f )p = ωδ0 /2 (Tf )p ωδ0 /2 (K)(·, y) f (y) dy Rd

p

sup ωδ0 /2 (K)(· + z, z) f p z∈Rd

1

= r0 f p .

(2.18)

For any discrete set Γ with 1 AΓ (δ0 ) BΓ (δ0 ) < ∞, we define {uγ }γ ∈Γ as in (2.14). Then δ0d δ0d uγ 1 BΓ (δ0 ) AΓ (δ0 )

for all γ ∈ Γ.

(2.19)

From (2.1), (2.18) and Lemma 2.4, we obtain the estimates in (2.2) for p = ∞. On the other hand, from (2.1), (2.18), (2.19) and Lemma 2.4, we get the following estimate for 1 p < ∞:


f (γ )p

1/p

γ ∈Γ

2431

1/p p −d 1/p f (γ ) uγ 1 δ0 BΓ (δ0 ) γ ∈Γ

1/p δ0−d BΓ (δ0 ) (1 + r0 )f p

and

f (γ )p

1/p

γ ∈Γ

1/p 1/p f (γ )p uγ 1 δ0−d AΓ (δ0 ) γ ∈Γ

1/p δ0−d AΓ (δ0 ) (1 − r0 )f p .

This proves (2.2) for 1 p < ∞.

2

Proof of Theorem 2.3. Similar argument used in the proof of Theorem 2.1 can be applied to prove Theorem 2.3. We leave the detailed proof for the interested readers. 2 3. Iterative approximation–projection reconstruction algorithm In this section, we show that signals in a reproducing kernel subspace of Lp (Rd ) associated with an idempotent integral operator can be reconstructed, via an iterative approximation– projection reconstruction algorithm, from its samples taken on a relatively-separated set with sufficiently small gap. Theorem 3.1. Let 1 p ∞, T be an idempotent integral operator whose kernel K satisfies (1.5) and (1.6), V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T , and δ0 > 0 be so chosen that (2.1) holds. Set r0 := sup ωδ /2 (K)(· + z, z) . z∈Rd

0

L1 (Rd )

Then for any relatively-separated subset Γ with gap δ0 and c0 = (c0 (γ ))γ ∈Γ ∈ p (Γ ), the sequence {fn }∞ n=0 of signals in V defined by ⎧ ⎪ f0 (x) = c0 (γ )T uγ (x), ⎪ ⎪ ⎨ γ ∈Γ (3.1) ⎪ ⎪ fn (x) = f0 (x) + fn−1 (x) − fn−1 (γ )T uγ (x) for n 1, ⎪ ⎩ γ ∈Γ

converges exponentially, precisely fn − f∞ Lp (Rd ) T f0 Lp (Rd ) r0n+1 /(1 − r0 )

for some f∞ ∈ V ,

(3.2)

where U := {uγ }γ ∈Γ is a BUPU in (2.11). The sample of the limit signal f∞ and the given initial data c0 are related by c0 (γ ) − f∞ (γ ) T uγ (x) ≡ 0. (3.3) γ ∈Γ

2432


Furthermore the iterative algorithm (3.1) is consistent, i.e., if the given initial data c0 = (g(γ ))γ ∈Γ is obtained by sampling a signal g ∈ V then the sequence {fn }∞ n=0 in the iterative algorithm (3.1) converges to g. Proof. Define a bounded operator QΓ,U on Lp by

QΓ,U f (x) :=

(Tf )(γ )uγ (x) − (Tf )(x)

γ ∈Γ

=

Rd

(3.4)

uγ (x)K(γ , y) − K(x, y) f (y) dy,

f ∈ Lp .

γ ∈Γ

Then QΓ,U T = QΓ,U

(3.5)

by (1.2), and QΓ,U f p r0 f p

for all f ∈ Lp

by the following estimate for the integral kernel of the operator QΓ,U : uγ (x)K(γ , y) − K(x, y) sup ωδ0 /2 (K) x − y + z , z .

(3.6)

(3.7)

z ∈Rd

γ ∈Γ

Define the approximation–projection operator PΓ,U by PΓ,U = T QΓ,U + T .

(3.8)

Then it follows from (1.2), (3.5) and (3.6) that PΓ,U T = T PΓ,U = PΓ,U , (T − PΓ,U )n = (−1)n T QnΓ,U

(3.9)

for all n 1,

(3.10)

and (T − PΓ,U )n T r n 0

for all n 1.

(3.11)

By (3.1), (3.4) and (3.8), fn+1 − fn = (T − PΓ,U )(fn − fn−1 ) = ··· = (T − PΓ,U )n (f1 − f0 ) = (T − PΓ,U )n+1 f0 ,

n 0.

(3.12)

This together with (3.11) proves the exponential convergence of fn , n 0, and the estimate (3.2).


2433

Eq. (3.3) follows easily by taking limit on both sides of (3.1) and applying (2.2). Define RAP := T +

∞ (T − PΓ,U )n .

(3.13)

n=1

Then it follows from (3.9) and (3.11) that RAP is a bounded operator on Lp and a pseudo-inverse of the operator PT ,U , i.e., RAP PΓ,U = PΓ,U RAP = T ,

(3.14)

and moreover it satisfies RAP T = T RAP = RAP . Applying (3.12) iteratively leads to

n k fn = T + (T − PΓ,U ) f0

for all n 1,

(3.15)

k=1

which together with (3.13) implies that f∞ = lim fn = RAP f0 . n→∞

(3.16)

In the case that the initial data c0 is the sample of a signal g ∈ V , the initial signal f0 in the iterative algorithm (3.1) and the signal g are related by f0 = PΓ,U g.

(3.17)

Combining (3.14), (3.16) and (3.17) proves the consistency of the iterative algorithm (3.1).

2

From the proof of Theorem 3.1, we have the following result for the operator RAP in (3.13). Corollary 3.2. Let 1 p ∞, T be an idempotent integral operator whose kernel K satisfies (1.5) and (1.6), V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T , δ0 > 0 be so chosen that (2.1) holds, Γ be a relatively-separated subset with gap δ0 , U := {uγ }γ ∈Γ is a BUPU in (2.11), and RAP be as in (3.13). Then RAP is a bounded integral operator on Lp (Rd ) and its kernel KAP satisfies (1.5), (1.6), and K(x, z1 )KAP (z1 , z2 )K(z2 , y) dz1 dz2 for all x, y ∈ Rd . (3.18) KAP (x, y) = Rd Rd

Remark 3.1. If the initial sample c0 in the iterative approximation–projection reconstruction algorithm (3.1) is the corrupted sample of a signal g ∈ V , i.e., c0 = g(γ ) + (γ ) γ ∈Γ

2434


for some noise = ( (γ ))γ ∈Γ , then the Lp norm of the original signal g and the recovered signal f∞ via the iterative approximation–projection reconstruction algorithm (3.1) is bounded by the p norm of the noise . More precisely, from (3.11) and (3.12) we obtain fn − gp

∞ T Qk

n T Qk h0 (f − h ) + 0 p Γ,U 0 Γ,U p

k=n+1

T

k=0

∞

r0k f0 − h0 p + T

k=n+1

n

r0k h0 p

k=0

T (1 − r0 ) f0 p r0n+1 + h0 p 1/p c0 p r0n+1 + p T 2 (1 − r0 )−1 sup uγ 1 −1

γ ∈Γ

(3.19)

and f∞ − gp T (1 − r0 )−1 h0 p 1/p T 2 (1 − r0 )−1 sup uγ 1 p , γ ∈Γ

(3.20)

where h0 = γ ∈Γ (γ )T uγ and fn , n 0, are given in the approximation–projection reconstruction algorithm (3.1). Define the sample-to-noise ratio in the logarithmic decibel scale, a term for the power ratio between a sample and the background noise, by SNR(dB) = 20 log10

c0 p . p

(3.21)

The estimate in (3.19) suggests that the stopping step n0 for the iterative approximation– projection reconstruction algorithm (3.1) is n0 =

SNR(dB) , 20 ln10 (1/r0 )

(3.22)

where [x] denotes the integral part of a real number x. In this case, 1/p p , fn0 − gp 2T 2 (1 − r0 )−1 sup uγ 1 γ ∈Γ

(3.23)

and the error between the resulting signal fn0 and the original signal g is about twice the error due to the noise in the initial sample data. Remark 3.2. Given the initial data c0 = (c0 (γ ))γ ∈Γ , define Fn = fn (γ ) γ ∈Γ ,

n 0,

(3.24)


2435

and A = (T uγ )(γ ) γ ,γ ∈Γ ,

(3.25)

where fn , n 0, is given in the iterative approximation–projection reconstruction algorithm (3.1). This leads to the discrete version of the iterative approximation–projection reconstruction algorithm (3.1):

F0 = Ac0 , Fn = F0 + (I − A)Fn−1 ,

(3.26)

n 1.

Exponential convergence. Now let us consider the exponential convergence of the sequence Fn , n 0, when (1.5), (1.6) and (2.1) hold. By (3.26), we have Fn − Fn−1 = (I − A)n F0 = (I − A)n Ac0 ,

n 1.

(3.27)

Define cp,U

c(γ ) uγ =

p

γ ∈Γ

for c = c(γ ) γ ∈Γ ,

(3.28)

where 1 p ∞.For c = (c(γ ))γ ∈Γ with cp,U < ∞, write (I − A)n Ac = (dn (γ ))γ ∈Γ and define cΓ,U (x) = γ ∈Γ c(γ )uγ (x). Similar to Eq. (3.11) we have dn (γ ) = (−1)n T QnΓ,U cΓ,U (γ ).

(3.29)

This together with (3.6) implies that (I − A) Acp,U n

n uγ (·) K(γ , z) QΓ,U cΓ,U (z) dz γ ∈Γ

p

Rd

n K(·, z) + ωδ0 /2 (K)(·, z) QΓ,U cΓ,U (z) dz

p

Rd

C0 r0n cp,U

(3.30)

where C0 = sup K(· + z, z) + sup ωδ0 /2 (K)(· + z, z) . z∈Rd

1

z∈Rd

1

(3.31)

Hence the exponential convergence of the sequence Fn in the · p,U norm follows from (3.27) and (3.30).

2436


Numerical stability and stopping rule. Next let us consider the numerical stability of the iterative algorithm (3.26). Assume that the numerical error in n-th iterative step in the iterative algorithm (3.26) is n , n 0, i.e.,

F˜0 = Ac0 + 0 , Fñ = F˜0 + (I − A)Fñ−1 + n ,

(3.32)

n 1.

Let Fn = (fn (γ ))γ ∈Γ , n 0, where fn , n 0, are given in the iterative approximation– projection reconstruction algorithm (3.1) with initial data c0 . By induction, we obtain Fñ − Fn = −

n−1 (I − A)n−1−k A˜ k + ñ ,

(3.33)

k=0

where ˜0 = 0 and ˜k = (k + 1) 0 + 1 + · · · + k for k 1. Therefore Fñ − Fn p,U

n−1 (I − A)n−1−k A˜ k + ñ p,U p,U k=0

n−1

C0 r0n−1−k ˜k p,U + ñ p,U

k=0

C0

n−1

r0n−1−k

(k + 1) 0 p,U +

k

j p,U

j =1

k=0

+ (n + 1) 0 p,U +

n

j p,U

j =1

n 1 − r0 + C0 j p,U . (n + 1) 0 p,U + 1 − r0

(3.34)

j =1

Denote the limit of Fn as n tends to infinity by F∞ . By (3.27) and (3.30) we have Fn − F∞ p,U

∞

C0 r0k+1 c0 p,U

k=n

C0 r0 n r c0 p,U . 1 − r0 0

(3.35)

Define the sample-to-numerical-error ratio (SNER) of the iterative algorithm (3.32) in the logarithmic decibel scale by SNER(dB) = 20 inf log10 n1

nc0 p,U . n 0 p,U + nj=1 j p,U

(3.36)

Then Fñ − Fn p,U

1 − r0 + C0 (n + 1)10−SNER(dB)/20 c0 p,U , 1 − r0

(3.37)


2437

which together with (3.35) implies that Fñ − F∞ p,U

1 − r0 + C n+1 r0 + (n + 1)10−SNER(dB)/20 c0 p,U . 1 − r0

(3.38)

This suggests that a reasonable stopping step n1 in the iterative algorithm (3.26) is n1 =

SNER(dB) log10 (ln(1/r0 )) − −1 , 20 log10 (1/r0 ) log10 1/r0

(3.39)

as the function f (y) = r0 + y10−SNER(dB)/20 attains the absolute minimum at y

y0 :=

log10 (ln(1/r0 )) SNER(dB) − . 20 log10 (1/r0 ) log10 1/r0

(3.40)

4. Iterative frame reconstruction algorithm In this section, we study the convergence and consistency of the iterative frame algorithm for reconstructing a signal in the reproducing kernel subspace of Lp (Rd ) associated with an idempotent integral operator from its samples taken a relatively-separated set with sufficient small gap. The readers may refer to [13,14] for an introduction to frame theory, and [4,9,11,15, 30,35,49] for various frame algorithms to reconstruct a signal from its samples. Theorem 4.1. Let 1 p ∞, T be an idempotent integral operator whose kernel K satisfies (1.5) and (1.6), V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T , and δ1 > 0 be so chosen that r2 := (2r1 + r0 )r0 < 1,

(4.1)

where r0 := sup ωδ1 /2 (K)(· + z, z) z∈Rd

L1 (Rd )

and r1 := sup K(· + z, z) z∈Rd

L1 (Rd )

.

Let Γ be a relatively-separated subset of Rd with gap δ1 , U = {uγ }γ ∈Γ be a BUPU associated with the covering {γ + [−δ1 /2, δ1 /2]d }γ ∈Γ , and SΓ,U f (x) :=

γ ∈Γ

(Tf )(γ )uγ L1 (Rd ) K(x, γ ),

f ∈ Lp R d

(4.2)

2438


be the preconditioned frame operator on Lp (Rd ). Given a sequence c0 = (c0 (γ ))γ ∈Γ ∈ p (Γ ), we define the iterative frame reconstruction algorithm by ⎧ ⎨ f0 = c0 (γ )uγ L1 (Rd ) K(·, γ ), ⎩

(4.3)

γ ∈Γ

fn = f0 + fn−1 − SΓ,U fn−1 ,

n 1.

Then the iterative algorithm (4.3) converges to f∞ exponentially and is consistent. Moreover, f ∞ = RF f 0 ,

(4.4)

where RF := T +

∞ (T − SΓ,U )n

(4.5)

n=1

defines a bounded integral operator on Lp (Rd ) and is a pseudo-inverse of the preconditioned frame operator SΓ,U , i.e., RF T = T RF = RF

and RF SΓ,U = SΓ,U RF = T .

(4.6)

Furthermore, the kernel KF (x, y) of the integral operator RF satisfies (1.5), (1.6), and KF (x, y) =

K(x, z1 )KF (z1 , z2 )K(z2 , y) dz1 dz2

for all x, y ∈ Rd .

(4.7)

Rd Rd

Proof. Define an integral operator CΓ,U by CΓ,U f (x) = K(x, γ ) − K(x, z) uγ (z) K(γ , y) − K(z, y) f (y) dy dz γ ∈Γ

Rd Rd

for all f ∈ Lp ,

(4.8)

and let Q∗Γ,U be the adjoint of the integral operator QΓ,U in (3.4), i.e., Q∗Γ,U f (x) =

K(γ , x) − K(y, x) uγ (y) f (y) dy Rd

for all f ∈ Lp .

(4.9)

γ ∈Γ

Then SΓ,U − T = T QΓ,U + Q∗Γ,U T + CΓ,U , which implies that

(4.10)


SΓ,U f − Tf p T QΓ,U f p + Q∗Γ,U Tf p + CΓ,U f p T hδ1 /2 (· − y) f (y) dy + hδ1 /2 (z − ·) Tf (z) dz p

Rd

Rd

2439

p

dy dz f (y) + h (· − z)h (z − y) δ1 /2 δ1 /2

p

Rd Rd

r2 f p

for all f ∈ V ,

(4.11)

where hδ = supz ∈Rd ωδ (K)(· + z , z ). By the iterative algorithm (4.3), fn = f0 +

n (T − SΓ,U )k f0

for all n 1.

(4.12)

k=1

This together with (4.11) proves the exponential convergence of fn , n 0, and the limit function f∞ is given by (4.4). By (1.2), (4.2) and Theorem A.1 in Appendix A, we have SΓ,U T = T SΓ,U = SΓ,U .

(4.13)

This together with the exponential convergence of the right-hand side of Eq. (4.5) establishes that RF is a bounded operator and satisfies (4.6), and hence it is the pseudo-inverse of SΓ,U . The consistency of the frame iterative algorithm (4.3) follows from (4.4) and the fact that f0 = SΓ,U g if the initial data c0 = (g(γ ))γ ∈Γ is the sample of g ∈ V taken on the set Γ . From (1.5), (4.1), (4.8), (4.9) and (4.10), it follows that ∞ (r2 )n < ∞. sup KF · + z , z sup K · + z , z + 1

z ∈Rd

z ∈Rd

1

n=1

Hence KF satisfies the off-diagonal decay property (1.5). The reproducing equality (4.7) follows from T R F T = RF by (4.6). The regularity property (1.6) for the kernel KF holds because of the off-diagonal decay property (1.5) for the kernel F , the regularity property (1.6) for the kernel K of the idempotent operator T , and the following estimate ωδ (K)(x, z1 )KF (z1 , z2 ) K(z2 , y) + ωδ (K)(z2 , y) dz1 dz2 ωδ (KF )(x, y) Rd Rd

+ Rd Rd

by (4.7).

2

K(x, z1 )KF (z1 , z2 )ωδ (K)(z2 , y) dz1 dz2

2440


5. Asymptotic pointwise error estimates for reconstruction algorithms In this section, we discuss the asymptotic pointwise error estimate for reconstructing a signal from its samples corrupted by white noises, as the maximal gap of the sampling set tends to zero. Theorem 5.1. Let 1 p ∞, T be an idempotent integral operator whose kernel K satisfies (1.5) and (1.6), and V be the reproducing kernel subspace of Lp (Rd ) associated with the operator T . Let Γ be a relatively-separated subset of Rd with gap δ, U := {uγ }γ ∈Γ be a BUPU associated with the covering {γ + [−δ/2, δ/2]d }γ ∈Γ , and R := {Rγ (x)}γ ∈Γ be either the displayer {(uγ L1 (Rd ) )−1 RAP uγ }γ ∈Γ in the approximation–projection reconstruction algorithm or the displayer {RF K(·, γ )}γ ∈Γ in the frame reconstruction algorithm where the operators RAP and RF are defined in (3.13) and (4.5) respectively. Assume that (γ ), γ ∈ Γ , are bounded i.i.d. noises with zero mean and σ 2 variance, i.e.,

(γ ) ∈ [−B, B],

E (γ ) = 0,

and

Var (γ ) = σ 2

(5.1)

for some positive constant B, and that the initial data c0 is the sample of a signal g ∈ V taken on Γ corrupted by random noise := ( (γ ))γ ∈Γ , i.e., c0 = g(γ ) + (γ ) γ ∈Γ .

(5.2)

E g(x) − Rc0 (x) = 0

(5.3)

Then for any x ∈ Rd

and 2 uγ 2L1 (Rd ) Rγ (x) Var g(x) − Rc0 (x) = γ ∈Γ

σ 2 sup uγ L1 (Rd ) γ ∈Γ

K(x, z)2 dz + o(1) as δ → 0,

(5.4)

Rd

where Rc0 (x) =

γ ∈Γ

c0 (γ )uγ L1 (Rd ) Rγ (x)

for all c0 = c0 (γ ) γ ∈Γ ∈ ∞ (Γ ).

(5.5)

Furthermore if uγ L1 (Rd ) = α(δ) 1 + o(1) as δ → 0

(5.6)

for some positive numbers α(δ) independent of γ , then the inequality in (5.4) becomes an equality, i.e.,


Var g(x) − Rc0 (x) = α(δ)σ 2

K(x, z)2 dz + o(1)

2441

(5.7)

Rd

as δ tends to zero. Remark 5.1. The error estimate (5.7) is established in [5] for reconstructing signals in a finitelygenerated shift-invariant subspace of L2 (Rd ) from corrupted uniform sampling data via the frame reconstruction algorithm. More precisely, Γ = δZd , uγ (x) = χ[−δ/2,δ/2]d (x − γ ) for γ ∈ Γ , the idempotent operator T is defined in (2.4), and the range space associated with the idempotent operator T is the shift-invariant space V2 (φ1 , . . . , φr ) in (2.5). Remark 5.2. By the definition of a BUPU associated with the covering {γ + [−δ/2, δ/2]d }γ ∈Γ of Rd , we have uγ L1 (Rd ) δ d .

(5.8)

The above inequality becomes an equality when Γ = δZd and uγ = χ[−δ/2,δ/2]d . It is expensive to find the operators RAP and RF when the sampling set has very small gap δ. As noticed in the proof of Theorem 5.1, both operators are close to the idempotent operator T when the sampling set has very small gap. Then a natural replacement of the displayer Rγ in (5.5) is either (uγ L1 (Rd ) )−1 T uγ or K(·, γ ). In both cases, the variance estimates in (5.4) and (5.7) still hold, but the unbiased condition (5.4) does not. To prove Theorem 5.1, we need several technical lemmas. The first lemma is a slight generalization of Theorem 5.1. Lemma 5.2. Let the operator T , the kernel K, the reproducing kernel space V , the sampling set Γ , the bounded uniform partition of unity U = {uγ }γ ∈Γ , the random noise , and the variance σ of the noise be as in Theorem 5.1, and let the displayer R := {Rγ (x)}γ ∈Γ satisfy g(x) =

g(γ )uγ L1 (Rd ) Rγ (x)

for all g ∈ V ,

(5.9)

γ ∈Γ

and lim sup

sup

δ→0 γ ∈Γ z∈γ +[−δ/2,δ/2]d

Rγ (· + z) − K(· + z, z)

L1 (Rd )

= 0.

(5.10)

Then (5.3), (5.4) and (5.7) hold. Proof. Set hδ (x) = sup

sup

γ ∈Γ z∈γ +[−δ/2,δ/2]d

Rγ (x + z) − K(x + z, z).

(5.11)

2442


By (1.5), (5.10) and (5.11), we have

uγ 1 Rγ (x)

γ ∈Γ

uγ (z) K(x, z) + hδ (x − z) dz

γ ∈Γ

Rd

sup K(· + z, z) + hδ 1 < ∞. 1

z∈Rd

(5.12)

This together with (5.1) and (5.9) leads to E g(x) − Rc0 (x) = E

(γ )uγ 1 Rγ (x) =

γ ∈Γ

E (γ ) uγ 1 Rγ (x) = 0,

(5.13)

γ ∈Γ

and the unbiased property (5.3) for the reconstruction process in (5.5) follows. By (5.1), (5.3) and (5.12), we obtain 2 Var g(x) − Rc0 (x) = E

(γ )uγ 1 Rγ (x) γ ∈Γ

= σ2

2 uγ 21 Rγ (x) .

γ ∈Γ

Therefore 2 2 Var g(x) − Rc0 (x) σ sup uγ 1 uγ 1 Rγ (x) γ ∈Γ

σ2

σ

2

γ ∈Γ

γ ∈Γ

2 K(x, z) + hδ (x − z) dz sup uγ 1

sup uγ 1

γ ∈Γ

Rd

K(x, z)2 dz + o(1) ,

(5.14)

Rd

where we have used (5.10) and (5.11) to obtain the last two estimates. Hence the variance estimate (5.4) for the reconstruction process in (5.5) is established. By (5.6), (5.10) and (5.14), we get 2 2 Var g(x) − Rc0 (x) = σ α(δ) + o(1) uγ 1 Rγ (x) γ ∈Γ


2443

2 K(x, z) + O hδ (x − z) dz = σ 2 α(δ) + o(1) Rd

2 2 K(x, z) dz + o(1) , = σ α(δ)

(5.15)

Rd

and hence (5.7) is proved.

2

Lemma 5.3. Let the operator T , the kernel K, the reproducing kernel space V , the sampling set Γ , the bounded uniform partition of unity U = {uγ }γ ∈Γ , the random noise , and the variance σ of the noise be as in Theorem 5.1, and let the displayer R = {Rγ }γ ∈Γ be defined by −1 Rγ = uγ 1 RAP uγ ,

γ ∈Γ

(5.16)

where RAP is given in (3.13). Then the above displayer R satisfies (5.9) and (5.10). Proof. By (3.13), (3.16) and (3.17), the reconstruction formula (5.9) holds for the displayer R in (5.16). Denote the kernel of the integral operators RAP − T by K˜ AP . By (1.2), (3.7), (3.10), (3.13) and (3.18), we have K˜ AP (x, y) =

K(x, z1 )K˜ AP (z1 , z2 )K(z2 , y) dz1 dz2 ,

(5.17)

Rd Rd

and sup K˜ AP · + z , z z ∈Rd

1

∞ n →0 sup K · + z , z sup ωδ/2 (K) · + z , z 1

d n=1 z ∈R

z ∈Rd

1

as δ → 0.

(5.18)

This together with (1.5) and (1.6) implies that sup

sup

γ ∈Γ z ∈γ +[−δ/2,δ/2]d

uγ 1 −1 RAP uγ · + z − K · + z , z

sup ωδ (K) · + z , z z ∈Rd

+ sup

z∈Rd

→0

1

1

K(· + z, z1 )K˜ AP (z1 , z2 ) K(z2 , z) + ωδ (K)(z2 , z) dz1 dz2

1

Rd Rd

as δ → 0.

Hence (5.10) follows.

(5.19) 2

2444


Lemma 5.4. Let the operator T , the kernel K, the reproducing kernel space V , the sampling set Γ , the bounded uniform partition of unity U = {uγ }γ ∈Γ , the random noise , and the variance σ of the noise be as in Theorem 5.1, and let the displayer R = {Rγ }γ ∈Γ be defined by Rγ = RF K(·, γ ),

γ ∈Γ

(5.20)

where RF is given in (4.5). Then the above displayer R satisfies (5.9) and (5.10). Proof. The reconstruction formula (5.9) follows from Theorem 4.1. Denote the integral kernel of the integral operator RF − T by K˜ F . Then K(x, z1 )K˜ F (z1 , z2 )K(z2 , y) dz1 dz2 , K˜ F (x, y) =

(5.21)

Rd Rd

and sup K˜ F (· + z, z) z∈Rd

1

∞ n 2 n 2 sup K(· + z, z) + sup ωδ (K)(· + z, z) sup ωδ (K)(· + z, z)

→0

1

z∈Rd

n=1

1

z∈Rd

z∈Rd

as δ → 0

(5.22)

by (1.6), (4.5), and (4.10). Therefore sup

Rd

sup

γ ∈Γ z∈γ +[−δ/2,δ/2]d

RF K(·, γ ) (x + z) − K(x + z, z) dx

sup

+

sup

γ ∈Γ z∈γ +[−δ/2,δ/2]d

Rd

sup

Rd

sup

K(x + z, γ ) − K(x + z, z) dx ˜ K(x + z, z ) K (z , z )K(z , γ ) dz d 1 F 1 2 2 1 2 dx d

γ ∈Γ z∈γ +[−δ/2,δ/2]

Rd Rd

sup ωδ/2 (K) x + z , z dx

z ∈Rd

Rd

+

Rd Rd Rd

×

1

sup K˜ F z1 − z2 + z , z sup K x − z1 + z , z

z ∈Rd

z ∈Rd

sup K z2 + z , z + sup ωδ/2 (K) z2 + z , z dz1 d2 dx

z ∈Rd

z ∈Rd

→ 0 as δ → 0. Then (5.10) is established for the displayer R in (5.20).

2


2445

Proof of Theorem 5.1. The conclusions in Theorem 5.1 follows directly from Lemmas 5.2, 5.3 and 5.4. 2 Acknowledgments The authors would like to thank Professors Akram Aldroubi, Deguang Han, Karlheinz Gröchenig, Wai-Shing Tang, and Jun Xian for their discussion and suggestions. The authors also thank the referee for his/her suggestion on the presentation of this paper. Rd ) associated with idempotent Appendix A. Reproducing kernel subspaces of Lp (R integral operators The range space associated with an idempotent operator T on Lp (Rd ) whose kernel satisfies (1.5) and (1.6) include the space of all p-integrable non-uniform splines of order n satisfying n −1 continuity conditions at each knot (Example A.3), and the space introduced in [43] for modeling signals with finite rate of innovation (Example A.4). In this appendix, we establish some properties of such range spaces, particularly the reproducing kernel property in Theorem A.1 and the frame property in Theorem A.2. A.1. Reproducing kernel property In this subsection, we show that the range space of an idempotent operator on Lp (Rd ) whose kernel satisfies (1.5) and (1.6) has some properties similar to the ones for a reproducing kernel Hilbert subspace of L2 (Rd ). Theorem A.1. Let T be an idempotent integral operator on Lp (Rd ) whose kernel K satisfies (1.5) and (1.6), and V be the range space of the operator T . Set aδ (q) = δ −d+d/q sup K(· + z, z) z∈Rd

× sup K(· + z, z) z∈Rd

1/q L1 (Rd )

L1 (Rd )

+ sup ωδ (K)(· + z, z) z∈Rd

1−1/q L1 (Rd )

and 1−1/q −d+d/q δ bδ (q) = 6d + 1 sup ωδ (K)(· + z, z) z∈Rd

for δ > 0 and 1 q ∞. Then (i) V is a reproducing kernel subspace of Lp (Rd ). Moreover, f (x) aδ p/(p − 1) f for any f ∈ V and δ > 0.

Lp (Rd )

L1 (Rd )

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(ii) The kernel K satisfies the “reproducing kernel property”: K(x, z)K(z, y) dz = K(x, y)

for all x, y ∈ Rd .

(A.1)

Rd

(iii) K(·, y) ∈ V for any y ∈ Rd . (iv) The functions K(x, ·), K(·, y), ωδ (K)(x, ·) and ωδ (K)(·, y) belong to Lq (Rd ) for all x, y ∈ Rd and 1 q ∞, and their Lq (Rd )-norms are uniformly bounded. Moreover, max sup K(x, ·)Lq (Rd ) , sup K(·, y)Lq (Rd ) aδ (q)

(A.2)

max sup ωδ (K)(x, ·)Lq (Rd ) , sup ωδ (K)(·, y)Lq (Rd ) bδ (q).

(A.3)

x∈Rd

y∈Rd

and

x∈Rd

y∈Rd

Proof. (iv): By the definition of the modulus of continuity,

K(x, y) δ −d

K(x, z) + ωδ (K)(x, z) dz

(A.4)

kδ+[−δ/2,δ/2]d

where y, z ∈ kδ + [−δ/2, δ/2]d and x ∈ Rd . Thus sup K(x, ·)∞ δ −d sup K(· + z, z) + sup ωδ (K)(· + z, z)

x∈Rd

1

z∈Rd

1

z∈Rd

and sup K(x, ·)1 sup K(· + z, z) .

x∈Rd

z∈Rd

1

Interpolating the above estimates for the L1 and L∞ norms of K(x, ·) yields sup K(x, ·)q aδ (q).

x∈Rd

Similarly, we have that supy∈Rd K(·, y)q aδ (q). Therefore (A.2) follows. The estimate (A.3) for ωδ (K) can be established by similar argument used in the proof of the estimate (A.2) except replacing (A.4) by the following two inequalities: ωδ (K)(x, y) δ

−d

kδ+[−δ/2,δ/2]d

ωδ (K)(x, z) + ω2δ (K)(x, z) dz


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for any x ∈ Rd , y ∈ kδ + [−δ/2, δ/2]d and k ∈ Zd , and

ω2δ (K)(x, y)

ωδ (K) x + δ, y + δ

(A.5)

, ∈{−1,0,1}d

for all x, y ∈ Rd . (i): By (1.4) and (A.2), we have that |f (x)| K(x, ·)p/(p−1) f p aδ (p/(p − 1))f p for all x ∈ Rd and f ∈ V . Then (A.1) holds and V is a reproducing kernel subspace of Lp . (ii): Noting that Rd

2 sup K(x + z, z) dx < ∞, sup K(x + z, y)K(y, z) dy dx

z∈Rd

Rd

Rd

z∈Rd

we then have that the kernel A(x, y) := Rd K(x + z, y)K(y, z) dy − K(x, y) of the linear operator T 2 − T satisfies supz∈Rd |A(· + z, z)|1 < ∞. This together with (1.2) proves (A.1). (iii): The conclusion that K(·, y) ∈ V for any y ∈ Rd follows from (A.1) and (A.2). 2 A.2. Frame property In this subsection, we show that the range space of an idempotent integral operator whose kernel satisfies (1.5) and (1.6) has localized frames. Let 1 p ∞, V ⊂ Lp and W ⊂ Lp/(p−1) . We say that the p-frame Φ˜ = {φ˜ λ }λ∈Λ ⊂ W for V and the p/(p − 1)-frame Φ = {φλ }λ∈Λ ⊂ V for W form a dual pair if the following reconstruction formulae hold: f=

f, φ˜ λ φλ

for all f ∈ V ,

(A.6)

g, φλ φ˜ λ

for all g ∈ W.

(A.7)

λ∈Λ

and g=

λ∈Λ

Here we denote by f, g the standard action (1.11) between a function f ∈ Lp and a function g ∈ Lp/(p−1) . Theorem A.2. Let 1 p ∞, T be an idempotent integral operator on Lp (Rd ) whose kernel K satisfies (1.5) and (1.6), T ∗ be the adjoint of the idempotent operator T , i.e., ∗

T g(x) =

K(y, x)g(y) dy

for all g ∈ Lp/(p−1) Rd ,

(A.8)

Rd

and let V and V ∗ be the range spaces of the operator T on Lp (Rd ) and the operator T ∗ on Lp/(p−1) (Rd ) respectively. Then there exist a relatively-separated subset Λ, and two families Φ := {φλ }λ∈Λ of functions φλ ∈ V and Φ˜ := {φ˜ λ }λ∈Λ of functions φ˜ λ ∈ V ∗ such that

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(i) Both Φ and Φ˜ are localized in the sense that φλ (x) + φ˜ λ (x) h(x − λ), ωδ (φλ )(x) + ωδ (φ˜ λ )(x) hδ (x − λ) (ii) (iii) (iv)

for all λ ∈ Λ and x ∈ Rd ,

where h and hδ are integrable functions with limδ→0 hδ 1 = 0. Φ˜ is a p-frame for V and Φ is a p/(p − 1)-frame for V ∗ . Φ and Φ˜ form a dual pair. Both V and V ∗ are generated by Φ and Φ˜ respectively in the sense that p V = Vp (Φ) := c(λ)φλ c(λ) λ∈Λ ∈ (Λ) ,

(A.9)

(A.10)

λ∈Λ

and ˜ := V ∗ = Vp/(p−1) (Φ)

˜ c(λ) ˜ φ˜ λ c(λ) ∈ p/(p−1) (Λ) . λ∈Λ

(A.11)

λ∈Λ

Remark 5.3. The space Vp (Φ) was introduced in [43] to model signals with finite rate of innovations. From Theorem A.2, we see that signals in a reproducing kernel subspace associated with an idempotent operator on Lp (Rd ) with its kernel satisfying (1.5) and (1.6) have finite rate of innovation. Proof of Theorem A.2. Let δ0 > 0 be a sufficiently small positive number chosen later. Define the operator Tδ0 by Tδ0 f (x) = Kδ0 (x, y)f (y) dy f ∈ Lp Rd , (A.12) Rd

where Kδ0 (x, y) = δ0−d

[−δ0 /2,δ0

/2]d

[−δ0 /2,δ0

K(x, λ + z1 )K(λ + z2 , y) dz1 dz2 .

(A.13)

λ∈δ0 Zd

/2]d

Then Tδ0 T = T T δ0 = Tδ0

(A.14)

by (1.2), and Kδ (x, y) − K(x, y) 0

K(x, z)ωδ (K)(z, y) dz 0

(A.15)

Rd

by Theorem A.1. Therefore Tδ0 f − Tf p r1 (δ0 )f p

for all f ∈ Lp ,

(A.16)


2449

where r1 (δ0 ) = supz∈Rd |K(· + z, z)|1 supz∈Rd ωδ0 (K)(· + z, z)1 . Let δ0 > 0 be so chosen that r1 (δ0 ) < 1. The existence of such a positive number follows from (1.5) and (1.6). Then n it follows from (A.14), (A.15) and (A.16) that the operator Tδ†0 := T + ∞ n=1 (T − Tδ0 ) is a bounded integral operator with the property that Tδ†0 Tδ0 = Tδ0 Tδ†0 = T and that the kernel KD,δ0

of the operator Tδ†0 satisfies

sup KD,δ0 (· + z, z) < ∞ 1

z∈Rd

and lim sup ωδ (KD,δ0 )(· + z, z) = 0. 1

δ→0 z∈Rd

Define ⎧ −d/p ⎪ ⎪ φ (x) = δ λ ⎪ 0 ⎪ ⎪ ⎨ Rd

KD,δ0 (x, z1 )K(z1 , λ + z2 ) dz2 dz1 , [−δ0 /2,δ0 /2]d

⎪ −d+d/p ⎪ ⎪ φ˜ λ (x) = δ0 ⎪ ⎪ ⎩

(A.17) K(λ + z, x) dz

[−δ0 /2,δ0 /2]d

for all λ ∈ δ0 Zd , and set Φ = {φλ }λ∈δ0 Zd and Φ˜ = {φ˜ λ }λ∈δ0 Zd . Then one may verify that the above two families Φ and Φ˜ of functions satisfy all required properties. We leave the detailed verification for the interested readers. 2 A.3. Examples In this subsection, we present two examples of a reproducing kernel space associated with an idempotent integral operator on Lp . Example A.3. (See [38].) Let n 1, Λ = {λk }k∈Z be a bi-infinite increasing sequence of real numbers with 0 < infk∈Z (λk+1 − λk ) supk∈Z (λk+1 − λk ) < ∞, and Snn−1 (Λ) = f ∈ C n−1 (R): f |[λk ,λk+1 ] is a polynomial having degree at most n for each k ∈ Z .

(A.18)

Let Bi be the normalized B-spline associated with the knots λi , . . . , λi+n+1 , and define its autocorrelation matrix A = (Bi , Bj )i,j ∈Z . Then the infinite matrix A is invertible and its inverse B = (bij )i,j ∈Z has exponential off-diagonal decay, that is, there exist constants C and such that |bij | C exp(− |i − j |) for all i, j ∈ Z. Define K(x, y) =

i,j ∈Z

Bi (x)bij Bj (y)

2450


and Tf (x) =

K(x, y)f (y) dy. R

Then one may verify that the above integral operator T is an idempotent operator on Lp (R), the kernel K of the operator T satisfies (1.5) and (1.6), and Snn−1 (Λ) ∩ Lp (R) is the range of the operator T on Lp (R). The spline model has many practical advantages over the bandlimited model in Shannon’s sampling theory, and has been well-studied (see [44,46,48] and the references therein). Example A.4. (See [43].) Let Λ be a relatively-separated subset of Rd with positive gap, Φ = {φλ }λ∈Λ and Φ˜ = {φ˜ λ }λ∈Λ be two families of functions such that φλ (x) + φ˜ λ (x) h(x − λ), x ∈ Rd , and ωδ (φλ )(x) + ωδ (φ˜ λ )(x) hδ (x − λ),

x ∈ Rd ,

hold for all λ ∈ Λ and δ > 0, where h and hδ are functions in the Wiener amalgam space W with limδ→0 hδ W = 0. Then one may verify that the kernel function K(x, y) :=

φλ (x)φ˜ λ (y)

(A.19)

λ∈Λ

satisfies (1.5) and (1.6). If we further assume that Φ and Φ˜ satisfy φλ (x)φ˜ λ (x) dx = δλ,λ for all λ, λ ∈ Λ, Rd

where δλ,λ stands for the Kronecker symbol, then the operator T with the kernel K in (A.19) is an idempotent operator on L2 . In this case,

c(λ)2 < ∞ V2 (Φ) := c(λ)φλ (x) (A.20) λ∈Λ

λ∈Λ

is the range space of the operator T on L2 and hence a reproducing kernel subspace of L2 . A special case of the above space V2 (Φ) is the finitely-generated shift-invariant space V2 (φ1 , . . . , φr ) in (2.5), see [1,4,8,30] and references therein. References [1] A. Aldroubi, A.G. Baskakov, I. Krishtal, Slanted matrices, Banach frames, and sampling, J. Funct. Anal. 255 (2008) 1667–1691. [2] A. Aldroubi, H. Feichtinger, Exact iterative reconstruction algorithm for multivariate irregularly sampled functions in spline-like spaces: the Lp theory, Proc. Amer. Math. Soc. 126 (1998) 2677–2686.


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[3] A. Aldroubi, K. Gröchenig, Beurling–Landau-type theorems for non-uniform sampling in shift invariant spline spaces, J. Fourier Anal. Appl. 6 (2000) 93–103. [4] A. Aldroubi, K. Gröchenig, Nonuniform sampling and reconstruction in shift-invariant space, SIAM Rev. 43 (2001) 585–620. [5] A. Aldroubi, C. Leonetti, Q. Sun, Error analysis of frame reconstruction from noisy samples, IEEE Trans. Signal Process. 56 (2008) 2311–2325. [6] A. Aldroubi, Q. Sun, W.-S. Tang, p-Frames and shift invariant subspaces of Lp , J. Fourier Anal. Appl. 7 (2001) 1–21. [7] A. Aldroubi, Q. Sun, W.-S. Tang, Non-uniform average sampling and reconstruction in multiply generated shiftinvariant spaces, Constr. Approx. 20 (2004) 173–189. [8] A. Aldroubi, Q. Sun, W.-S. Tang, Convolution, average sampling, and a Calderon resolution of the identity for shift-invariant spaces, J. Fourier Anal. Appl. 22 (2005) 215–244. [9] A. Aldroubi, M. Unser, Sampling procedure in functions spaces and asymptotic equivalence with Shannon’s sampling theory, Numer. Funct. Anal. Optim. 15 (1994) 1–21. [10] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68 (1950) 337–404. [11] J.J. Benedetto, Irregular sampling and frames, in: C.K. Chui (Ed.), Wavelet: A Tutorial in Theory and Applications, Academic Press, California, 1992, pp. 445–507. [12] N. Bi, M.Z. Nashed, Q. Sun, Reconstructing signals with finite rate of innovation from noisy samples, Acta Appl. Math. 107 (2009) 339–372. [13] P.G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000) 129–201. [14] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhäuser, 2003. [15] W. Chen, S. Itoh, J. Shiki, Irregular sampling theorems for wavelet subspaces, IEEE Trans. Inform. Theory 44 (1998) 1131–1142. [16] J.G. Christensen, G. Olafsson, Examples of co-orbit spaces for dual pairs, Acta Appl. Math. 107 (2009) 25–48. [17] P.L. Dragotti, M. Vetterli, T. Blu, Sampling moments and reconstructing signals of finite rate of innovation: Shannon meets Strang-Fix, IEEE Trans. Signal Process. 55 (2007) 1741–1757. [18] Y.C. Eldar, M. Unser, Nonideal sampling and interpolation from noisy observations in shift-invariant spaces, IEEE Trans. Signal Process. 54 (2006) 2636–2651. [19] H.G. Feichtinger, Generalized amalgams with application to Fourier transform, Canad. J. Math. 42 (1990) 395–409. [20] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions I, J. Funct. Anal. 86 (1989) 307–340. [21] H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations II, Monatsh. Math. 108 (1989) 129–148. [22] H.G. Feichtinger, K. Gröchenig, Iterative reconstruction of multivariate band-limited functions from irregular sampling values, SIAM J. Math. Anal. 231 (1992) 244–261. [23] M. Fornasier, L. Gori, Sampling theorems on bounded domains, J. Comput. Appl. Math. 221 (2008) 376–385. [24] M. Fornasier, H. Rauhut, Continuous frames, function spaces, and the discretization problem, J. Fourier Anal. Appl. 11 (2005) 245–287. [25] K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math. 112 (1991) 1–42. [26] K. Gröchenig, Reconstructing algorithms in irregular sampling, Math. Comp. 59 (1992) 181–194. [27] D. Han, M.Z. Nashed, Q. Sun, Sampling expansions in reproducing kernel Hilbert and Banach spaces, Numer. Funct. Anal. Optim. 26 (2009) 971–987. [28] J.A. Jerri, The Shannon sampling theorem – its various extensions and applications: A tutorial review, Proc. IEEE 65 (1977) 1565–1596. [29] V.G. Kurbatov, Functional Differential Operators and Equations, Kluwer Academic Publishers, 1999. [30] Y. Liu, G.G. Walter, Irregular sampling in wavelet subspaces, J. Fourier Anal. Appl. 2 (1996) 181–189. [31] I. Maravic, M. Vetterli, Sampling and reconstruction of signals with finite rate of innovation in the presence of noise, IEEE Trans. Signal Process. 53 (2005) 2788–2805. [32] P. Marziliano, M. Vetterli, T. Blu, Sampling and exact reconstruction of bandlimited signals with shot noise, IEEE Trans. Inform. Theory 52 (2006) 2230–2233. [33] C. van der Mee, M.Z. Nashed, S. Seatzu, Sampling expansions and interpolation in unitarily translation invariant reproducing kernel Hilbert spaces, Adv. Comput. Math. 19 (2003) 355–372. [34] M.Z. Nashed, Q. Sun, W.-S. Tang, Average sampling in L2 , C. R. Math. Acad. Sci. Paris Ser. I 347 (2009) 1007– 1010. [35] M.Z. Nashed, G.G. Walter, General sampling theorems for functions in reproducing kernel Hilbert spaces, Math. Control Signals Systems 4 (1991) 363–390.

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[36] M. Pawlak, E. Rafajlowicz, A. Krzyzak, Postfiltering versus prefiltering for signal recovery from noisy samples, IEEE Trans. Inform. Theory 49 (2003) 3195–3212. [37] G.K. Rohde, C.A. Berenstein, D.M. Healy Jr., Measuring image similarity in the presence of noise, in: Proc. SPIE, vol. 5747, 2005, pp. 132–143. [38] L.L. Schumaker, Spline Functions: Basic Theory, John Wiley & Sons, New York, 1981. [39] C.E. Shannon, Communications in the presence of noise, Proc. IRE 37 (1949) 10–21. [40] S. Smale, D.X. Zhou, Shannon sampling and function reconstruction from point values, Bull. Amer. Math. Soc. 41 (2004) 279–305. [41] Q. Sun, Local reconstruction for sampling in shift-invariant spaces, Adv. Comp. Math., doi:10.1007/s10444-0089109-0, in press. [42] Q. Sun, Wiener’s lemma for localized integral operators, Appl. Comput. Harmon. Anal. 25 (2008) 148–167. [43] Q. Sun, Frames in spaces with finite rate of innovation, Adv. Comput. Math. 28 (2008) 301–329. [44] Q. Sun, Non-uniform sampling and reconstruction for signals with finite rate of innovations, SIAM J. Math. Anal. 38 (2006) 1389–1422. [45] W. Sun, X. Zhou, Characterization of local sampling sequences for spline subspaces, Adv. Comput. Math. 30 (2009) 153–175. [46] M. Unser, Sampling – 50 years after Shannon, Proc. IEEE 88 (2000) 569–587. [47] M. Vetterli, P. Marziliano, T. Blu, Sampling signals with finite rate of innovation, IEEE Trans. Signal Process. 50 (2002) 1417–1428. [48] G. Wahba, Spline Models for Observational Data, in: CBSM-NSF Regional Conf. Ser. Appl. Math., vol. 59, SIAM, Philadelphia, 1990. [49] K. Yao, Applications of reproducing kernel Hilbert spaces – bandlimited signal models, Inform. Control 11 (1967) 429–444.


The role of restriction theorems in harmonic analysis on harmonic NA groups ✩ Pratyoosh Kumar a , Swagato K. Ray a , Rudra P. Sarkar b,∗ a Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur 208016, India b Stat-Math Unit, Indian Statistical Institute, 203 B. T. Rd., Calcutta 700108, India

Received 1 July 2009; accepted 4 January 2010 Available online 15 January 2010 Communicated by H. Brezis

Abstract We shall obtain inequalities for Fourier transform via moduli of continuity on NA groups. These results in particular settle the conjecture posed in a recent paper by W.O. Bray and M. Pinsky in the context of noncompact rank one symmetric spaces. These problems naturally demand versions of Fourier restriction theorem on these spaces which we shall prove. We shall also elaborate on the connection between the restriction theorem and the Kunze–Stein phenomena on NA groups. For noncompact Riemannian symmetric spaces of rank one analogues of all the results follow the same way. © 2010 Elsevier Inc. All rights reserved. Keywords: Harmonic NA groups; Moduli of continuity; Spherical mean; Kunze–Stein phenomenon

1. Introduction This article was originally inspired by a recent paper of Bray and Pinsky [3] on growth properties of the Fourier transform on Rn and noncompact rank one Riemannian symmetric spaces G/K (see also [4] for improvements on these results). In the context of symmetric spaces the authors in [3,4] deal with the growth properties of the Helgason Fourier transform f(λ, k) and asked about the validity of one of their results without the assumption of K-finiteness ✩

First author is supported by a research fellowship of CSIR, India. This work is also partially supported by a research grant (No. 48/1/2006-R&DII/1488) of NBHM, India. * Corresponding author. E-mail addresses: [email protected] (P. Kumar), [email protected] (S.K. Ray), [email protected] (R.P. Sarkar). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.001

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P. Kumar et al. / Journal of Functional Analysis 258 (2010) 2453–2482

[3, Conjecture 16]. (See Sections 2 and 6 for notation.) Authors in [3] also conjectured about the analogue of these results for harmonic NA groups. Apart from certain nontrivial estimates of the Bessel function and the elementary spherical function φλ the results in [3,4] depend crucially on the Hausdorff–Young (HY) inequality (proved in [12,13]) and the following inequality available in [25,28]: for λ ∈ R, and 1 p < 2 f(λ, ·)

L1 (K)

Cf p .

(1.1)

However (1.1) is far from being the best possible estimate as it does not accommodate the following estimates for λ ∈ R which are not difficult to prove (see [26]), f(λ + iρ, ·) 1 f(λ, ·) 2 Cf , Cf 1 , 1 L (K) L (K) f(λ − iρ, ·) ∞ Cf 1 . L (K)

(1.2)

This necessitates to revisit these results with the hope to prove the best possible version of an inequality of the form f(λ, ·)

Lq (K)

Cf p

(1.3)

as well as its analogue for NA groups. In the context of Rn an inequality of the form (1.3) reads f(λ, ·)

Lq (S n−1 )

Cf p

(where f(λ, ω) is the Euclidean Fourier transform of f in polar coordinates and S n−1 is the (n − 1)-dimensional sphere) and is available in the literature on restriction conjecture (see e.g. [30,29]). As an analogy the inequalities of the form (1.3) will be called restriction theorems. As on Rn one can use duality in (1.3) to prove estimates of the following kind for the Poisson transform Pλ (see [19] for definition): Pλ F Lp (G/K) CF p ,

(1.4)

which are really size estimates of certain matrix entries of the class-1 principal series representation of the underlying semisimple Lie group G. However, the structure of the semisimple Lie group crucially intervenes in these results. An important difference arises because of the analytic continuation of the Fourier transform. It is known that, like radial functions the domain of the Fourier transform f(·, k) of a general Lp function on G/K is a strip Sp in the complex plane (see [25], or Section 2) and hence one needs to deal with complex λ ∈ Sp in the inequality (1.3). It turns out that the exponent q appearing in (1.3) depends on the imaginary part of λ. This is only to be expected in view of (1.2). In the context of symmetric spaces all these are rooted in the Kunze–Stein phenomenon. The fundamental works in this direction are of Herz, Lohoué, Lohoué and Rychener, Cowling, Cowling and Haggerup, etc. (see [20,23,24,5,9]). As we move towards harmonic NA groups, we face fresh difficulties. Harmonic NA groups are also known as Damek–Ricci (DR) spaces and we shall use these names interchangeably. We recall that they are solvable Lie groups. Despite being the most distinguishable prototypes, the rank one noncompact Riemannian symmetric spaces, which sit inside them as Iwasawa NA groups, account for a very thin subclass (see [1]). From the geometric point of view the major


2455

difference of DR spaces with the symmetric spaces is the lack of symmetry (as reflected by the fact that the geodesic inversion is not isometry). On the other hand being a solvable group S is amenable and nonunimodular. It is well known that a noncompact amenable group cannot have Kunze–Stein property (see e.g. [11]). Therefore it is not a priori clear if the results analogous to the symmetric spaces would be true here. But we will see that these differences can be overwhelmed to obtain analogous inequalities for DR spaces. The analogue of the Helgason Fourier transform on DR spaces S = NA has been defined in [2] and involves integration against suitable powers of the Poisson kernel on N . In this paper we shall prove the following version of the restriction theorem. Theorem 1.1 (Restriction on line). Let f be a measurable function on S and α ∈ R. (i) For f ∈ Lp,1 (S), 1 p < 2 and p q p ,

f(α + iγq ρ, n)q dn

1/q

Cp,q f ∗p,1 ,

C1,q = 1.

N

(ii) For f ∈ Lp,∞ (S), 1 < p < 2, p < q < p ,


1/q

Cp,q f ∗p,∞ .

N

The constants Cp,q are independent of α ∈ R. Estimates (i) and (ii) are sharp. We may point out that the use of the Lorentz norms in the theorem above is essential and can be motivated recalling the facts that when 1 < p < 2 then for λ in the boundary of the strip Sp , φλ ∈ Lp ,∞ (S) \ Lp (S), while for λ in the interior of the strip Sp , φλ ∈ Lp ,1 (S). To emphasize the importance of this result it is only fair to say that the inequality corresponding to (i) for G/K is solely responsible for the initiation of the study of the Lorentz space version of the Kunze– Stein phenomenon on semisimple Lie groups (see [6,21] and the references therein). After settling the issue of restriction theorem we turn our attention towards HY inequality. It is clear from (1.2) that, unlike that on Rn , one cannot expect a norm inequality on symmetric spaces and DR spaces. Indeed in these spaces the HY inequality will involve mixed norms in (λ, k) which will change as λ varies over the strip in a manner mentioned above. This generalizes the corresponding result for radial functions on symmetric spaces (see [8]). Results of this genre were initiated in [26]. Here we shall further generalize these results to accommodate our need. After accomplishing these we shall go back to apply these results to obtain new analogues for NA groups of the results in [3,4]. Our next aim is to understand the role of restriction theorem mentioned above in the Kunze– Stein type normed inequalities in these non-Kunze–Stein groups, namely the DR spaces. A precursor to this is an interesting result in [1] which proves certain Kunze–Stein type relations for the right radial convolutors on DR spaces. As an application of the restriction theorems, we will try to prove all the expected results of Kunze–Stein type on Lorentz spaces in the NA groups. The paper is organized as follows. After explaining the preliminaries (in Section 2), in Section 3 we obtain restriction theorem and Hausdorff–Young inequality. In Section 4 we apply these results to obtain analogues of results in [3,4] on growth properties of Fourier transform on NA

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groups. In Section 5 we establish the link between the restriction theorem and the Kunze–Stein phenomenon on these groups. In Section 6 we revisit the symmetric spaces. 2. Preliminaries Most of the preliminaries can be found in [2,1,26]. To make the article self-contained we shall gather only those results which are required for this paper. For a detailed account we refer to [26]. We will follow the standard practice of using the letter C for constant, whose value may change from one line to another. Occasionally the constant C will be suffixed to show its dependency on important parameters. The letters C and R will denote the set of complex and real numbers respectively. We will reserve the notation ·,· for the real L2 -inner product on S. For any measurable function f on S, we define f ∗ by f ∗ (x) = f (x −1 ) for all x ∈ S. Unless mentioned otherwise notation f p will mean the Lp -norm of f on S. A subset relation between normed spaces will always mean the corresponding norm inequality. Everywhere in this article for any p ∈ [1, ∞), p = p/(p − 1) and γp = 2/p − 1, γ∞ = −1. We note that γp = −γp . For a complex number z, we will use z and z to denote respectively the real and imaginary parts of z. For p ∈ [1, 2] we define

Sp = z ∈ C | z| γp ρ . We recall that S1 is the well-known Helgason–Johnson strip. By Sp◦ and ∂Sp we denote respectively the interior and the boundary of the strip Sp . Before entering into the preliminaries of the harmonic NA group let us briefly introduce the Lorentz spaces (see [18,31,26] for details). Let (M, m) be a σ -finite measure space, f : M → C be a measurable function and p ∈ [1, ∞), q ∈ [1, ∞]. We define f ∗p,q

=

( pq

∞ 0

[f ∗ (t)t 1/p ]q dtt )1/q

supt>0 t df

when q < ∞, when q = ∞.

(t)1/p

Here df is the distribution function of f and f ∗ is the nonincreasing rearrangement of f . That is, for α > 0,

df (α) = m x f (x) > α

and f ∗ (t) = inf s df (s) t .

We take Lp,q (M) to be the set of all measurable f : M → C such that f ∗p,q < ∞. For 1 p < ∞, Lp,p (M) = Lp (M) and · ∗p,p = · p . By L∞,∞ (M) and · ∗∞,∞ we mean respectively the space L∞ (M) and the norm · ∞ . The space Lp,∞ (S) is known as the weak Lp -space. Following properties of the Lorentz spaces will be required:

(i) For 1 < p, q < ∞, the dual space of Lp,q (S) is Lp ,q (S) and dual of Lp,1 (S) is Lp ,∞ (S). (ii) If q1 q2 ∞ then Lp,q1 (S) ⊂ Lp,q2 (S) and f ∗p,q2 f ∗p,q1 . Let n = v ⊕ z be an H -type Lie algebra where v and z are vector spaces over R of dimensions m and l respectively. Indeed z is the centre of n and v is its ortho-complement with respect to the inner product of n. One knows that m is even. Let N = exp n. We shall identify v and z and N with Rm , Rl and Rm × Rl respectively. Elements of A will be identified with at = et , t ∈ R. A acts on N by nonisotropic dilation: δat (X, Y ) = (e−t X, e−2t Y ) = at na−t for at ∈ A and


2457

n = (X, Y ) ∈ N (X ∈ v, Y ∈ z). For convenience henceforth we shall write δt for δat . Let S = NA be the semidirect product of N and A under the action above. Then S is a solvable, connected and simply connected Lie group with Lie algebra s = v ⊕ z ⊕ R. It is well known that S is a nonunimodular amenable Lie group. The homogeneous dimension of S is Q = m/2 + l. For convenience we shall also use the symbol ρ for Q/2. An element x = na = n(X, Y )a ∈ S can be written as (X, Y, a), X ∈ v, Y ∈ z, a ∈ A. Precisely (X, Y, a) is identified with exp(X + Y )a. We shall use the notation A(x) = A(nat ) = t. A function f on S is called radial if for all x, y ∈ S, f (x) = f (y) if d(x, e) = d(y, e), where d is the metric induced by the canonical left invariant Riemannian structure of S and e is the identity element of S. It follows that d(at , e) = |t| for any t ∈ R. For a radial function f we shall also use f (t) to mean f (at ). For a function space L(S) on S we denote its subspace of radial functions by L(S)# . For a suitable function f on S its radialization Rf is defined as (2.1) Rf (x) = f (y) dσν (y), Sν

where ν = r(x) and dσν is the surface measure induced by the left invariant Riemannian metric on the geodesic sphere Sν = {y ∈ S | d(y, e) = ν} normalized by Sν dσν (y) = 1. It is clear that Rf is a radial function and if f is radial then Rf = f . From definition it follows that Rf ∞ f ∞ . Also as S f (x) dx = R+ Rf (t)J (t) dt where J (t) is the Jacobian of the polar decomposition we have Rf 1 f 1 . From these using interpolation (see [31, p. 197]) we have Rf ∗q,r f ∗q,r ,

1 < q < ∞, 1 r ∞.

The Poisson kernel P : S × N → R is given by P(nat , n1 ) = Pat (n−1 1 n) where Q Pat (n) = Pat (V , Z) = Cat

The value of C is adjusted so that the following: (1) (2) (3) (4) (5)

N

|V |2 at + 4

2 + |Z|

2

−Q ,

n = (V , Z) ∈ N.

(2.2)

Pa (n) dn = 1 and P1 (n) 1 (see [2, (2.6)]). We also need

Pa (n) = Pa (n−1 ). Pat (n) = P1 (a−t nat )e−Qt . P(x, n) = P(n1 at , n) = Pat (n−1 n1 ) = Pat (n−1 1 n). Pλ (x, n) = P(x, n)1/2−iλ/Q = P(x, n)−(iλ−ρ)/Q . R(Pλ (·, n))(x) = φλ (x)Pλ (e, n), R(e(iλ−ρ)A(·) )(x) = φλ (x).

The action of the analogue of class-1 principal series representation πλ , λ ∈ C realized on functions on N is given by:

t (iλ−ρ)

π−λ (n1 at )φ (n) = φ a−t n−1 . 1 nat e 1/2−iλ/Q

From this it is easy to verify that (π−λ (x)P1

)(n) = Pλ (x, n).

2458


The elementary spherical function φλ (x) is given by 1/2−iλ/Q 1/2−iλ/Q φλ (x) = πλ (x)P1 , P1 = L2 (N )

Pλ (x, n)P−λ (e, n) dn. N

It follows that φλ is a radial eigenfunction of the Laplace–Beltrami operator L of S with eigenvalue −(λ2 + ρ 2 ) satisfying φλ (x) = φ−λ (x), φλ (x) = φλ (x −1 ) and φλ (e) = 1. As P−iρ (x, n) ≡ 1 for all x ∈ S and n ∈ N and Piρ (x, n) = P(x, n), φ−iρ (x) =

Piρ (e, n) =

N

P1 (n) dn = 1. N (α,β)

For α = m+l−1 and β = l−1 with the ideal 2 2 , φλ is identical with the Jacobi function φλ situation of α > β > − 12 (see [1]). We define the spherical Fourier transform f of a suitable radial function f as f(λ) =

f (x)φλ (x) dx, S

whenever the integral converges. It is clear that the spherical Fourier transform is indeed the Jacobi transform. The left invariant Haar measure on S decomposes as

f (x) dx =

f (nat )e−Qt dt dn,

N ×A

S

where dn(X, Y ) = dX dY and dX, dY , dt are Lebesgue measures on v, z and R respectively. Jacobians of the following transformations will be required for our computations. For y ∈ S let Ry be the right-translation operator, i.e. for a measurable function f on S, Ry f (x) = f (xy). For t ∈ R and a measurable function F on N , the dilation is defined as: δt F (n) = F (δt (n)) = F (at na−t ). Then, (a) N f (δt (n)) dn = N f (n)e−Qt dn. (b) S Ry f (x) dx = S f (x) dx eQA(y) , i.e. the modular function (y) = e−QA(y) . (c) S f (x −1 ) dx = S f (x)eQA(x) dx and S f (x −1 )eQA(x) dx = S f (x) dx. For two measurable functions f and g on S we define their convolution as (see [15, p. 51]): f ∗ g(x) = S

f (y)g y −1 x dy =

S

f y −1 g(yx) y −1 dy =

f xy −1 g(y) y −1 dy.

S

If g is radial then g ∗ = g as d(x, e) = d(x −1 , e). It is easy to see that for measurable functions f, g, h on S, f ∗ g, h = f, h ∗ g ∗ if both sides make sense.


2459

For a measurable function f on S we define its Fourier transform (which is an analogue of the Helgason Fourier transform on the symmetric space) by f(λ, n) =

f (x)Pλ (x, n) dx, S

whenever the integral converges. If f is radial then using (5) above we see that f(λ, n) = f(λ)Pλ (e, n). The Poisson transform of a function F on N is defined as (see [2]) Pλ F (x) =

1/2−iλ/Q F (n)Pλ (x, n) dn = πλ (x)P1 ,F .

N

We note that for a function f on S, a function F on N and for λ ∈ C, f(λ, ·)Lq (N ) Cf Lp (S) is equivalent to Pλ F Lp (S) CF Lq (N ) by duality (see [26]). We have the following asymptotic estimate of φλ when iλ > 0 [1]: lim e−(iλ−ρ)t φλ (at ) = c(λ)

t→∞

where c(λ) is the Harish-Chandra c-function. From this and the continuity of φλ we have: φiρ is bounded and the following pointwise estimate for 1 < p < 2,

φ−iγp ρ (x) c(−iγp ρ)eQ d(x)/p c(iγp ρ)eQ d(x)/p ,

(2.3)

where d(x) = d(x, e) is the distance of x from the identity. An upshot of (2.3) is that for 1 < p < 2, φλ ∈ Lp ,1 (S) (respectively Lp ∞ (S)) if and only if λ ∈ Sp◦ (respectively Sp ). We also have if λ ∈ S2 = R then φλ /(1 + d(·)) ∈ L2,∞ (S). The geodesic inversion σ : S → S is an involutive, measure preserving, diffeomorphism which is explicitly given by [7,27]: σ (V , Z, at ) =

|V |2 e + 4 t

2

−1 |V |2 t − e + + JZ V , −Z, at . + |Z| 4 2

(2.4)

The following properties of σ will be important for us: (i) (ii) (iii) (iv)

P(σ (x), e) = CeQA(x) , x ∈ S for x ∈ S. σ (at ) = at−1 = a−t for at ∈ A. σ (at n) = σ (at )σ (n) for n ∈ N , at ∈ A. σ (at x) = σ (at )σ (x) for x ∈ S, at ∈ A.

Property (i) is proved in [26]. Properties (ii) and (iii) follow from a straightforward computation using the defining formula (2.4) above (an intermediate step for proving (iii) could be σ (at−1 nat ) = at σ (nat )). Property (iv) follows from (iii). We conclude this section by quoting two theorems from [26].

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Theorem 2.1. Let f be a measurable function on S. (i) If f ∈ Lp,1 (S), 1 p < 2, then there exists a subset Np of N of full Haar measure, depending only on f , such that f(λ, n) exists for all n ∈ Np and λ ∈ Sp . (ii) If f ∈ Lp,∞ (S), 1 < p < 2, then there exists a subset Np of N of full Haar measure, depending only on f , such that f(λ, n) exists for (λ, n) ∈ Sp◦ × Np and also for every n ∈ Np and almost every λ ∈ ∂Sp . For f ∈ L1,∞ (S), f(λ, n) does not exist. Theorem 2.2 (Riemann–Lebesgue lemma). Let 1 p < 2. If f ∈ Lp,1 (S) then for almost every fixed n ∈ N the map λ → f(λ, n) is continuous on Sp and analytic on Sp◦ . Furthermore lim f(ξ + iη, n) = 0

|ξ |→∞

uniformly in η ∈ [−γp ρ, γp ρ]. For functions in Lp,∞ (S), the assertions above remain valid for λ ∈ Sp◦ and for η ∈ [−(γp ρ − δ), (γp ρ − δ)] for any 0 < δ < γp . 3. Restriction theorems and Hausdorff–Young inequalities We shall first prove the restriction theorem mentioned in the Introduction (Theorem 1.1). Starting from the mapping property of the spectral projection f → f ∗ φλ , an Lp –Lq restriction theorem for NA groups was proved in [26]. Here we shall obtain the results which are best possible in this setup. This will naturally involve Lorentz spaces. The drastic change of the Lorentz norm as the argument of the Fourier transform moves from boundary to the interior of the strip Sp is motivated in the Introduction. In particular this theorem will extend a fundamental result proved for G/K in [24] where the role of the space Lp,1 (G/K) was first recognized (though from a different perspective). Cowling, Meda and Setti [6] exploited the behaviour of the intertwining operators to improve the result. These ideas culminate into the Lorentz space version of the Kunze–Stein phenomenon for noncompact semisimple Lie groups of real rank one (see [6]). The end point estimate for this phenomenon was obtained by Ionescu in [21] and its analogue for the convolution corresponding to the Jacobi transform was obtained by Liu in [22]. Apart from its application to the growth properties of the Fourier transform in Section 4, this result we obtain here will play a key role in Section 5. To proceed towards restriction theorem the following result of Folland and Stein (see [16, pp. 62–66]) will be required. For f, φ ∈ Cc∞ (N ), the radial maximal function Mφ0 is defined by Mφ0 f (n) =

sup

−∞ Q, then 0 M f ACp,β f p φ p

for all f ∈ Lp (N ), p > 1.


2461

Proof of Theorem 1.1. (i) When p = 1, then f ∈ L1 (S) and 1 q ∞. For q < ∞, we note that f(α + iγq ρ, ·)q S |f (x)|( N P(x, n) dn)1/q dx and N Pa (n) dn 1. For q = ∞, |Pα+iγq (x, n)| P−iρ (x, n) = 1. Hence the result follows with C1,q = 1. We will deal now with 1 < p < 2. First we take q = p for which it is enough to prove Piγp ρ F Lp ,∞ (S) CF Lp (N ) for all F ∈ Cc∞ (N), as the assertion follows from this by duality. We have for x = n1 at , Piγp ρ F (x) =

F (n)Piγp ρ (x, n) dn N

=

F (n)Piγp ρ (n1 at , n) dn N

=

F (n)Piγp ρ at , n−1 1 n dn

N

=

F (n1 n)Piγp ρ at , n−1 dn

N

= eQt/p

− Qt F (n1 n)Piγp ρ at , n−1 e p dn.

N

Let φ(n) = P1 (n)1/p . Let φt (n) = e−Qt φ(δ−t (n)). Then φt (n) = e−Qt/p Piγp ρ (at , n). Hence Piγp ρ F (n1 at ) = (F ∗ φt )(n1 )eQt/p . Therefore Piγ ρ F (n1 at ) eQt/p p

sup

−∞ α p φ p

= x = n1 at e−Qt < Mφ0 F (n1 )α −1

0 p p = Mφ F (n1 ) α −p dn1 α −p F p

L (N )

.

N

In the last step we have used Theorem 3.1, noting that φ satisfies the hypothesis. Indeed |φ(n)| C(1 + n)−2Q/p and 2Q/p > Q as p < 2. Therefore

1/p F Lp (N ) . sup α n1 at Piγp ρ F (n1 at ) > α

α>0

2462


For q = p we need to prove Piγp ρ F ∗p ,∞ CF Lp (N ) which is same as proving πiγ

p ρ

1/p

(·)P1

∗ , F p ,∞ CF Lp (N ) .

We need to define the intertwining operator Ip : Lp (N ) → Lp (N ) for 1 p 2 such that: (a) Ip φ, ψ = φ, Ip ψ for measurable functions φ and ψ on N for which both sides make sense. (b) Ip (πiγp (nat )φ)(n1 ) = πiγp (nat )(Ip φ)(n1 ) for all n, n1 ∈ N , t ∈ R and φ ∈ Cc∞ (N ). (c) Ip is strong type (p, p ). 1/p

(d) Ip (P1

1/p

)(n) CP1

(n).

We recall (see Section 2) that for a suitable function φ on N ,

πλ (n)φ(n1 ) = φ n−1 n1

and πλ (at )φ(n1 ) = e−Qt/p φ δ−t (n1 ) .

We also recall that N f (δr (n)) dn = e−Qr N f (n) dn. It is clear that δr (V , Z) = er (V , Z) where n is the homogeneous norm of n ∈ N defined above. We define

Ip φ = φ ∗ · −2Q/p . It is clear that Ip satisfies (a) and that Ip commutes with the translation on N which is the action of N under πλ for any λ. Therefore to prove (b) it is enough to show that Ip (πiγp (ar )φ) = πiγp (ar )(Ip φ) for all r ∈ R. For convenience let us temporarily use the symbol F (n) for

n−2Q/p It is clear that

πiγp (ar )(Ip φ)(n) = πiγp (ar )(φ ∗ F ) (n) =

−Qr/p φ(n1 )F n−1 . 1 a−r nar dn1 e

N

On the other hand

Ip πiγp (ar )φ (n) =

πiγp (ar )φ (n1 )F n−1 1 n dn1

N

=

−Qt/p φ(a−r n1 ar )F n−1 1 n dn1 e

N

= M

Qt −Qt/p φ(n2 )F ar n−1 2 a−r n dn2 e e


=

2463

2Qt/p Qt −Qt/p φ(n2 )F n−1 e e 2 a−r nar dn2 e

N

=

−Qr/p φ(n2 )F n−1 . 2 a−r nar dn2 e

N

In one of the steps above we have used

−1

2Qr/p F ar n−1 F n−1 2 a−r nar = F a−r ar n2 a−r n ar = e 2 a−r n . This proves property (b) of Ip . We note that (see [14, Cor. 1.6]) for any nonzero complex number α and positive number b,

(V , Z)α−Q dV dZ = cα bα .

0(V ,Z)b

Therefore (taking α = Q) we conclude that the measure of the set {(V , Z) | (V , Z)−2Q/p > b} is equal to CQ b−p /2 . Thus · −2Q/p ∈ Lp /2,∞ (N ) and hence by Young’s inequality [14, Prop. 1.10] Ip φLp (N ) Cp f Lp (N ) . This proves property (c) of Ip .

1/p

For (d) we first note that P1 1/p

P1

(n) · −2Q/p . Therefore

∗ · −2Q/p (n) P1

1/p

1/p

∗ P1

(n)

1/p

= φiγp ρ (n) e−Q d(n)/p P1

(n).

Thus 1/p

P1

1/p

(n) CP1

1/p ∗ · −2Q/p (n) = CIp P1 (n).

Now using the operator Ip we have 1/p

πiγp ρ (x)P1

1/p

1/p = CIp πiγp ρ (x)P1 . = Cπiγp ρ (x)Ip P1

Hence 1/p 1/p 1/p πiγp ρ (·)P1 , F = C Ip πiγp ρ (·) P1 , F = C πiγp ρ (·)P1 , Ip (F ) . Thus πiγ

p ρ

1/p

(·)P1

∗ ∗ 1/p , F p ,∞ = C πiγp ρ (·)P1 , Ip (F ) p ,∞ C Ip (F )Lp (N ) CF Lp (N ) .

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This completes (i) except for the cases p < q < p which are taken care of in (ii) as f ∗p,∞ f ∗p,1 . (ii) We take p1 , p2 1 such that p1 < p < p2 < q < p . Using the result in [26, Theorem 4.2] we have f(α + iγq ρ, ·)

Lq (N )

Cp1 ,q f p1

which is equivalent to the following by duality: Pα+iγq ρ ξ p1 Cp1 ,q ξ Lq (N ) . Through similar arguments we also get Pα+iγq ρ ξ p2 Cp2 ,q ξ Lq (N ) . We interpolate between the two results above [18, p. 64, 1.4.2] to get Pα+iγq ρ ξ ∗p ,1 Cp1 ,p2 ,p,q ξ Lq (N ) as p2 < p < p1 . The last result is equivalent to (by duality) f(α + iγq ρ, ·)

Lq (N )

Cp1 ,p2 ,p,q f ∗p,∞ .

To complete we shall now show that the norm estimates obtained above are sharpest possible. For α ∈ R and r ∈ R using dilation on N we get Pα+iγq ρ (δr F )(x) =

F (n)Pα+iγq ρ a−r n−1 ar x, e dn e−Qr .

N

Since σ (ar x) = σ (ar )σ (x) for ar ∈ A and x ∈ S (see Section 2) we have

−1 Pα+iγq ρ a−r n−1 ar x, e = e(Q/q−iα)A(σ (a−r n ar x)) −1

= e(Q/q−iα)r e(Q/q−iα)A(σ (n ar x))

= eQr/q e−iαr Pα+iγq ρ n−1 ar x, e = eQr/q e−iαr Pα+iγq ρ (ar x, n). Thus Pα+iγq ρ (δr F )(x) = e

−Qr/q −iαr

=e

−Qr/q −iαr

e

F (n)Pα+iγq ρ (ar x, n) dn N

e

Pα+iγq ρ (F )(ar x).


2465

Using the left invariance of the Haar measure we thus get Pα+iγ ρ (δr F )∗ = e−Qr/q Pα+iγ ρ F ∗ q q u,v u,v for any suitable u, v. On the other hand it is easy to prove that δr F Ls (N ) = eQr/s F Ls (N ) . Therefore if we assume an inequality of the form Pα+iγq ρ F ∗u,v F Ls (N ) then substituting F by δr F we get for all r ∈ R,

e−Qr/q Pα+iγq ρ F ∗u,v e−Qr/s F Ls (N ) and that implies s = q. Going to the dual picture we have proved that the only inequalities we can have is of the form: f(α + iγq ρ, ·)

Lq (N )

f ∗u ,v .

(3.1)

1/q But if f is radial then f (iγq ρ, n) = f (iγq ρ)P1 ∗ (n). Thus applying (3.1) on positive radial functions f we get S f (x)φiγq ρ (x) dx Cf u ,v . Hence without assuming radiality and positivity of the function f we have

f (x)φα+iγ ρ (x) dx R |f | (x)φiγ ρ (x) dx C R |f | ∗ Cf ∗ . q q u ,v u ,v S

S

This implies φα+iγq ρ ∈ Lu,v (S) for all α ∈ R. We recall (see Section 2) that the two best possible values of (u, v) are (p , 1) and (q , ∞). Therefore inequality (3.1) is restricted to be either f(α + iγq ρ, ·) q f ∗p,∞ L (N )

or f(α + iγq ρ, ·)Lq (N ) f ∗q,1 .

This establishes the sharpness of the estimates.

(3.2)

2

Remark 3.2. Following remarks are in order. (1) For Iwasawa NA groups (in other words for the rank one symmetric spaces) the fact that 1/p 1/p Ip (P1 ) = c(iγp ρ)P1 is proved in [9, p. 530] through nontrivial steps using representations of the Heisenberg groups. A step by step adaptation of the proof yields the result for general NA groups. This result is of independent interest and much stronger than property (d) of Ip used in the proof above. (2) It is clear that for p, q, α as in the second part of the theorem above and 1 r < ∞, if f ∈ Lp,r (S) then N


1/q

Cp,q f ∗p,r .

2466


(3) If we restrict only to the radial functions then it is possible to have sharper estimates. We have shown inside the proof of Theorem 1.1(i) that for any q 1 the function n → n−2Q/q is V 4

in Lq/2,∞ (N) where n = (V , Z) = ( 16Rm + Z2Rl )1/2 . It is clear from the definition that Piγq ρ (e, n) = P1 (n)1/q n−2Q/q and hence Piγq ρ (e, ·) ∈ Lq/2,∞ (N ). Using the fact that for a radial function f , f(λ, n) = f(λ)Pλ (e, n) we thus get: 1/q ∗ 1/q ∗ f(iγq ρ, ·)∗ = f(iγq ρ)P1 (·)q/2,∞ P1 (·)q/2,∞ φiγq ρ q ,∞ f q,1 q/2,∞ 1/q ∗ or P1 (·)q/2,∞ φiγq ρ q,∞ f q ,1 , depending on whether q 2 or q > 2. As P1 ∈ L∞ (N ), we also have f(iγq ρ, ·)∞ Cf q,1 or f q ,1 depending on whether q 2 or q > 2. Similarly we also get for 1 p < q < p , f(iγq ρ, ·)∗

1/q ∗ P1 (·)q/2,∞ φiγq ρ p ,1 f p,∞ f(iγq ρ, ·) Cf p,∞ . ∞

q/2,∞

and

Coming back to the restriction theorem of not necessarily radial functions on S, using Hölder’s inequality we have for 1 q1 < q, α ∈ R

q f(α + iγq ρ, n)q1 P1 (n)1− q1 dn

1/q1

f(iγq ρ, ·)q .

(3.3)

N

This leads to another set of results. Note that for a fixed 1 q1 < 2, and for q1 q q1 {α + iγq ρ | α ∈ R} is a straight line parallel to R on the strip Sq1 . We notice from (3.3) that the weight in the inequality will change with q and will vary from 1 to P1 (n)2−q1 . (One can compare the situation with the case of symmetric spaces (see Section 6) where restriction theorem is considered in the compact picture.) As P1 (n) 1 for all n ∈ N , one can use the uniform weight P1 (n)2−q1 when q1 is fixed. It is evident that to have a norm estimate of f(λ, ·) which is uniform over the strip Sq1 , and where the weight does not depend on q1 we have to consider a weighted measure space (N, P1 (n) dn). We get the following result whose proof is omitted as it goes along the line of [26, Corollary 4.4]. Note that (N, P1 (n) dn) is a finite measure space. Corollary 3.3 (Restriction on strip). Let L (N, P1 ) = f measurable on N q

q f (n) P1 (n) dn < ∞ . N

(a) Let 1 p < q 2 and 1 r q. If f ∈ Lp,∞ (S) then f(λ, ·)

Lr (N,P1 )

Cp,q f ∗p,∞

for any λ in the strip Sq = {z ∈ C | | z| γq ρ}.


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(b) Let 1 p < q < 2 and f ∈ Lp,∞ (S). Then f(λ, ·)

Lq,1 (N,P1 )

Cλ,p,q f ∗p,∞

for all λ ∈ Sq◦ .

(c) For p < q < q1 2, λ ∈ R f(λ + iγq , ·)∗ q,1 Cp,q,q1 f ∗p,∞ . 1 L (N,P ) 1

The next goal is to obtain an analogue of the Hausdorff–Young inequality. For radial functions on symmetric spaces an analogue of this inequality appears in [8]. In [26] an inequality of this genre was first proved. We consider the product measure space (Y, dy) = (N, dn) × (R, |c(λ)|−2 dλ). Theorem 3.4 (Hausdorff–Young inequality). Let 1 p 2. Then: (a) For p q p , R

f(λ + iγq ρ, n)q dn

p q

c(λ)−2 dλ

1 p

Cp,q f p .

N

(b) For 1 < p < q < p , p r ∞ and 1 s ∞ 1/q ∗ Cp,q,r f ∗ f(• + iγq , n)q p,s N

r,s

where by ( N |f(• + iγq , n)|q )1/q we mean the function λ → ( N |f(λ + iγq , n)|q )1/q on the measure space (R, |c(λ)|−2 dλ). Proof. Part (a) is proved in [26, Theorem 4.6]. For (b) we fix p, q satisfying the condition in the hypothesis and consider the operator T between the measure spaces (S, dx) and (R, |c(λ)|−2 dλ) defined by: Tf (λ) = f(λ + iγq ρ, ·)q . We choose p1 and p2 such that 1 p1 < p < p2 < q. Then by [26, Corollary 4.7] Tf p1 Cp,q f p1 and Tf p2 Cp,q f p2 . The result (b) then follows by interpolation [31, p. 197]. 2 4. Growth of Fourier transform and moduli of continuity For motivation of this section we refer the readers to [3,4,17]. Let σt be the normalized surface measure of the geodesic sphere of radius t. Then σt is a nonnegative radial measure. For a suitable function f on S we define the spherical mean operator Mt f = f ∗ σt . Using the radialization

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operator R (see Section 2) then

Mt f (x) = R xf (t) where xf = Rx f is the right-translation of f by x. We will see that Mt is strong type (p, p), 1 p ∞ and Mt Lp →Lp = φiγp ρ (t). Here Mt f Lp →Lp is the operator norm of Mt from Lp (S) to Lp (S). Instead of the particular measure σt let us first consider an arbitrary radial nonnegative finite measure μ and define the operator Tμ by Tμ (f ) = f ∗ μ. Then for all f ∈ Cc∞ (S) we have Tμ (f )(y) =

f (yz) dμ(z) = lim

f (yz)kn (z) dz

n

S

S

= lim n

=

f yz−1 eQA(z) kn (z) dz

S

f yz−1 eQA(z) dμ(z),

S

where {kn } is a sequence of radial measurable functions which converges weakly to μ. The following proposition now follows exactly as the proof of Theorem 3.3 in [1]. Proposition 4.1. If μ is a radial nonnegative finite measure on S, then Tμ is strong type (p, p) and p p T L →L = φiγp ρ (z) dμ(z). S

Thus using the proposition above we have Mt f p φiγp ρ (at )f p

and Mt Lp →Lp = φiγp ρ (t)

for p ∈ [1, ∞].

(4.1)

Since for t > 0, φiγp ρ (at ) e−(Q/p )t for 1 p 2 and φiγp ρ = φiγp ρ , we have from above

Mt f Lp →Lp e−(Q/p )t or e−(Q/p)t depending on p 2 or p > 2. Using interpolation [31, p. 197] we also have Mt f ∗p,s Cp,s f ∗p,s

for p ∈ (1, ∞), s ∈ [1, ∞].

We recall that for a function f on S its radialization R(f ) is given by R(f )(at ) = t (λ) = φλ (at ), we S f (x) dσt (x). Since R(Pλ (·, n))(t) = φλ (t)Pλ (e, n) (see Section 2) and σ have (4.2) σt (λ, n) = Pλ (x, n) dσt (x) = φλ (at )Pλ (e, n). S

Using this we get: Mt φλ (x) = φλ (at )φλ (x) which is an analogue of the functional equation for the elementary spherical functions on the symmetric spaces. We will sketch the proof.


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Mt φλ (x) = φλ ∗ σt (x) = φλ (xy) dσt (y) S

=

Pλ x −1 , n P−λ (y, n) dn dσt (y)

S N

=

P−λ (y, n) dσt (y) Pλ x −1 , n dn

N S

=

σt (−λ, n)Pλ x −1 , n dn

N

=

σt (λ)P−λ (e, n)Pλ x −1 , n dn

N

= φλ (at )φλ (x). The functional equation above and (4.1) suggest an alternative proof of the following wellknown result: Corollary 4.2. For any p ∈ [1, 2) if λ ∈ Sp then |φλ (at )| φiγp ρ (at ). Proof. When λ ∈ ∂Sp , it is clear that |φλ (at )| φiγp ρ (at ). Therefore we take λ ∈ Sp◦ . Then φλ ∈ Lp (S). Using (4.1) we have that Mt φλ p φiγp (at )φλ p and hence by the functional equation |φλ (at )|φλ p φiγp (at )φλ p . Thus |φλ (at )| φiγp (at ) for all λ ∈ Sp◦ . 2 From (4.2) through easy computation we also obtain

σt (λ, ·) (n) = φλ (at )Pλ (z, n). Mt Pλ (·, n) (z) = πλ (z)

(4.3)

Therefore for a suitable function f on S, M t f (λ, n) = f ∗ σt , Pλ (·, n) = f, Pλ (·, n) ∗ σt = f, φλ (at )Pλ (·, n) = f(λ, n)φλ (at ),

(4.4)

whenever both sides make sense. Proposition 4.3. For f ∈ L1 (S), Mt f converges to f in L1 (S) as t → 0. Also for all f ∈ Lp,q (S), 1 < p < ∞, 1 q ∞, Mt f converges to f in Lp,q (S) as t → 0. A standard argument involving dominated convergence theorem and approximation by functions in Cc∞ (S) proves the result for Lp -spaces. If f ∈ Lp,q (S) with p, q as above, then there exist f1 ∈ L1 (S), f2 ∈ Lr (S) with r ∈ (p, 2] such that f = f1 + f2 (see [26]). The use of this decomposition gives the result for the Lorentz spaces.

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Following [3] we define the spherical modulus of continuity for p = 1 as Ω1 [f ](r) = sup Mt f − f ∗1 . 0 −1/2. Then there are positive constants C1,α and C2,α such that

C1,α min 1, (λt)2 1 − jα (λt) C2α min 1, (λt)2 . Lemma 4.5. Let α > −1/2, −1/2 β α and t0 > 0. Then for |η| ρ, μ ∈ R, there exists a positive constant C = Ct0 ,α,β such that 1 − φ (α,β) (at ) C 1 − jα (μt) μ+iη

for all 0 t t0 . This has the following two consequences. (a) For t0 , μ and C as above, 2 1 1 − φ (α,β) z dz C min 1, μ μ±iρ r r 0

for all r 1/t0 . (b) For a fixed 0 < η0 < ρ, there exists a positive constant C = Cα,β,η0 such that for all |η| η0 , μ ∈ R and t > 0,

1 − φ (α,β) (t) C min 1, (μt)2 . μ+iη

The lemma above indicates that the results which involve boundaries of the strip S1 have to be treated differently. In fact only for those results spherical modulus of continuity has to be used. It is now straightforward that Theorem 1.1, Corollary 3.3(a) and the arguments of Theorem 12 in [3] prove the following versions of [3, Theorem 12], [4, Theorem 14]: Theorem 4.6. (a) Let 1 0, 1/q q

2 sup min 1, (λt) f (λ + iγq ρ, n) dn Cp,q Mt f − f ∗p,1 .

λ∈R

N


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(b) Let 1 0, 1/q

f(λ + iγq ρ, n)q dn Cp,q Mt f − f ∗p,∞ . sup min 1, (λt)2

λ∈R

N

(c) Let 1 0,

sup min 1, (λt)2

λ∈R

f(λ + iη, n)q P1 (n) dn

1/q

Cp,q Mt f − f ∗p,∞ .

N

(d) Let 1 < q < ∞. Then for f ∈ L1 (S) and t > 0,

1/q q

2 Cq Mt f − f 1 . sup min 1, (λt) f (λ + iγq ρ, n) dn λ∈R

N

One can also prove similar results using parts (b) and (c) of Corollary 3.3. As a consequence of Theorem 3.4 we have the following result which accommodates [3, Conjecture 16]. Theorem 4.7. (a) Let 1 < p 2 and p q p . Then for f ∈ Lp (S)

p /q 1/p −2 q

2p c(λ) dλ min 1, (λt) Cp,q Mt f − f p . f (λ + iγq ρ, n) dn

R

N

(b) Let 1 < p 2, p < q 0 be fixed. Then for f ∈ L1 (S) and for all r r0 , sup λ∈R, n∈N

1 2 min 1, (λ/r) f (λ − iρ, n) Cr0 Ω1 [f ] r

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and

f(λ + iρ, n) dn Cr Ω1 [f ] 1 . sup min 1, (λ/r)2 0 r λ∈R N

Proof. (b) We shall use here the notation explained in Theorem 3.4(b). We apply Theorem 3.4(b) to the function Mt f − f ∈ Lp,s (S) to obtain for all t > 0, 1/q ∗ Cp,q,s Mt f − f ∗ . (Mt f − f )(• + iγq ρ, n)q dn p,s u,s

N

Since M t f (λ + iγq ρ, n) = φλ+iγq ρ (at )f(λ, n), from above we get for all t > 0 1/q ∗ φ•+iγ ρ (at ) − 1 Cp,q,s Mt f − f ∗ . f(• + iγq ρ, n)q dn q p,s u,s

N

From this using Lemmas 4.4 and 4.5(b) we get the assertion. (c) We apply Theorem 3.4(a) for p = 1 and q = 1 to the function Mt f − f to obtain: sup λ∈R

(Mt f − f )(λ + iρ, n) dn CMt f − f 1 .

N

Hence sup sup 01/t


1/q

Cp,q Mt f − f ∗p,∞ .

N

(c) If 1 1/t


p /q

c(λ)−2 dλ

1/p Cp,q Mt f − f p .

N

(d) Let r0 > 0 be fixed. Then for f ∈ L1 (S), λ ∈ R and for all r r0 , f(λ − iρ, n) Cr Ω1 [f ] 1 and 0 r |λ|>r,n∈N 1 . sup f(λ + iρ, n) dn Cr0 Ω1 [f ] r |λ|>r sup

N

(e) If 1 < q < ∞ and f ∈ L1 (S) then sup

|λ|>1/t

f(λ + iγq , n)q dn

1/q Cq Mt f − f 1 .

N

Remark 4.10. One may be curious to know if it is possible to have analogues of (i) and (ii) at least for Lp (Rn ) by appealing to available versions of Euclidean restriction theorem. It appears to us that this approach fails there because in the case of Euclidean spaces the constants appearing in the restriction inequalities for spheres depend on the radius of the sphere. 5. Kunze–Stein phenomenon on NA groups Purpose of this section is to relate restriction theorems to the Kunze–Stein phenomenon. This may sound contradictory as Damek–Ricci spaces are not Kunze–Stein groups. Indeed we have the following result related to the left and right convolutors on S:

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Proposition 5.1. We suppose that p 1. Then: (a) For no q 1 the following holds true: Lp (S) ∗ Lq (S) ⊂ Lp (S). (b) Lq (S) ∗ Lp (S) ⊂ Lp (S) if and only if q = 1. p p Proof. It is implicit in [1, Theorem 3.3] that for a function f ∈ L1 (S)+ loc , if L (S) ∗ f ⊂ L (S) then S f (x)eQA(x)/p dx < ∞. Note that the function x → eQA(x)/p is in no Lq (S) as it is independent of N . This proves (a). It is easy to verify that L1 (S) ∗ Lp (S) ⊂ Lp (S). To prove the converse part in (b) first we note that (f ∗ g)∗ = g ∗ ∗ f ∗ . Suppose that f is a nonnegative left convolutor on Lp (S); i.e. f ∗ gp Cgp for any g ∈ Lp (S). Now f ∗ gp = (g ∗ ∗ f ∗ )∗ p = (g ∗ ∗ f ∗ )e(Q/p)A(·) p = g ∗ fp where g (x) = g ∗ (x)e(Q/p)A(x) and f = f ∗ (x)e(Q/p)A(x) . Then −1 p QA(x) dx = S |g(x)|p dx = gp . g p = G |g(x )| e Thus we have g ∗ f p g p . As for any function g, g = g this implies that fis a nonnegative right convolutor. Applying the argument of [1, Theorem 3.3] mentioned in part (a) above on f we get S f (x) dx = S f(x)e(Q/p )A(x) dx < ∞. That is, f ∈ L1 (S). 2

Remark 5.2. The theorem above is not surprising since S is an amenable as well as nonunimodular group. We note that for a nonunimodular group nothing more than the classical Young’s inequality is expected (see [10, Lemma A.4]) and hence in particular a nonunimodular group does not have Kunze–Stein property. It is also known that noncompact amenable groups are not Kunze–Stein groups. We recall that the most distinguished prototypes of harmonic NA groups are the rank one Riemannian symmetric spaces X = G/K, where G is a noncompact connected semisimple Lie group with finite centre and K is a maximal compact subgroup of G. More precisely, X = G/K can be realized through the Iwasawa decomposition G = NAK as X = NA and therefore X is called Iwasawa NA group. Functions on X have the natural identification with right K-invariant functions on G. The right convolution of a K-biinvariant function f2 on G with a function f1 on G/K is the same as their convolution (in that order) realized as functions on S = NA. Precisely, f1 ∗G f2 = f1 ∗S f2 . Thus for the Iwasawa NA groups one observes a Kunze–Stein phenomenon (when the right convolutor is radial) as a consequence of the corresponding phenomenon in the underlying semisimple group G [5,6]. Results below will reflect that so far the Kunze–Stein phenomenon with a radial right convolutor is concerned the vein runs from the Iwasawa NA groups to the general one-dimensional solvable extension of the H -type groups. Decomposing the space S in horocycles we have the following integral formula: Lemma 5.3. For f ∈ L1 (S) we have

f (x) dx = f nσ (n1 at ) dn1 dt dn = f nat σ (n1 ) e−Qt dn1 dt dn N R N

S

= S N

f xσ (n) dn dx.

N R N


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Proof. We recall that σ (at n) = σ (at )σ (n) and σ (at ) = a−t (see Section 2). Therefore

f nσ (n1 at ) dt dn dn1 =

N ×R×N

f na−t σ (a−t n1 at ) dt dn dn1

N ×R×N

f na−t σ (n1 ) eQt dt dn dn1

= N ×R×N

f nσ (at n1 ) eQt dt dn dn1 .

= N ×R×N

Now,

f nat σ (n1 ) e−Qt dn1 dt dn =

N R N

As

N

S N

eQA(σ (n)) dn =

N

Piρ (n, e) dn =

N

f xσ (n) dn dx =

f (x) dx eQA(σ (n)) dn. N S

P1 (n) dn = 1 the lemma follows.

2

The following lemma relates restriction theorems to the Kunze–Stein phenomenon. Indeed this is the heart of the phenomenon. Let Λ be the left regular representation of S, i.e. (Λ(x)f )(y) = f (x −1 y) for a measurable function f on S. We note that Λ(x)ξ, η = η ∗ ξ ∗ and when ξ is radial then Λ(x)ξ, η = η ∗ ξ . We also note that Λ(x)ξ, η = ξ, Λ(x −1 )η.

Lemma 5.4. Let p 1. For ξ ∈ Lp (S) and η ∈ Lp (S) there exist ξp ∈ Lp (N ) and ηp ∈ Lp (N ) such that ξ p = ξp Lp (N ) , ηp = ηp Lp (N ) and |Λ(x)ξ, η| πiγp ρ (x)ξp , ηp . If ξ is radial then |η ∗ ξ(x)| = |Λ(x)ξ, η| ξ p Piγp ρ (ηp )(x). If both ξ and η are radial functions then |η ∗ ξ(x)| = |Λ(x)ξ, η| φiγp ρ (x)ξ p ηp . Proof. For ξ ∈ Lp we define

ξ nat σ (n1 ) p e−Qt dn1 dt

ξp (n) =

1/p .

R×N

We also define ηp similarly. It is clear from Lemma 5.3 that ξp ∈ Lp (N ) and ηp ∈ Lp (N ). Let x = n2 as . Then

Λ(x)ξ, η = ξ x −1 y η(y) dy S

S

ξ(y)η(xy) dy

= N ×R×N

ξ nat σ (n1 ) η xnat σ (n1 ) e−Qt dn1 dt dn

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ξ nat σ (n1 ) p e−Qt dn1 dt

N

R×N

η xnat σ (n1 ) p e−Qt dn1 dt

× =

1/p

1/p dn

R×N

ξp (n)ηp (n2 as na−s )eQs/p dn N

=

Qs/p ξp a−s n−1 dn 2 nas ηp (n)e

N

=

πiγp ρ (x)ξp (n)ηp (n) dn

N

= πiγp ρ (x)ξp , ηp . Now ξp (n) =

ξ nσ (n1 at ) p dn1 dt

R×N

=

ξ(nx)p eQA(σ (x)) dx

1/p

1/p

S

=

ξ(x)p eQA(σ (n−1 x)) dx

1/p

S

=

ξ(x)p Piρ (x, n) dx

1/p ,

S

where we have substituted x for σ (n1 at ). If ξ is radial, then so is |ξ |p . We also recall that R(Piρ (·, n))(x) = φiρ (x)P(e, n) = P1 (n). Therefore from above we get ξp (n) = ( S |ξ(x)|p dx)1/p P1 (n)1/p = ξ p P1 (n)1/p . Thus if ξ is radial then from above we get Λ(x)ξ, η ξ p πiγ ρ (x)P 1/p , ηp = ξ p Piγ ρ (ηp )(x). p p 1

1/p

If η is also radial then ηp (n) = ηp P1 (n)1/p = ηp P−iγp ρ (e, n). As Piγp ρ (P1 φiγp ρ (x), the last assertion follows. 2 We have an immediate corollary of the lemma above:

)(x) =


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Corollary 5.5. (a) If a locally integrable function f satisfies f (x)Piγ ρ (α)(x) dx Cα p p L (N )

for all α ∈ Lp (N )+ ,

S

then Λ(f )Lp (S)# →Lp (S) < ∞, where for a measurable function ξ on S, Λ(f )ξ = f (x)Λ(x)ξ dx = f ∗ ξ . S (b) If f ∈ Lq (S)+ , 1 q < p and f satisfies f (x)Piγ ρ (α)(x) dx Cf Lq (S) α p , p L (N ) S

then Λ(f )Lp (S)# →Lp (S) Cf q . Theorem 5.6 (Kunze–Stein phenomenon). We have the following subset relations and the corresponding norm inequalities. (a) For 1 p < q < p , Lq (S) ∗ Lp,∞ (S)# ⊂ Lq (S). (b) For 1 p < 2, Lp (S) ∗ Lp (S)# ⊂ Lp ,∞ (S). (c) For 1 < p < 2 and 1 r, s, t ∞ with 1/r + 1/s 1 + 1/t, Lp,r (S) ∗ Lp,s (S)# ⊂ Lp,t (S). (d) (e) (f) (g) (h) (i)

For 1 < p < 2 and 1 r ∞, Lp ,r (S) ∗ Lp,1 (S)# ⊂ Lp ,r (S). For 1 p < 2 and 1 r ∞, Lp ,r (S) ∗ Lp,r (S)# ⊂ Lp ,∞ (S). (This accommodates (b).) ,∞ p p # p (S). For 1 p < 2, L (S) ∗ L (S) ⊂ L For 1 p < 2, Lp,1 (S) ∗ Lp (S)# ⊂ Lp (S). For 1 < p < 2 and p < q < p , Lq (S) ∗ Lq (S)# ⊂ Lp ,1 (S). p,∞ q # (S) ∗ L (S) ⊂ Lq (S). For 1 2. (b) We take ξ ∈ Lp (S)# and η ∈ Lp (S). By Lemma 5.4 η ∗ ξ(x) = Λ(x)ξ, η ξ p Piγ ρ (ηp )(x). p Therefore by the first part of Theorem 1.1(i) ∗ η ∗ ξ ∗p ,∞ ξ p Piγp ρ (ηp )p ∞ Cp ξ p ηp Lp (N ) = Cp ξ p ηp .

(c) We take f ∈ Lp,1 (S), g ∈ Lp (S)# and h ∈ Lp (S), use f ∗ g, h = f, h ∗ g and (b) to get p,1 L (S) ∗ Lp (S)# ⊂ Lp (S). From this using Zafran’s multilinear interpolation [32,6] we get (c). (d) Taking r = t, s = 1 in (c) we get Lp,r (S) ∗ Lp,1 (S)# ⊂ Lp,r (S) and then use duality.

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(e) Taking r = 1, s = t in (c) we get Lp,1 (S) ∗ Lp,s (S)# ⊂ Lp,s (S) and then use duality. (f) Thorough steps similar to that of (b) this follows from Lemma 5.4 and the second part of Theorem 1.1(i). (g) This follows from (f) by duality. (h) Thorough steps similar to that of (b) this follows from Lemma 5.4 and Theorem 1.1(iii). (i) This follows from (h) by duality. 2 Remark 5.7. (i) For noncompact semisimple Lie groups G of real of rank one Ionescu [21] proved the following end-point estimate: L2,1 (G) ∗ L2,1 (G) ⊂ L2,∞ (G). A step by step adaptation of the proof yields the result L2,1 (S) ∗ L2,1 (S)# ⊂ L2,∞ (S). This is a result of independent interest. From this some of the results of Theorem 5.6 follow. For instance, multilinear interpolation between Ionescu’s result and the fact L1 (S) ∗ L1 (S)# ⊂ L1 (S) implies (c). However at this moment it is not clear to us whether it is possible to obtain Theorem 5.6 without appealing to Theorem 1.1. (ii) Some of the special cases of Theorem 5.6(c) are interesting in themselves, e.g. Lp,1 (S) ∗ p,1 L (S)# ⊂ Lp,1 (S) which we get taking r = s = t = 1. (iii) Theorem 5.6(b) has the following corollary (see [24,6] for the result in symmetric spaces): Let A, B be two measurable subsets of S of which B is symmetric (i.e. indicator function of B is radial) then m(A.B) m(A).m(B). Here m denotes the left invariant Haar measure of S. We include a sketch. We have χA (x) (m(B))−1 χA.B ∗ χB −1 where B −1 = {x −1 | x ∈ B}. As χB −1 is radial using the relation Lp (S) ∗ Lp (S)# ⊂ Lp ∞ (S), 1 < p < 2 we get

m(A)1/p = χA p ,∞

C χA.B p χB −1 p = Cm(AB)1/p m(B)−1/p m(B)

which implies m(AB) Cm(A)m(B). (iv) It is implicit in the proof of Theorem 5.1(b) that it is not possible to have a Kunze–Stein phenomenon for left radial convolutor. We conclude this section with the following remark on the existence of the operator-valued Fourier transform πλ (f ) where πλ is a class-1 principal series representation and f ∈ Lp (S), 1 p < 2. Remark 5.8. It is clear from the action of πλ (see Section 2) that for 1 p ∞ and x ∈ S, πiγp ρ (x) is an isometry on Lp (N ). Therefore πiγp ρ (x)Lp (N )→Lp (N ) is uniformly bounded on S. Thus for f ∈ L1 (S) and λ ∈ S1 , πλ (f ) = S f (x)πλ (x) is a bounded operator from 1/2−iλ/Q )(n) and for λ = α + iγp ρ, Lp (N ) to Lp (N ) where λ = γp ρ. As f(λ, n) = (πλ (f )P1 1/2−iλ/Q 1/p p 1 | = P1 ∈ L (N ), existence of f (λ, n) for f ∈ L (S) and λ ∈ S1 can also be estab|P1 lished through this. Thus for f ∈ L1 (S) the situation is the same as that of the rank one symmetric space X. However there is a drastic change when we consider f ∈ Lp (S), 1 < p < 2. For symmetric spaces X using Kunze–Stein phenomenon and a result due to Herz and Lohoué (see [23, Lemma 2]) one can show that for f ∈ Lp (X) and λ ∈ Sp◦ , πλ (f ) exists as a bounded linear operator from Lq (N) to Lq (N ) where q > p and λ = γq ρ. On the other hand we shall see that the


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same is not true when f ∈ Lp (S), 1 p and λ = iγq ρ. Suppose for any f ∈ Lp (S) and λ = α + iγq ρ, πλ (f ) is a bounded operator from Lq (N ) to Lq (N). This implies |πλ (f )F1 , F2 | < CF1 q F2 q for all F1 ∈ Lq (N ) and F2 ∈ Lq (N ) and hence πλ (·)ξq , ηq Cξ q ηq p

for all ξ ∈ Lq (S) and η ∈ Lq (S) where ξq , ηq are related to ξ, η as in Lemma 5.4. Using Lemma 5.4 we thus have Λ(·)ξ, ηp Cξ q ηq , i.e. η ∗ ξ ∗ p Cξ q ηq . From this using duality we get f1 ∗ f2 q Cf1 p f2 q for all f1 ∈ Lp (S) and f2 ∈ Lq (S) which contradicts Proposition 5.1. 6. Symmetric spaces, revisited In this section we consider the noncompact Riemannian symmetric spaces X of rank one. Most of the notations are standard and can be found in [19,3]. It is not difficult to see that all the theorems proved for Damek–Ricci spaces will have analogue for symmetric spaces where N will be replaced by K and the Fourier transform defined in Section 2 will be substituted by the usual Helgason Fourier transform. As K is compact and hence a finite measure space, some of the statements will look simpler here, e.g. P1 (n) will be substituted by 1. We shall give here a brief sketch. Let the symmetric space X be the quotient space G/K where G is a connected noncompact semisimple Lie group with finite centre and with real rank one. Using the Iwasawa decomposition G = NAK we identify X with NA and thus X is realized as an NA group and known as Iwasawa NA group. Let θ be the Cartan involution on G. For x ∈ X, and k ∈ K/M we shall −1 denote the kernel of the Helgason Fourier transform e(iλ−ρ)H (x k) by Qλ (x, k). We recall that the Helgason Fourier transform on X is defined by f(λ, k)X = X f (x)Qλ (x, k) dx and the ele mentary spherical function on X is φλX (x) = K Qλ (x, k) dk. Viewing X as an NA group through Iwasawa decomposition and noting that the Jacobian of the transformation k → k(θ (n)) is given by dk = P1 (n) dn and that Pλ (x, n)/P(e, n) = Qλ (θ (x), k(θ (n))) (see [2, pp. 418–419]), we have

Pλ (x, n) P1 (n) dn = Qλ θ (x), k θ (n) P1 (n) dn Pλ (x, n)P−λ (e, n) dn = Pλ (e, n) N

N

=

N

Qλ θ (x), k dk.

K

The left-most side is φλ (x) and the right-most is φλX (θ (x)) = φλX (x), where θ is the Cartan involution on G. Thus φλX coincides with φλ . It is also clear that

f(λ, n) = Pλ (e, n)fθ λ, k θ (n) X , where fθ (x) = f (θ (x)).

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For λ = α + iγq with α ∈ R and q 1 we have using |Pλ (e, n)| = P1 (n)1/q , f(λ, ·) q = L (N )

fθ λ, k θ (n) q Pλ (e, n)q dn X

N

=

fθ λ, k θ (n) q P1 (n) dn X

N

=

fθ (λ, k)X q dk

1/q

1/q

1/q .

K

We note that for any p, fθ p = f p . Thus using Theorem 1.1 we obtain a version of that result for the symmetric space in compact picture: Theorem 6.1 (Restriction on line). Let f be a measurable function on X and α ∈ R. (i) For f ∈ Lp,1 (X), 1 p < 2 and p q p ,

f(α + iγq ρ, k)X q dk

1/q

Cp,q f ∗p,1 ,

C1,q = 1.

K

(ii) For f ∈ Lp,∞ (X), 1 < p < 2, p < q < p ,

f(α + iγq ρ, k)X q dk

1/q

Cp,q f ∗p,∞ .

K

As K is compact from (i) above we get for any 1 q1 < q and λ = α + iγq ρ,

f(λ, k)X q1 dk

1/q1

f p,1 .

(6.1)

K

On the other hand as above we can show that N

q f(λ, n)q1 P1 (n)1− q1 dn =

fθ (λ, k)X q1 dk.

K

This establishes the link between (6.1) and (3.3). Next we consider the spherical mean operator Mt . On X it takes a simpler form as the radialization can be obtained here through K-action. More precisely, for a function f on X, let Mt f (x) = K f (xkat ) dk. It is clear that Mt f is a right K-invariant function and hence a function on X. We will give a direct proof of the fact that Mt is a bounded operator from Lp (X) to Lp (X) for every p 1.


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Proposition 6.2. Mt is strong type (p, p) and Mt f p f p φiγp ρ (at ). Proof. If x = nak then Mt f (x) = Mt f (na). Thus Mt f (x)p dx = Mt f (n2 as )p e−Qs dn2 ds. N ×R

G

Therefore

f (xkat )p dx

Mt f p K

dk

X

= K

1/p

f (n2 as kat )p e−Qs dn2 ds

1/p dk.

(6.2)

N ×R

For the inside integral we put kat = n1 ar k1 . (Then H (at−1 k −1 ) = −r.)

f (n2 as kat )p e−Qs dn2 ds =

N ×R

f (n2 as n1 ar )p e−Qs dn2 ds

N ×R

=

f (n2 n3 as+r )p e−Qs dn2 ds

where n3 = as n1 a−s

N ×R

=

f (nas )p e−Qs eQr dn ds

N ×R −1 −1 k )

= f p eQr = f p e−QH (at p

p

We put this back in (6.2) to get Mt f p f p φiγp ρ (at ).

.

2

A line by line adaptation of the arguments in Sections 3 and 4 will yield all the results for X. Acknowledgments Authors would like to thank the referees for several valuable suggestions and criticisms which improved the paper. References [1] J.-P. Anker, E. Damek, C. Yacoub, Spherical analysis on harmonic AN groups, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 23 (4) (1996) 643–679 (1997), MR1469569 (99a:22014). [2] F. Astengo, R. Camporesi, B. Di Blasio, The Helgason Fourier transform on a class of nonsymmetric harmonic spaces, Bull. Austral. Math. Soc. 55 (3) (1997) 405–424, MR1456271 (98j:22008). [3] W.O. Bray, M.A. Pinsky, Growth properties of Fourier transforms via moduli of continuity, J. Funct. Anal. 255 (9) (2008) 2265–2285, MR2473257. [4] W.O. Bray, M.A. Pinsky, Growth properties of Fourier transforms, http://arxiv.org/abs/0910.1115v1.

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[5] M. Cowling, The Kunze–Stein phenomenon, Ann. of Math. (2) 107 (2) (1978) 209–234, MR0507240 (58 #22398). [6] M. Cowling, Herz’s “principe de majoration” and the Kunze–Stein phenomenon, in: Harmonic Analysis and Number Theory, in: CMS Conf. Proc., vol. 21, Amer. Math. Soc., Providence, RI, 1997, pp. 73–88, MR1472779 (98k:22040). [7] M. Cowling, A. Dooley, A. Korányi, F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type, J. Geom. Anal. 8 (2) (1998) 199–237. [8] M. Cowling, S. Giulini, S. Meda, Lp –Lq estimates for functions of the Laplace–Beltrami operator on noncompact symmetric spaces. I, Duke Math. J. 72 (1) (1993) 109–150, MR1242882 (95b:22031). [9] M. Cowling, U. Haagerup, Completely bounded multipliers of the Fourier algebra of a simple Lie group of real rank one, Invent. Math. 96 (3) (1989) 507–549, MR0996553 (90h:22008). [10] M. Cowling, S. Meda, A.G. Setti, An overview of harmonic analysis on the group of isometries of a homogeneous tree, Expo. Math. 16 (5) (1998) 385–423, MR1656839 (2000i:43005). [11] J. Dixmier, Les algèbres d’opérateurs dans l’espace hilbertien, Gauthier–Villars, Paris, 1969, MR1451139 (98a:46065). [12] M. Eguchi, S. Koizumi, S. Tanaka, A Hausdorff–Young inequality for the Fourier transform on Riemannian symmetric spaces, Hiroshima Math. J. 17 (1) (1987) 67–77, MR0886982 (88h:22015). [13] M. Eguchi, K. Kumahara, An Lp Fourier analysis on symmetric spaces, J. Funct. Anal. 47 (2) (1982) 230–246, MR0664337 (84e:43010). [14] G.B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (2) (1975) 161–207, MR0494315 (58 #13215). [15] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, Boca Raton, FL, 1995, MR1397028 (98c:43001). [16] G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes, vol. 28, Princeton University Press, Princeton, NJ, 1982, MR0657581 (84h:43027). [17] D. Gioev, Moduli of continuity and average decay of Fourier transforms: two-sided estimates, in: Contemp. Math., vol. 458, Amer. Math. Soc., Providence, RI, 2008, pp. 377–391, MR2411919. [18] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., New Jersey, 2004. [19] S. Helgason, Geometric Analysis on Symmetric Spaces, Math. Surveys Monogr., vol. 39, Amer. Math. Soc., Providence, RI, 1994, MR1280714 (96h:43009). [20] C. Herz, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971) 69–82, MR0272952 (42 #7833). [21] A.D. Ionescu, An endpoint estimate for the Kunze–Stein phenomenon and related maximal operators, Ann. of Math. (2) 152 (1) (2000) 259–275, MR1792296 (2001m:22017). [22] J. Liu, The Kunze–Stein phenomenon associated with Jacobi transforms, Proc. Amer. Math. Soc. 133 (6) (2005) 1817–1821, MR2120282 (2006d:33015). [23] N. Lohoué, Estimations Lp des coefficients de représentation et opérateurs de convolution, Adv. Math. 38 (2) (1980) 178–221, MR0597197 (82m:43004). [24] N. Lohoué, Th. Rychener, Some function spaces on symmetric spaces related to convolution operators, J. Funct. Anal. 55 (2) (1984) 200–219, MR0733916 (85d:22024). [25] 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, MR2040178 (2005b:43005). [26] S.K. Ray, R.P. Sarkar, Fourier and Radon transform on harmonic NA groups, Trans. Amer. Math. Soc. 361 (8) (2009) 4269–4297. [27] F. Rouvière, Espaces de Damek–Ricci, géométrie et analyse, in: Analyse sur les groupes de Lie et théorie des représentations, Kénitra, 1999, in: Sémin. Congr., vol. 7, Soc. Math. France, Paris, 2003, pp. 45–100, MR2038648 (2004m:22015). [28] R.P. Sarkar, A. Sitaram, The Helgason Fourier transform for symmetric spaces, in: A Tribute to C.S. Seshadri, in: Trends Math., Birkhäuser, Basel, 2003, pp. 467–473. [29] E.M. Stein, Harmonic analysis on R n , in: Studies in Harmonic Analysis, in: MAA Stud. Math., vol. 13, Math. Assoc. Amer., Washington, DC, 1976, pp. 97–135, MR0461002 (57 #990). [30] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., vol. 43, Princeton University Press, Princeton, NJ, 1993, MR1232192 (95c:42002). [31] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser., vol. 32, Princeton University Press, Princeton, NJ, 1971, MR0304972 (46 #4102). [32] M. Zafran, A multilinear interpolation theorem, Studia Math. 62 (2) (1978) 107–124, MR0499959 (80h:46119).


Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales Turdebek N. Bekjan a,1 , Zeqian Chen b,2 , Mathilde Perrin c,∗,3 , Zhi Yin b,d a College of Mathematics and Systems Sciences, Xinjiang University, Urumqi 830046, China b Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, PO Box 71010, 30 West Strict,

Xiao-Hong-Shan, Wuhan 430071, China c Laboratoire de Mathématiques, Université de Franche-Comté, 25030 Besançon Cedex, France d Graduate School, Chinese Academy of Sciences, Wuhan 430071, China

Received 7 July 2009; accepted 15 December 2009 Available online 8 January 2010 Communicated by N. Kalton

Abstract We prove that atomic decomposition for the Hardy spaces h1 and H1 is valid for noncommutative martingales. We also establish that the conditioned Hardy spaces of noncommutative martingales hp and bmo form interpolation scales with respect to both complex and real interpolations. © 2009 Elsevier Inc. All rights reserved. Keywords: Noncommutative Lp -spaces; Noncommutative martingales; Atoms; Interpolation; Hardy spaces; Square functions

0. Introduction Atomic decomposition plays a fundamental role in the classical martingale theory and harmonic analysis. For instance, atomic decomposition is a powerful tool for dealing with duality theorems, interpolation theorems and some fundamental inequalities both in martingale theory and harmonic analysis. Atoms for martingales are usually defined in terms of stopping times. * Corresponding author.

E-mail address: [email protected] (M. Perrin). 1 Partially supported by NSFC grant No. 10761009. 2 Partially supported by NSFC grant No. 10775175. 3 Partially supported by ANR 06-BLAN-0015.


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T.N. Bekjan et al. / Journal of Functional Analysis 258 (2010) 2483–2505

Unfortunately, the concept of stopping times is, up to now, not well defined in the generic noncommutative setting (there are some works on this topic, see [1] and references therein). We note, however, that atoms can be defined without help of stopping times. Let us recall this in classical martingale theory. Given a probability space (Ω, F , μ), let (Fn )n1 be an increasing filtration of σ -subalgebras of F such that F = σ ( n Fn ) and let (En )n1 denote the corresponding family of conditional expectations. An F -measurable function a ∈ L2 is said to be an atom if there exist n ∈ N and A ∈ Fn such that (i) En (a) = 0; (ii) {a = 0} ⊂ A; (iii) a2 μ(A)−1/2 . Such atoms are called simple atoms by Weisz [21] and are extensively studied by him (see [20] and [21]). Let us point out that atomic decomposition was first introduced in harmonic analysis by Coifman [3]. It is Herz [4] who initiated atomic decomposition for martingale theory. Recall that we denote by H1 (Ω) the space of martingales f with respect to (Fn )n1 such that the quadratic variation S(f ) = ( n |dfn |2 )1/2 belongs to L1 (Ω), and by h1 (Ω) the space of martingales f such that the conditioned quadratic variation s(f ) = ( n En−1 |dfn |2 )1/2 belongs to L1 (Ω). We say that a martingale f = (fn )n1 is predictable in L1 if there exists an adapted sequence (λn )n0 of non-decreasing, non-negative functions such that |fn | λn−1 for all n 1 and such that supn λn ∈ L1 (Ω). We denote by P1 (Ω) the space of all predictable martingales. In a disguised form in the proof of Theorem A∞ in [4], Herz establishes an atomic description of P1 (Ω). Since P1 (Ω) = H1 (Ω) for regular martingales, this gives an atomic decomposition of H1 (Ω) in the regular case. Such a decomposition is still valid in the general case but for h1 (Ω) instead of H1 (Ω), as shown by Weisz [20]. In this paper, we will present the noncommutative version of atoms and prove that atomic decomposition for the Hardy spaces of noncommutative martingales is valid for these atoms. Since there are two kinds of Hardy spaces, i.e., the column and row Hardy spaces in the noncommutative setting, we need to define the corresponding two type atoms. This is a main difference from the commutative case, but can be done by considering the right and left supports of martingales as being operators on Hilbert spaces. Roughly speaking, replacing the supports of atoms in the above (ii) by the right (resp. left) supports we obtain the concept of noncommutative right (resp. left) atoms, which are proved to be suitable for the column (resp. row) Hardy spaces. On the other hand, due to the noncommutativity some basic constructions based on stopping times for classical martingales are not valid in the noncommutative setting, our approach to the atomic decomposition for the conditioned Hardy spaces of noncommutative martingales is via the h1 − bmo duality. Recall that the duality equality (h1 )∗ = bmo was established independently by [8] and [13]. However, this method does not give an explicit atomic decomposition. The other main result of this paper concerns the interpolation of the conditioned Hardy spaces hp . Such kind of interpolation results involving Hardy spaces of noncommutative martingales first appear in Musat’s paper [11] for the spaces Hp . We will present an extension of these results to the conditioned case. Note that our method is much simpler and more elementary than Musat’s arguments. It seems that even in the commutative case, our method is simpler than all existing approaches to the interpolation of Hardy spaces of martingales. The main idea is inspired by an equivalent quasinorm for hp , 0 < p 2 introduced by Herz [5] in the commutative case. We translate this quasinorm to the noncommutative setting to obtain a new characteriza-


2485

tion of hp , 0 < p 2, which is more convenient for interpolation. By this way we show that (bmo, h1 )1/p = hp for any 1 < p < ∞. The study of the Hardy spaces of noncommutative martingales Hp and hp in the discrete case is the starting point for the development of an Hp -theory for continuous time. In a forthcoming paper by Marius Junge and the third named author, it appears that the spaces hp are much easier to be handled than Hp . It seems that their use is unavoidable for problems on the spaces Hp at the continuous time. The remainder of this paper is divided into four sections. In Section 1 we present some preliminaries and notation on the noncommutative Lp -spaces and various Hardy spaces of noncommutative martingales. The atomic decomposition of the conditioned Hardy space h1 (M) is presented in Section 2, from which we deduce the atomic decomposition of the Hardy space H1 (M) by Davis’ decomposition. In Section 3 we define an equivalent quasinorm for hp (M), 0 < p 2, and discuss the description of the dual space of hp (M), 0 < p 1. Finally, using the results of Section 3, the interpolation results between bmo and h1 are proved in Section 4. Any notation and terminology not otherwise explained, are as used in [18] for theory of von Neumann algebras, and in [15] for noncommutative Lp -spaces. Also, we refer to a recent book by Xu [24] for an up-to-date exposition of theory of noncommutative martingales. 1. Preliminaries and notations Throughout this paper, M will always denote a von Neumann algebra with a normal faithful normalized trace τ . For each 0 < p ∞, let Lp (M, τ ) or simply Lp (M) be the associated noncommutative Lp -spaces. We refer to [15] for more details and historical references on these spaces. For x ∈ Lp (M) we denote by r(x) and l(x) the right and left supports of x, respectively. Recall that if x = u|x| is the polar decomposition of x, then r(x) = u∗ u and l(x) = uu∗ . r(x) (resp. l(x)) is also the least projection e such that xe = x (resp. ex = x). If x is selfadjoint, r(x) = l(x). Let us now recall the general setup for noncommutative martingales. In the sequel, we always denote by (Mn )n1 an increasing sequence of von Neumann subalgebras of M such that the union of Mn ’s is w∗ -dense in M and En the conditional expectation of M with respect to Mn . A sequence x = (xn ) in L1 (M) is called a noncommutative martingale with respect to (Mn )n1 if En (xn+1 ) = xn for every n 1. If in addition, all xn ’s are in Lp (M) for some 1 p ∞, x is called an Lp -martingale. In this case we set xp = sup xn p . n1

If xp < ∞, then x is called a bounded Lp -martingale. Let x = (xn ) be a noncommutative martingale with respect to (Mn )n1 . Define dxn = xn − xn−1 for n 1 with the usual convention that x0 = 0. The sequence dx = (dxn ) is called the martingale difference sequence of x. x is called a finite martingale if there exists N such that dxn = 0 for all n N . In the sequel, for any operator x ∈ L1 (M) we denote xn = En (x) for n 1.

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Let us now recall the definitions of the square functions and Hardy spaces for noncommutative martingales. Following [14], we introduce the column and row versions of square functions relative to a (finite) martingale x = (xn ): Sc,n (x) =

n

1/2 |dxk |

2

,

Sc (x) =

k=1

∞

1/2 |dxk |

;

2

k=1

and Sr,n (x) =

n ∗ 2 dx

1/2 ,

k

Sr (x) =

k=1

∞ ∗ 2 dx k

1/2 .

k=1

Let 1 p < ∞. Define Hpc (M) (resp. Hpr (M)) as the completion of all finite Lp -martingales under the norm xHpc = Sc (x)p (resp. xHpr = Sr (x)p ). The Hardy space of noncommutative martingales is defined as follows: if 1 p < 2, Hp (M) = Hpc (M) + Hpr (M) equipped with the norm xHp = inf yHpc + zHpr , where the infimum is taken over all y ∈ Hpc (M) and z ∈ Hpr (M) such that x = y + z. For 2 p < ∞, Hp (M) = Hpc (M) ∩ Hpr (M) equipped with the norm xHp = max xHpc , xHpr . The reason that Hp (M) is defined differently according to 1 p < 2 or 2 p ∞ is presented in [14]. In that paper Pisier and Xu prove the noncommutative Burkholder–Gundy inequalities which imply that Hp (M) = Lp (M) with equivalent norms for 1 < p < ∞. We now consider the conditioned version of Hp developed in [10]. Let x = (xn )n1 be a finite martingale in L2 (M). We set sc,n (x) =

n

1/2 Ek−1 |dxk |

2

,

sc (x) =

k=1

∞

1/2 Ek−1 |dxk |

2

;

k=1

and sr,n (x) =

n

2 Ek−1 dx ∗ k

k=1

1/2 ,

sr (x) =

∞

2 Ek−1 dx ∗ k

k=1

1/2 .


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These will be called the column and row conditioned square functions, respectively. Let 0 < p < ∞. Define hcp (M) (resp. hrp (M)) as the completion of all finite L∞ -martingales under the (quasi)norm xhcp = sc (x)p (resp. xhrp = sr (x)p ). For p = ∞, we define hc∞ (M) (resp. hr∞ (M)) as the Banach space of the L∞ (M)-martingales x such that k1 Ek−1 |dxk |2 (respectively k1 Ek−1 |dxk∗ |2 ) converge for the weak operator topology. We also need p (Lp (M)), the space of all sequences a = (an )n1 in Lp (M) such that ap (Lp (M)) =

p

an p

1/p 1 (this example has been with a function f ∈ C ∞ (Ω), such that ∇ 2 Gf ∈ further refined by D. Jerison and C.E. Kenig in [41] where the authors have constructed a bounded / L1 (Ω)). domain ∂Ω ∈ C 1 and f ∈ C ∞ (Ω) with ∇ 2 Gf ∈ It has long been understood that this regularity issue is intimately linked to the analytic and geometric properties of the underlying domain Ω. To illustrate this point, let us briefly consider the case when Ω ⊂ R2 is a polygonal domain with at least one re-entrant corner. In this scenario, let ω1 , . . . , ωN be the internal angles of Ω satisfying π < ωj < 2π , 1 j N , and denote by P1 , . . . , PN the corresponding vertices. Then the solution to the Poisson problem (1.1) with a datum f ∈ L2 (Ω) permits the representation u=

N

λj vj + w,

λj ∈ R,

(1.3)

j =1

where w ∈ W 2,2 (Ω) has zero boundary trace and, for each j , vj is a function exhibiting a singular behavior at Pj of the following nature. Given j ∈ {1, . . . , N}, choose polar coordinates (rj , θj ) taking Pj as the origin and so that the internal angle is spanned by the half-lines θj = 0 and θj = ωj . Then π/ωj

vj (rj , θj ) = φj (rj , θj )rj

sin(πθj /ωj ),

1 j N,

(1.4)

where φj is a C ∞ -smooth cut-off function of small support, which is identically one near Pj . / W 1+(π/ωj ),2 (Ω). This In this scenario, vj ∈ W s,2 (Ω) for every s < 1 + (π/ωj ), though vj ∈ implies that the best regularity statement regarding the solution of (1.1) is u ∈ W s,2 (Ω)

for every s < 1 +

π , max{ω1 , . . . , ωN }

(1.5)

and this fails for the critical value of s. In particular, this provides a geometrically quantifiable way of measuring the failure of the membership of u to W 2,2 (Ω) for Lipschitz, piece-wise C ∞ domains exhibiting inwardly directed irregularities. For more details on the theory of elliptic regularity in domains with isolated singularities, the interested reader is referred to, e.g., [23,36, 48] and the references therein. The issue of identifying those Sobolev–Besov spaces within which the natural correlation between the smoothness of the data and that of the solutions is preserved when the domain in question has a Lipschitz boundary was considered by D. Jerison and C.E. Kenig in the 1990s. In their ground breaking work [41], they were able to produce such an optimal ‘well-posedness region’ for the Poisson problem with Dirichlet boundary condition for the scalar, flat-space Laplacian in bounded, Euclidean Lipschitz domains in the context of Sobolev–Besov spaces. The main estimate in [41] is Gf B p,p (Ω) Cf Bαp,p (Ω) α+2

for a suitable range R of indices (α, 1/p),

(1.6)

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D. Mitrea et al. / Journal of Functional Analysis 258 (2010) 2507–2585

plus a similar inequality involving fractional Sobolev spaces. Here R depends exclusively on the Lipschitz character of the domain Ω (which, in the case of domains with corners, essentially amounts to the aperture of the smallest angle). See Section 4.1 for a discussion in this regard. After switching homogeneities, so that one seeks a harmonic function with a prescribed trace (in a Besov space on the boundary), the main step in [41] is establishing an atomic estimate in a certain end-point case. It is in this step that the authors rely on harmonic measure estimates. The full range of indices is then arrived at via interpolation with other, known results. The counterexamples in [41] show that the range R appearing in (1.6) is optimal, but only if p,p one insists that p 1 (when all spaces involved are Banach). However, the Besov scale Bα naturally continues below p = 1, though the corresponding spaces are no longer locally convex. The p,q p,q consideration of the entire scales Bα , Fα , 0 < p, q < ∞, is also natural both because Hardy spaces occur precisely when p 1 on the Triebel–Lizorkin scale, and because Besov spaces with p < 1 offer a natural framework for certain types of numerical approximation schemes (a point eloquently made by R.A. DeVore and collaborators in a series of papers [22,24–26]). The work in [41] has been extended in [28,66] to allow Neumann boundary condition and varip,q p,q able coefficient operators, and further, in [50,51], to allow data from Bα , Fα , 0 < p, q < ∞, α ∈ R for an optimal range of indices. In the case of Dirichlet boundary condition, these results are presented in Theorem 4.1. The reader is referred to (4.1)–(4.2) in Section 4.1 for a precise description of this sharp range of indices. Here we only wish to single out a corollary of this theorem to the effect that if Ω is Lipschitz, then the operators in (1.2) are bounded if X = hp (Ω) for 1 − ε 0 depends on the Lipschitz character of Ω. This provides a solution to a conjecture made by D.-C. Chang, S.G. Krantz and E.M. Stein in [13,14]. Roughly speaking, the goal of the present paper is to explore the extent to which the range R in (1.6) becomes larger if the underlying Lipschitz domain satisfies a uniform exterior ball condition (UEBC) or, somewhat more restrictively, is a convex domain. We wish to point out that it has been recently proved in [61] that the former class coincides with the class of semiconvex domains. The issue of regularity of Green potential associated with the Laplacian in these classes of domains has already received considerable attention. For example, according to the literature on this subject, the operators in (1.2) are bounded if Ω is a bounded Lipschitz domain and, in addition, Ω is convex and X = L2 (Ω) [42,78],

(1.8)

Ω satisfies a UEBC and X = L2 (Ω) [2],

(1.9)

Ω is convex and X = Lp (Ω) with 1 < p 2 [3,31],

(1.10)

Ω satisfies a UEBC and X

= Lp (Ω)

with 1 < p 2 [39],

p,2

Ω satisfies a UEBC and X = Fα (Ω) with −1 α 0 and p,2

Ω ⊂ R2 is convex and X = Fα (Ω) with 0 < α < 1 and p,2

Ω is convex and X = Fα (Ω) with −1 α < 1 and

α+1 2

α+1 2

α+1 2

0. In addition, the solutions admit the integral representations in Ω: u=D

1 I +K 2

−1 f ,

v=D

1 I +K 2

−1 g = S S −1 g .

(2.37)

See [20,40,84] for a proof. In closing, we briefly recall the Newtonian volume potential for the Laplacian. Specifically, given a function f ∈ L1 (Ω), we set

Πf (x) :=

E(x − y)f (y) dy,

x ∈ Rn ,

(2.38)

Ω

and note that Πf = f

in Ω.

(2.39)

2.2. Smoothness spaces in the Euclidean setting Here we briefly review the Besov and Triebel–Lizorkin scales in Rn . One convenient point of view is offered by the classical Littlewood–Paley theory (cf., e.g., [72,80,81]). More specifically, let Ξ be the collection of all systems {ζj }∞ j =0 of Schwartz functions with the following properties: (i) there exist positive constants A, B, C such that

supp(ζ0 ) ⊂ x: |x| A ; supp(ζj ) ⊂ x: B2j −1 |x| C2j +1

if j ∈ N;

(2.40)


2521

(ii) for every multi-index α there exists a positive, finite constant Cα such that sup sup 2j |α| ∂ α ζj (x) Cα ;

(2.41)

x∈Rn j ∈N

(iii) ∞

ζj (x) = 1 for every x ∈ Rn .

(2.42)

j =0 n Fix some family {ζj }∞ j =0 ∈ Ξ . Also, let F and S (R ) denote, respectively, the Fourier transp,q n form and the class of tempered distributions in R . Then the Triebel–Lizorkin space Fs (Rn ) is defined for s ∈ R, 0 0

Then the Hardy–Lorentz space hp,q (Rn ), 0 < p < ∞, 0 < q ∞, is defined as the collection of tempered distributions u in Rn for which urad belongs to Lp,q (Rn ), and uhp,q (Rn ) := urad uLp,q (Rn ) . In the Euclidean setting, these spaces have been studied in [29] (for q = ∞) and [1] (for 0 < q ∞).


2523

Following [29], hp,∞ (Rn ) will be referred to as a weak Hardy space. As is well known, while the space hp,∞ (Rn ) coincides with Lp,∞ (Rn ) for 1 < p < ∞, these two scales are not comparable when 0 < p 1. For nice functions f in Rn it is nonetheless true that f L1,∞ (Rn ) Cn f h1,∞ (Rn ) .

(2.58)

See the discussion in [29]. Furthermore, the following interpolation result holds p n p n

h 0 R , h 1 R θ,∞ = hp,∞ Rn ,

(2.59)

granted that 0 < p0 , p1 < ∞, θ ∈ (0, 1) and 1/p = (1 − θ )/p0 + θ/p1 ; see [30]. Here and everywhere else in the paper, (·,·)θ,q stands for the real interpolation brackets. Returning to Hardy spaces, we also have p,q n p,q n

h 1 R , h 2 R θ,q = hp,q Rn ,

(2.60)

provided 0 < p 1, 0 < q0 , q1 ∞, θ ∈ (0, 1) and 1/q = (1 − θ )/q0 + θ/q1 ; see [1]. Finally, it is also more or less folklore (see, e.g., [29]) that h1 Rn → L1 Rn → h1,∞ Rn .

(2.61)

2.3. Besov and Hardy spaces on Lipschitz boundaries Here we discuss the adaptation of certain smoothness classes to the situation when the Euclidean space is replaced by the boundary of a Lipschitz domain Ω. For a ∈ R set (a)+ := max{a, 0}. Consider three parameters p, q, s subject to 0 < p, q ∞,

1 −1 <s 0. Call a function a ∈ L (∂Ω) an inhomogeneous (p, po )-atom if for some surface ball r ⊆ ∂Ω supp a ⊆ r ,

aLpo (∂Ω) r

(n−1)( p1o − p1 )

either r = η, or r < η and

,

and

a dσ = 0.

(2.73)

∂Ω p

We then define hat (∂Ω) as the p -span of inhomogeneous (p, po )-atoms, and equip it with the natural infimum-type quasi-norm. One can check that this is a “local” quasi-Banach space, in the sense that 1 p hat (∂Ω) is a module over C α (∂Ω) for any α > (n − 1) −1 . (2.74) p Different choices of the parameters po , η lead to equivalent quasi-norms and p

∗ (n−1)( p1 −1) hat (∂Ω) = C (∂Ω).

(2.75)

We now proceed to discuss regular Hardy spaces defined on the boundary of a bounded Lipschitz domain. Assume that Ω is a bounded Lipschitz domain in Rn , and assume that p (n − 1)/n < p 1 < po ∞ are fixed. A function a ∈ L1 o (∂Ω) is called a regular (p, po )atom if there exists a surface ball r for which supp a ⊆ r ,

∇tan aLpo (∂Ω) r

(n−1)( p1o − p1 )

(2.76)

.

With 0 < η < diam(Ω) fixed, we next define 1,p hat (∂Ω) :=

f ∈ Lip(∂Ω) : f = λj aj , (λj )j ∈ p and aj regular (p, po )-atom j

supported in a surface ball of radius η for every j ,

(2.77)

where the series converges in Lip(∂Ω) , and equip it with the natural infimum quasi-norm. We conclude this subsection by recording a useful Sobolev space-like characterization of the regular Hardy space (cf. [63] for a proof). Proposition 2.3. Let Ω ⊂ Rn be a bounded Lipschitz domain, and assume that that p ∗ ∈ (1, ∞) is such that 1 1 1 . = − p∗ p n − 1 Also, assume that 1 < q p ∗ . Then

n−1 n

< p 1 and

(2.78)

2526


1,p p hat (∂Ω) = f ∈ Lq (∂Ω): ∂τj k f ∈ hat (∂Ω), 1 j, k n

(2.79)

and, in addition, f h1,p (∂Ω) ≈ f at

Lq (∂Ω)

n

+

∂τj k f hp (∂Ω) . at

(2.80)

j,k=1

2.4. Besov and Triebel–Lizorkin spaces in Lipschitz domains p,q

In this subsection we review how the Besov and Triebel–Lizorkin spaces Bs (Rn ), 0 < p, q ∞, s ∈ R, originally considered in the entire Euclidean setting, can be defined on arbitrary open subsets of Rn . Concretely, given an arbitrary open subset Ω of Rn , we denote by f |Ω the restriction of a distribution f in Rn to Ω. For 0 < p, q ∞ and s ∈ R, both p,q p,q Bs (Rn ) and Fs (Rn ) are spaces of (tempered) distributions, hence it is meaningful to define p,q Fs (Rn ),

p,q p,q As (Ω) := f distribution in Ω: ∃g ∈ As Rn such that g|Ω = f , p,q n p,q R , g|Ω = f , f ∈ As (Ω), := inf gAp,q f Ap,q n : g ∈ As s (Ω) s (R )

(2.81)

where A = B, or A = F . Throughout the paper, the subscript loc appended to one of the function spaces already introduced indicates the local version of that particular space. The existence of a universal extension operator for Besov and Triebel–Lizorkin spaces in an arbitrary Lipschitz domain Ω ⊂ Rn has been established by V.S. Rychkov in [73]. This allows transferring a number of properties of the Besov–Triebel–Lizorkin spaces in the Euclidean space Rn to the setting of a bounded Lipschitz domain Ω ⊂ Rn . Here, we only wish to mention a few of these properties. First, if 0 < p ∞, 0 < q < ∞ and s ∈ R, then p,min(p,q)

Bs

p,q

(Ω) → Fs

p,max(p,q)

(Ω) → Bs

(Ω),

(2.82)

and, with A ∈ {B, F }, p,q

p,q

p,q0

p,q1

if s1 < s0 , 0 < p, q0 , q1 ∞,

(2.83)

if 0 < q0 < q1 ∞, 0 < p ∞.

(2.84)

As0 0 (Ω) → As1 1 (Ω) As

(Ω) → As

(Ω)

Furthermore, p ,p

p,q

Bs00 (Ω) → Fs if 0 < p0 < p < p1 ∞ and

1 p0

−

s0 n

=

p ,q0

Fs 0 0 if 0 < p0 p ∞, 0 < q0 , q ∞,

1 p

−

if 0 < p0 p ∞, 0 < q0 , q ∞,

1 p

−

s n

=

−

s1 n,

p,q

(Ω)

1 p1

(Ω) → Fs

s n

p ,q0

Bs00

−

1 p

p ,p

(Ω) → Bs11 (Ω)

1 p0

−

s0 n,

s n

>

1 p0

−

s0 n.

whereas (2.86)

and

p,q

(Ω) → Bs

(2.85)

(Ω)

(2.87)


2527

Second, if k is a nonnegative integer and 1 < p < ∞, then p,2 Fk (Ω) = W k,p (Ω) := f ∈ Lp (Ω): ∂ α f ∈ Lp (Ω), |α| k ,

(2.88)

the classical Sobolev spaces in Ω. Third, if k ∈ N0 and 0 < s < 1, then ∞,∞ Bk+s (Ω) = C k+s (Ω),

(2.89)

where C k+s (Ω) := u ∈ C k (Ω): with uC k+s (Ω) < ∞, where uC k+s (Ω)

k j ∇ u := j =0

L∞ (Ω)

+

|α|=k

|∂ α u(x) − ∂ α u(y)| sup . |x − y|s x=y∈Ω

(2.90)

We conclude this subsection by recording a couple of useful lifting results on Besov and Triebel–Lizorkin spaces on bounded Lipschitz domains. The following has been proved in [60]. Proposition 2.4. Let 1 < p, q < ∞, k ∈ N and s ∈ R. Then for any distribution u in the bounded Lipschitz domain Ω ⊂ Rn , the following implication holds: p,q

∂ α u ∈ As (Ω),

∀α: |α| = k

⇒

p,q

u ∈ As+k (Ω),

(2.91)

where, as usual, A ∈ {B, F }. Going further, for 0 < p, q ∞, s ∈ R, we set

p,q p,q As,0 (Ω) := f ∈ As Rn : supp f ⊆ Ω , f Ap,q (Ω) := f Ap,q n , s (R ) s,0

p,q

f ∈ As,0 (Ω),

(2.92) p,q

p,q

where we use the convention that either A = F and p < ∞, or A = B. Thus, Bs,0 (Ω), Fs,0 (Ω) p,q p,q are closed subspaces of Bs (Rn ) and Fs (Rn ), respectively. Second, for 0 < p, q ∞ and s ∈ R, we introduce p,q p,q As,z (Ω) := f distribution in Ω: ∃g ∈ As,0 (Ω) with g|Ω = f , p,q p,q := inf gAp,q f ∈ As,z (Ω) f Ap,q n : g ∈ As,0 (Ω), g|Ω = f , s,z (Ω) s (R )

(2.93)

(where, as before, A = F and p < ∞ or A = B) and, in keeping with earlier conventions, p p,2 p,2 Ls,z (Ω) := Fs,z (Ω) = f distribution in Ω: ∃g ∈ Fs,0 (Ω) with g|Ω = f

(2.94)

if 1 < p < ∞, s ∈ R. For further use, let us also make the simple yet important observation that the operator of restriction to Ω induces linear, bounded mappings in the following settings

2528

D. Mitrea et al. / Journal of Functional Analysis 258 (2010) 2507–2585 p,q n

RΩ : As

R

p,q

−→ As (Ω)

p,q

p,q

and RΩ : As,0 (Ω) −→ As,z (Ω)

(2.95)

for 0 < p, q ∞, s ∈ R. For further use, let us now record a useful extension result, proved in [73], pertaining to the existence of a universal, linear extension operator. More specifically, we have Proposition 2.5. If Ω is a bounded Lipschitz domain in Rn , then there exists a linear operator Ex mapping Cc∞ (Ω) into distributions on Rn , and such that for any numbers 0 < p, q ∞, s ∈ R, and A ∈ {B, F }, p,q n

p,q

Ex : As (Ω) −→ As

R

(2.96)

boundedly, and p,q

RΩ ◦ Ex = I,

the identity operator on As (Ω).

(2.97)

Moving on, if 1 < p, q < ∞ and 1/p + 1/p = 1/q + 1/q = 1, then p,q

∗ p ,q As,z (Ω) = A−s (Ω)

if s > −1 +

p,q

∗ p ,q As (Ω) = A−s,z (Ω)

if s
0 with the property that f L1,∞ (Ω) Cf h1,∞ (Ω) ,

(2.122)

p h 0 (Ω), hp1 (Ω) θ,∞ = hp,∞ (Ω),

(2.123)

for nice functions f in Ω. Also,

provided 0 < p0 , p1 < ∞, θ ∈ (0, 1) and 1/p = (1 − θ )/p0 + θ/p1 , and

p,q h 1 (Ω), hp,q2 (Ω) θ,q = hp,q (Ω),

(2.124)

granted that 0 < p 1, 0 < q0 , q1 ∞, θ ∈ (0, 1) and 1/q = (1 − θ )/q0 + θ/q1 . Finally, h1 (Ω) → L1 (Ω) → h1,∞ (Ω),

(2.125)

in a bounded fashion. In closing, let us also point out that it is possible to provide an inn 0. Define the radial maximal function of a distribution u ∈ D (Ω) as uΩ,rad (x) := sup (ϕt ∗ u)(x): 0 < t < dist(x, ∂Ω) ,

x ∈ Ω.

(2.126)

n One can then use the results in [69,71,70] to show that if n+1 < p < ∞ and 0 < q ∞, then u ∈ D (Ω) belongs to hp,q (Ω) if and only if uΩ,rad ∈ Lp,q (Ω), and uhp,q (Ω) ≈ uΩ,rad Lp,q (Ω) .

2.5. Envelopes of nonlocally convex spaces Let X be a quasi-normed space and, for each 0 0: x/λ ∈ BX,p }.

(2.128)

Then, for each quasi-normed space X whose dual separates its points, we denote by Ep (X) the p-envelope of X, defined as the completion of X in the quasi-norm ||| · |||p . The case p = 1 corresponds to taking the Banach envelope, i.e. the minimal enlargement of the space in question to a Banach space; cf. [45] for a discussion. Several results are going to be of importance for us here. The first one essentially asserts that for a linear operator, being bounded, and being onto are stable properties under taking envelopes.

2532


Proposition 2.9. Let X, Y be two quasi-normed spaces and let T : X −→ Y be a bounded, linear operator. Then, for each 0 < p 1, this extends to a bounded, linear operator T : Ep (X) −→ Ep (Y ). Furthermore, if T is onto, then so is T. See [57] for a proof in the case p = 1 which readily adapts (cf. [51]) to the above setting. Our next result explicitly identifies the envelopes of regular Hardy spaces on boundaries of Lipschitz domains. Theorem 2.10. Let Ω ⊂ Rn be a bounded Lipschitz domain, 1,p

q,q Eq hat (∂Ω) = Bs (∂Ω),

n−1 n

< p q 1. Then

1 1 − . where s := 1 + (n − 1) q p

(2.129)

Again, see [57] for the case q = 1, and [51] for the general case. Moving on, let Ω ⊂ Rn be Lipschitz and assume that L is a constant coefficient, elliptic differential operator. For 0 < p < ∞, α ∈ R, introduce the space Hpα (Ω; L) := u ∈ D (Ω): Lu = 0 in Ω and uHpα (Ω;L) < ∞ ,

(2.130)

where, with ρ(x) := dist(x, ∂Ω), x ∈ Rn , we set α−1 ∇ j u p . uHpα (Ω;L) := ρ α−α |∇ α u|Lp (Ω) + L (Ω)

(2.131)

j =0

Above, α will denote the smallest nonnegative integer greater than or equal to α, and ∇ j u will stand for the vector of all mixed-order partial derivatives of order j of the components of u. The following theorem has been proved in [51]. Theorem 2.11. If L is an elliptic, homogeneous, constant coefficient differential operator and Ω is a bounded Lipschitz domain in Rn , then Hpα (Ω; L) = u ∈ Fαp,2 (Ω): Lu = 0 in Ω , Hpα (Ω; L) = u ∈ Bαp,p (Ω): Lu = 0 in Ω

(2.132) (2.133)

for each α ∈ R and each 0 < p < ∞. Later on, we shall also make use of the fact that, given a null-solution of an elliptic PDE in a Lipschitz domain, its membership to a Triebel–Lizorkin space is unaffected by the selection of the second integrability index for this scale of spaces. More precisely, the following result has been proved in [43]. Theorem 2.12. Let L be an elliptic, homogeneous, constant coefficient differential operator, and fix a bounded Lipschitz domain Ω ⊂ Rn . Then for each p ∈ (0, ∞) and α ∈ R, p,q

the space Fα (Ω) ∩ Ker L is independent of q ∈ (0, ∞).

(2.134)


2533

Finally, we conclude with a useful envelope identification result from [38]. Theorem 2.13. Let L be a second-order, elliptic, homogeneous, differential operator with real, constant coefficients, and let Ω be a bounded Lipschitz domain in Rn . Then, for any n−1 n 0

(3.1)

◦

(with W 1,1 (Ω) denoting the closure of Cc∞ (Ω) in W 1,1 (Ω)), and

∇x G(x, y), ∇ϕ(x) dx = ϕ(y),

∀ϕ ∈ Cc∞ (Ω).

(3.2)

Ω

Thus, G(x, y)|x∈∂Ω = 0 for every y ∈ Ω, and −G(·, y) = δy

for each fixed y ∈ Ω,

(3.3)

where the restriction to the boundary is taken in the sense of Sobolev trace theory, and δy is the Dirac distribution in Ω, with mass at y. See, e.g., [37] and [46]. As is well know, the Green function is symmetric, i.e., G(x, y) = G(y, x),

∀x, y ∈ Ω,

(3.4)

so that, by the second line in (3.3), −G(x, ·) = δx

for each fixed x ∈ Ω.

(3.5)

2534


Lemma 3.1. Let Ω be a bounded Lipschitz domain in Rn . Then for each y ∈ Ω fixed, ∇x G(x, y) x∈∂Ω

exists for a.e. x ∈ ∂Ω,

(3.6)

where the boundary trace is taken in the sense of (2.14). Proof. Upon recalling the fundamental solution for the Laplacian in Rn from (2.23), we observe that for each y ∈ Ω we may write G(x, y) = −E(x − y) + v y (x),

x ∈ Ω,

(3.7)

where v y solves (cf. Theorem 2.1) ⎧ y v = 0 in Ω, ⎪ ⎨ v y ∂Ω = E(· − y)|∂Ω ∈ L21 (∂Ω), ⎪ ⎩ y N ∇v ∈ L2 (∂Ω).

(3.8)

Indeed, since by (2.13) and (2.23) we have 2n

v y ∈ W 1, n−1 (Ω) ∩ C(Ω)

1,1 n R ∩ C ∞ Rn \ {0} , and E ∈ Wloc

(3.9)

and v y = E(· − y) on ∂Ω, (3.1) follows. Also, (3.2) is a direct consequence of (3.7) and the fact that v y is harmonic in Ω. Finally, since for every r > 0 the function −E(· − y) + v y is harmonic in Ω \ Br (y) and is 0 on ∂(Ω \ Br (y)) if r > 0 is small enough, the maximum principle ensures that −E(· − y) + v y 0 on Ω \ Br (y), granted that r > 0 is small enough. Thus, −E(· − y) + v y : Ω → [0, +∞], so (3.7) is indeed the Green function for Ω. Going further, (3.7) allows us to write

∇x G(x, y) = −(∇E)(x − y) + ∇v y (x),

x, y ∈ Ω,

and since ∇v y |∂Ω exists a.e. on ∂Ω, we can conclude that (3.6) holds.

(3.10)

2

As a consequence of Lemma 3.1, ∂ν(x) G(x, y)|x∈∂Ω := ν(x) ·

!

" lim ∇z G(z, y)

z∈Γκ (x) z→x

(3.11)

exists for a.e. x ∈ ∂Ω. Our next result provides a useful integral representation formula for the normal derivative of the Green function. Lemma 3.2. Let Ω ⊂ Rn be a bounded Lipschitz domain with outward unit normal ν and, as before, let G(·,·) denote the associated Green function. Then for any fixed y ∈ Ω, −∂ν(x) G(x, y) =

1 I +K 2

−1 ∗

∂ν(·) E(· − y) (x)

for σ -a.e. x ∈ ∂Ω, where both adjunction and inverse are taken in the sense of L2 (∂Ω).

(3.12)


2535

Proof. In general, for any two reasonably well-behaved functions u, v in Ω, the following Green formula holds

uv − uv = ∂ν uv dσ − u∂ν v dσ. (3.13) Ω

Ω

∂Ω

∂Ω

Fix x, y ∈ Ω, x = y and use (3.13) for u := E(x − ·) and v := E(· − y). Then, since u = δx and v = δy , (3.13) becomes

∂ν(z) E(x − z) E(z − y) dσ (z) =

∂Ω

E(x − z)∂ν(z) E(z − y) dσ (z),

(3.14)

∂Ω

that is,

D E(· − y)|∂Ω (x) = S ∂ν(·) E(· − y) (x).

(3.15)

Keeping now y ∈ Ω fixed, and letting x go nontangentially to the boundary, (3.15) implies

1 I + K E(· − y)|∂Ω = S ∂ν(·) E(· − y) , 2

∀y ∈ Ω.

(3.16)

If we now apply − 12 I + K to both sides of (3.16), we obtain that

1 1 1 − I + K S ∂ν(·) E(· − y) = − I + K I + K E(· − y)|∂Ω , 2 2 2

∀y ∈ Ω.

(3.17)

To continue, we further specialize (3.13) to the case when u := Sf , v := Sg, where f, g ∈ L2 (∂Ω) are arbitrary. On account of (2.26) and (2.27), this yields

1 1 ∗ ∗ − I + K f Sg dσ = Sf − I + K g dσ. 2 2 ∂Ω

(3.18)

∂Ω

Given that S : L2 (∂Ω) → L2 (∂Ω) is self-adjoint, and f, g are arbitrary, this readily implies KS = SK ∗

as operators on L2 (∂Ω).

(3.19)

Consequently, S

−1

1 1 ∗ ± I + K = ± I + K S −1 , 2 2

(3.20)

where S −1 is regarded as a bounded mapping from L21 (∂Ω) onto L2 (∂Ω), while K and K ∗ are bounded epimorphisms of L21 (∂Ω) and L2 (∂Ω), respectively. With (3.19)–(3.20) in hand, (3.17) becomes

2536


1 1 1 − I + K ∗ ∂ν(·) E(· − y) = I + K ∗ − I + K ∗ S −1 E(· − y)|∂Ω , 2 2 2

∀y ∈ Ω. (3.21)

We now apply ( 12 I + K ∗ )−1 to (3.21) and obtain that, for each y ∈ Ω, −1

1 1 ∗ ∗ I +K ∂ν(·) E(· − y) = − I + K S −1 E(· − y)|∂Ω . ∂ν(·) E(· − y) − 2 2 (3.22) Recall (3.7) to first note that −∂ν(·) G(·, y) = ∂ν(·) E(· − y) − ∂ν(·) v y .

(3.23)

Moreover, since v y solves (3.8), we know from (2.37) that

v y = S S −1 E(·, y)|∂Ω ,

(3.24)

so that the term in the right-hand side of (3.22) is equal to ∂ν v y . Combining now (3.22), (3.23), and (3.24) we obtain that −∂ν(·) G(·, y) =

1 I + K∗ 2

−1

∂ν(·) E(· − y) ,

(3.25)

2

from which (3.12) follows easily.

It is useful to note that, as (3.12) and Theorem 2.1 show, for every y ∈ Ω fixed, we have −∂ν(·) G(·, y) ∈ Lp (∂Ω),

∀p ∈ (1, 2 + ε),

(3.26)

whenever Ω ⊂ Rn is a bounded Lipschitz domain and ε = ε(∂Ω) > 0 is as in Theorem 2.1. Proposition 3.3. Let Ω ⊂ Rn be a bounded Lipschitz domain with outward unit normal ν and, as before, let G(·,·) denote the associated Green function. Also, recall the parameter ε = ε(∂Ω) > 0 from Theorem 2.1. Then for any f ∈ Lp (∂Ω), 2 − ε < p < ∞, the unique solution of the Dirichlet problem

(D)p

⎧ ⎨ u = 0 in Ω, u| = f on ∂Ω, ⎩ ∂Ω p N u ∈ L (∂Ω),

(3.27)

can be expressed as

u(y) = − ∂Ω

∂ν(x) G(x, y)f (x) dσ (x),

y ∈ Ω.

(3.28)


2537

Proof. By virtue of (2.37) and (3.12), for each y ∈ Ω fixed, we can write u(y) = D

1 I +K 2

−1 f (y)

∂ν(·) E(· − y)

= ∂Ω

= ∂Ω

1 I +K 2

−1 ∗

1 I +K 2

−1 f (x) dσ (x)

∂ν(·) E(· − y) (x)f (x) dσ (x)

=−

(3.29)

∂ν(x) G(x, y)f (x) dσ (x),

∂Ω

proving (3.28).

2

Remark. Let Ω ⊂ Rn be a bounded Lipschitz domain. Then, as is well known, there exists a family of probability measures {ωy }y∈Ω on ∂Ω with the property that the unique solution of the classical Dirichlet problem u = 0

in Ω,

u ∈ C 2 (Ω) ∩ C 0 (Ω),

u|∂Ω = f ∈ C 0 (∂Ω),

(3.30)

can be represented as

u(y) =

f (x) dωy (x),

y ∈ Ω.

(3.31)

∂Ω

Thus, from (3.30)–(3.31) and (3.28) it follows that, for each y ∈ Ω, the harmonic measure with pole at y, i.e., dωy , is absolutely continuous with respect to the surface measure, and the Radon– Nikodym derivative of the former with respect to the latter is given by dωy = −∂ν(·) G(·, y), dσ

y ∈ Ω.

(3.32)

3.2. Dirichlet and Regularity problem with nontangential maximal function estimates We debut by making the following definition. Definition 3.4. An open set Ω ⊂ Rn is said to satisfy a uniform exterior ball condition (henceforth abbreviated by UEBC), if there exists r > 0 with the following property: For each x ∈ ∂Ω, there exists a point y = y(x) ∈ Rn such that Br (y)\{x} ⊆ Rn \Ω

and x ∈ ∂Br (y).

(3.33)

The largest radius r satisfying the above property will be referred to as the UEBC constant of Ω.

2538


Parenthetically, we note that a bounded open set Ω has a C 1,1 boundary if and only if Ω and Rn \ Ω satisfy a UEBC. However, UEBC alone does allow the boundary to develop irregularities which are “outwardly directed.” It is also important to note that any bounded, open convex subset of Rn satisfies a UEBC with constant r for any r > 0. We continue to review a series of definitions and basic results. Definition 3.5. Let O be an open set in Rn . The collection of semiconvex functions on O consists of continuous functions u : O → R with the property that there exists C > 0 such that 2u(x) − u(x + h) − u(x − h) C|h|2 ,

∀x, h ∈ Rn with [x − h, x + h] ⊆ O.

(3.34)

The best constant C above is referred to as the semiconvexity constant of u. Some of the most basic properties of the class of semiconvex functions are collected in the next two propositions below. Proofs can be found in, e.g., [10]. Proposition 3.6. Assume that O is an open, convex subset of Rn . Given a function u : O → R and a finite constant C > 0, the following conditions are equivalent: (i) u is semiconvex with semiconvexity constant C; (ii) u satisfies

λ(1 − λ) u λx + (1 − λ)y − λu(x) − (1 − λ)u(y) C |x − y|2 , 2

(3.35)

for all x, y ∈ O and all λ ∈ [0, 1]; (iii) the function O x → u(x) + C|x|2 /2 ∈ R is convex in O; (iv) there exist two functions, u1 , u2 : O → R such that u = u1 + u2 , u1 is convex, u2 ∈ C 2 (O) and ∇ 2 u2 L∞ (O) C; (v) for any v ∈ S n−1 , the (distributional) second-order directional derivative of u along v, i.e., Dv2 u, satisfies Dv2 u C in O, in the sense that

u(x) Hessϕ (x)v · v dx C

O

ϕ(x) dx,

∀ϕ ∈ Cc∞ (O), ϕ 0,

(3.36)

O 2

ϕ )1j,kn is the Hessian matrix of the function ϕ; where Hessϕ := ( ∂x∂j ∂x k (vi) the function u can be represented as u(x) = supi∈I ui (x), x ∈ O, where {ui }i∈I is a family of functions in C 2 (O) with the property that ∇ 2 ui L∞ (O) C for every i ∈ I ; (vii) the same as (vi) above except that, this time, each function ui is of the form ui (x) = ai + wi · x + C|x|2 /2, for some number ai ∈ R and vector wi ∈ Rn .

We also have Proposition 3.7. Suppose that O is an open subset of Rn and that u : O → R is a semiconvex function. Then the following assertions hold:


2539

(1) The function u is locally Lipschitz in O. (2) The gradient of u (which, by Rademacher’s theorem exists a.e. in O) belongs to BVloc (O, Rn ). (3) The function u is twice differentiable a.e. in O (Alexandroff’s theorem). More concretely, for a.e. point x0 in O there exists an n × n symmetric matrix Hu (x0 ) with the property that lim

x→x0

u(x) − u(x0 ) − (x − x0 ) · ∇u(x0 ) + 2−1 (Hu (x0 )(x − x0 )) · (x − x0 ) = 0. |x − x0 |2

(3.37)

Definition 3.8. A nonempty, proper, bounded open subset Ω of Rn is called semiconvex provided there exist b, c > 0 with the property that for every x0 ∈ ∂Ω there exist an (n − 1)-dimensional affine variety H ⊂ Rn passing through x0 , a choice N of the unit normal to H , and cylinder C as in (2.1) and some semiconvex function ϕ : H → R satisfying (2.2)–(2.4) as well as (2.5). It is then clear from Proposition 3.7, the definition of a Lipschitz domain at the beginning of Section 2.1, and Definition 3.8 that bounded semiconvex domains form a subclass of the class of bounded Lipschitz domains. The key feature which distinguishes the former from the latter is described in the theorem below, recently proved in [61]. Theorem 3.9. Let Ω ⊆ Rn be a nonempty, bounded, open set. Then the following conditions are equivalent: (i) Ω is a Lipschitz domain satisfying a UEBC; (ii) Ω is a semiconvex domain. Our aim now is to show that in the case when Ω is a bounded semiconvex domain (hence, a Lipschitz domain satisfying a UEBC), then (3.28) solves (D)p for every p ∈ (1, ∞). The wellposedness of (D)p for each p ∈ (1, ∞) has been established in the case when ∂Ω ∈ C 1 by E.B. Fabes, M. Jodeit and N.M. Rivière in [27] by relying on the method of layer potentials. By way of contrast, in the current setting, we shall use (3.28) and a key ingredient in this regard is an estimate proved by M. Grüter and K.-O. Widman (see Theorem 3.3(v) in [37]) to the effect that if Ω ⊂ Rn is a domain which satisfies a UEBC then the Green function satisfies ∇x G(x, y) C dist(y, ∂Ω)|x − y|−n ,

∀x, y ∈ Ω,

(3.38)

where C depends only on n and the UEBC constant of Ω. Based on this, we shall prove the following. Theorem 3.10. Assume that Ω ⊂ Rn is a bounded semiconvex domain and suppose that p ∈ (1, ∞) is arbitrary and fixed. Then for every datum f ∈ Lp (∂Ω) the Dirichlet problem (D)p (cf. (3.27)) is uniquely solvable. Moreover, the solution u satisfies N uLp (∂Ω) Cf Lp (∂Ω) , and can be expressed as in (3.28).

(3.39)

2540


Proof. In concert with Lemma 3.1, (3.38) yields (by taking the nontangential limit of x to an arbitrary boundary point) that for each fixed y ∈ Ω, ∇x G(x, y) C dist(y, ∂Ω)|x − y|−n ,

for a.e. x ∈ ∂Ω.

(3.40)

Note that this implies that −∂ν(·) G(·, y) ∈ L∞ (∂Ω),

∀y ∈ Ω.

(3.41)

As a consequence, if 1 < p < ∞ and f ∈ Lp (∂Ω) is fixed, then u in (3.28) is a well-defined, harmonic function in Ω (for the latter claim see (3.5)). At the moment, our goal is to show that Nu ∈ Lp (∂Ω). To this end, we remark that from (3.40) and (3.28) we have u(y) C

∂Ω

dist(y, ∂Ω) dσ (x), f (x) |x − y|n

∀y ∈ Ω.

(3.42)

To continue, fix κ > 0 and consider an arbitrary boundary point, y0 ∈ ∂Ω. We claim that there exists a constant C = C(∂Ω, κ) > 0 such that

|x − y| C dist(y, ∂Ω) + |x − y0 | ,

∀y ∈ Γκ (y0 ), x ∈ ∂Ω.

(3.43)

Indeed, the fact that |x − y| dist(y, ∂Ω) is immediate. Next, pick z ∈ ∂Γκ (y0 ) ⊂ Ω such that dist(x, Γκ (y0 )) = |x − z|. We therefore have |y0 − z| = (1 + κ) dist(z, ∂Ω), hence we can write |x − y0 | |x − z| + |z − y0 | = |x − z| + (1 + κ) dist(z, ∂Ω) (2 + κ)|x − z|.

(3.44)

Thus, |x − y0 | (2 + κ) dist(x, Γκ (y0 )) (2 + κ)|x − y|, and (3.43) follows. Going further and making use of (3.43) in (3.42) gives that u(y) C

∂Ω

dist(y, ∂Ω) f (x) dσ (x), n (dist(y, ∂Ω) + |x − y0 |)

∀y ∈ Γκ (y0 ).

(3.45)

Let us fix y ∈ Γκ (y0 ) and set r := dist(y, ∂Ω). We make use of#a familiar argument based on decomposing ∂Ω into a (finite) family of dyadic annuli ∂Ω = N j =0 Rj (y0 ), where R0 (y0 ) = 2r (y0 ) and Rj (y0 ) := 2j +1 r (y0 ) \ 2j r (y0 ) for 1 j N . Estimating we can conclude that

Rj (y0 )

C r f (x) dσ (x) n n−1 (r + |x − y0 |) r 2j n

|f | dσ C2−j Mf (y0 ),

(3.46)

2j +1 r (y0 )

uniformly in j 1, where M denotes the Hardy–Littlewood maximal function on ∂Ω (cf. (2.9)). Also,

R0 (y0 )

r f (x) dσ (x) C n (r + |x − y0 |) r n−1

|f | dσ CMf (y0 ), r (y0 )

(3.47)


2541

thus, on account of (3.45)–(3.47), we obtain that (Nu)(y0 ) C(Mf )(y0 ). Since y0 was arbitrarily selected in ∂Ω, this proves that N u CMf pointwise on ∂Ω. Hence, for every 1 < p < ∞, N uLp (∂Ω) C(Ω, p)f Lp (∂Ω) ,

(3.48)

by the boundedness of M on Lp (∂Ω). In summary, to show that u defined as in (3.28) is a solution of (3.27), we are left with proving that its nontangential boundary trace exists and equals the given datum f a.e. on ∂Ω. Since we have proved that u defined as in (3.28) is harmonic and verifies (3.48), a Fatou-type theorem proved by B. Dahlberg in [18] gives that u|∂Ω exists a.e. on ∂Ω. Consequently, if we consider the linear assignment T : Lp (∂Ω) → Lp (∂Ω), given by Lp (∂Ω) f → Tf := u|∂Ω ∈ Lp (∂Ω),

(3.49)

then T is well defined and bounded, thanks to (3.48). In addition, Proposition 3.3 ensures that Tf = f whenever f ∈ L2 (∂Ω) ∩ Lp (∂Ω). By density, we may therefore conclude that Tf = f for every f ∈ Lp (∂Ω), 1 < p < ∞. For the proof of the uniqueness part in the statement of Theorem 3.10, we shall need the fact that for each p ∈ (1, ∞), the Regularity problem

(R)p

⎧ ⎨ u = 0 in Ω, p u| = f ∈ L1 (∂Ω), ⎩ ∂Ω N (∇u) ∈ Lp (∂Ω)

(3.50)

has a solution u satisfying N (∇u)

Lp (∂Ω)

Cf Lp (∂Ω) , 1

(3.51)

p

whenever f ∈ L1 (∂Ω) ∩ L21 (∂Ω). From Theorem 2.1, we know that this is the case when 1 < p 2, so it suffices to treat the case when 2 < p < ∞. Assuming that 2 < p < ∞, fix p an arbitrary f ∈ L1 (∂Ω) → L21 (∂Ω) and, by relying on Theorem 2.1, we let u solve (R)2 in Ω with datum f . In particular, N (∇u), Nu ∈ L2 (∂Ω).

(3.52)

Next, fix a boundary point z = (z , zn ) ∈ ∂Ω and assume that ϕ : Rn−1 → R is a Lipschitz function such that ϕ(z ) = zn and, for some T , R > 0, ΣR :=

x , ϕ x : x ∈ Rn−1 , x − z < R ⊂ ∂Ω,

OR,T := {x + ten : x ∈ ΣR , 0 < t < T } ⊂ Ω.

(3.53)

For ε ∈ (0, εo ), with εo > 0 small, let us also set OR,T ,ε := {x + ten : x ∈ ΣR−ε , ε < t < T − ε}.

(3.54)

2542


Without loss of generality, we can assume that the origin in Rn belongs to OR,T ,ε and that OR,T is star-like with respect to this point. With p ∈ (1, ∞) denoting the conjugate exponent for p, pick h denote the extension of h an arbitrary h ∈ Lp (∂Ω) ∩ L2 (∂Ω) such that supp h ⊆ ΣR , and let $ h. by zero to a function in Lp (∂OR,T ) ∩ L2 (∂OR,T ) Also, let v solve (D)p in OR,T with datum $ This solution is constructed according to the recipe presented in (3.28). Since the boundary datum belongs to Lp (∂OR,T ) ∩ L2 (∂OR,T ), the first part in the present proof ensures that

N v ∈ Lp (∂OR,T ) ∩ L2 (∂OR,T ).

(3.55)

Moving on, set $ v (x) := v(x) − v(0), for x ∈ OR,T . Then $ v (0) = 0, so that

1 $ v (tx)

w(x) :=

dt , t

x ∈ OR,T ,

(3.56)

0

is well defined (note that the properties of v ensure that the integral is convergent) and, with ∇η := nj=1 xj ∂j denoting the radial derivative, w is a harmonic function which is a normalized radial anti-derivative for $ v . That is, w(0) = 0 and w = 0,

v in OR,T . ∇η w = $

(3.57)

Going further, with NR,T denoting the nontangential maximal operator associated with the Lipschitz domain OR,T , we have NR,T (∇w)

Lp (∂ OR,T )

C NR,T (∇η w)Lp (∂ O

= CNR,T $ v Lp (∂ OR,T ) hLp (∂ OR,T ) + C v(0) CNR,T vLp (∂ OR,T ) + C v(0) C$ R,T )

ChLp (∂Ω) .

(3.58)

Above, the first inequality is implied by Lemma 2.12 from [64] which, in turn, is inspired by earlier work in [77,84]. Also, the next-to-last inequality is a consequence of the fact that v h, while the last inequality can be easily justified based on (2.13) and solves (D)p with datum $ the mean-value theorem for the harmonic function v. In a similar fashion, we also have that NR,T (∇w) ∈ L2 (∂OR,T ).

(3.59)

Then, for any Lipschitz domain D, with outward unit normal ν, such that D is a relatively compact subset of OR,T , we have n

v= ∂ν v = ∂ν$

νj ∂j (xk ∂k w) =

j,k=1

= ∂ν w +

n

n

νj ∂j w +

j =1

n

j =1 ∂j ∂j w

= 0 in D. We now proceed by writing

νj xk ∂k ∂j w

j,k=1

xk (νj ∂k − νk ∂j )∂j w = ∂ν w +

j,k=1

since

n

n j,k=1

xk ∂τj k ∂j w

in D,

(3.60)


∂ν uh dσ = ∂Ω

∂ OR,T

= lim

= lim

=

ε→0+ ∂ OR,T ,ε

u(· + ten )∂ν v dσ

ε→0+ ∂ OR,T ,ε

u(· + ten ) ∂ν w +

lim

u∂ν w +

n

xk ∂τj k ∂j w dσ

j,k=1

ε→0+ ∂ OR,T ,ε

lim

t→0+

t→0+

∂ν u(· + ten ) v dσ

lim

∂ν u(·, t + en ) v dσ

∂ OR,T

∂ OR,T ,ε

t→0+

t→0+

∂ OR,T

ε→0+

= lim

∂ν uv dσ = lim

lim

t→0+

= lim

∂ν u $ h dσ =

2543

n

u(· + ten )∂ν w +

n

∂τkj xk u(· + ten ) ∂j w dσ

j,k=1

(3.61)

∂τkj (xk u)∂j w dσ.

j,k=1

∂ OR,T

The third equality above is implied by (3.52) and the fact that h ∈ L2 (∂Ω), while the fourth uses (3.55) and the observation that for each small, fixed t > 0, the function u(· + ten ) is C ∞ in a neighborhood of OR,T . Next, the fifth equality is a consequence of Green’s formula for u, v which are harmonic in a neighborhood of OR,T ,ε , whereas the sixth equality is based on (3.60) (written for D := OR,T ,ε ). The seventh equality uses an integration by parts on the boundary and, finally, the last equality is implied by (3.52) and (3.59). Next, based on (3.61), (3.58) and Hölder’s inequality we may estimate

∂ν uh dσ C ∇tan uLp (∂ O ) + uLp (∂ O ) h p R,T R,T L (∂Ω) ∂Ω

C uLp (∂Ω) +

1/p

|∇u|p dσ

1

∂ OR,T \∂Ω

1/p

+

hLp (∂Ω) .

|u| dσ p

(3.62)

∂ OR,T \∂Ω

By raising both sides to the p-th power and suitably averaging the resulting estimate in R, we arrive at

1/p

1/p p p ∂ν uh dσ C u p hLp (∂Ω) . + |∇u| dx + |u| dx L1 (∂Ω) ∂Ω

Ω

Ω

(3.63)

2544


To continue, fix some ε > 0 (to be specified later), and consider the compact set Dε := x ∈ Ω: dist(x, ∂Ω) ε .

(3.64)

Then

|∇u|p + |u|p dx =

Ω

Dε

|∇u|p + |u|p dx +

|∇u|p + |u|p dx

Ω\Dε

|∇u|p + |u|p dx + εC

Dε

N (∇u)p + |N u|p dσ, (3.65)

∂Ω

where C = C(∂Ω) > 0 is a finite constant which only depends on the Lipschitz character of the domain Ω. To further bound the second-to-the-last integral above, employ interior estimates, the solvability of (R)2 (and the fact that p > 2) in order to write

|∇u|p + |u|p dx

1/p Cε uL2 (∂Ω) Cε uLp (∂Ω) . 1

1

(3.66)

Dε

When used in concert with (3.63) and (3.66), estimate (3.65) yields

1/p

p 1/p p ∂ν uh dσ Cε u p N (∇u) + |N u| dσ hLp (∂Ω) , C L1 (∂Ω) + ε ∂Ω

∂Ω

(3.67) with C independent of ε. In turn, the above estimate shows that there exists C > 0 such that for every ε > 0 there exists Cε > 0 with the property that ∂ν uLp (ΣR ) Cε uLp (∂Ω) + εC N (∇u)Lp (∂Ω) + N uLp (∂Ω) . 1

(3.68)

By covering ∂Ω with finitely many boundary patches ΣR and adding up the corresponding estimates we therefore arrive at ∂ν uLp (∂Ω) Cε uLp (∂Ω) + εC N (∇u)Lp (∂Ω) + N uLp (∂Ω) , 1

(3.69)

where C is independent of ε. Next, recall the integral representation formula u = D(u|∂Ω ) − S(∂ν u)

in Ω,

(3.70)

where the boundary traces are taken using (3.52). Based on this and well-known algebraic manipulations (which involve integrating by parts on the boundary), for every j ∈ {1, . . . , n} we then obtain ∂j u =

n k=1

∂k S(∂τj k u) − ∂j S(∂ν u)

in Ω,

(3.71)


2545

so that, by standard Calderón–Zygmund estimates, N(∇u)

Lp (∂Ω)

+ N uLp (∂Ω) C uLp (∂Ω) + ∂ν uLp (∂Ω) , 1

(3.72)

where C = C(Ω, p) > 0. Combining this with (3.69), choosing ε > 0 sufficiently close to zero and absorbing the terms with small coefficients in the left-hand side then gives N (∇u)

Lp (∂Ω)

+ N uLp (∂Ω) CuLp (∂Ω) , 1

(3.73)

where C = C(∂Ω, p) > 0. This proves (3.51). Having established that the Regularity problem (3.50) has a solution u satisfying (3.51) whenp ever p ∈ (1, ∞) and the boundary datum f belongs to L1 (∂Ω) ∩ L21 (∂Ω), we can now proceed with the proof of the uniqueness stated in Theorem 3.10. To get started, consider a family {Ωj }j ∈N of domains in Rn satisfying the following properties: (i) Each Ωj is a bounded Lipschitz domain satisfying a UEBC (hence, a semiconvex domain), whose Lipschitz character and UEBC constant are bounded uniformly in j ∈ N; # (ii) For every j ∈ N one has Ωj ⊂ Ωj +1 ⊂ Ω, and Ω = j ∈N Ωj ; (iii) There exist bi-Lipschitz homeomorphisms Λj : ∂Ω → ∂Ωj , j ∈ N, such that Λj (x) → x as j → ∞, in a nontangential fashion; (iv) There exist nonnegative, measurable functions ωj on ∂Ω which are bounded away from zero and infinity uniformly in j ∈ N, and which have the property that for each integrable function g : ∂Ωj → R the following change of variable formula holds

g dσj = ∂Ωj

g ◦ Λj ωj dσ,

(3.74)

∂Ω

where σj is the canonical surface measure on ∂Ωj . Such a family of approximating domains has been constructed in [61]. We denote by Gj (·,·) the Green function corresponding to each Ωj , j ∈ N. As seen in the proof of Lemma 3.1, if y ∈ Ω is arbitrary and fix, then for j ∈ N large enough we have Gj (x, y) = −E(x − y) + v y,j (x),

x ∈ Ωj ,

(3.75)

where v y,j is the unique solution of the problem ⎧ y,j v = 0 in Ωj , ⎪ ⎪ ⎨ p v y,j ∂Ω = E(· − y)|∂Ωj ∈ L1 (∂Ωj ) ∩ L21 (∂Ωj ), j ⎪ ⎪ ⎩ y,j Nj ∇v ∈ Lp (∂Ωj ),

(3.76)

where 1/p + 1/p = 1, and Nj is the nontangential maximal operator relative to Ωj . That this Regularity problem has at least one solution is guaranteed by our earlier considerations pertaining

2546


to (3.50). In addition, the membership Nj (∇v y,j ) ∈ Lp (∂Ωj ) is uniform in j , which further yields that for each fixed y ∈ Ω,

Nj x → ∇x Gj (x, y) ∈ Lp (∂Ωj )

uniformly, for j ∈ N large enough.

(3.77)

If now we take u to be a null-solution for the Dirichlet problem (D)p , then u ∈ C ∞ (Ω j ). Hence, for each y ∈ Ω fixed, by Proposition 3.3, (3.28) and (iii) in the above enumeration, we can write

u(y) = −

νj (x) · (∇x Gj )(x, y)(u|∂Ωj )(x) dσj (x)

(3.78)

∂Ωj

which, by (3.77), gives that u(y) C

1/p |u| dσj p

,

(3.79)

∂Ωj

for some finite constant C > 0 which is independent of j ∈ N. In concert with property (iv) listed above, this allows us to estimate u(y)p C

u Λj (x) p dσ (x),

(3.80)

∂Ω

for some C > 0 independent of j . By Lebesgue Dominated Convergence Theorem (with the uniform domination provided by N u; here, property (iii) is also used) we have that u ◦ Λj → 0 in Lp (∂Ω) as j → ∞. Thus, by letting j → ∞ we obtain that u(y) = 0. Given that y ∈ Ω was arbitrary, we may therefore conclude that u = 0, as desired. 2 We conclude this subsection with the following companion result for Theorem 3.10, dealing with the well-posedness of the Regularity problem (R)p . Theorem 3.11. Assume that Ω ⊂ Rn is a bounded semiconvex domain and suppose that p ∈ p (1, ∞) is arbitrary and fixed. Then for every datum f ∈ L1 (∂Ω) the Regularity problem (R)p (cf. (3.50)) is uniquely solvable. Moreover, the solution u satisfies N (∇u)

Lp (∂Ω)

Cf Lp (∂Ω) , 1

(3.81)

and can be expressed as in (3.28). Proof. Uniqueness follows from the corresponding theory in Lipschitz domains (in which case (3.50) is well posed for 1 0) and the fact that Lp (∂Ω) ⊆ L2 (∂Ω) if p 2 (as ∂Ω has finite measure). p Fix now 1 < p < ∞ and f ∈ L1 (∂Ω), and let u be as in (3.28). Then, applying Theorem 3.10, p we have that u = 0, N u ∈ L (∂Ω) and u|∂Ω exists and equals f σ -a.e. on ∂Ω. As such, to show that u is a solution of (3.50), all we have to prove is that N (∇u) ∈ Lp (∂Ω). To this end,

D. Mitrea et al. / Journal of Functional Analysis 258 (2010) 2507–2585 p

2547

p

we pick fj ∈ L1 (∂Ω) ∩ L21 (∂Ω) such that fj → f in L1 (∂Ω) as j → ∞ and, for each j ∈ N, we let uj be as in (3.50)–(3.51) when f is replaced by fj . Then (2.13) gives u − uj

pn L n−1 (Ω)

C N (u − uj )Lp (∂Ω) Cf − fj Lp (∂Ω) ,

(3.82)

where for the last inequality in (3.82) we have used (3.48). Since the sequence {fj }j converges np to f in Lp (∂Ω), from (3.82) we conclude that uj → u in L n−1 (Ω) as j → ∞. Now take O an arbitrary compact subset of Ω and let d := dist(O, ∂Ω). Then, using interior estimates for harmonic functions, for each n-tuple α ∈ Nn0 (recall that N0 = N ∪ {0}), and for each x ∈ O we have

− |u − uj |

α ∂ (u − uj )(x) C d |α|

Bd/2 (x)

n−1

np np n−1 C(O) |u − uj | → 0 as j → ∞.

(3.83)

Ω

Consequently, from (3.83) we can conclude that ∂ α uj → ∂ α u

uniformly on compact sets of Ω, ∀α ∈ Nn0 .

(3.84)

Fix κ > 0 and, for ε > 0, consider the nontangential approach region truncated away from the boundary Γκε (y0 ) := {y ∈ Γκ (y0 ): |y − y0 | > ε}, where y0 ∈ ∂Ω. Then clearly ∇uL∞ (Γκε (y0 )) ∇uj L∞ (Γκε (y0 )) + ∇u − ∇uj L∞ (Γκε (y0 )) ,

(3.85)

so by taking lim infj →∞ in (3.85) and using (3.84) we obtain ∇uL∞ (Γκε (y0 )) lim inf ∇uj L∞ (Γκε (y0 )) lim inf N (∇uj )(y0 ). j →∞

j →∞

(3.86)

Since y0 ∈ ∂Ω is arbitrary, (3.86) implies that for each p ∈ (1, ∞), p p N (∇u)(y) lim inf N (∇uj )(y) , j →∞

∀y ∈ ∂Ω.

(3.87)

Employing now Fatou’s lemma in concert with (3.87) we can write that N(∇u)p p

L (∂Ω)

p p p lim infN (∇uj )Lp (∂Ω) C lim inf fj Lp (∂Ω) = Cf Lp (∂Ω) , j →∞

j →∞

1

(3.88)

1

p

where for the second inequality in (3.88) we have used the fact that fj ∈ L1 (∂Ω) ∩ L21 (∂Ω) and that the uj ’s are as in (3.50)–(3.51) with f replaced by fj . This completes the proof of the well-posedness of (3.50). 2

2548


Remark. In the proof of Theorems 3.10–3.11, we have established that, in the case of the Laplace operator, the solvability of (D)p implies the solvability of (R)p , 1/p + 1/p = 1, in a bounded Lipschitz domain (assumed to satisfy a UEBC, although this condition did not play a crucial role for this particular task). This was done in a direct, fairly self-contained fashion. It should be noted, however, that, recently, a closely related result, valid for more general operators, has been proved in [75]. 4. The Poisson problem on Besov and Triebel–Lizorkin spaces This section is divided into four parts. In Section 4.1 we review the well-posedness results for the Poisson problem for the Laplacian with a Dirichlet boundary condition on the scales of Besov and Triebel–Lizorkin spaces in arbitrary Lipschitz domains. In Section 4.2 we then proceed to study similar issues in the class of semiconvex domains. Here we deal with the case when the boundary Besov spaces are Banach. As a preamble to completing this study, in Section 4.3 we establish the well-posedness of the Regularity problem with atomic data, and nontangential maximal function estimates, in semiconvex domains. Finally, in Section 4.4, we deal with the most general version of the Poisson problem formulated on the scales of Besov and Triebel–Lizorkin spaces. 4.1. Known results in the class of Lipschitz domains This section contains a description of known results in the class of Lipschitz domains. The following result appears in [50,51]. Theorem 4.1. For each bounded, connected Lipschitz domain Ω in Rn there exists ε = ε(Ω) ∈ 1 (0, 1] with the following significance. Assume that n−1 n < p ∞, (n − 1)( p − 1)+ < s < 1, are such that either one of the four conditions (I): (II): (III): (IV):

n−1 1 < p 1 and (n − 1) − 1 + 1 − ε < s < 1; n−1+ε p 1p

2 1+ε

and

2 − 1 − ε < s < 1; p

2 2 p and 0 < s < 1; 1+ε 1−ε 2 2 p ∞ and 0 < s < + ε, 1−ε p

(4.1)

is satisfied, if n 3, and either one of the following three conditions I : II : III :

2 2 p 1+ε 1−ε 2 2 0 such that the operator ∂xj ∂xk G : hp (Ω) −→ hp (Ω),

1 j, k n,

(4.9)

2550


Fig. 3.

Fig. 4.

is bounded, provided that 1 − ε < p < 1. This result, which provides a solution to a conjecture made by D.-C. Chang, S.G. Krantz and E.M. Stein (cf. [13,14]), is sharp in the class of Lipschitz domains. As pointed out in Section 1, B. Dahlberg [19] has constructed a Lipschitz domain for which (4.9) fails for the entire Lp scale, 1 < p < ∞ (cf. also [41] for a refinement of this counterexample). Nonetheless, when Ω is a bounded convex domain, we shall show that the operators in (4.9) n are actually bounded for n+1 < p 2. This will eventually be obtained in Section 5.1, after the build-up in the remainder of Section 4. 4.2. The Dirichlet problem on Besov and Triebel–Lizorkin spaces For the purpose of studying the Poisson problem in Besov and Triebel–Lizorkin spaces (a task which we take up later, in Section 4.4), it is important to be able to relate nontangential maximal function estimates to membership to these smoothness spaces. A first result in this regard is recalled below (cf. [68] for a proof). To state it, given a function u and k ∈ N, set |∇ k u| := γ |γ |k |∂ u|. Theorem 4.2. Let Ω be a bounded Lipschitz domain in Rn and assume that L is a homogeneous, constant (real) coefficient, symmetric, strongly elliptic system of differential operators of p,q order 2m. Then if u ∈ Fm−1+1/p (Ω) for some n−1 n < p 2, 0 < q < ∞ and Lu = 0 in Ω, it m−1 p u) ∈ L (∂Ω) and a natural estimate holds. follows that N(∇


2551

Our next theorem addresses the converse direction, i.e., passing from nontangential maximal function estimates to membership to Besov spaces. Theorem 4.3. Let Ω be a bounded Lipschitz domain in Rn and assume that 1 < p 2. Also, let L be a homogeneous, constant (real) coefficient, symmetric, strongly elliptic system of differential operators of order 2m. If the function u is such that Lu = 0 in Ω and N (∇ m−1 u) ∈ Lp (∂Ω), p,2 then u ∈ Bm−1+1/p (Ω), plus a natural estimate. Proof. Let u be as in the hypotheses of the theorem and, for an arbitrary multi-index γ with |γ | = p,2 m − 1, set v := ∂ γ u. Our aim is to show that v ∈ B1/p (Ω) since, granted this, Proposition 2.4 yields the desired conclusion. The strategy is to combine the techniques in [41] (where the case L = has been treated), with the results from [21]. To get started, we briefly recall the socalled trace method of interpolation. Given a compatible couple of Banach spaces A0 , A1 and 1 p < ∞, θ ∈ (0, 1), set (A0 , A1 )θ,p for the intermediate space obtained via the standard real interpolation method (cf., e.g., [6, Chapter 3]). Then, if 1 p0 , p1 < ∞ are such that 1/p = (1 − θ )/p0 + θ/p1 , we have w(A0 ,A1 )θ,p

∞ 1/p0 ∞ 1/p1 p0 dt θ θ p1 dt t f (t) t f (t) ≈ inf + , A0 t A1 t 0

(4.10)

0

uniformly for w ∈ (A0 , A1 )θ,p , where the infimum is taken over all functions f : (0, ∞) → A0 + A1 with the property that f is locally A0 -integrable, f (taken in the sense of distributions) is locally A1 -integrable, and such that limt→0+ f (t) = w in A0 + A1 . See Theorem 3.12.2 on p. 73 in [6]. In our case, since for every 1 0 such that the infimum of

∞ 2 1−1/p t f (t)

W 1,p (Ω)

0

dt + t

∞ 1−1/p 2 t f (t) p

L (Ω)

dt t

(4.12)

0

taken over all functions f : (0, ∞) → Lp (Ω) + W 1,p (Ω) with the property that f is locally W 1,p (Ω)-integrable, f (taken in the sense of distributions) is locally Lp (Ω)-integrable, and satisfying limt→0+ f (t) = v in Lp (Ω) + W 1,p (Ω), is bounded by CN (∇ m−1 u)2Lp (∂Ω) . Note that since v belongs to W 1,p (O) for any O ⊂⊂ Ω, we only need to prove the corresponding estimate for a small, fixed neighborhood of the boundary. Since the domain Ω is Lipschitz, it suffices to prove the latter in the case when the boundary point is the origin and for some constant r > 0, depending only on Ω, Br (0) ∩ ∂Ω is part of the graph of a Lipschitz function ϕ with ϕ(0) = 0 (otherwise we just use a suitable rigid motion). In addition, we can assume that

dist x , y , ∂Ω ≈ y − ϕ x , Next, we choose

∀x = x , y ∈ Br (0).

(4.13)

2552


η ∈ Cc∞ Br (0) , |η| 1, η(x) = 1 for |x| r/2,

and

θ ∈ Cc∞ ((−r, r)), θ (t) = 1

|θ | 1,

for |t| < r/2.

(4.14)

If we now consider the function f such that (f (t))(x , y) := η(x)v(x , y + t)θ (t) if x = (x , y), then clearly f is locally W 1,p (Ω)-integrable, f is locally Lp (Ω)-integrable, and limt→0+ f (t) = ηv. Thus, it is enough to prove that

∞ 2 1−1/p t f (t)

W 1,p (Ω)

dt + t

0

∞ 1−1/p 2 t f (t) p

L (Ω)

2 dt C N ∇ m−1 u Lp (∂Ω) . t

(4.15)

0

To this end, first we note that since 2 − 2/p − 1 > −1 (and Ω has finite measure),

∞

r 2 1−1/p dt I := t f (t) Lp (Ω) t 0

0

p v x , y dx dy

2/p t 2−2/p−1 dt

B2r (0)∩Ω

Cv2Lp (Ω) Cv2Lpn/(n−1) (Ω) 2 C N (v)Lp (∂Ω) ,

(4.16)

where the last step uses (2.13). Second, we have

∞

II :=

η(x)∇v x , y + t θ (t)t 1−1/p p dx

2/p

dt t

Ω

0

r C

t

r

2−2/p |x | 0 independent of f such that N (∇u)

Lp (∂Ω)

Cf h1,p (∂Ω) .

(4.52)

at

1,p

Proof. To begin with, note that Proposition 4.9 ensures that the trace in (4.51) exists in hat (∂Ω), so the problem is meaningfully formulated. To proceed, fix n−1 n 0 such that if u is the solution of (R)2 with datum f then


N (∇u)

Lp (∂Ω)

C.

2559

(4.53)

This estimate also proves (4.52). To justify it, pick x0 ∈ ∂Ω and r > 0 such that supp f ⊆ r (x0 ) and ∇tan f L∞ (∂Ω) r

−(n−1) p1

.

(4.54)

.

(4.55)

In particular, we also have that f L∞ (∂Ω) Cr

1−(n−1) p1

Following a standard technique, we shall prove (4.53) by estimating separately the Lp -norm of N(∇u) near x0 , and away from x0 . Near x0 , we make use of Hölder’s inequality and the well-posedness of the L2 -Regularity problem combined with (4.54) in order to write

N (∇u)p dσ C

100r (x0 )

100r (x0 )

Cr

N (∇u)2 dσ

p/2 · r (n−1)(1−p/2)

p N (∇u)L2 (∂Ω)

(n−1)(1−p/2)

p

Cr (n−1)(1−p/2) ∇tan f L2 (∂Ω) C.

(4.56)

To estimate the contribution from integrating N (∇u)p away from x0 , fix x ∈ Ω and let dωx denote the harmonic measure of Ω with pole at x (cf. (3.30)–(3.31)). Making use of (4.55), we have

1

u(x) = f dωx Cr 1−(n−1) p ωx r (x0 ) . (4.57) ∂Ω

Let T (x0 , r) := Br (x0 ) ∩ Ω be the “tent” region above r (x0 ) and let Ar (x0 ) be the corresponding “corkscrew” point, i.e., a point in T (x0 , r) satisfying dist(Ar (x0 ), ∂Ω) = r ≈ |Ar (x0 ) − x0 |. Then, if G(·,·) is the Green function for the Laplacian with homogeneous Dirichlet boundary condition, we have ωx (r (x0 )) ≈ r n−2 , G(x, Ar (x0 ))

∀x ∈ Ω \ T (x0 , 2r).

(4.58)

This has been proved in [18], [47, p. 476] using the so-called comparison principle (for nonnegative harmonic functions vanishing on a portion of the boundary) and Harnack’s inequality. Combining (4.57) and (4.58) we obtain that 1

u(x) Cr −(n−1)( p −1) G x, Ar (x0 ) ,

∀x ∈ Ω \ T (x0 , 2r).

(4.59)

Bring in the following estimate, proved by M. Grüter and K.-O. Widman (cf. Theorem 3.3(ii) in [37]), G(x, y) C dist(y, ∂Ω)|x − y|1−n ,

∀x, y ∈ Ω,

(4.60)

2560


where C depends only on n and the UEBC constant of Ω. Based on this, for α ∈ (0, 1) fixed and x ∈ Ω \ T (x0 , 2r) arbitrary we have 1−n

G x, Ar (x0 ) C dist Ar (x0 ), ∂Ω x − Ar (x0 ) Cr|x − x0 |1−n Cα r α |x − x0 |2−n−α .

(4.61)

Hence, (4.61) can be further used in (4.59) to conclude that 1 u(x) Cα r α−(n−1)( p −1) |x − x0 |2−n−α ,

∀x ∈ Ω \ T (x0 , 2r), ∀α ∈ (0, 1).

(4.62)

Next, consider the annuli Rj (x0 ) := 2j +1 r (x0 ) \ 2j r (x0 ), j ∈ N and, for each j , introduce the following truncated nontangential maximal functions: j,r N w (x) := sup w(y): y ∈ Γ (x) with |y − x| 2j +1 r , (Nj,r w)(x) := sup w(y): y ∈ Γ (x) with |y − x| < 2j +1 r ,

x ∈ ∂Ω,

(4.63)

x ∈ ∂Ω.

(4.64)

Combining interior estimates for u with (4.62) and the fact that for each α ∈ (0, 1) one has 1 − n − α < 0, we have that for x ∈ Rj (x0 ) ∇u(y)

C j (2 r)n+1

u(z) dz

B2j r (y)

Cα α−(n−1)( p1 −1) j n+1 r (2 r)

|z − x0 |2−n−α dz

B2j r (y)

Cα r

α−(n−1)( p1 −1)

|x − x0 |1−n−α ,

∀y ∈ Γ (x) with |y − x| 2j +1 r.

(4.65)

Thus, on account of (4.65) we obtain that for each fixed α ∈ (0, 1), N j,r (∇u)(x) Cα r

α−(n−1)( p1 −1)

|x − x0 |1−n−α ,

∀ x ∈ Rj (x0 ), ∀j ∈ N.

(4.66)

As a consequence, if α ∈ (0, 1) is fixed, then for every j ∈ N we have

j,r N (∇u)(x)p dσ (x) Cα r αp−(n−1)(1−p) 2j r (1−n−α)p 2j r n−1

Rj (x0 )

(n−1)(1−p)−αp = Cα 2 j . Since

n−1 n

(4.67)

< p 1, it follows that 0 (n − 1)( p1 − 1) < 1 and if we choose 1 − 1 < α < 1, (n − 1) p

then (n − 1)(1 − p) − αp < 0.

(4.68)


2561

Going further, for each j ∈ N, consider a Lipschitz domain Dj (x0 , r) whose Lipschitz character is bounded by a constant independent of j , r and x0 , and such that Dj (x0 , r) ⊆ Ω,

diam Dj (x0 , r) ≈ 2j r,

Rj (x0 ) ⊆ ∂Dj (x0 , r) ∩ ∂Ω.

(4.69)

Then we have

p Nj,r (∇u)p dσ C 2j r (n−1)(1− 2 )

Rj (x0 )

Nj,r (∇u)2 dσ

p/2

Rj (x0 )

(n−1)(1− p ) 2 C 2j r

Nj,r (∇u)2 dσ

p/2

∂Dj (x0 ,2r)

(n−1)(1− p ) 2 C 2j r

p/2

|∇tan u|2 dσ

∂Dj (x0 ,2r)\∂Ω

(n−1)(1− p ) 2 C 2j r

p/2

|∇u| dσ 2

,

(4.70)

∂Dj (x0 ,2r)\∂Ω

where the first inequality in (4.70) is Hölder’s, the second one is trivial while the third one uses the well-posedness of the L2 -Regularity problem in Dj (x0 , 2r) and the fact that ∇tan u = 0 on ∂Ω \ 2r (x0 ). Let us now momentarily digress and point out that, in general, if D is a Lipschitz domain in Rn with diameter diam(D), and u is a solution of the L2 -Regularity problem for in D, then

N (∇u)2 dσ C

∂D

C |∇tan u| dσ + (diam(D))2

|u|2 dσ,

2

∂D

(4.71)

∂D

with the constant in (4.71) depending only on the Lipschitz character of D. This can be seen by first assuming that D has diam(D) = 1 and then rescaling the estimate to obtain the general case.Now (4.71) continues to hold if u is replaced bu u − c, for c ∈ R. In particular, by choosing c := − ∂D u dσ , and recalling Poincaré’s inequality:

u − − u dσ ∂D

L2 (∂D)

C diam(D)∇tan uL2 (∂D) ,

(4.72)

where C again depends only on the Lipschitz character of D, we obtain that

∂D

N (∇u)2 dσ C

|∇tan u|2 dσ,

(4.73)

∂D

with C depending only on the Lipschitz character of D. It is precisely (4.73) (with D := Dj (x0 , 2r)) which is used to prove the third inequality in (4.70).

2562


Raising the resulting inequality from (4.70) to the power averaging for r/2 ρ 2r, we arrive at

2 p,

re-denoting r by ρ, and then

2/p Nj,r (∇u)(x)p dσ (x)

Rj (x0 )

(n−1)( 2 −1)−1 p C 2j r

∇u(x)2 dx.

(4.74)

{x∈Ω: |x−x0 |≈2j r, dist(x,∂Ω)C2j r}

Next, since u = 0 on ∂Ω \ r (x0 ), we may invoke boundary Cacciopolli’s inequality to further bound the last term in (4.74) and obtain

Nj,r (∇u)p dσ

2/p

(n−1)( 2 −1)−3 p C 2j r

u(x)2 dx

{x∈Ω: |x−x0 |≈2j r, dist(x,∂Ω)C2j r}

Rj (x0 )

(n−1)( 2 −1)−3 2[α−(n−1)( 1 −1)] j 2(2−n−α) j n p p Cα 2 j r 2 r 2 r r 2[(n−1)( 1 −1)−α] p = Cα 2 j , (4.75) where for the second inequality we have used (4.62). At this point, we select α as in (4.68) and combine (4.67) and (4.75) to conclude that

∞ N(∇u)p dσ C

j,r N (∇u)p dσ +

j =1 R (x ) j 0

∂Ω\2r (x0 )

C

∞

j (n−1)( 1 −1)−α p 2 +C

j =1

= C < +∞.

Nj,r (∇u)p dσ

Rj (x0 ) ∞

j 2[(n−1)( 1 −1)−α] p 2

j =1

(4.76)

Having established this, (4.53) now follows from (4.56) and (4.76). As mentioned earlier, this shows that (4.51) has a solution, which also satisfies (4.52). There remains to prove that this solution is unique. To this end, assume that the function u solves the homogeneous version of (4.51). Then (2.13) implies that u ∈ W 1,q (Ω) where q := pn/(n − 1) > 1. With this in hand, the desired conclusion then follows from the uniqueness part p,2 in Theorem 4.5, after embedding W 1,q (Ω) into Fs+1/p (Ω) for some p ∈ (1, ∞) and s ∈ (0, 1) sufficiently small. 2 4.4. The fully inhomogeneous problem The first-order of business is to convert the well-posedness result from Theorem 4.10 in which the size of the solution is measured using the nontangential maximal operator, into a well-posedness result on Besov and Triebel–Lizorkin scales.


2563

Theorem 4.11. Let Ω be a bounded semiconvex domain in Rn and assume that n−1 n < p ∞, 1 0 < q ∞ and (n−1)( p −1)+ < s < 1. Then the Dirichlet problem (4.28) has a unique solution which, in addition, satisfies (4.29). 1 Moreover, if n−1 n < p < ∞, 0 < q < ∞ and (n − 1)( p − 1)+ < s < 1 then the Dirichlet problem (4.30) also has a unique solution, which satisfies (4.31). Proof. Of course, the novel case here is when n−1 n < p 1 since otherwise the claims are covered by Theorem 4.5 and Theorem 4.6. Assume that n−1 n < p < 1 and recall the Poisson integral operator PI introduced in the course of the proof of Theorem 4.5. Theorem 4.10 then ensures that 1,p PI : hat (∂Ω) −→ u ∈ C ∞ (Ω): u = 0 in Ω, N (∇u) ∈ Lp (∂Ω) is well defined, linear and bounded. Consequently, if

n−1 n

(4.77)

< p < q 1 then

1,p

PI : Eq hat (∂Ω) −→ Eq u ∈ C ∞ (Ω): u = 0 in Ω, N (∇u) ∈ Lp (∂Ω)

(4.78)

is, by Proposition 2.9, linear and bounded. Based on this, (2.129), (2.135) and (2.132), we may therefore conclude that the operator q,q

q q,2 (Ω; ) = F 1 (Ω) ∩ ker s+ q1 s+ q

PI : Bs (∂Ω) −→ H

(4.79)

is well defined, linear and bounded, provided s := 1 + (n − 1)( q1 − p1 ). Finally, by bringing in (2.134) and observing that (n − 1)( q1 − 1) < s < 1 as q,q

PI : Bs (∂Ω) −→ F

n−1 n

< p < q 1, we obtain that

q,r (Ω) s+ q1

(4.80)

1 is well defined, linear and bounded, whenever n−1 n < q 1, (n − 1)( q − 1) < s < 1 and 0 < r < ∞. This ensures that the Dirichlet problem (4.30) has a solution which satisfies (4.31). The uniqueness of such a solution is then a consequence of the uniqueness part in Theorem 4.5 and p,q p ,q the fact that Bs+1/p (Ω) embeds into some space Bs00+1/p0 (Ω) with 1 < p0 < ∞ and s0 ∈ (0, 1).

Moving on, the fact that (4.80) is a linear and bounded operator for n−1 n < q 1, 1 (n − 1)( q − 1) < s < 1 and 0 < r < ∞, implies, via the real interpolation formulas (2.69) and (2.115) that p,q

p,q (Ω) s+ q1

PI : Bs (∂Ω) −→ B

(4.81)

1 is well defined, linear and bounded, whenever n−1 n < p 1, (n − 1)( p − 1) < s < 1 and 0 < q ∞. In particular, the Dirichlet problem (4.28) has a solution which satisfies (4.29). Finally, the uniqueness of such a solution is then proved much as before. 2

The main result in this subsection is the theorem below, dealing with the fully inhomogeneous problem for the Laplacian.

2564


Theorem 4.12. Let Ω be a bounded semiconvex domain in Rn . Then for every 0 < q ∞ and (n − 1)( p1 − 1)+ < s < 1, the Dirichlet problem

n−1 n

⎧ p,q u = f ∈ B 1 (Ω), ⎪ ⎪ s+ p −2 ⎪ ⎨ p,q Tr u = g ∈ Bs (∂Ω), ⎪ ⎪ p,q ⎪ ⎩ u ∈ B 1 (Ω)

 0 such that the solution u of (4.82) satisfies uB p,q

1 (Ω) s+ p

Cf B p,q

1 −2 s+ p

(Ω)

+ CgBsp,q (∂Ω) .

(4.83)

Similar results are also valid on the Triebel–Lizorkin scale. More precisely, if 0 < q < ∞ and (n − 1)( p1 − 1)+ < s < 1, then the Dirichlet problem

n−1 n

< p < ∞,

⎧ p,q u = f ∈ F 1 (Ω), ⎪ ⎪ s+ p −2 ⎪ ⎨ p,p

Tr u = g ∈ Bs (∂Ω), ⎪ ⎪ p,q ⎪ ⎩ u ∈ F 1 (Ω)

(4.84)

s+ p

has a unique solution, which also satisfies uF p,q

1 (Ω) s+ p

Cf F p,q

1 −2 s+ p

(Ω)

+ CgBsp,p (∂Ω) .

(4.85)

Proof. We look for a solution to (4.82) in the form u := [Πfo ]|Ω + v, where fo is an extension p,q of f to a compactly supported distribution in B 1 (Rn ), and v solves s+ p −2

v = 0 in Ω,

p,q

Tr v = g − Tr[Πfo ] ∈ Bs

(∂Ω),

p,q (Ω). s+ p1

v∈B

(4.86)

That this problem is well formulated and such a function v exists, then follows from the fact p,q that [Πfo ]|Ω ∈ B 1 (Ω), Theorem 2.7 and Theorem 4.5. This proves that (4.82) always has a s+ p

solution which satisfies (4.83). The uniqueness of such a solution follows from the corresponding uniqueness part in Theorem 4.5. Altogether, we have that (4.82) is well posed, and the argument for (4.84) is analogous. 2 Consider the (open) pentagonal region in Fig. 5. Then Theorem 4.12 states that the inhomogeneous problems (4.82), (4.84) are well posed if 0 < q ∞ whenever the point with coordinates (s, 1/p) belongs to the open shaded region in Fig. 2 (with the convention that q = ∞ for the Triebel–Lizorkin scale, while for the Besov scale the bottom segment is also included).


2565

Fig. 5.

We conclude this subsection by formally stating a result about the mapping properties of the Poisson integral operator

PI f (x) := −

∂ν(x) G(x, y)f (x) dσ (x),

y ∈ Ω,

(4.87)

∂Ω

whose proof is implicit in what we established so far. Theorem 4.13. Let Ω be a bounded semiconvex domain in Rn and assume that and (n − 1)( p1 − 1)+ < s < 1. Then the operators p,q

p,p

0 such that

wW 1,2 (Ω) C wL2 (Ω) + div wL2 (Ω) + curl wL2 (Ω) ,

(5.12)

whenever ν × w = 0. We now proceed to discuss a special well-posedness result which, in the case of domains satisfying more restrictive conditions than semiconvexity, is due to J. Kadlec and G. Talenti in the 60s; see, e.g., [2,3,31,36,39,42,65,78], for other related versions. Theorem 5.5. Let Ω be a bounded semiconvex domain in Rn . Then the boundary value problem ⎧ 2 ⎪ ⎨ u = f ∈ L (Ω), u ∈ W 2,2 (Ω), ⎪ ⎩ Tr u = 0 on ∂Ω,

(5.13)

is well posed. In particular, there exists a finite constant C = C(Ω) > 0 such that uW 2,2 (Ω) Cf L2 (Ω) . As a corollary,

(5.14)

2568


G : L2 (Ω) −→ W 2,2 (Ω)

(5.15)

is well defined, linear and bounded. Proof. Given f ∈ L2 (Ω) → W −1,2 (Ω), Lax–Milgram’s lemma ensures that there exists a unique function u ∈ W01,2 (Ω) such that u = f . Thus, it suffices to show that the vector field w := ∇u ∈ L2 (Ω)n actually belongs to W 1,2 (Ω)n (plus a natural estimate). This, in turn, is a direct consequence of (5.11)–(5.12) upon noticing that div w = u = f ∈ L2 (Ω), curl w = 0, and that ν × w = 0. The latter equality is (in light of (5.9) and (5.10)) a consequence of the fact that if u ∈ W01,2 (Ω) then n

(∂k u)∂j (Ψj k − Ψkj ) dx = 0,

(5.16)

j,k=1 Ω

for any family of functions Ψj k from W 1,2 (Ω). Indeed, (5.16) is readily verified when u ∈ Cc∞ (Ω) (integrating by parts and using simple symmetry considerations), so a simple density argument concludes the proof. 2 The proof of the above theorem makes essential use of Proposition 5.4 which, in turn, involves successive integrations by parts (cf. [67]). Thus, it makes essential use of the Hilbert character of L2 and, as such, it does not readily extend to Lp -based Sobolev space with p = 2. Our goal is to establish the following extension. Theorem 5.6. Let Ω be a bounded semiconvex domain in Rn . Assume that α ∈ R and 0 < p ∞ are such that either p = ∞ and −2 < α < −1, or 1 n+1+α α + 1 (n + 1)α 1 , + < < min α + 2, . max 0, 2 2 2 p n

(5.17)

In geometrical terms, condition (5.17) is equivalent to the membership of the point with coordinate (α, 1/p) to the interior of the open pentagonal region in Fig. 2 (in Section 1). Then, with A ∈ {B, F }, the Green operator p,q

G : Ap,q α (Ω) −→ Aα+2 (Ω)

(5.18)

is well defined, linear and bounded for any 0 < q ∞ when A = B, and any 0 < q < ∞ when A = F. Proof. This follows from Corollary 5.1, Theorem 5.5 and complex interpolation.

2

In the special case when Ω is actually a convex domain, Theorem 5.6 can be further refined as follows. Theorem 5.7. Let Ω be a bounded convex domain in Rn . Assume that α ∈ R and 0 < p ∞ are such that either p = ∞ and −2 < α < −1, or


1 n+1+α n−1−α α+1 < < min α + 2, , . max 0, 2 p n n−2

2569

(5.19)

Geometrically, (5.19) is equivalent to the membership of the point with coordinate (α, 1/p) to the interior of the open pentagonal region in Fig. 1 (in Section 1). Then the Green operators p,q

0 < q ∞,

(5.20)

p,q

0 < q < ∞,

(5.21)

G : Bαp,q (Ω) −→ Bα+2 (Ω), G : Fαp,q (Ω) −→ Fα+2 (Ω),

are well defined, linear and bounded (assuming p < ∞ in (5.21)). Proof. In the special case in which 0
s, then the spaces appearing in (5.43) further take the more familiar p,q form B1+s (∂Ω). (ii) Although, as proved above, the trace operator in (5.41) is onto, this does not, generally speaking, have a universal, linear, bounded, right-inverse, even in the class of convex domains. p p∗ p,p Indeed, if this were the case, then Xs (∂Ω) := {g ∈ L1 (∂Ω): ∇tan g ∈ [Bs (∂Ω)n ]tan } would be p,q p a retract of F1+s+1/p (Ω). In turn, this would imply that {Xs (∂Ω)}p,s is a complex interpolation p0 p1 p scale (in the sense that [Xs0 (∂Ω), Xs1 (∂Ω)]θ = Xs (∂Ω) if θ ∈ (0, 1) and 1/p = (1 − θ )/p0 + p,q p,q p θ/p1 ). Now, (, Tr) : F1+s+1/p (Ω) → Fs+1/p−1 (Ω) ⊕ Xs (∂Ω) is a bounded, linear operator for all indices and, granted the current working assumption, would be an isomorphism when p = q = 2 and s = 1/2, whenever Ω is a bounded convex domain (here Theorem 5.5 is also used). As a consequence of the fact that being an isomorphism is a stable property on complex interpolation scales (cf., e.g., [44] for general results of this type), we would then be able to conclude (specializing the above discussion to the case when s = 1 − 1/p and q = 2) that the problem u = f ∈ Lp (Ω), u ∈ W 2,p (Ω), Tr u = 0 on ∂Ω, continues to be uniquely solvable for some p > 2. This, however, contradicts the counterexamples in [31,4]. To proceed, it is natural to make the following definition (extending earlier considerations in [34]).

2574


Definition 5.11. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and denote by ν the outward unit normal to ∂Ω. Also, suppose that 1 < p, q < ∞ and 0 < s < 1. Then introduce p,q p,q NBs (∂Ω) := g ∈ Lp (∂Ω): gνj ∈ Bs (∂Ω), 1 j n ,

(5.44)

where the νj ’s are the components of ν. This space is equipped with the natural norm gNBp,q := s (∂Ω)

n j =1

gνj Bsp,q (∂Ω) .

(5.45)

As we shall see momentarily, the above spaces are closely related to the standard Besov scale on ∂Ω (the acronym NB stands for “new Besov”), to which they reduce if the domain is sufficiently smooth. Lemma 5.12. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and fix 1 < p, q < ∞, 0 < p,q s < 1. Then NBs (∂Ω) is a reflexive Banach space which embeds continuously into Lp (∂Ω). Furthermore, if the multiplication operators by the components of the unit normal are p,q p,q p,q bounded on Bs (∂Ω), then NBs (∂Ω) = Bs (∂Ω). Thus, in particular, this is the case when Ω is a bounded domain whose boundary is of class C 1,ε , with ε > 1/p. Proof. Obviously we have g=

n

νj (gνj )

for any function g ∈ Lp (∂Ω),

(5.46)

j =1

. Consequently, we obtain that the natural incluso that, in particular, gLp (∂Ω) ngNBp,q s (∂Ω) p,q p,q p sion NBs (∂Ω) → L (∂Ω) is bounded. If {gk }k∈N is a Cauchy sequence in NBs (∂Ω) then, p,q for each j ∈ {1, . . . , n}, {gk νj }k∈N is a Cauchy sequence in Bs (∂Ω) and, from what we have proved so far, {gk }k∈N converges in Lp (∂Ω) to some g ∈ Lp (∂Ω). It follows that {gk νj }k∈N converges in Lp (∂Ω) to gνj for each j ∈ {1, . . . , n}. With this in hand, it is then easy to conclude p,q p,q that g is the limit of {gk }k∈N in NBs (∂Ω). This proves that NBs (∂Ω) is Banach. Next, if we consider p,q n p,q Φ : NBs (∂Ω) −→ Bs (∂Ω) ,

Φ(g) := (gνj )1j n ,

(5.47)

p,q

it follows that Φ is an isometric embedding, which allows identifying NBs (∂Ω) with a closed p,q p,q subspace of the reflexive space [Bs (∂Ω)]n . As is well known, this implies that NBs (∂Ω) is also reflexive. Finally, the claims in the second part in the statement of the lemma are direct consequences of (5.46). 2 p,q

Our interest in the space NBs (∂Ω), 1 < p, q < ∞, 0 < s < 1, stems from the fact that this arises naturally when considering the Neumann trace operator acting from

p,q p,q p,q u ∈ A1+s+1/p (Ω): γN (u) = 0 = A1+s+1/p (Ω) ∩ As+1/p,z (Ω),

A ∈ {B, F }, (5.48)


2575

p,q

considered as a closed subspace of A1+s+1/p (Ω) (hence, a Banach space when equipped with the inherited norm). More specifically, we have Lemma 5.13. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and that 1 < p, q < ∞ and 0 < s < 1. Then the Neumann trace operator γN considered in the contexts of p,q

p,q

p,q

(5.49)

p,q

p,q

p,p

(5.50)

γN : B1+s+1/p (Ω) ∩ Bs+1/p,z (Ω) −→ NBs (∂Ω), γN : F1+s+1/p (Ω) ∩ Fs+1/p,z (Ω) −→ NBs (∂Ω),

is, in each case, well defined, linear, bounded, onto and with a linear, bounded right inverse. p,q In addition, the null-spaces of γN in (5.49) and (5.50) are, respectively, B1+s+1/p,z (Ω) and p,q F1+s+1/p,z (Ω), so that, in particular, p,q

p,q NBs (∂Ω)

is isomorphic to

p,q

B1+s+1/p (Ω) ∩ Bs+1/p,z (Ω) p,q

B1+s+1/p,z (Ω) p,q

p,p NBs (∂Ω)

is isomorphic to

(5.51)

,

p,q

F1+s+1/p (Ω) ∩ Fs+1/p,z (Ω) p,q

F1+s+1/p,z (Ω)

(5.52)

.

p,q

p,q

Proof. To prove the well-definiteness of (5.49), note that if u ∈ B1+s+1/p (Ω) ∩ Bs+1/p,z (Ω) then (2.109) and Theorem 5.9 give

p,q

n p 0, γN (u) = γ (u) ∈ (g0 , g1 ) ∈ L1 (∂Ω) ⊕ Lp (∂Ω): ∇tan g0 + g1 ν ∈ Bs (∂Ω)

(5.53)

p,q

from which we deduce that γN (u) ∈ NBs (∂Ω) and γN (u)NBp,q CuB p,q for s (∂Ω) 1+s+1/p (Ω) some C = C(Ω, p, q, s) > 0 independent of u. This shows that (5.49) is well defined, linear and bounded. Moving on, denote by E a linear, bounded, right inverse for γ in (5.34). Then, if p,q

n p,q p ι : NBs (∂Ω) → (g0 , g1 ) ∈ L1 (∂Ω) ⊕ Lp (∂Ω): ∇tan g0 + g1 ν ∈ Bs (∂Ω)

(5.54)

p,q

is the injection given by ι(g) := (0, g), for every g ∈ NBs (∂Ω), it follows that the composip,q p,q p,q tion E ◦ ι : NBs (∂Ω) → B1+s+1/p (Ω) ∩ Bs+1/p,z (Ω) is a linear, bounded, right inverse for the operator γN in (5.49). As a consequence, this operator is onto. Finally, the fact that the nullp,q space of γN in (5.49) is precisely B1+s+1/p,z (Ω) follows from its definition and the last part in the statement of Theorem 5.9. The proof on the Triebel–Lizorkin scale is analogous and this finishes the proof of the lemma. 2 Our goal is to use the above Neumann trace result in order to extend the action of the trace p,q p,q operator Tr from Theorem 2.7 to the space {u ∈ A1/p−s (Ω): u ∈ A1/p−s (Ω)}, which we consider equipped with the graph norm u → uAp,q (Ω) + uAp,q (Ω) , A ∈ {B, F }. To state 1/p−s 1/p−s our next result, we agree that, given a Banach space X , the pairing X ∗ Λ, XX is the duality matching between a functional Λ ∈ X ∗ and a vector X ∈ X . In order to streamline notation, let us also make the following definition.

2576


Definition 5.14. Given a bounded Lipschitz domain Ω ⊂ Rn and 1 < p, q < ∞, 0 < s < 1, set p ,q

∗ p,q NB−s (∂Ω) := NBs (∂Ω) ,

(5.55)

where 1/p + 1/p = 1/q + 1/q = 1. We have Theorem 5.15. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and that 1 < p, q < ∞, 1/p +1/p = 1/q +1/q = 1, and 0 < s < 1. Then there exists a unique linear, bounded operator p,q p,q p,q γD : u ∈ B1/p−s (Ω): u ∈ B1/p−s (Ω) −→ NB−s (∂Ω)

(5.56)

which is compatible with the trace Tr from Theorem 2.7, in the sense that, for each smoothness index α ∈ (1/p, 1 + 1/p), one has γD (u) = Tr u

p,q

for every u ∈ Bαp,q (Ω) with u ∈ B1/p−s (Ω).

(5.57)

Furthermore, this extension of the trace operator has dense range and it allows for the following generalized integration by parts formula p ,q

NBs

γN (w), γD (u) NBp,q (∂Ω)

(∂Ω)

−s

= (B p,q

1/p−s

(Ω))∗

w, uB p,q

1/p−s (Ω)

− (B p,q

1/p−s (Ω))

∗

w, uB p,q

1/p−s (Ω)

(5.58)

,

valid for every p,q

u ∈ B1/p−s (Ω)

p,q

with u ∈ B1/p−s (Ω)

p ,q

p ,q

and w ∈ B1+s+1/p (Ω) ∩ Bs+1/p ,z (Ω).

(5.59)

Finally, similar results are valid for the Triebel–Lizorkin scale, in which case p,q p,q p,p γD : u ∈ F1/p−s (Ω): u ∈ F1/p−s (Ω) −→ NB−s (∂Ω),

(5.60)

in a linear and bounded fashion. p,q

p,q

Proof. Let u ∈ B1/p−s (Ω) be such that u ∈ B1/p−s (Ω). We attempt to define a funcp q

p q

tional γD (u) ∈ (NBs (∂Ω))∗ as follows. Assume an arbitrary function g ∈ NBs (∂Ω) has p ,q p q been given. By Lemma 5.13, there exists some w ∈ B1+s+1/p (Ω) ∩ Bs+1/p ,z (Ω) such that γN (w) = g and w p ,q Cg p q for some constant C = C(Ω, p, q, s) > 0, NBs

B1+s+1/p (Ω)

(∂Ω)

independent of g. We then set p ,q

NBs

(∂Ω)

g, γD (u)

p ,q

(Ns

(∂Ω))∗

:= (B p,q

1/p−s (Ω))

− (B p,q

∗

w, uB p,q

1/p−s (Ω))

1/p−s (Ω)

∗

w, uB p,q

1/p−s (Ω)

.

(5.61)


2577

The first-order of business is to show that the above definition does not depend on the particular choice of w, with the properties listed above. By linearity, this comes down to proving the folp ,q p ,q lowing claim: If u is as before and w ∈ B1+s+1/p (Ω) ∩ Bs+1/p ,z (Ω) is such that γN (w) = 0 then (B1/p−s (Ω))∗ w, uB1/p−s (Ω) p,q

p,q

= (B p,q

1/p−s (Ω))

∗

w, uB p,q

1/p−s (Ω)

.

(5.62)

p q

However, since Lemma 5.13 gives that w ∈ B1+s+1/p ,z (Ω), it follows that w can be approxip ,q

mated with functions form Cc∞ (Ω) in the norm of B1+s+1/p (Ω), by Proposition 2.6. With this in hand, (5.62) follows via a standard limiting argument. Thus, formula (5.61) yields a well-defined p ,q functional γD (u) ∈ (NBs (∂Ω))∗ which satisfies γD (u)

p,q

NB−s (∂Ω)

C uB p,q

1/p−s (Ω)

+ uB p,q

1/p−s (Ω)

.

(5.63)

Thus, the operator (5.56) is well defined, linear and bounded. Also, by definition, this operator will satisfy (5.58). Next, we will show that (5.57) is valid whenever α ∈ (1/p, 1 + 1/p). Fix such a number α p,q p,q p,q along with some function u ∈ Bα (Ω) with u ∈ B1/p−s (Ω). In particular, Tr u ∈ Bα−1/p (∂Ω). We shall make use of a density result to the effect that if 1 < p, q < ∞,

−1 + 1/p < β < 1/p,

α < 2 − β,

(5.64)

densely,

(5.65)

then p,q C ∞ (Ω) → u ∈ Bαp,q (Ω): u ∈ Bβ (Ω)

where the latter space is equipped with the natural graph norm u → uBαp,q (Ω) + uB p,q (Ω) . β

On the scale of L2 -based Sobolev spaces, this appears as Lemma 1.5.3.9 on p. 60 of [36] when α = 1, β = 0, and in [17] when α < 2, β = 0. The case of Besov spaces considered here is dealt with similarly. Then (5.57) follows as soon as we show that

Lp (∂Ω)

γN (w), Tr u Lp (∂Ω) = (B p,q

1/p−s (Ω))

− (B p,q

∗

w, uB p,q

1/p−s (Ω))

1/p−s (Ω)

∗

w, uB p,q

1/p−s (Ω)

(5.66)

p ,q

whenever u ∈ C ∞ (Ω) and w ∈ B1+s+1/p (Ω) with Tr w = 0. To this end, consider wj ∈ p ,q

C ∞ (Ω), j ∈ N, such that wj → w in B1+s+1/p (Ω). Then passing to the limit j → ∞ in Green’s formula

γN (wj ) Tr u dσ − Tr wj γN (u) dσ = wj u dx − uwj dx, (5.67) ∂Ω

∂Ω

Ω

readily yields (5.66). This concludes the proof of (5.57).

Ω

2578


Next, we aim to establish the uniqueness of an operator satisfying (5.56)–(5.57). This, however, is again a consequence of (5.64)–(5.65). There remains to show that the trace operator (5.56) has dense range. With this goal in mind, granted (5.64)–(5.65), it suffices to show that p,q u|∂Ω : u ∈ C ∞ (Ω) is a dense subspace of NB−s (∂Ω).

(5.68)

Going further, (5.68) will follow as soon as we prove that p,q

∗ if Λ ∈ NB−s (∂Ω) vanishes on Tr C ∞ (Ω) then necessarily Λ = 0.

(5.69)

To this end, fix a functional Λ as in the first part of (5.69), recall (5.55) and note that p ,q since NBs (∂Ω) is a reflexive Banach space, continuously embedded into Lp (∂Ω) (cf. p ,q Lemma 5.12), we may conclude that Λ ∈ NBs (∂Ω) → Lp (∂Ω). Together with Lemma 5.13, p ,q p ,q this further shows that there exists a function w ∈ B1+s+1/p (Ω) ∩ Bs+1/p ,z (Ω) with the property that γN (w) = Λ. Consequently, p ,q

NBs

(∂Ω)

γN (w), Tr u NBp,q (∂Ω) = 0 for all u ∈ C ∞ (Ω). −s

(5.70)

With (5.70) in hand, the integration by parts formula (5.58) in Theorem 5.15 then yields (B1/p−s (Ω))∗ w, uB1/p−s (Ω) p,q

p,q

= (B p,q

1/p−s (Ω))

∗

for all u ∈ C ∞ (Ω).

w, uB p,q

1/p−s (Ω)

(5.71)

p ,q

On the other hand, since w ∈ B1+s+1/p (Ω), for every u ∈ C ∞ (Ω) we may write

p,q (B1/p−s (Ω))∗

w, u

p,q B1/p−s (Ω)

=

γN (w) Tr u dσ −

∂Ω

Tr wγN (u) dσ

∂Ω

+ (B p,q

1/p−s (Ω))

∗

w, uB p,q

1/p−s (Ω)

.

(5.72)

p ,q

Upon recalling that we also have w ∈ Bs+1/p ,z (Ω) and Λ = γN (w), based on (5.71)–(5.72) we deduce that

Λ Tr u dσ = 0 for all u ∈ C ∞ (Ω). (5.73) ∂Ω

Since, as is well known, Tr C ∞ (Ω) is dense in Lp (∂Ω),

(5.74)

it follows from (5.73) that Λ = 0 in Lp (∂Ω), completing the justification of (5.69). The case of the operator (5.60) is treated analogously and this finishes the proof of the theorem. 2 We conclude this section with a discussion aimed at showing that, in the class of bounded p,q Lipschitz domains Ω ⊂ Rn , the space NBs (∂Ω) is nontrivial. One line of reasoning is to note that, thanks to (5.51)–(5.52), this comes down to checking whether

D. Mitrea et al. / Journal of Functional Analysis 258 (2010) 2507–2585 p,q

p,q

p,q

p,q

p,q

p,q

2579

B1+s+1/p,z (Ω) = B1+s+1/p (Ω) ∩ Bs+1/p,z (Ω), F1+s+1/p,z (Ω) = F1+s+1/p (Ω) ∩ Fs+1/p,z (Ω).

(5.75)

Take the second relation in (5.75) in the case when n = 2, p = q = 2 and s = 1/2. In this setting, consider the sector Ωα := {reiθ : 0 < θ < α, 0 < r < 1} where α ∈ (0, π). Also, pick ψ ∈ C ∞ (R2 ) with ψ(z) = 1 for |z| 1/4 and ψ(z) = 0 for |z| > 1/2. Then the function u(z) := ψ(z) Im(zπ/α ) = ψ(reiθ )r π/α sin(πθ/α), if z = reiθ , satisfies

1/2 |∇ 2 u| dx dy C r 2(π/α−2)+1 dr < +∞,

Ωα

(5.76)

0

since α < π . Consequently, for α ∈ (0, π), we have that u ∈ W 2,2 (Ωα ). Since also Tr u = 0 on ∂Ωα , we may conclude that 2,2 u ∈ F22,2 (Ωα ) ∩ F1,z (Ωα ).

(5.77)

2,2 (Ωα ). To jusAs such, the desired conclusion follows as soon as we show that u ∈ / F2,z π/α tify this, note that near the origin u(z) = Re(z ), a harmonic function in Ωα , which has v(z) := Im(zπ/α ) as a harmonic conjugate in Ωα . Thus, by the Cauchy–Riemann equations, we have |γN (u)| = |∇tan (v|∂Ωα )| a.e. near the point 0 ∈ ∂Ωα . Now, since v|∂Ωα is not constant near 0 ∈ ∂Ωα , its tangential gradient does not vanish identically, so γN (u) = 0 on ∂Ωα . Thus, 2,2 (Ωα ), as wanted. In general, the aforementioned nontriviality statement can be seen u∈ / F2,z from the following result.

Corollary 5.16. Assume that Ω ⊂ Rn is a bounded Lipschitz domain and that 1 < p, q < ∞, 0 < s < 1. Then

p,q u|∂Ω : u ∈ C ∞ (Ω) is a dense subspace of NB−s (∂Ω).

Proof. Given that the map (5.56) has dense range, this is a consequence of (5.64)–(5.65).

(5.78) 2

5.3. Noncanonical Poisson problems Having extended the action of the trace operator beyond the canonical range of Theorem 2.7, here the goal is to establish the well-posedness of the Poisson problem, for the Laplacian with Dirichlet boundary condition, in settings when one either has more smoothness (cf. Theorem 5.17), or less smoothness (cf. Theorem 5.19), than in Theorem 4.12. Theorem 5.17. Assume that Ω ⊂ Rn is a bounded domain and that 0 < p, q ∞, (n − 1)( p1 − 1)+ < s < 1. Set α := 1/p + s − 1. In addition, suppose that either (i) the domain Ω is convex and (5.19) holds, or (ii) the domain Ω is semiconvex and (5.17) holds.

2580


Then the boundary value problem ⎧ p,q ⎪ ⎨ u = f ∈ Bs+1/p−1 (Ω), p,q u ∈ B1+s+1/p (Ω), ⎪ ⎩ Tr u = g on ∂Ω,

(5.79)

has a solution if and only if p∗

g ∈ L1 (∂Ω)

p,q and ∇tan g ∈ Bs (∂Ω)n tan ,

(5.80)

s −1 ∗ where p ∗ := ( p1 − n−1 ) ∈ (1, n−1 n−2 ) if p 1, and p := p if p > 1. Furthermore, in the case when g is as in (5.80), the solution u of (5.79) is unique. Finally, a similar statement is valid on the Triebel–Lizorkin scale, i.e. for

⎧ p,q ⎪ ⎨ u = f ∈ Fs+1/p−1 (Ω), p,q u ∈ F1+s+1/p (Ω), ⎪ ⎩ Tr u = g on ∂Ω

(5.81)

(assuming that, in addition, p, q < ∞). In this case, (5.81) is solvable, if and only if p∗

g ∈ L1 (∂Ω)

p,p and ∇tan g ∈ Bs (∂Ω)n tan ,

(5.82)

and the solution is unique, whenever the boundary datum is as in (5.82). Proof. In one direction, if (5.79) has a solution u, then Theorem 5.10 gives that g = Tr u satisp,q fies (5.80). Conversely, take g as in (5.80), and let v ∈ B1+s+1/p (Ω) be such that Tr v = g (this is possible by Theorem 5.10). If we now set w := G(v − f ) then by Theorem 5.6 when Ω is a semiconvex domain, and by Theorem 5.7 when Ω is convex, it follows that

p,q

w = v − f ∈ Bs+1/p−1 (Ω), p,q

w ∈ B1+s+1/p (Ω),

(5.83)

Tr w = 0.

Hence, u := v − w solves (5.79). Uniqueness for (5.79) follows from the uniqueness part in Theorem 4.12 and standard embedding results on the Besov scale in Ω. Finally, the case of (5.81) is treated analogously. 2 Specializing the above theorem to the case when 1 < p 2 and s = 1 − 1/p then yields the following. Corollary 5.18. Assume that Ω ⊂ Rn is a bounded semiconvex domain. Then for each 1 < p 2, the problem u = f ∈ Lp (Ω),

u ∈ W 2,p (Ω),

Tr u = g

on ∂Ω,

(5.84)


2581

has a unique solution if and only if p,p and ∇tan g ∈ Bs (∂Ω)n tan .

p

g ∈ L1 (∂Ω)

(5.85)

In the case when g = 0, this corollary is well known; cf. [42,3,31,50]. The case when the Poisson problem is formulated with the Dirichlet boundary condition interpreted in the sense of Theorem 5.15 is discussed next. Before doing so, we note that this theorem can be regarded as the dual statement corresponding to Theorem 5.17 (this will become more apparent from an examination of its proof). Theorem 5.19. Assume that Ω ⊂ Rn is a bounded domain and that 1 < p, q < ∞, 0 < s < 1. Denote by p , q the conjugate exponents of p, q, and set α := 1/p + s − 1. In addition, suppose that either (i) the domain Ω is convex and (5.19) holds, or (ii) the domain Ω is semiconvex and (5.17) holds. Then each of the following boundary value problems ⎧ p ,q ⎪ u = f ∈ F1/p −s (Ω), ⎪ ⎪ ⎨

⎧ p ,q ⎪ u = f ∈ B1/p −s (Ω), ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎩

⎪ ⎪ ⎪ ⎩

p ,q

u ∈ F1/p −s (Ω),

p ,p

γD (u) = g ∈ NB−s (∂Ω),

p ,q

u ∈ B1/p −s (Ω),

(5.86)

p ,q

γD (u) = g ∈ NB−s (∂Ω)

has a unique solution, which in addition satisfies a natural estimate. Proof. Consider the first problem in (5.86) (the second one is treated similarly), and observe that, by subtracting a suitable Newtonian potential, there is no loss of generality in assuming that f = 0. To continue, assume that g ∈ {ψ|∂Ω : ψ ∈ C ∞ (Ω)} and set

u(y) := −

y ∈ Ω.

∂ν(x) G(x, y)g(x) dσ (x),

(5.87)

∂Ω

Since u|∂Ω = g and u = 0 in Ω, via a density argument based on Corollary 5.16 and Theorem 5.15, it suffices to show that there exists a finite constant C = C(Ω, p, q, s) > 0 such that u

p ,q

F1/p −s (Ω)

Cg

p ,q

NB−s (∂Ω)

.

(5.88)

With this goal in mind, we proceed by duality and note that for every v ∈ C ∞ (Ω), we have

u(y)v(y) dy = − Ω

∂Ω

= ∂Ω

G(x, y)v(y) dy dσ (x)

g(x)∂ν(x) Ω

g∂ν (Gv) dσ = Ω

(γN ◦ G)∗ g(y)v(y) dy,

(5.89)

2582


so that, further, u = (γN ◦ G)∗ g.

(5.90)

Above, (γN ◦ G)∗ is the adjoint of the composition G

p,q

p,q

p,q

γN

p,q

γN ◦ G : Fs+1/p−1 (Ω) −→ F1+s+1/p (Ω) ∩ Fs+1/p,z (Ω) −→ NBs (∂Ω).

(5.91)

Since by Theorem 5.6, Theorem 5.7 and Lemma 5.13 each of the above arrows is a bounded assignment, it follows from this, Definition 5.14 and (2.103) that p ,q

p ,q

(γN ◦ G)∗ : NB−s (∂Ω) −→ F1/p −s (Ω)

(5.92)

is a well-defined, linear and bounded operator. Wit this in hand, (5.88) follows from (5.90). At this stage, there remains to establish uniqueness. Thus, assume that u satisfies p ,q

u = 0 in Ω,

u ∈ F1/p −s (Ω), p ,q

p ,q

γD (u) = 0 in NB−s (∂Ω).

(5.93)

Then, for an arbitrary v ∈ (F1/p −s (Ω))∗ = F1/p+s−1 (Ω), consider w := Gv in Ω and note p,q p,q that w ∈ F1+s+1/p (Ω) ∩ Fs+1/p,z (Ω) by the aforementioned mapping properties of the Green operator. Then the version of the integration by parts formula (5.58)–(5.59) written for the Triebel–Lizorkin scale (and the conjugate integrability indices) gives that p ,q

(F1/p −s (Ω))∗

=

v, u

p ,q

(F1/p −s (Ω))∗

p,q

p ,q

F1/p −s (Ω)

w, u

p ,q

F1/p −s (Ω)

γ (w), = NBp,p γD (u) s (∂Ω) N = 0 + 0 = 0.

p ,p

NB−s (∂Ω)

+

p ,q

(F1/p −s (Ω))∗

w, u

p ,q

F1/p −s (Ω)

(5.94)

p ,q

In turn, since v ∈ (F1/p −s (Ω))∗ was arbitrary, the Hahn–Banach theorem gives that u = 0, as desired. 2 In closing, we note that Theorem 1.6 is obtained by specializing the above theorem to the case when 1 < p 2 and s = 1/p (and readjusting notation). References [1] W. Abu-Shammala, A. Torchinsky, The Hardy–Lorentz spaces H p,q (Rn ), Studia Math. 182 (3) (2007) 283–294. [2] V. Adolfsson, L2 -integrability of second-order derivatives for Poisson’s equation in nonsmooth domains, Math. Scand. 70 (1) (1992) 146–160. [3] V. Adolfsson, Lp -integrability of the second order derivatives of Green potentials in convex domains, Pacific J. Math. 159 (2) (1993) 201–225. [4] V. Adolfsson, D. Jerison, Lp -integrability of the second order derivatives for the Neumann problem in convex domains, Indiana Univ. Math. J. 43 (4) (1994) 1123–1138.


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Réflexivité d’une extension d’un opérateur sous-normal par un opérateur algébrique M’hammed Benlarbi Delaï, Omar El-Fallah ∗ , Khalid Elhachimi Faculté des Sciences, Département de Mathématiques, Université Mohammed V, BP 1014 Rabat, Maroc Reçu le 17 juillet 2008 ; accepté le 4 décembre 2009 Disponible sur Internet le 29 janvier 2010 Communiqué par N. Kalton

Résumé Un opérateur linéaire borné T sur un espace de Hilbert est dit réflexif si les opérateurs qui laissent invariant les sous-espaces invariants pour T sont limites, pour la topologie faible des opérateurs, de polynômes en T . Dans ce papier nous donnons une condition nécessaire et suffisante pour que l’extension d’un opérateur sous-normal par un opérateur algébrique soit réflexive. Nous donnons également une formule du défaut de réflexivité de telles extensions. © 2010 Publié par Elsevier Inc. Abstract A bounded linear operator on a Hilbert space is said to be reflexive if the operators which leave invariant the invariant subspaces of T are wot-limits of polynomials in T . In this paper we give a necessary and sufficient condition for an extension of a subnormal operator by an algebraic one to be reflexive.We also give a formula for the reflexivity defect of such extensions. © 2010 Publié par Elsevier Inc. Mots-clés : Décomposition de Sarason ; Opérateur algébrique ; Opérateur réflexif ; Opérateur sous-normal ; Point d’évaluation borné ; Sous-espaces invariants Keywords: Algebraic operator; Bounded point evaluation; Invariant subspaces; Reflexive operator; Sarason decomposition; Subnormal operator

* Auteur correspondant.

Adresses e-mail : [email protected] (M. Benlarbi Delaï), [email protected] (O. El-Fallah), [email protected] (K. Elhachimi). 0022-1236/$ – see front matter © 2010 Publié par Elsevier Inc. doi:10.1016/j.jfa.2009.12.010

M. Benlarbi Delaï et al. / Journal of Functional Analysis 258 (2010) 2586–2600

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1. Introduction Soient H un espace de Hilbert complexe séparable et B(H) l’algèbre des opérateurs linéaires bornés sur H. Pour T ∈ B(H), nous désignons par Lat T le treillis des sous-espaces fermés de H invariants pour T , AlgLat T l’algèbre des opérateurs S ∈ B(H) tels que Lat T ⊂ Lat S et W (T ) la fermeture faible de l’algèbre des polynômes en T . Un opérateur T est dit réflexif si AlgLat T = W (T ). Si p est un polynôme, nous désignons par WT (p(T )) la fermeture faible de WOT l’idéal engendré par p(T ) dans W (T ) (i.e. WT (p(T )) = {p(T )q(T ): q ∈ C[z]} ) et par {T } le commutant de T . La notion de réflexivité fut initiée par D. Sarason dans [10], où il montre que tout opérateur normal est réflexif. De nombreux auteurs ont depuis mis en évidence d’autres classes d’opérateurs réflexifs et on peut citer par exemple [6–8] (voir aussi [3]). Dans [2] les auteurs ont caractérisé les extensions des opérateurs normaux par des opérateurs nilpotents qui sont réflexives. Dans ce papier, nous étudions la réflexivité d’une extension T d’un opérateur sousnormal A par un opérateur algébrique. Nous montrons que la réflexivité d’une extension d’un opérateur sous-normal par un opérateur algébrique se ramène à celle d’un nombre fini d’extensions de A par des opérateurs nilpotents. Nous caratérisons ensuite les extensions des opérateurs sous-normaux par des opérateurs nilpotents qui sont réflexives et nous calculons leurs défauts de réflexivité, ce qui généralise les résultats obtenus dans [2]. La démarche adoptée ici est basée sur la description de la fermeture faible des polynômes dans L∞ (μ) dûe à D. Sarason [9], sur l’existence des points d’évaluations analytiques pour un opérateur sous-normal pur cyclique [12] et aussi sur le théorème de factorisation de Olin–Thomson [11]. Soient H et K deux espaces de Hilbert complexes séparables. Soient A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ, G(μ) l’intérieur de Sarason et R ∈ de l’enveloppe ni . Soit T une extension (z − λ ) B(K) un opérateur algébrique de polynôme minimal m = i=r i i=1 de A par R, l’opérateur T s’écrit suivant la décomposition H ⊕ K : A X T= où X ∈ B(K, H). 0 R Comme λ∈σ (R) ker(A − λI ) est réduisant pour A, on peut supposer sans perdre de généralité que m(A) est injectif pour l’étude de la réfléxivité de T . Soient R = i=r i=1 Ri la décomposition i=r primaire de R sur K = i=1 Ki avec pour tout i ∈ {1, . . . , r}, Ki = Ker(R − λi )ni , Ri = R|Ki et Ti = T|H⊕Ki . Nous montrons dans un premier temps que T est réflexif si et seulement si, Ti est réflexif pour tout i = 1, . . . , r. Ceci nous ramène à l’étude de la réflexivité de T lorsque R est nilpotent. Dans ce cas nous montrons que T est réflexif si et seulement si l’une des trois / Im A (R n = 0, e ∈ K conditions suivantes est satisfaite (i) 0 ∈ G(μ), (ii) R est réflexif, (iii) T n e ∈ tel que R n−1 e = 0). Dans toute la suite, si S ∈ B(H), nous désignons par σ (S) le spectre de S, σp (S) le spectre ponctuel de S. Si y1 , y2 , . . . , yr ∈ H, Vect(y1 , y2 , . . . , yr ) est le sous-espace vectoriel de H engendré par y1 , y2 , . . . , yr , VectS (y1 , y2 , . . . , yr ) est le sous-espace vectoriel de H invariant pour S engendré par y1 , y2 , . . . , yr ; et VectS (y1 , y2 , . . . , yr ) la fermeture de VectS (y1 , y2 , . . . , yr ) dans H. 2. Préliminaires Dans cette section nous présentons les principaux résultats qui nous seront utiles dans la suite de ce travail.

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Soit μ une mesure positive à support compact dans C. Pour tout polynôme p, nous notons par P ∞ (p, μ) l’adhérence faible dans L∞ (μ) de l’idéal {p(z)q(z): q ∈ C[z]} et par P 2 (μ) la fermeture dans L2 (μ) de l’algèbre des polynômes de la variable complexe z. Si p = 1, P ∞ (p, μ) sera noté simplement P ∞ (μ). En vertu du théorème de Sarason [9], la mesure μ se décompose de manière unique sous la forme μ = μa ⊕ μs avec μa ⊥ μs et P ∞ (μ) = P ∞ (μa ) ⊕ L∞ (μs ) où P ∞ (μa ) s’identifie canoniquement à H ∞ (G(μ)) (G(μ) est l’intérieur de l’enveloppe de Sarason et H ∞ (G(μ)) est l’algèbre des fonctions holomorphes bornées sur G(μ)). D’une part il est bien connu que si λ ∈ / G(μ), alors P ∞ (λ − z, μ) = P ∞ (μ) ; d’autre part si λ ∈ G(μ), alors P ∞ (λ − z, μ) = (λ − z)P ∞ (μa ) ⊕ L∞ (μs ). Le lemme suivant découle immédiatement des considérations précédentes : Lemme 2.1. Soient μ une mesure positive à support compact dans C et G(μ) l’intérieur de l’enveloppe de Sarason. Soit p = p1 p2 un polynôme tel que les racines de p1 sont dans G(μ) et celles de p2 sont à l’extérieur de G(μ). On a alors P ∞ (p, μ) = P ∞ (p1 , μ) = p1 P ∞ (μa ) ⊕ L∞ (μs ). Dans ce qui suit, H et K désignent deux espaces de Hilbert complexes séparables. A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ, R ∈ B(K) un opérateur algébrique de polynôme minimal m et T=

A 0

une extension de A par R. Pout tout p =

p(A) p(T ) = 0

X R

où X ∈ B(K, H)

i=n

i=0 αi z

i=n

i=0 αi Xi

p(R)

i,

on a

où Xi =

i−1

Ai−j −1 XR j .

j =0

Dans [2] les auteurs ont donné une description complète de W (T ) lorsque T est une extension d’un opérateur normal par un opérateur nilpotent. Cette description s’étend dans le cas d’une extension T d’un opérateur sous-normal par un opérateur algébrique. En effet lorsque A est un opérateur sous-normal de mesure spectrale scalaire μ et G(μ) est l’intérieur de l’enveloppe de Sarason associé à μ. D’une part il est bien connu (voir par exemple [4]) que pour tout polynôme p, WA (p(A)) s’identifie à P ∞ (p, μ) ; d’autre part si p = p1 p2 avec les racines de p1 sont dans G(μ) et celles de p2 sont à l’extérieur de G(μ), alors on obtient en vertu du lemme 2.1 que WA (p2 (A)) = W (A) et WA (p(A)) = WA (p1 (A)). En tenant compte de ces deux considérations et en procédant de la même manière que dans la preuve du théorème 6 dans [2], on obtient le lemme suivant : Lemme 2.2. Si A ∈ B(H) est sous-normal, R ∈ B(K) est un opérateur algébrique un opérateur i de degré n tel que m(A) est injectif et a z de polynôme minimal m = i=n i i=0 T=

A 0

X R

où X ∈ B(K, H)


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une extension de A par R. Alors

W (T ) = Cn−1 [T ] ⊕ WT m(T ) et

WT m(T ) =

B 0

n

Y : B ∈ WA m(A) et B ai Xi = m(A)Y . 0 i=0

− λi )ni . Posons Soit R ∈ B(K) un opérateur algébrique de polynôme minimal m = ri=1 (z n pour tout i ∈ {1, . . . , r}, Ki = Ker(R − λi ) i et Ri = R|Ki . Comme AlgLat R = ri=1 AlgLat Ri et le polynôme minimal de Ri est égal à (z − λi )ni , la description de AlgLat R se ramène au cas nilpotent. Le théorème suivant donne une description explicite de AlgLat N dans le cas où N est nilpotent. Cette description est une adaptation triviale de celle qui figure dans [5]. Théorème 2.3. Soient N ∈ B(K) un opérateur nilpotent d’ordre n 1, e ∈ K tel que N n−1 e = 0 et F ∈ Lat N tel que K = VectN (e) ⊕ F . Soit zm le polynôme minimal de N|F . Les assertions suivantes sont équivalentes (1) Q ∈ AlgLat N . (2) Il existe p ∈ C[z] et D ∈ B(K) tel que Q = p(N ) + D avec De = 0, D|F = 0 et pour tout 0 i n − 1, D(N i e) ∈ VectN (N m+i e). Remarque 2.4. 1. Dans le théorème précédent, l’existence du sous-espace F invariant pour N et supplémentaire de VectN (e), lorsque N est nilpotent, provient du lemme 1 dans [1]. Ce lemme peut-être étendu sans difficulté au cas algébrique. Autrement dit, si R ∈ B(K) est un opérateur algébrique de polynôme minimal m et e ∈ K est un vecteur de polynôme R-annulateur m (i.e. m est le générateur de l’idéal {p ∈ C[z]: p(R)e = 0}), alors VectR (e) admet un sous-espace supplémentaire invariant pour R. 2. Soient R ∈ B(K) un opérateur algébrique de polynôme minimal m, e ∈ K un vecteur de polynôme R-annulateur m et F ∈ Lat R tels que K = VectR (e) ⊕ F . Si Lat R ⊂ Lat Q, alors il existe en vertu du théorème 2.3, un polynôme p et D ∈ B(K) tels que Q = p(R) + D avec De = 0 et D|F = 0. 3. Description de l’algèbre AlgLat T et réflexivité L’objectif de cette section est de donner une description de l’algèbre AlgLat T et d’étudier la réflexivité de T lorsque T=

A X 0 R

où X ∈ B(K, H)

est une extension d’un opérateur sous-normal A ∈ B(H) par un opérateur algébrique R ∈ B(K). Nous commençons par donner quelques lemmes essentiels qui nous seront utiles pour la suite. Lemme 3.1. Soient A ∈ B(H) tel que AlgLat A ⊂ {A} , R ∈ B(K) un opérateur algébrique de polynôme minimal m et

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T=

A 0

X R

où X ∈ B(K, H)

une extension de A par R. Soit p un polynôme divisible par m. Si pour tout h ∈ H, alors AlgLat T ⊂ {p(T )} . Démonstration. Posons p(z) = sous la forme

n

i=0 αi z

i

x∈H VectA (p(A)x + h) = {0}

de degré n et soit S ∈ AlgLat T . L’opérateur S s’écrit

B S= 0

Y D

avec B ∈ AlgLat A, D ∈ AlgLat nPour montrer que S ∈ {p(T )} il faut et n R et Y ∈ B(K, H). il suffit de montrer que B i=0 αi Xi = p(A)Y + i=0 αi Xi D. Soient donc x ∈ H et y ∈ K. Comme S(x + y) ∈ VectT (x + y) et

VectT (x + y) = q(T )(x + y): deg(q) < n + VectA p(A)x +

n

αi Xi y ,

i=0

il existe un polynôme q(z) = tel que

k bk z

k

(dépendant de x et y) de degré inférieur ou égal à n − 1

S(x + y) − q(T )(x + y) ∈ VectA p(A)x +

n

αi Xi y .

i=0

Ceci implique que

q(R)y = Dy, Bx + Yy − q(A)x − k bk Xk y ∈ VectA (p(A)x + ni=0 αi Xi y).

En appliquant p(A) à cette dérnière équation (et en tenant compte de l’hypothèse AlgLat A ⊂ {A} ) on obtient : n n

αi Xi y + p(A) Yy − bk Xk y + q(A) − B αi Xi y B − q(A) p(A)x + i=0

∈ VectA p(A)x +

n

k

i=0

αi Xi y .

i=0

Comme B ∈ AlgLat A, on déduit que : n n

p(A) Yy − bk Xk y + q(A) − B αi Xi y ∈ VectA p(A)x + αi Xi y . k

i=0

Le vecteur x étant arbitraire, on obtient en vertu de l’hypothèse :

i=0


n

p(A) Yy − bk Xk y + q(A) − B αi Xi y = 0. k

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(∗)

i=0

D’autre part de l’égalité p(T )q(T ) = q(T )p(T ), on obtient : p(A)

bk X k y +

k

n

αi Xi q(R)y = q(A)

i=0

n

αi Xi y +

bk Xk p(R)y.

k

i=0

Comme q(R)y = Dy et m divise p, cette dérnière égalité devient : p(A)

bk X k y +

k

αi Xi Dy = q(A)

i=0

En reportant l’expression de q(A)

B

n

n

n

i=0 αi Xi y

αi Xi y.

i=0

dans (∗), on obtient

αi Xi = p(A)Y +

i=0

n

n

2

αi Xi D.

i=0

La preuve du théorème suivant repose en partie sur la notion de points d’évaluations analytiques bornés et le théorème de Thomson [12]. Plus précisément si A = Mz (l’opérateur de multiplication par la variable complexe z sur P 2 (μ)) est sous-normal pur, alors il existe un sousensemble ouvert Ω(μ) ⊂ C (appelé ensemble des points d’évaluations bornés analytiques) tel que pour tout λ ∈ Ω(μ) : (1) ∃kλ ∈ P 2 (μ) tel que p(λ) = pkλ dμ. (2) ∀f ∈ P 2 (μ), la fonction z → f kz dμ est analytique sur un voisinage de λ. De plus P 2 (μ) = Vect{kλ : λ ∈ Ω(μ)}. Théorème 3.2. Si A ∈ B(H) est un opérateur sous-normal, R ∈ B(K) est un opérateur algébrique de polynôme minimal m et

A X T= 0 R

où X ∈ B(K, H)

une extension de A par R, alors AlgLat T ⊂ {p(T )} pour tout polynôme p divisible par m et tel que 0 ∈ / σp (p(A)). Démonstration. Notons d’abord que si A = N ⊕ S est la décomposition de A sur H = H0 ⊕ H1 en une partie N normale sur H0 et une partie S sous-normale pure sur H1 ; on a pour tout h = h0 ⊕ h1 ∈ H0 ⊕ H1 et tout polynôme p : x=x0 +x1

VectA p(A)x + h ⊂ VectA p(N )x0 + h0 ⊕ VectA p(S)x1 + h1 . x0 ∈H0

x1 ∈H1

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Donc pour démontrer le théorème il suffit, en vertu du lemme 3.1, de prouver que

VectA p(A)x + h = {0},

∀h ∈ H,

x∈H

d’une part pour un opérateur A normal et d’autre part pour un opérateur sous-normal pur. Soient h ∈ H et p un polynôme divisible par m. Cas 1. A est normal. Posons Λ = {w ∈ C: p(w) = 0} et considérons, pour chaque k ∈ N∗ , k = {z ∈ σ (A): |z − λ| k1 , ∀λ ∈ Λ}. Soit la fonction fk définie par fk (z) =

1 p(z) ,

0,

si z ∈ k ; si z ∈ σ (A)\k .

En posant xk = −fk (A)h, on obtient p(A)xk = −E(K )h où E est la mesure spectrale de A. SOT

Comme 0 ∈ / σp (p(A)), E(k ) −→ I et donc p(A)xk −→ −h, ce qui prouve le résultat du premier cas. Cas 2. A est sous-normal pur. Soient F0 = VectA (h) et A0 = A|F0 . Comme A0 est un opérateur sous-normal pur cyclique, on peut supposer que A0 = Mz l’opérateur de multiplication par la variable complexe z sur F0 = P 2 (μ). Pour λ ∈ Ω(μ) tel que p(λ) = 0, considérons λ xλ = − h,k kλ . Comme {kλ : λ ∈ Ω(μ)} est dense dans F0 et p(A0 )xλ + h, kλ = 0, on ob p(λ)

tient x∈F0 VectA0 (p(A0 )x + h) = {0} et donc x∈H VectA (p(A)x + h) = {0}, ce qui achève la démonstration. 2 Un des piliers de la théorie des opérateurs sous-normaux est le théorème de factorisation des formes linéaires w ∗ -continues dû à Olin et Thomson [8]. Le lemme suivant est une variante de ce résultat et dont la preuve peut se faire en suivant la démarche proposée dans [4]. Rappelons qu’une sous algèbre A de L∞ (μ) contenant 1 est antisymétrique si les seules fonctions réelles de A sont les fonctions constantes. Grâce à la décomposition de Sarason, on vérifie que P ∞ (μ) est antisymétrique si et seulement si G(μ) est connexe et la mesure singulière μs qui figure dans la décomposition de Sarason μ = μa ⊕ μs est nulle. Dans la suite pour u, v ∈ H, la notation u ⊗ v désignera la forme linéaire sur B(H) définie par u ⊗ v(S) = Su, v pour tout S ∈ B(H). Lemme 3.3. Soit A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ et tel que P ∞ (μ) est antisymétrique. Soient L une forme linéaire w ∗ -continue sur W (A) et h ∈ H. Alors pour tout λ ∈ G(μ) et tout entier n, il existe x et y dans H tels que L = ((A − λ)n x + h) ⊗ y. Lemme 3.4. Soient A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ et B ∈ W (A). Soient h ∈ H, λ ∈ G(μ) et n 1. Si, pour tout x ∈ H, on a

B (A − λ)n x + h ∈ VectA (A − λ)n (A − λ)n x + h , alors B ∈ WA ((A − λ)n ).


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Démonstration. Puisque B ∈ W (A), il existe un polynôme q de degré inférieur ou égal à n − 1 tel que B − q(A) ∈ WA ((A − λ)n ). Nous allons montrer que q est identiquement nul. Soit G0 la composante connexe de G(μ) contenant λ. On peut écrire H = H0 ⊕ H1 avec H0 et H1 invariant par A. De plus si on note respectivement par μ0 et μ1 les mesures spectrales de A|H0 et A|H1 , alors μ = μ0 ⊕ μ1 , P ∞ (μ) = P ∞ (μ0 ) ⊕ P ∞ (μ1 ) et P ∞ (μ0 ) s’identifie canoniquement à H ∞ (G0 ). Donc quitte à considérer A|H0 , on peut supposer que P ∞ (μ) est antisymétrique. Soit k ∈ {1, . . . , n} et considérons la forme linéaire Lk définie sur les polynômes en A par Lk (p(A)) = p (k) (λ) (p (k) est la dérivée k-ème de p). Puisque λ ∈ G(μ), Lk se prolonge en une forme linéaire w ∗ -continue sur W (A). D’après le lemme 3.3, il existe x et y dans H tels que

Lk = (A − λ)n x + h ⊗ y. Comme B((A − λ)n x + h) ∈ VectA [(A − λ)n ((A − λ)x + h)] et compte tenu du fait que B − q(A) ∈ WA ((A − λ)n ), on a

q(A) (A − λ)n x + h ∈ VectA (A − λ)n (A − λ)n x + h . Ainsi on obtient q (k) (λ) = Lk (q(A)) = q(A)((A − λ)n x + h), y = 0. Comme deg(q) n − 1, q = 0 et par suite B ∈ WA ((A − λ)n ). 2 Lemme 3.5. Soient A ∈ B(H) un opérateur sous-normal, R ∈ B(K) un opérateur algébrique de polynôme minimal m et e ∈ K un vecteur de polynôme R-annulateur m. Soit T=

A X 0 R

où X ∈ B(K, H)

une extension de A par R telle que m(A) est injectif. Soit S=

B 0

Y D

tel que De = 0.

Si S ∈ AlgLat T , alors B ∈ WA (m(A)). Démonstration. Supposons que S ∈ AlgLat T . Soient μ la mesure spectrale scalaire de A et G(μ) l’intérieur de l’enveloppe de Sarason associée à μ. Posons m = ri=1 (z − λi )ni , J = {i ∈ {1, . . . , r}: λi ∈ / G(μ)}. Pour montrer il suffit de montrer, en vertu du

que B ∈ WA (m(A)) ni ), que pour un certain h ∈ H on a Lemme 3.4 et du fait que WA (m(A)) = i ∈J W ((A − λ ) A i / pour tout x ∈ H et i ∈ /J :

B (A − λi )ni x + h ∈ VectA (A − λi )ni (A − λi )ni x + h . ni Soient i ∈ / J et x ∈ H. Posons Ki = Ker(R −n λi ) , Ri = R|Ki . Soient ei ∈ Ki , la projection de j e sur Ki parallélement à j =i Ker(R − λj ) . Comme S(x + ei ) ∈ VectT (x + ei ) et ei admet (z − λi )ni comme polynôme Ri -annulateur, il existe un polynôme q (dépendant de x et ei ) avec deg(q) < ni tel que

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q(R)ei = Dei = 0, Bx + Y ei − q(A)x − q(T )ei ∈ VectA ((A − λi )ni x + (T − λi )ni ei ).

Ainsi q = 0 et Bx + Y ei ∈ VectA ((A − λi )ni x + (T − λi )ni ei ). Ceci entraine

(A − λi )ni (Bx + Y ei ) ∈ VectA (A − λi )ni (A − λi )ni x + (T − λi )ni ei .

(3.1)

Mais d’aprés le théorème 3.2 on a (A − λi )ni Y ei = B(T − λi )ni ei . L’équation (3.1) devient alors :

B (A − λi )ni x + (T − λi )ni ei ∈ VectA (A − λi )ni (A − λi )ni x + (T − λi )ni ei . Ceci achève la démonstration.

2

Lemme 3.6. Soient A ∈ B(H) un opérateur sous-normal, R ∈ B(K) un opérateur algébrique de polynôme minimal m = nj=0 aj zj tel que m(A) est injectif et

A T= 0

X R

où X ∈ B(K, H)

une extension de A par R. Soient e ∈ K un vecteur de polynôme R-annulateur m et F ∈ Lat R tel que K = VectR (e) ⊕ F . On a :

0 C : D ∈ AlgLat R, De = 0, D|F = 0, 0 D n aj X j D = 0 . m(A)C +

AlgLat T ⊂ W (T ) ⊕

j =0

Démonstration. Posons n 0 C : D ∈ AlgLat R, De = 0, D|F = 0, m(A)C + aj X j D = 0 . E= 0 D j =0

C On a E ∩ W (T ) = {0}, car si V = 00 D ∈ W (T ), alors D ∈ W (R) soit D = q(R) pour un certain polynôme q ; en particulier q(R)e = 0, d’oú l’on déduit (d’après l’hypothèse initiale sur e) que q est un mutiple de m et finalement D = q(R) = 0. L’équation m(A)C + nj=0 aj Xj D = 0 et l’injectivité de m(A) conduisent maintenant à C = 0. Ainsi V = 0. Soit S ∈ AlgLat T . D’après le théorème 3.2 et la réflexivité de A, S s’écrit sous la forme S=

B 0

Y D


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avec B ∈ W (A), D ∈ AlgLat R et B nj=0 aj Xj = m(A)Y + nj=0 aj Xj D. D’après la remarque 2.4, et quite à retrancher de S un polynôme convenable en T , on peut supposer que De = 0 et D|F = 0. Par suite on a B ∈ WA (m(A)) par le lemme 3.5. Soit C ∈ B(K, H) défini par :

C|F = 0, CR k e = BXk e − Ak Y e + Y R k e,

1 k n − 1.

Posons S0 =

Y −C 0

B 0

et S1 =

0 C . 0 D

= m(A)Y + nj=0 aj Xj D et m(T )T k = T k m(T ), on obtient m(A)C + j =0 aj Xj D = 0, en effet on a :

De B n

n

j =0 aj Xj

m(A)CR k e = m(A) BXk e − Ak Y e + Y R k e n n n k k = B m(A)Xk e − A aj X j e + aj X j R e − aj Xj DR k e j =0

j =0

= B m(T )T k − T k m(T ) e −

n

j =0

aj Xj DR k e

j =0

=−

n

aj Xj DR k e.

j =0

n

Comme B j =0 aj Xj = m(A)(Y − C) et B ∈ WA (m(A)), S0 ∈ W (T ) par le lemme 2.2, ce qui achève la démonstration. 2 Lemme 3.7. Soient A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ, R ∈ B(K) un opérateur algébrique de polynôme minimal m = (z − λ)n . Supposons que A − λ est injectif et soit T=

A X 0 R

où X ∈ B(K, H)

une extension de A par R. Si λ ∈ G(μ), alors T est réflexif. Démonstration. Soit S ∈ AlgLat T et posons m = supposer que S s’écrit sous la forme S=

0 C 0 D

n

i=0 ai z

i.

D’après le lemme 3.6, on peut

avec m(A)C + ni=0 ai Xi D = 0. Soient x ∈ H et y ∈ K. Comme S(x + y) ∈ VectT (x + y), il existe un polynôme q(z) = k bk zk (dépendant de x, y) avec deg(q) < deg(m) tel que

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Dy = q(R)y, Cy − q(A)x − k bk Xk y ∈ VectA (m(A)x + ni=0 ai Xi y).

(3.2)

D’autre part ; en tenant compte de ce qui précède et de l’identité m(T )q(T ) = q(T )m(T ), on a : n m(A) Cy − q(A)x − bk Xk y = −m(A)q(A)x − ai Xi q(R)y + m(A) bk X k y k

k

i=0

= −m(A)q(A)x − q(A)

n

ai X i y

i=0

= −q(A) m(A)x +

n

ai X i y .

i=0

Par conséquent on a q(A) m(A)x +

n

ai Xi y ∈ VectA m(A) m(A)x +

i=0

n

ai X i y

.

i=0

Comme λ ∈ G(μ), q(A) ∈ WA (m(A)) par le lemme 3.4 et par suite q = 0 puisque deg(q) < deg(m). Donc D = 0 et comme m(A) est injectif, C = 0, ce qui implique que S = 0. 2 Le théorème suivant réduit l’étude de la réflexivité d’une extension d’un opérateur sousnormal par un opérateur algébrique à celle de la réflexivité d’une extension d’un opérateur sous-normal par un opérateur nilpotent. Théorème 3.8. Soient A ∈ B(H)un opérateur sous-normal et R ∈ B(K) un opérateur algébrique de polynôme minimal m = ri=1 (z − λi )ni . Supposons que m(A) est injectif et soit T=

A 0

X R

où X ∈ B(K, H)

une extension de A par R. Posons pour i = 1, . . . , r, Ki = Ker(R − λi )ni et Ti = T|H⊕Ki . Alors on a T est réflexif si et seulement si pour tout i = 1, . . . , r, Ti est réflexif. Démonstration. Supposons que Ti est réflexif pour tout i ∈ {1, . . . , r}. Montrons que T est réflexif. Soient S ∈ AlgLat T , e ∈ K un vecteuri de polynôme R-annulateur m et F ∈ Lat R tel que K = VectR (e) ⊕ F . Posons m = i=n i=0 ai z . D’aprés le lemme 3.6, on peut supposer que S s’écrit sous la forme : S=

0 Y 0 D

où Y ∈ B(K, H)

avec D ∈ AlgLat R, De = 0, D|F = 0 et m(A)Y + nj=0 aj Xj D = 0. Posons pour tout i = 1, . . . , r, Ri = R|Ki , Di = D|Ki et Si = S|H⊕Ki . Il est clair que pour tout i = 1, . . . , r


Si =

0 0

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Yi ∈ AlgLat(Ti ). Di

Puisque Ti est réflexif, Si ∈ W (Ti ) et par suite Di ∈ W (Ri ). Si on note ei la projection de e sur Ki paralléllement à j =i Ker(R − λj )nj , on obtient Di ei = 0 puisque De = 0. Ainsi Di = 0, donc D = 0, d’où S = 0. Il en résulte que T est réflexif. Inversement, supposons qu’il existe i0 ∈ {1, . . . , r} tel que Ti0 n’est pas réflexif. On peut écrire

A X i0 Ti 0 = 0 Ri 0

avec X i0 ∈ B(Ki0 , H). D’après le lemme 3.6, il existe

0 Si0 = 0

Yi0 ∈ AlgLat(Ti0 ) Di 0

/ W (Ti0 ). Soient μ la mesure spectrale scalaire de A et G(μ) l’intérieur de l’envetel que Si0 ∈ / G(μ) par le lemme 3.7, donc le polynôme (z − λi0 )ni0 loppe de Sarason associé à μ. On a λi0 ∈ divise le polynôme mext (mext est le facteur, du polynôme minimal m, dont les racines sont à l’extérieur de G(μ)). Si on pose mext (z) = i βi zi , on obtient en vertu du théorème 3.2 mext (A)Yi0 +

βi Xi Di0 = 0.

(3.3)

i

Soient maintenant D ∈ B(K) et Y ∈ B(K, H) définis par

D = Di 0 , D = 0,

sur Ki0 , sur j =i0 Kj ,

et

Y = Yi0 , Y = 0,

sur Ki0 , sur j =i0 Kj .

Considérons maintenant l’opérateur S ∈ B(H ⊕ K) défini par S=

0 Y . 0 D

/ W (Ti0 ), S ∈ / W (T ). Montrons que S ∈ AlgLat T . Soient x ∈ H et y ∈ K. Comme Si0 ∈ Puisque Di0 ∈ AlgLat Ri0 et AlgLat R = i=r i=1 AlgLat Ri , D ∈ AlgLat R. Il existe donc un polynôme p (dépendant de y) tel que Dy = p(R)y. Il est clair que mext (A)Y + i βi Xi D = 0 et mext (R)D = 0, ce qui implique que mext (T )S = 0. Il en résulte que mext (T )(S − p(T ))(x + y) = −mext (T )p(T )(x + y) ∈ VectT (x + y). Comme I ∈ WA (mext (A)), (S − p(T ))(x + y) ∈ VectT (x + y) et donc S(x + y) ∈ VectT (x + y). Ceci montre que S ∈ AlgLat(T ). Or S ∈ / W (T ) et donc T n’est pas réflexif. D’où le résultat. 2 Théorème 3.9. Soient A ∈ B(H) un opérateur sous-normal de mesure spectrale scalaire μ et G(μ) l’intérieur de l’enveloppe de Sarason. Soient R ∈ B(K) un opérateur nilpotent d’ordre n et e ∈ K tel que R n−1 e = 0. Supposons que A est injectif et soit T=

A X 0 R

où X ∈ B(K, H)

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une extension de A par R. Alors T est réflexif si et seulement si, 0 ∈ G(μ) ou R est réflexif ou / Im A. T ne ∈ Démonstration. Si 0 ∈ G(μ), alors T est réflexif d’après le lemme 3.7. Supposons R est réflexif / Im(A) et montrons que T est réflexif. Si S ∈ AlgLat T , alors, d’après le lemme 3.6, on ou T n e ∈ peut supposer que

0 Y S= 0 D

où Y ∈ B(K, H)

avec D ∈ AlgLat R, An Y + Xn D = 0, De = 0 et D|F = 0. Si R est réflexif, il existe p polynôme tel que D = p(R), donc D = 0 et ainsi S = 0. / Im(A). Comme D ∈ AlgLat R, il existe pour tout k = 0, . . . , n, pk polySupposons T n e ∈ nôme tels que DR k e = pk (R)e. De pk (T )T n = T n pk (T ) on obtient pk (A)Xn e − Xn pk (R)e ∈ Im An et de An Y + Xn D = 0, on obtient Xn pk (R)e = Xn DR k e = An Y R k e. Ceci implique que, pour tout k = 0, . . . , n, pk (A)Xn e ∈ Im(An ) et par suite zn divise pk , donc DR k e = 0, donc D = 0 et par suite S = 0. Inversement, supposons que 0 ∈ / G(μ), R n’est pas réflexif et T n e ∈ Im(A). Il existe donc un n vecteur y0 ∈ H tel que T e = Ay0 et, par le théorème 2.3, l’ordre de nilpotence p de R|F est inférieur ou égale à n − 2. Considérons l’opérateur D ∈ B(K) défini par ⎧ D|F = 0, ⎪ ⎪ ⎨ De = 0, ⎪ DR k e = 0, ⎪ ⎩ DR k e = R n−1 e,

si k n − p, si 1 k n − p − 1.

Notons que D ∈ / W (R) et D ∈ AlgLat R par le théorème 2.3. Soit Y ∈ B(K, H) défini par ⎧ Y|F = 0, ⎪ ⎪ ⎨ Y e = 0, ⎪ Y R k e = 0, ⎪ ⎩ Y R k e = Xn−1 e − y0 ,

si k n − p, si 1 k n − p − 1.

De T n T k = T k T n (1 k n − p − 1), on obtient An Y R k e = An Xn−1 e − An y0 = An Xn−1 e − An−1 Xn e

= An Xn−1 − An−1 Xn e = −Xn R n−1 e = −Xn DR k e. On obtient donc An Y + Xn D = 0. Soit S ∈ B(H ⊕ K) défini par S=

0 Y . 0 D


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Il est clair que S ∈ / W (T ) puisque D ∈ / W (R). Montrons que S ∈ AlgLat T . Soient x ∈ H et y ∈ K. Soit p polynôme dépendant de y tel que Dy = p(R)y. On a T n S = 0 en vertu de An Y + Xn D = 0 et T n (S − p(T ))(x + y) = −T n p(T )(x + y) ∈ VectT (x + y). Comme I ∈ WA (A), (S −p(T ))(x +y) ∈ VectT (x +y) et ainsi S ∈ AlgLat T et T est non réflexif. D’où le résultat. 2 4. Défaut de réflexivité Le défaut de réflexivité d’un opérateur T est défini par la quantité

δ(T ) = dim AlgLat T /W (T ) . Il est clair qu’un opérateur T est réflexif si et seulement si δ(T ) = 0. En adoptant la démarche entreprise dans la preuve du théorème 3.8, on peut obtenir aisément le théorème suivant : Théorème 4.1. Soient A ∈ B(H)un opérateur sous-normal et R ∈ B(K) un opérateur algéni brique de polynôme minimal m = i=r i=1 (z − λi ) . Supposons que m(A) est injectif et soit

A X T= 0 R

où X ∈ B(K, H)

une extension de A par R. Posons pour i = 1, . . . , r, Ki = Ker(R − λi )ni et Ti = T|H⊕Ki . On a δ(T ) =

i=r

δ(Ti ).

i=1

Ainsi le calcul de défaut de réflexivité d’une extension T d’un opérateur sous-normal par un opérateur algébrique est réduit donc au calcul de défaut de réflexivité d’une extension d’un opérateur sous-normal par un opérateur nilpotent. On a le résultat suivant qui peut s’obtenir en adoptant la même démarche que dans la preuve du théorème 3.9. Théorème 4.2. Soient A ∈ B(H) un opérateur sous-normal et N ∈ B(K) un opérateur nilpotent d’ordre n. Soit A X T= où X ∈ B(K, H) 0 N une extension de A par N . Supposons que A est injectif. Soient e ∈ K tel que N n−1 e = 0, F ∈ Lat N tel que K = VectN (e) ⊕ F . Soient zp le polynôme minimal de N|F , h = Max{k 0: T n e ∈ Ak (H)} et l = Min(n − p − 1, h). On a δ(T ) = l(n − p − l) +

l(l − 1) . 2

Remerciements Les auteurs tiennent à remercier le referee pour ses remarques et commentaires.

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Références [1] M. Barraa, B. Charles, Sous-espaces invariants d’un opérateur nilpotent sur un espace de Banach, Linear Algebra Appl. 153 (1991) 177–182. [2] M. Benlarbi Delai, O. El-Fallah, Réflexivité d’une extension d’un opérateur normal par un opérateur nilpotent, J. Funct. Anal. 223 (2005) 28–43. [3] J.B. Conway, A Course in Operator Theory, Grad. Stud. Math., vol. 21, AMS, Providence, RI, 2000. [4] J.B. Conway, The Theory of Subnormal Operators, Math. Surveys Monogr., vol. 36, AMS, Providence, RI, 1991. [5] J.A. Deddens, P.A. Fillmore, Reflexive linear transformations, Linear Algebra Appl. 10 (1975) 89–93. [6] D. Hadwin, C. Laurie, Reflexive binormal operators, J. Funct. Anal. 123 (1994) 99–108. [7] J. McCarthy, Reflexivity of subnormal operators, Pacific J. Math. 161 (2) (1993) 359–370. [8] R.F. Olin, J.E. Thomson, Algebras of subnormal operators, J. Funct. Anal. 37 (1980) 271–301. [9] D. Sarason, Weak-star density of polynomials, J. Reine Angew. Math. 252 (1972) 1–15. [10] D. Sarason, Invariant subspaces and unstarred algebras, Pacific J. Math. 17 (1966) 511–517. [11] J.E. Thomson, Factorisation over algebras of subnormal operators, Indiana Univ. Math. J. 37 (1988) 191–199. [12] J.E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991) 477–507.


Majorization in de Branges spaces I. Representability of subspaces Anton Baranov a , Harald Woracek b,∗ a Department of Mathematics and Mechanics, Saint Petersburg State University, 28, Universitetski pr.,

198504 Petrodvorets, Russia b Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstr. 8-10/101,

A-1040 Wien, Austria Received 6 June 2009; accepted 14 January 2010 Available online 1 February 2010 Communicated by Paul Malliavin

Abstract In this series of papers we study subspaces of de Branges spaces of entire functions which are generated by majorization on subsets D of the closed upper half-plane. The present, first, part is addressed to the question which subspaces of a given de Branges space can be represented by means of majorization. Results depend on the set D where majorization is permitted. Significantly different situations are encountered when D is close to the real axis or accumulates to i∞. © 2010 Elsevier Inc. All rights reserved. Keywords: de Branges subspace; Majorant; Beurling–Malliavin Theorem

1. Introduction In the paper [7] L. de Branges initiated the study of Hilbert spaces of entire functions, which satisfy specific additional axioms. These spaces can be viewed as a generalization of the classical Paley–Wiener spaces PW a , which consist of all entire functions of exponential type at most a whose restriction to the real line is square-integrable. The theory of de Branges spaces can be viewed as a generalization of classical Fourier analysis. For example, their structure theory gives rise to generalizations of the Paley–Wiener Theorem, which identifies PW a as the Fourier im* Corresponding author.

E-mail addresses: [email protected] (A. Baranov), [email protected] (H. Woracek). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.016

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A. Baranov, H. Woracek / Journal of Functional Analysis 258 (2010) 2601–2636

age of all square-integrable functions supported in the interval [−a, a]. De Branges spaces also appear in many other areas of analysis, like the theory of Volterra operators and entire operators in the sense of M.G. Kre˘ın, the shift operator in the Hardy space, V.P. Potapov’s J -theory, the spectral theory of Schrödinger operators, Stieltjes or Hamburger power moment problems, or prediction theory of Gaussian processes, cf. [9,11–13,21,24]. The present paper is the first part of a series, in which we investigate the aspect of majorization in de Branges spaces. Such considerations have a long history in complex analysis, going back to the Beurling–Malliavin Multiplier Theorem, cf. [5]. In recent investigations by V.P. Havin and J. Mashreghi, results of this kind were proven in the more general setting of shift-coinvariant subspaces of the Hardy space, cf. [16,17]. All these considerations, as well as our previous work [3], deal with majorization along the real line. Having these concepts in mind, a general notion of majorization in de Branges spaces evolves: 1.1. Definition. Let H be a de Branges space, and let m : D → [0, ∞) where D ⊆ C+ ∪ R. Set Rm (H) := F ∈ H: ∃C > 0: F (z), F # (z) Cm(z), z ∈ D , and define Rm (H) := closH Rm (H). It turns out that, provided Rm (H) = {0} and m satisfies a mild regularity condition, the space Rm (H) is a de Branges subspace of H, i.e. is itself a de Branges space when endowed with the inner product inherited from H. The following questions related to this concept come up naturally. Which de Branges subspaces L of a given de Branges space H can be realized as L = Rm (H) with some majorant m? If L is of the form Rm (H) with some m, how big or how small can m be chosen such that still L = Rm (H)? Let us point out the two aspects of the second question. If L = Rm (H), we have available a dense linear subspace of L which consists of functions with limited growth on the domain D of m, namely Rm (H). This knowledge becomes stronger, the smaller m is. On the other hand, the equality L = Rm (H) also says that an element of H already belongs to L if it is majorized by m. This knowledge becomes stronger, the bigger m is. Answers to these questions will, of course, depend on the set D where majorization is permitted. Up to now, only majorization along R has been considered. For this case, the first question has been answered completely in [3]. The “how small”–part of the second question is related to the deep investigations in [16,17]. In this paper we give some answers to the first question, and to the “how big”–part of the second question. As domains D of majorization we consider, among others, rays contained in the closed upper half-plane, lines parallel to the real axis contained in the closed upper half-plane, or combinations of such types of sets. For example, it turns out that each de Branges subspace L of any given de Branges space H can be realized as Rm (H), when majorization is allowed on R ∪ i[0, ∞). Even more, one can choose for m a majorant which is naturally associated to L,


2603

does not depend on the external space H, is quite big, and actually gives L = Rm (H). It is an interesting and, on first sight, maybe surprising consequence of de Branges’ theory, that the main strength of majorization is contributed by boundedness along the imaginary half-line, and not along R. In fact, if we permit majorization only on some ray i[h, ∞) where h > 0, then all de Branges subspaces subject to an obvious necessary condition can be realized in the way stated above. Similar phenomena, where growth restrictions on the imaginary half-axis imply a certain behaviour along the real line, have already been experienced in the classical theory, see, e.g., [1, Theorem 2] or [8, Theorem 26]. Let us close this introduction with an outline of the organization of this paper. In order to make the presentation as self-contained as possible, we start in Section 2 with recalling some basic definitions and collecting some results which are essential for what follows, among them, the definition of de Branges spaces of entire functions, their relation to entire functions of Hermite– Biehler class, and the structure of de Branges subspaces. In Section 3, we make precise under which conditions on m the space Rm (H) becomes a de Branges subspace of H, and discuss some examples of majorants. Sections 4 and 5 contain the main results of this paper. First we deal with representation of de Branges subspaces by majorization along rays not parallel to the real axis. Then we turn to spaces Rm (H) obtained when majorization is required on a set close to the real axis, for example a line parallel to R. The paper closes with two appendices. In Appendix A we prove an auxiliary result on model subspaces generated by inner functions, which is employed in Section 5. We decided to move this theorem out of the main text, since it is interesting on its own right and independent of the presentation concerning de Branges spaces. In Appendix B, we are summing up the representation theorems for de Branges subspaces obtained in Sections 4 and 5 in tabular form. 2. Preliminaries I. Mean type and zero divisors We will use the standard theory of Hardy spaces in the half-plane as presented, e.g., in [10] or [25]. In this place, let us only recall the following notations. We denote by: (i) N = N (C+ ) the set of all functions of bounded type, that is, of all functions f analytic in C+ , which can be represented as a quotient f = g −1 h of two bounded and analytic functions g and h. (ii) N+ = N+ (C+ ) the Smirnov class, that is, the set of all functions f analytic in C+ , which can be represented as f = g −1 h with two bounded and analytic functions g and h where in addition g is outer. (iii) H 2 = H 2 (C+ ) the Hardy space, that is, the set of all functions f analytic in C+ , which satisfy 2 sup f (x + iy) dx < ∞. y>0

R

If f ∈ N , the mean type of f is defined by the formula mt f := lim sup y→+∞

1 logf (iy). y

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Then mt f ∈ R, and the radial growth of f is determined by the number mt f in the following sense: For every a ∈ R and 0 < α < β < π , there exists an open set a,α,β ⊆ (0, ∞) with finite logarithmic length, such that lim

r→∞ r ∈ / a,α,β

1 logf a + reiθ = mt f · sin θ, r

(2.1)

uniformly for θ ∈ [α, β]. If, for some > 0, the angle [α − , β + ] does not contain any zeros of f (a + z), then one can choose a,α,β = ∅. we understand by the logarithmic length of a subset M of R+ the value of the integral Here −1 dx. When speaking about logarithmic length of a set M, we always include that M should Mx be measurable. 2.1. Definition. Let m : D → C be a function defined on some subset D of the complex plane. (i) By analogy with (2.1) we define the mean type of m as

1 1 lim sup logm a + reiθ ∈ [−∞, +∞], mtH m := inf sin θ r→∞ r r∈M

where the infimum is taken over those values a ∈ R, θ ∈ (0, π), and those sets M ⊆ R+ of infinite logarithmic length, for which {a + reiθ : r ∈ M} ⊆ D. Thereby we understand the infimum of the empty set as +∞. (ii) We associate to m its zero divisor dm : C → N0 ∪ {∞}. If w ∈ C, then dm (w) is defined as the infimum of all numbers n ∈ N0 , such that there exists a neighbourhood U of w with the property inf

z∈U ∩D |z−w|n =0

|m(z)| > 0. |z − w|n

Note that in general mt m may take the values ±∞. However, the above definition ensures that mt m coincides with the classical notion in case m ∈ N . A similar remark applies to dm . If D is open, and m is analytic, then dm |D is just the usual zero divisor of m, i.e. dm (w) is the multiplicity of the point w as a zero of m whenever w ∈ D. / D. Moreover, note that the definition of dm is made in such a way that dm (w) = 0 whenever w ∈ II. Axiomatics of de Branges spaces of entire functions Our standard reference concerning the theory of de Branges spaces of entire functions is [8]. In this and the following two subsections we will recall some basic facts about de Branges spaces. Our aim is not only to set up the necessary notation, but also to put emphasis on those results which are significant in the context of the present paper. We start with the axiomatic definition of a de Branges space. 2.2. Definition. A de Branges space is a Hilbert space H, (·,·) , H = {0}, with the following properties:


2605

(dB1) The elements of H are entire functions, and for each w ∈ C the point evaluation F → F (w) is a continuous linear functional on H. (dB2) If F ∈ H, also F # (z) := F (z) belongs to H and F # = F . (dB3) If w ∈ C \ R and F ∈ H, F (w) = 0, then z−w F (z) ∈ H z−w

z − w = F . and F (z) z − w

By (dB1) a de Branges space H is a reproducing kernel Hilbert space. We will denote the kernel corresponding to w ∈ C by K(w, ·) or, if it is necessary to be more specific, by KH (w, ·). A particular role is played by the norm of reproducing kernel functions. We will denote ∇H (z) := K(z, ·)H ,

z ∈ C.

This norm can be computed, e.g., as 1/2 ∇H (z) = sup F (z): F H = 1 = K(z, z) . Let us explicitly point out that every element of H is majorized by ∇H : By the Schwarz inequality we have F (z) F ∇H (z),

z ∈ C, F ∈ H.

(2.2)

2.3. Remark. Let H be a de Branges space. For a subset L ⊆ H we define dL : C → N0 as dL (w) := min dF (w). F ∈L

Due to the axiom (dB3), we have dH (w) = 0, w ∈ C \ R. In fact, if F ∈ H and w is a nonreal zero of F , then (z − w)−1 F (z) ∈ H. This need not be true for real points w. However, one can show that, if w ∈ R and dF (w) > dH (w), then (z − w)−1 F (z) ∈ H. 2.4. Remark. Let H be a de Branges space, and let m : D → C be a function defined on some subset D of the complex plane. We define the mean type of m relative to H by mtH m := mt

m . ∇H

If L is a subset of H, the mean type of L relative to H is mtH L := sup mtH F. F ∈L

Note that, by (2.2), we have mtH L 0. For each α 0 the set {F ∈ H: mtH F α} is closed, cf. [18]. This implies that always mtH closH L = mtH L.

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2.5. Remark. For a de Branges space H let SH denote the operator of multiplication by the independent variable. That is, dom SH := F ∈ H: zF (z) ∈ H . (SH F )(z) := zF (z), The relationship between de Branges spaces and entire operators in the sense of M.G. Kre˘ın is based on the fact that SH is a closed symmetric operator with defect index (1, 1) for which every complex number is a point of regular type. 2.6. Remark. Taking up the operator theoretic viewpoint, the role played by functions associated to H can be explained neatly. For a de Branges space H, the set of functions associated to H can be defined as Assoc H := G1 (z) + zG2 (z): G1 , G2 ∈ H . Clearly, Assoc H is a linear space which contains H. The space Assoc H can be used to describe the extensions of SH by means of difference quotients. We have F ∈ Assoc H

⇐⇒

∀G ∈ H, w ∈ C:

F (z)G(w) − F (w)G(z) ∈ H. z−w

Moreover, for each F ∈ Assoc H and w ∈ C, F (w) = 0, the difference quotient operator ρF,w : G →

G(z) −

G(w) F (w) F (z)

z−w

is a bounded linear operator of H into itself, actually, the resolvent of some extension of SH . Let us note that, if F is not only associated to H but belongs to H, we have ρF,w H = dom SH , F (w) = 0. III. De Branges spaces and Hermite–Biehler functions It is a basic fact that a de Branges space H is completely determined by a single entire function. 2.7. Definition. We say that an entire function E belongs to the Hermite–Biehler class HB, if # E (z) < E(z), z ∈ C+ . If E ∈ HB, define

F F# H(E) := F entire: , ∈ H 2 C+ , E E and (F, G)E := R

F (t)G(t) dt, |E(t)|2

F ∈ H(E).


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−1 # Instead of E −1 F, E −1 F # ∈ H 2 one could, equivalently, require that E −1 F and E F are of bounded type and nonpositive mean type in the upper half-plane, and that R |E −1 (t)F (t)|2 dt < ∞. This is, in fact, the original definition in [7]. The relation between de Branges spaces and Hermite–Biehler functions is established by the following fact:

2.8. De Branges spaces via HB. For every function E ∈ HB, the space H(E), (·,·)E is a de Branges space, and conversely every de Branges space can be obtained in this way. The function E ∈ HB which realizes a given de Branges space H, (·,·) as H(E), (·,·)E is not unique. However, if E1 , E2 ∈ HB and H(E1 ), (·, ·)E1 = H(E2 ), (·,·)E2 , then there exists a constant 2 × 2-matrix M with real entries and determinant 1, such that (A2 , B2 ) = (A1 , B1 )M. Here, and later on, we use the generic decomposition of a function E ∈ HB as E = A − iB with A :=

E + E# , 2

B := i

E − E# . 2

(2.3)

For each two function E1 , E2 ∈ HB with H(E1 ), (·,·)E1 = H(E2 ), (·,·)E2 , there exist constants c, C > 0 such that cE1 (z) E2 (z) C E1 (z),

z ∈ C+ ∪ R.

The notion of a phase function is important in the theory of de Branges spaces. For E ∈ HB, a phase function of E is a continuous, increasing function ϕE : R → R with E(t) exp(iϕE (t)) ∈ R, t ∈ R. A phase function ϕE is by this requirement defined uniquely up to an additive constant which belongs to πZ. Its derivative is continuous, positive, and can be computed as ϕ (t) = π

|Im zn | K(t, t) = a + , |E(t)|2 |t − zn |2 n

(2.4)

where zn are zeros of E listed according to their multiplicities, and a := − mt(E −1 E # ). 2.9. Remark. Let H, (·,·) be a de Branges space, and let E ∈ HB be such that H, (·,·) = H(E), (·,·)E . Then all information about H can, theoretically, be extracted from E. In general this is a difficult task, however, for some items it can be done explicitly. For example: (i) The reproducing kernel K(w, ·) of H is given as K(w, z) =

E(z)E # (w) − E(w)E # (z) . 2πi(w − z)

In particular, this implies that E ∈ Assoc H. (ii) We have dH = dE . This equality even holds if we only assume that H = H(E) as sets, i.e., without assuming equality of norms.

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(iii) The function ∇H is given as

∇H (z) =

−|E(z)| 1/2 ( |E(z)| ) , 4π Im z

z ∈ C \ R,

π −1/2 |E(z)|(ϕE (z))1/2 ,

z ∈ R.

2

2

(2.5)

In particular, we have d∇H = dH . (iv) We have mtH F = mt

F , E

F ∈ H.

This follows from the estimates (with w0 ∈ C+ fixed) 0) |E(w0 )|(1 − | E(w E(w0 ) |)

2π∇H (w0 )

1 1 ∇H (z) 1 , √ √ |z − w0 | |E(z)| 2 π Im z

z ∈ C+ ,

(2.6)

which are deduced from the inequality |K(w0 , z)| = |(K(w0 , ·), K(z, ·))| ∇H (w0 )∇H (z) and (2.5). IV. Structure of dB-subspaces The, probably, most important notion in the theory of de Branges spaces is the one of de Branges subspaces. 2.10. Definition. A subset L of a de Branges space H is called a dB-subspace of H, if it is itself, with the norm inherited from H, a de Branges space. We will denote the set of all dB-subspaces of a given space H by Sub H. If d : C → N0 , we set Subd H := {L ∈ Sub H: dL = d}. Since dB-subspaces with dL = dH appear quite frequently, we introduce the shorthand notation Sub∗ H := SubdH H. It is apparent from the axioms (dB1)–(dB3) of Definition 2.2 that a subset L of H is a dBsubspace if and only if the following three conditions hold: (i) L is a closed linear subspace of H; (ii) If F ∈ L, then also F # ∈ L; (iii) If F ∈ L and z0 ∈ C \ R is such that F (z0 ) = 0, then

F (z) z−z0

∈ L.

2.11. Example. Some examples of dB-subspaces can be obtained by imposing conditions on real zeros or on mean type. If d : C → N0 , supp d ⊆ R, is a function such that dF0 d for some F0 ∈ H \ {0}, then Hd := {F ∈ H: dF d} ∈ Sub H. We have dHd = max{d, dH }.


2609

If α 0 is such that mtH F0 , mtH F0# α for some F0 ∈ H \ {0}, then H(α) := F ∈ H: mtH F, mtH F # α ∈ Sub∗ H, and we have mtH H(α) = α. Those dB-subspaces which are defined by mean type conditions will in general not exhaust all of Sub∗ H. However, sometimes, this also might be the case. Trivially, the set Sub H, and hence also each of the sets Subd H, is partially ordered with respect to set-theoretic inclusion. One of the most fundamental and deep results in the theory of de Branges spaces is the Ordering Theorem for subspaces of H, cf. [8, Theorem 35] where even a somewhat more general version is proved. 2.12. De Branges’ Ordering Theorem. Let H be a de Branges space and let d : C → N0 . Then Subd H is totally ordered. The chains Subd H have the following continuity property: For a dB-subspace L of H, set {K ∈ SubdL H: K L}, if L = H, L˜ := closH {K ∈ SubdL H: K L}, if dim L > 1. L˘ :=

(2.7)

Then ˘ dim(L/L) 1 and

˜ 1. dim(L/L)

2.13. Example. Let us explicitly mention two examples of de Branges spaces, which show in some sense extreme behaviour. (i) Consider the Paley–Wiener space PW a where a > 0. This space is a de Branges space. It can be obtained as H(E) with E(z) = e−iaz . The chain Sub∗ (PW a ) is equal to Sub∗ PW a = {PW b : 0 < b a}. Apparently, we have PW b = (PW a )(b−a) , and hence in this example all dB-subspaces are obtained by mean type restrictions. (ii) In the study of the indeterminate Hamburger moment problem de Branges spaces occur which contain the set of all polynomials C[z] as a dense linear subspace, see, e.g., [2,6], [9, §5.9]. If H is such that H = closH C[z], then the chain Sub∗ H has order type N. In fact, Sub∗ H = C[z]n : n ∈ N0 ∪ {H}, where C[z]n denotes the set of all polynomials whose degree is at most n. Examples of de Branges spaces H for which the chain Sub∗ H has all different kinds of order types can be constructed using canonical systems of differential equations, see, e.g., [8, Theorems 37, 38], [11], or [14].

2610


With help of the estimates (2.6), it is easy to see that mt

KH (w, ·) = 0. ∇H (z)

This implies that for any dB-subspace L of H the supremum in the definition of mtH L is attained (e.g., on the reproducing kernel functions KL (w, ·)). Moreover, we obtain mtH L = mtH ∇L . Also the fact whether a given de Branges space L is contained in H as a dB-subspace can be characterized via generating Hermite–Biehler functions. Choose E, E1 ∈ HB with H = H(E) and L = H(E1 ). Then L ∈ Sub∗ H if and only if there exists a 2 × 2-matrix function W (z) = (wij (z))i,j =1,2 such that the following four conditions hold: # = w , and det W (z) = 1. (i) The entries wij of W are entire functions, satisfy wij ij (ii) The kernel

W (z)J W (w)∗ − J , KW (w, z) := z−w

J :=

0 −1 , 1 0

is positive semidefinite. (iii) Write E = A − iB and E1 = A1 − iB1 according to (2.3). Then (A, B) = (A1 , B1 )W. (iv) Denote by K(W ) the reproducing kernel Hilbert space of 2-vector functions generated by the kernel KW (w, z). Then there exists no constant function uv ∈ K(W ) with uA1 + vB1 ∈ H(E1 ). Assuming that L ∈ Sub∗ H, the orthogonal complement of L in H can be described via the above matrix function W . In fact, the map ff+− → f+ A1 + f− B1 is an isometric isomorphism of K(W ) onto H L. Let us remark in this context that a function uA + vB can also be written in the form λ · (eiψ E + e−iψ E # ) and vice versa. A detailed discussion of the situation when such functions belong to H(E) and a criterion in terms of the zeros of E can be found, e.g., in [1]. Let us discuss in a bit more detail the particular situation that L ∈ Sub∗ H and dim(H/L) = 1, cf. [8, Theorem 29, Problem 87]. In this case L = closH (dom SH ). Choose E, E1 ∈ HB with H = H(E) and L = H(E1 ). Then the matrix W introduced in the above item is a linear polynomial of the form 1 − lz cos φ sin φ lz cos2 φ · M, W (z) = −lz sin2 φ 1 + lz cos φ sin φ where φ ∈ R, l > 0, and M is a constant 2 × 2-matrix φ with real entries and determinant 1. The space K(W ) is spanned by the constant function cos sin φ . We see that

cos φ −1 cos φ H L = span (A1 , B1 ) . = span (A, B)M sin φ sin φ


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3. Admissible majorants in de Branges spaces In what follows we will work with functions m which are defined on subsets of the closed upper half-plane and take nonnegative real values. To simplify notation we will often drop explicit notation of the domain of definition of the function m. If m1 , m2 : D → [0, ∞), we write m1 m2 if there exists a positive constant C, such that m1 (z) Cm2 (z), z ∈ D. Moreover, m1 m2 stands for “m1 m2 and m2 m1 ”. Using this notation, we can write Rm (H) = F ∈ H: F (z), F # (z) m(z), z ∈ D , compare with Definition 1.1. Our first aim is to show that Rm (H) will become a de Branges subspace of H, whenever Rm (H) = {0}, and m satisfies an obvious regularity condition. 3.1. Theorem. Let H be a de Branges space, and let m : D → [0, ∞) be a function with D ⊆ C+ ∪ R. Then Rm (H) ∈ Sub H if and only if m satisfies (Adm1) supp dm ⊆ R; (Adm2) Rm (H) contains a nonzero element. In this case we have dRm (H) = max{dm , dH } and

mtH Rm (H) mtH m.

(3.1)

Necessity of the conditions (Adm1) and (Adm2) is easy to see. In the proof of sufficiency we will employ the following elementary lemma. 3.2. Lemma. Let H be a de Branges space and let L be a nonzero linear subspace of H such that: (i) if F ∈ L, then also F # ∈ L; (ii) if F ∈ L and z0 ∈ C \ R with F (z0 ) = 0, then also

F (z) z−z0

∈ L.

Then closH L ∈ Sub H. Proof. The mapping F → F # is continuous on H. We have L# ⊆ L, and hence (closH L)# ⊆ closH (L# ) ⊆ closH L. Let F ∈ closH L, z0 ∈ C \ R with F (z0 ) = 0, be given. We have to show that z − z0 F (z) ∈ closH L, z − z0 or, equivalently, that

F (z) z−z0

∈ closH L. Choose an element F0 ∈ L with F0 (z0 ) = 1. Such a choice

is possible by (ii). The mapping ρF0 ,z0 : F → have ρF0 ,z0 L ⊆ L. Thus

F (z)−F (z0 )F0 (z) z−z0

is continuous. Moreover, by (ii), we

ρF0 ,z0 (closH L) ⊆ closH (ρF0 ,z0 L) ⊆ closH L.

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F (z) In particular, if F ∈ closH L and F (z0 ) = 0, then z−z ∈ closH L. 0 Together, we conclude that closH L ∈ Sub H. 2

Proof of Theorem 3.1. Assume first that Rm (H) ∈ Sub H. Then, clearly, Rm (H) = {0}, i.e. (Adm2) holds. Let w ∈ C, and choose F ∈ Rm (H) with dF (w) = dRm (H) (w). By analyticity, we have for some disk U centred at w, F (z) > 0. inf d (w) F z∈U (z − w) Since |F (z)| m(z), z ∈ U ∩ D, we obtain that dm (w) dF (w) = dRm (H) (w). It follows that dm takes only finite values and that supp dm is a discrete subset of R. In particular (Adm1) holds. Moreover, we see that dRm (H) max{dm , dH }. For the converse assume that m satisfies the conditions (Adm1) and (Adm2). We will apply Lemma 3.2 with L := Rm (H). The hypothesis (i) of Lemma 3.2 is satisfied by the definition of Rm (H). Let F ∈ Rm (H), w ∈ C, and assume that dF (w) > max{dm (w), dH (w)}. Then F (z) z−w ∈ H. Let U be a compact neighbourhood of w such that inf

z∈U ∩D |z−w|dm (w) =0

m(z) > 0. |z − w|dm (w)

For z ∈ / U , we have |z − w| 1, and hence |z − w|−1 |F (z)| |F (z)| m(z), z ∈ D \ U . The function (z − w)−dm (w)−1 F (z) is analytic, and hence bounded, on U . It follows that F (z) m(z) (z − w)dm (w)+1 |z − w|dm (w) ,

z∈

(D ∩ U ) \ {w}, dm (w) > 0, D ∩ U, dm (w) = 0,

and hence F (z) z − w m(z),

z ∈ U ∩ D.

F (z) # F (z) The same argument will show that |( z−w ) | m(z), z ∈ D, and hence z−w ∈ Rm (H). Since, by (Adm1), dm (w) = dH (w) = 0 for w ∈ C \ R, we conclude that Rm (H) satisfies the hypothesis (ii) of Lemma 3.2. Moreover,

min

F ∈Rm (H)\{0}

dF (w) max dm (w), dH (w) .

Hence Rm (H) ∈ Sub(H), and dRm (H) max{dm , dH }. We have proved the asserted equivalence and equality of divisors in (3.1). Let F ∈ Rm (H). Then |F (z)| m(z), z ∈ D, and hence mtH F mtH m. This proves the assertion concerning mean types. 2


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Theorem 3.1 justifies the following definition. 3.3. Definition. Let H be a de Branges space. A function m : D → [0, ∞) where D ⊆ C+ ∪ R, is called an admissible majorant for H if it satisfies the conditions (Adm1) and (Adm2) of Theorem 3.1. The set of all admissible majorants is denoted by Adm H. For the set of all those admissible majorants which are defined on a fixed set D, we write AdmD H. 3.4. Remark. (i) We allow majorization on a subset of the closed upper half-plane. Of course, the same definitions could be made for majorants m defined on just any subset of the complex plane. However, due to the symmetry with respect to the real line which is included into the definition of Rm , this would not be a gain in generality. (ii) Majorants defined on bounded sets give only trivial results: If D ⊆ C+ ∪ R is bounded and m ∈ AdmD H, then Rm (H) = Rm (H) = Hdm . In view of this fact, we shall once and for all exclude bounded sets D from our considerations. 3.5. Remark. As we have already noted after the definition of dm , we have dm (w) = 0 whenever w∈ / D. The first formula in (3.1) hence gives dRm (H) (x) = dH (x),

x ∈ R \ D.

It is worth to notice that this statement can also be read in a slightly different way: Assume that L ∈ Sub H is represented as L = Rm (H) with some m ∈ AdmD H. Then dL |R\D = dH . Assume that w ∈ R \ D and set d(z) =

0, dH (w) + 1,

z = w, z = w.

Unless dim H = 1, the subspace Hd will be a dB-subspace of H. It follows from the above notice that no subspace L ∈ Sub H with L ⊆ Hd can be realized as Rm (H) with some m ∈ AdmD H. Let us provide some standard examples of admissible majorants. We will mostly work with these majorants. 3.6. Example. An obvious, but surprisingly important, example of admissible majorants is provided by the functions ∇L |C+ ∪R , L ∈ Sub H. Since always L ⊆ R∇L |C+ ∪R (H), (Adm2) is satisfied. Also, it follows that d∇L dL and this yields (Adm1). Thus ∇L |C+ ∪R ∈ Adm H. The function ∇L is actually for several reasons a distinguished admissible majorant. This will be discussed in more detail later (see, also, the forthcoming paper [4]). 3.7. Example. Other examples of admissible majorants can be constructed from functions associated to the space H. Let S ∈ Assoc H, and assume that S does not vanish identically and

2614


does not satisfy dS (z) = dH (z), z ∈ C. Moreover, let D ⊆ C+ ∪ R be such that mS (z) = 0 for all z ∈ D \ R. Define mS (z) :=

max{|S(z)|, |S # (z)|} , |z + i|

z ∈ C+ ∪ R,

then mS |D ∈ Adm H. In fact, we have dmS |D (w) = dS (w), 0,

w ∈ R ∩ D, otherwise,

S(z) and, if z0 ∈ C is such that dS (z0 ) > dH (z0 ), then the function z−z belongs to Rm (H). 0 + Provided D contains a part of some ray in C with positive logarithmic density, we have mtH RmS |D (H) = mtH mS |D = max mtH S, mtH S # .

3.8. Example. Let H(E) be a de Branges space, and let L ∈ Sub H(E). Choose E1 ∈ HB with L = H(E1 ). Then E1 (z) ∈ Assoc H(E) and the conditions required in Example 3.7 are fulfilled for E1 . Since each kernel function KL (w, ·) of the space L belongs to RmE1 (H), we always have L ⊆ RmE1 |C+ (H). Note that the space RmE1 (H) does not depend on the particular choice of E1 with L = H(E1 ). In fact, if E1 and E2 both generate the space L, then we will have |E1 (z)| |E2 (z)| throughout C+ ∪ R, and hence also mE1 mE2 . Let us state explicitly how the majorants mE1 and ∇L are related among each other and with the space L. By (2.6) we have mE1 ∇L . Moreover, Rm (H) ⊆ ⊆ E1 L R∇L (H). ⊆ ⊆ R∇L (H) 4. Majorization on rays which accumulate to i∞ In this section we consider subspaces which are generated by majorization along rays being not parallel to the real axis. The following two statements are our main results in this respect. 4.1. Theorem. Consider D := i[h, ∞) where h > 0. Let H be a de Branges space, and let L ∈ Sub∗ H. Then L = R∇L |D (H). 4.2. Theorem. Consider D := eiπβ [h, ∞) where h > 0 and β ∈ (0, 12 ). Let H be a de Branges space, assume that each element of H is of zero type with respect to the order ρ := (2 − 2β)−1 , and let L ∈ Sub∗ H. Then L = R∇L |D (H). 4.3. Remark. (i) In both theorems we have D ∩ R = ∅. Hence, the requirement L ∈ Sub∗ H is necessary in order that L can be represented in the form Rm (H) with some m ∈ AdmD H, cf. Remark 3.5.


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(ii) With the completely similar proof, the analogue of Theorem 4.2 for rays D contained in the second quadrant holds true. In any case L ⊆ R∇L |D (H). Hence, in order to establish the asserted equality L = R∇L |D (H) in either Theorem 4.1 or Theorem 4.2, it is sufficient to show that R∇L |D (H) ⊆ L. For the proof of this fact, we will employ the same method as used in the proof of [8, Theorem 26]. Let us recall the crucial construction. Let F ∈ Assoc H and H ∈ H L be given. If G ∈ Assoc L, we may consider the function ΦF,H (w) :=

F (z) −

F (w) G(w) G(z)

z−w

, H (z) .

(4.1)

H

In the proof of [8, Theorem 26] it was shown that this function does not depend on the particular choice of G ∈ Assoc L, is entire, and of zero exponential type. First we treat a particular situation. 4.4. Lemma. Consider D := eiπβ [h, ∞) where h > 0 and β ∈ (0, 12 ]. Let H be a de Branges space, let L ∈ Sub∗ H, and assume that dim H/L = 1. Then L = R∇L |D (H). Proof. Let E, E1 ∈ HB be such that H = H(E) and L = H(E1 ). Assume for definiteness that the choice of E and E1 is made such that A := 12 (E + E # ) ∈ H and A1 := 12 (E1 + E1# ) = A. Then we have H = L ⊕ span{A1 }. / R∇L |D (H). Assume Since L ⊆ R∇L |D (H), the assertion R∇L |D (H) = L is equivalent to A1 ∈ on the contrary that |A1 (z)| ∇L (z), z ∈ D. We have 2 (z) = ∇L

|E1 (z)|2 − |E1# (z)|2 |E1 (z)| + |E1# (z)| |E1 (z)| − |E1# (z)| = 4π Im z 4π Im z

It follows that |A1 (z)|

|E1 (z)| |E1 (z) + E1# (z)| |E1 (z)| · |A1 (z)| = , 2π Im z π Im z |E1 (z)| Im z ,

1

z ∈ C+ .

z ∈ D, and hence

1 B1 (z) 1 E1 (z) = 1 − i , Im z A1 (z) Im z A1 (z)

z ∈ D.

Since A1 ∈ / L, we know from the proof of [8, Theorem 22] that lim

|z|→∞ z∈D

and have obtained a contradiction.

1 B1 (z) = 0, Im z A1 (z)

2

The general case will be reduced to this special case with the help of the next lemma. Recall the notation (2.7): L˘ := {K ∈ SubdL H: K L}.

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4.5. Lemma. Let H be a de Branges space, and let L ∈ Sub∗ H, L = H. Then L˘ = H ∩ Assoc L. Proof. Set M := H ∩ Assoc L. First we show that M ∈ Sub∗ H. Since both H and Assoc L are invariant with respect to F → F # and with respect to division by Blaschke factors, also M has this property. The crucial point is to show that M is closed in H. To this end, let (Fn )n∈N be a sequence of elements of M which converges to some element F ∈ H in the norm of H. Choose G0 ∈ L \ {0} and w ∈ C+ , G0 (w) = 0. Since the difference quotient operator ρG0 ,w is continuous, we have ρG0 ,w (Fn ) → ρG0 ,w (F ) in the norm of H. However, Fn ∈ Assoc L and G0 ∈ L, and therefore ρG0 ,w (Fn ) ∈ L. Since L is closed in H, we obtain ρG0 ,w (F ) ∈ L. The relation F (z) = (z − w)ρG0 ,w (F )(z) +

F (w) G0 (z) G0 (w)

gives F ∈ Assoc L. We conclude that M ∈ Sub H. The fact that L ⊆ M yields in particular that M ∈ Sub∗ H. Since we chose G0 ∈ L and L ⊆ M, we have dom SM = ρG0 ,w (M). However, since M ⊆ Assoc L, ρG0 ,w maps M into L and it follows that dom SM ⊆ L. By [8, Theorem 29], dim(M/ clos dom SM ) 1 and hence dim(M/L) 1. In case L˘ = L, this implies that M = L, and hence also the asserted equality L˘ = M holds true. Assume that L˘ L. Then there exist E1 , E˘ ∈ HB and a number l > 0, such that L = H(E1 ),

1 ˘ ˘ (A, B) = (A1 , B1 ) 0

˘ L˘ = H(E),

lz . 1

Moreover, with these choices, we have L˘ = L ⊕ span{A1 }. Since certainly A1 ∈ Assoc L, we ˘ 2 conclude that M L. It follows that also in this case M = L. Proof of Theorem 4.1. Fix E, E1 ∈ HB such that H = H(E) and L = H(E1 ). Let F ∈ R∇L |D (H) and H ∈ H L be given, and consider the function ΦF,H defined in (4.1). The basic estimate for our argument is obtained by writing out the inner product in the definition of ΦF,H as an L2 -integral, and applying the Schwarz inequality in L2 (R): For G ∈ Assoc L and w ∈ C \ R with G(w) = 0 we have F (w) G(t) F (t) − G(w) dt ΦF,H (w) = H (t) · t −w |E(t)|2 R

1 2 F (t) 2 dt E(t) |t − w|2 H H R

F (w) + G(w)

R

1 2 G(t) 2 dt E(t) |t − w|2 H H .

(4.2)


2617

Note that the integrals on the right side of this inequality converge, since F ∈ H and G ∈ Assoc L ⊆ Assoc H. Step 1: ΦF,H vanishes identically. Let us use the estimate (4.2) with G := E1 and w := iy, y h. From (2.6) and F ∈ R∇L |D (H), we obtain limy→∞ |E1−1 (iy)F (iy)| = 0. By the Bounded Convergence Theorem both integrals in (4.2) tend to 0 if y → ∞. In total, lim ΦF,H (iy) = 0.

y→∞

Similar reasoning applies with G := E1# and w := −iy, y h, in (4.2). It follows that also limy→−∞ |ΦF,H (iy)| = 0. Since ΦF,H is of zero exponential type, we may apply the Phragmén– Lindelöf Principle in the left and right half-planes separately, and conclude that the function ΦF,H vanishes identically. Step 2: End of proof. Since F ∈ R∇L |D (H) and H ∈ H L were arbitrary, we conclude that F (z) −

F (w) G(w) G(z)

z−w

∈ L,

F ∈ R∇L |D (H), G ∈ Assoc L.

˘ This just says that R∇L |D (H) ⊆ Assoc L, and Lemma 4.5 gives R∇L |D (H) ⊆ L. ˘ we are already done. Otherwise, applying Lemma 4.4 with the spaces L˘ and L ∈ If L = L, ˘ and using the already established fact R∇ |D (H) ⊆ L, ˘ gives Sub∗ L, L ˘ = L. R∇L |D (H) ⊆ R∇L |D (L)

2

Proof of Theorem 4.2. Again fix E, E1 ∈ HB such that H = H(E) and L = H(E1 ). Let F ∈ R∇L |D (H) and H ∈ H L be given, and consider the function ΦF,H . Step 1: ΦF,H is of order ρ and zero type. The function E1 belongs to HB and is of order ρ = (2 − 2β)−1 < 1. Hence |E1 (iy)| is a nondecreasing function of y > 0. Let > 0 be given, then there exists C > 0 with F (z) Ce|z|ρ ,

z ∈ C.

Using (4.2) with G := E1 and w := iy, y 1, we obtain ΦF,H (iy)

1/2 F (t) 2 dt · H H E(t) t 2 + 1 R

ρ

Cey · + |E1 (i)|

1/2 E1 (t) 2 dt ρ · H H = O ey , E(t) t 2 + 1

y 1.

R

ρ

Using G := E1# instead of E1 , gives the analogous estimate |ΦF,H (iy)| = O(e|y| ) for y < −1. ρ In total, we have |ΦF,H (iy)| = O(e|y| ), y ∈ R. Applying the Phragmén–Lindelöf Principle to the left and right half-planes separately, yields that ΦF,H is of order ρ. Since > 0 was arbitrary, ΦF,H is of zero type with respect to this order.

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Step 2: ΦF,H vanishes identically. Let C > 0 be such that |F (z)|, |F # (z)| Cm∇L |D (z), z ∈ D. We obtain from (2.6) and the estimate (4.2) used with G := E1 and w ∈ D, that ΦF,H (w)

1/2 F (t) 2 dt · H H E(t) |t − w|2 R

1/2 E1 (t) 2 dt + √ · H H , E(t) |t − w|2 2 π Im w C

w ∈ D.

R

Since R |E −1 F (t)|2 (1 + t 2 )−1 dt, R |E −1 E1 (t)|2 (1 + t 2 )−1 dt < ∞, and β ∈ (0, 12 ), both integrals tend to zero if |w| → ∞ within D. We obtain that lim ΦF,H (w) = 0.

|w|→∞ w∈D

The similar argument, applying (4.2) with G := E1# and w ∈ D, will give lim|w|→∞, w∈D |ΦF,H (w)| = 0. Consider the region G+ which is bounded by the ray D, its conjugate ray, and the line segment connecting these rays. Moreover, let G− := C \ G+ .

The opening of G+ is 2πβ < π < ρ −1 π . The opening of G− is π(2 − 2β) = ρ −1 π . Since ΦF,H is of order ρ zero type, we may apply the Phragmén–Lindelöf Principle to the regions G+ and G− separately, and conclude that ΦF,H vanishes identically. Step 3: End of proof. We repeat word by word the same reasoning as in Step 2 of the proof of Theorem 4.1, and obtain the desired assertion. 2 4.6. Remark. Consider D := eiπβ [h, ∞) where h > 0 and β ∈ (0, 12 ). We do not know at present, and find this an intriguing problem, whether Theorem 4.2 remains valid without any assumptions on the growth of elements of H. It is clear where the argument in the above proof breaks: If we merely know that ΦF,H is of zero exponential type, its smallness on the boundary of G+ does not imply that ΦF,H ≡ 0. On the other hand, we were not able to construct a counterexample. One reason for this will be explained later, cf. Remark 5.12.


2619

Having in mind Theorem 4.1 and the formula (3.1) for zero divisors, it does not anymore come as a surprise that all dB-subspaces of a given de Branges space can be realized by majorization on D := R ∪ i[0, ∞). 4.7. Corollary. Consider D := R ∪ i[0, ∞). Let H be a de Branges space, and let L ∈ Sub H. Then L = R∇L |D (H). Proof. We have d∇L = dL , and hence R∇L |D (H) ⊆ HdL . Since L ∈ Sub∗ HdL , we may apply Theorem 4.1, and obtain L ⊆ R∇L |D (H) ⊆ R∇L |D (HdL ) ⊆ R∇L |i[1,∞) (HdL ) ⊆ L.

2

Another statement which fits in the present context can be proved by a more elementary argument. Also this result is not much of a surprise, when thinking of [3, Theorem 3.4] (see Theorem 5.1 below) and the estimate (3.1) for mean type. 4.8. Proposition. Consider D := R ∪ eiπβ [0, ∞) where β ∈ (0, 12 ]. Let H be a de Branges space, and let L = H(E1 ) ∈ Sub H. Then L = RmE1 |D (H). Proof. In any case, we have L ⊆ RmE1 |D (H), cf. Example 3.8. Hence, in order to establish the asserted equality, it suffices to show that RmE1 |D (H) ⊆ L.

Let F ∈ RmE1 |D (H) be given, so that |E1−1 (z)F (z)| |z + i|−1 , z ∈ R ∪ eiπβ [0, ∞). Since

F ∈ H, the quotient E1−1 F is of bounded type in C+ . Since F is majorized by mE1 on the ray eiπβ [0, ∞), we have mt E1−1 F 0. The same arguments apply to F # . Finally, since F is majorized along the real axis, we have E1−1 F ∈ L2 (R). It follows that F ∈ H(E1 ) = L. 2

4.9. Remark. We would like to point out that, although seemingly very similar, Proposition 4.8 differs in some essential points from the previous results Theorems 4.1, 4.2 and Corollary 4.7. The obvious differences are of course that on the one hand mE1 |D ∇L |D , but on the other hand that also in case β ∈ (0, 12 ) there are no growth assumptions on H. The following two notices are not so obvious. First, in Proposition 4.8 we obtain only L = closH RmE1 |D (H), and taking the closure is in general necessary. In the previous statements, we had L = R∇L |D (H) and hence actually L = R∇L |D (H), cf. the chain of inclusions in Example 3.8. Secondly, the argument used to prove Proposition 4.8 relies mainly on majorization along R; majorization along the ray is only used to control mean type. Contrasting this, the argument used to deduce Corollary 4.7 from Theorem 4.1 relies mainly on majorization on the ray; majorization along R is only used to control dL . 5. Majorization on sets close to R In this section, we focus on majorization on sets D which are close to the real axis. If majorization is permitted only on R itself, we already know precisely which subspaces can be represented. Recall: 5.1. Theorem. (See [3, Theorem 3.4].) Consider D := R. Let H be a de Branges space, and let L = H(E1 ) ∈ Sub H. If mtH L = 0, then L = RmE1 |D (H). Conversely, if L = Rm (H) with some m ∈ AdmR H, then mtH L = 0.

2620


In view of this result, it is not surprising that we cannot capture mean type restrictions if D is too close to the real line. The following statement makes this quantitatively more precise. 5.2. Theorem. Let H be a de Branges space, and let ψ : R → R be a positive and even function which is increasing on [0, ∞) and satisfies ∞ 0

ψ(t) dt < ∞. t2 + 1

If D ⊆ {z ∈ C+ ∪ R: Im z ψ(Re z)} and m ∈ AdmD H, then mtH Rm (H) = 0. Proof. Let E be such that H = H(E), pick F ∈ Rm (H), F H = 1, and set a := mtH F . If a = 0, we are already done, hence assume that a < 0. Let ε ∈ (0, − a2 ), and let ψ˜ : R → R be a ˜ positive and even function which is increasing on [0, ∞), satisfies ψ(x) = o(ψ(x)), x → +∞, and is such that still ∞ ˜ ψ(t) dt < ∞. t2 + 1 0

It is a well-known consequence of the Beurling–Malliavin Theorem, that there exists a nonzero function f ∈ PW ε with f (x) exp −ψ(x) ˜ , x ∈ R, see, e.g., [15, p. 276] or [20, p. 159]. By the Phragmén–Lindelöf Principle the functions eiεz f and eiεz f # are bounded by 1 throughout C+ . Since ψ˜ is even and increasing on [0, ∞), we can estimate the Poisson integral for log |f | to obtain f (z) exp ε|Im z| − C ψ˜ |z| , z ∈ C, where the constant C > 0 does not depend z. Consider the function G(z) := F (z)f (z)ei(a+ε)z . Then |G(x)| |F (x)|, x ∈ R, and F G = mt + mt f − (a + ε) = mt f − ε 0, E E G# G# mt = mt + mt f # + (a + ε) a + 2ε < 0. E E mt

(5.1)

We conclude that G ∈ H. Next we show that G ∈ Rm (H). Indeed, if z ∈ D or z ∈ D, z = x + iy, ˜ then |y| ψ(x). Since ψ(x) = o(ψ(x)), x → +∞, we may estimate ˜ ˜ G(z) F (z)e(−a+2ε)|y|−C ψ(|z|) m(z)e(−a+2ε)ψ(x)−C ψ(x) m(z),

z ∈ D or z ∈ D.


2621

Finally, by the Phragmén–Lindelöf Principle, mt f −ε. Using (5.1), it follows that mt E −1 G −2ε and so mtH Rm (H) −2ε. Since ε ∈ (0, − a2 ) was arbitrary, we conclude that mtH Rm (H) = 0. 2 Theorem 5.2 can, of course, be viewed as a necessary condition for representability of a dBsubspace L as Rm (H), where m is defined on some set D close to R. We turn to the discussion of sufficient conditions for representability. To start with, let us discuss representability with the standard majorant ∇L restricted to R. 5.3. Theorem. Consider D := R. Let H be a de Branges space, and let L = H(E1 ) ∈ Sub H. If mtH L = 0 and supx∈R ϕE 1 (x) < ∞, then ˘ L ⊆ R∇L |D (H) ⊆ L. Thereby R∇L |D (H) = L, if and only if L˘ L and for some ϕ0 ∈ R we have cos ϕE (x) − ϕ0 ϕ (x) 1/2 , 1 E1

x ∈ R.

Proof. Let F ∈ R∇L |R (H) be given. Since mtH L = 0, we have mt E1−1 F = mt E −1 F 0. Thus E1−1 F ∈ N+ . Due to our present assumption on ϕE 1 , we have F (x) ∇L (x) = π −1/2 E1 (x) ϕ (x) 1/2 E1 (x), E1

x ∈ R.

The Smirnov Maximum Principle implies that |F (z)| |E1 (z)| throughout the half-plane C+ . It ˘ follows that F ∈ Assoc L. Using Lemma 4.5, we obtain R∇L |R (H) ⊆ L. ˘ By what we just proved, in case L = L, certainly also R∇L |R (H) = L. Hence, in order to characterize the situation that R∇L |R (H) = L, we may assume that L˘ = L. Let ϕ0 ∈ R be such that L˘ = L ⊕ span{S1 }, with S1 := eiϕ0 E1 + e−iϕ0 E1# . We have S1 (x) = 2E1 (x) cos ϕE1 (x) − ϕ0 , and hence S1 ∈ R∇L |R (H) if and only if |cos(ϕE1 (x) − ϕ0 )| (ϕE 1 (x))1/2 , x ∈ R. Since R∇L |R (H) = L is equivalent to S1 ∈ R∇L |R (H), the assertion follows. 2 5.4. Corollary. Consider D := R. Let H = H(E) be a de Branges space, assume that supx∈R ϕE (x) < ∞ and K˘ = K, K ∈ Sub∗ H, and let L ∈ Sub∗ H. If mtH L = 0, then L = R∇L |R (H). , L ∈ Sub∗ H, depends monotonically on L. By this Proof. By [8, Problem 154], the function ϕL we mean that (x) ϕK (x), ϕL

x ∈ R,

whenever L, K ∈ Sub∗ H,

The present assertion is now an immediate consequence of Theorem 5.3.

L ⊆ K. 2

Already the following simple remark shows that some additional conditions are really needed in order to obtain L = R∇L |R (H).

2622


5.5. Remark. Let H be a de Branges space, let L ∈ Sub H, and write L = H(E1 ). If infx∈R ϕE 1 (x) > 0, then L˘ ⊆ R∇L |R (H). This follows since the condition infx∈R ϕE 1 (x) > 0 certainly implies that every linear combination λE1 + μE1# is majorized by ∇L on the real axis. However, the situation can be really bad, if ϕE 1 grows fast. 5.6. Example. We are going to construct spaces H and L ∈ Sub∗ H, such that dim H/L = ∞ and R∇L |R (H) = H. As we see from the proof of Theorem 5.3, it will be a good start to construct the space L = H(E1 ) such that ϕE 1 (x) grows very fast. Let zn := (sign n) log |n| + i|n|−1 log−2 |n|, n ∈ Z, 1 # |n| 2. Since ∞ n=1 |Im zn | < ∞, there exists a function E1 ∈ HB with E1 (−z) = E1 (z) and

mt E1−1 E1# = 0, which has the points zn as simple zeros and does not vanish at any other point. For this function, we have

ϕE 1 (x) =

1 n log n[(x − log n)2 + n−2 log−4 n] 2

n∈Z |n|2

.

Let x ln 2 be given, and choose k ∈ N, k 2, such that x ∈ [log k, log(k + 1)). Then we have k log k (x − log k)2 +

1

2

k log k 2

k 2 log4 k

log(k + 1) − log k

2

+

1

k 2 log4 k

=log(1+ k1 ) k1

log2 k 1 1 log2 k . 1+ 1+ k k log4 k log4 2 It follows that ϕE 1 (x)

1+

1 log4 2

−1

−1 1 ex − 1 1+ , x2 log2 k log4 2 k

x ln 2.

(5.2)

Next, choose an entire matrix function W (z) = (wij (z))i,j =1,2 , W = I , of zero exponential type, # = w , W (0) = I , det W (z) = 1, such that the kernel K (w, z) is positive semidefinite with wij ij W and the reproducing kernel space K(W ) does not contain a constant vector function. Examples of such matrix functions can be obtained easily using the theory of canonical systems. Define a function E = A − iB by A(z), B(z) := A1 (z), B1 (z) W (z). Then E ∈ HB and H(E1 ) ∈ Sub∗ H(E). Moreover, the space K(W ) is isomorphic to the orthogonal complement H(E) H(E1 ) via the map f+ → f+ A1 + f− B1 . f−


2623

In particular, dim(H(E) H(E1 )) = dim K(W ) = ∞. Note here that K(W ) is certainly infinite dimensional, since it does not contain any constant. Finally, note that all elements ff+− of K(W ) are of zero exponential type. In view of (5.2) and the symmetry of E1 , this implies that f+ (x)A1 (x) + f− (x)B1 (x) E1 (x) ϕ (x) 1/2 = π 1/2 ∇H(E ) (x), E1 1

x ∈ R,

i.e. f+ A1 + f− B1 ∈ R∇H(E1 ) |R (H(E)). We conclude that R∇H(E1 ) |R (H(E)) = H(E). Our next aim is to obtain some information about majorization on lines D := R + ih, h > 0, parallel to the real axis. In order that a subspace L ∈ Sub H can be represented by majorization on D, it is not only necessary to have mtH L = 0, but also that L ∈ Sub∗ H. 5.7. Theorem. Consider D := R + ih where h > 0. Let H be a de Branges space, and let L = ˘ H(E1 ) ∈ Sub∗ H. If mtH L = 0, then L = RmE1 |D (H) and L ⊆ R∇L |D (H) ⊆ L. Proof. It is easy to see that, for the proof of the present assertion, we may assume without loss of generality that dH = 0. Step 1: Write H = H(E). We prove the following statement: If S ∈ Assoc H, and |E1−1 S| and are bounded on the line D = R + ih, then S ∈ Assoc L. The function E1−1 S is analytic on a domain containing the closed half-plane {z ∈ C: Im z h} and is of bounded type in the open half-plane H := {z ∈ C: Im z > h}. Also, we have |E1−1 S # |

mt

E S S = mt + mt 0. E1 E E 1 =0

Thus it belongs to the Smirnov class N+ (H) in the half-plane H. The Smirnov Maximum Principle hence applies, and we obtain that E1−1 S is bounded throughout H. The same argument applies with S # in place of S. Applying [8, Theorem 26] with the measure dμ(t) := |E(t)|−2 dt gives S ∈ Assoc L. Step 2: Let F ∈ R∇L |D (H). We have ∇L (z) =

|E1 (z)|2 − |E1# (z)|2 4πh

1/2

E1 (z),

z ∈ D,

and hence |E1−1 F | and |E1−1 F # | are bounded on D. By Step 1, it follows that F ∈ Assoc L. ˘ and we conclude that R∇ |D (H) ⊆ L. ˘ Lemma 4.5 implies that F ∈ L, L Step 3: Let F ∈ RmE1 |D (H). The function S(z) := zF (z) is associated to H, and E1−1 S

as well as E1−1 S are bounded on D. By Step 1, S ∈ Assoc L, and thus F ∈ L. We see that RmE1 |D (H) ⊆ L. The reverse inclusion holds in any case, cf. Example 3.8. 2 It is easy to give an example of de Branges spaces H and L ∈ Sub∗ H, mtH L = 0, such that R∇L |R+ih (H) = L.

2624


5.8. Example. Let D := R + ih where h > 0. Consider the space H := H(E) generated by the function E(z) := cos z − i(z cos z + sin z). The choice of E is made such that 1 A(z), B(z) = (cos z, sin z) 0

z . 1

Thus H(E) contains L := PW 1 as a dB-subspace with codimension 1, and H = L ⊕ span{cos z}. 2h 1/2 Since for z = x + ih ∈ D, ∇P W 1 (z) = ( sh , the function cos z belongs to R∇L |D (H). Thus 2πh ) R∇L |D (H) = H. Finally, we turn to majorization on rays parallel to the real axis. 5.9. Theorem. Consider D := iy0 + [h, ∞) where h ∈ R and y0 0. Let H be a de Branges space, assume that each element of H is of zero type with respect to the order ρ := 12 , and let L = H(E1 ) ∈ Sub∗ H. Then RmE1 |D (H) = L. Proof. Step 1: The case y0 > 0. We proceed similar as in the proof of Theorem 4.2. Fix E, E1 ∈ HB such that H = H(E) and L = H(E1 ). Let F ∈ RmE1 |D (H) and H ∈ H L be given, and consider the function ΦF,H defined as in (4.1). The argument which was carried out in Step 1 of the proof of Theorem 4.2, yields that ΦF,H is of zero type with respect to the order 12 . Let C > 0 be such that |F (z)|, |F # (z)| CmE1 |D (z), z ∈ D. Moreover, let z0 be a zero of E1 . The basic estimate (4.2), used with G(z) := (z − z0 )−1 E1 (z) and w ∈ D, gives ΦF,H (w)

1/2 F (t) 2 dt · H H E(t) |t − w|2 R

|w − z0 | +C |w + i|

1/2 G(t) 2 dt · H H , E(t) |t − w|2

w ∈ D.

R

However, since F ∈ H and G ∈ L ⊆ H, we have F, G ∈ L2 (|E(t)|−2 dt). Moreover, for w ∈ D, |t − w| y0 > 0. Hence, we may apply the Bounded Convergence Theorem to obtain lim ΦF,H (w) = 0.

|w|→∞ w∈D

Since ΦF,H is of order 12 and zero type, the Phragmén–Lindelöf Principle implies that ΦF,H vanishes identically. Since F ∈ RmE1 |D (H) and H ∈ H L were arbitrary, we conclude with the help of ˘ Thus, L ⊆ RmE |D (H) ⊆ L. ˘ Lemma 4.5 that RmE |D (H) ⊆ L. 1

1


2625

˘ Let In order to complete the proof, assume on the contrary that L L˘ and RmE1 |D (H) = L. α ∈ [0, π) be such that L˘ = L ⊕ span eiα E1 − e−iα E1# . Note that, in particular, eiα E1 − e−iα E1# ∈ / L. Since RmE1 |D (H) L, we can find a function F ∈ H(E1 ) and a constant λ ∈ C \ {0}, such that F + λ eiα E1 − e−iα E1# ∈ RmE1 |D (H). Set Θ := e−2iα E1−1 E1# and consider the associated model subspace KΘ := H 2 ΘH 2 of the Hardy space. Recall that the mapping F → F /E1 is a unitary transform of H(E1 ) onto KΘ . Then 1 −iα −1 e 1 , z ∈ D. E F +1 − Θ 1 λ |z + i| ∈K Θ

Theorem A.1 implies that 1 − Θ ∈ KΘ . This contradicts the fact that eiα E1 − e−iα E1# ∈ / L. Step 2: The case y0 = 0. We show that majorization remains present on each fixed ray iy + [h, ∞), y > 0. This reduces the case y0 = 0 to the case already settled in Step 1. Let F ∈ RmE1 |D (H) be given. Consider the function f (z) := E1−1 (z) · zF (z). Then f is of bounded type in C+ . Since F and E1 are entire functions of order 12 , we certainly have mt f = 0. Moreover, f has an analytic continuation to some domain which contains the closure of C+ . We conclude that f belongs to the Smirnov class N+ . Hence log |f | is majorized throughout the half-plane C+ by the Poisson integral of its boundary values: y logf (z) π

R

log |f (t)| dt, (t − x)2 + y 2

z = x + iy ∈ C+ .

Since F ∈ RmE1 |D (H), the function f is bounded on D = [h, ∞). Again, since F and E1 are of order 12 , we have R

| log |tF (t)|| dt < ∞, t2 + 1

R

| log |E1 (t)|| dt < ∞, t2 + 1

cf. [19, p. 50, Theorem]. Hence we may estimate y logf (z) π

[h,∞)

+

y π

log+ |f (t)| y dt + 2 2 π (t − x) + y

(−∞,h)

−

(−∞,h)

log |E1 (t)| dt, (t − x)2 + y 2

log+ |tF (t)| dt (t − x)2 + y 2

z = x + iy ∈ C+ .

2626


Since f is bounded on [h, ∞), the first summand is bounded independently of z ∈ C+ . The second and third summands are, for each fixed y > 0, nonincreasing and nonnegative functions of x h. In particular, they are bounded on each ray iy + [h, ∞), y > 0. It follows that, for each fixed positive value of y, we have |F (z)| mE1 (z), z ∈ iy + [h, ∞). 2 The following two examples show that the statement in Theorem 5.9 is in some ways sharp. 5.10. Example. There exists a space H = H(E) with E of order 12 and finite type, and a subspace L ∈ Sub∗ (H), such that RmE1 |D (H) = L. To show this we construct a matrix W with components of order 12 such that the space K(W ) exists and such that one of its rows is bounded on (0, ∞). Consider the two auxiliary functions G(z) :=

n∈N

√ n2 − i sin(π z + i ) =c , , c= √ n2 π z+i n∈N z ˜ 1− 2 . G(z) := n − in

z 1− 2 n −i

n∈N

Let x ∈ (k − 1/2, k + 1/2), k ∈ N. Then we can write 2 2 k 2 − x 2 + i 1 − x , G x = k2 − i n2 − i n =k

and it is easy to see that 2 k 2 − x 2 + i | sin πx| G x k2 − i · x

x 2 −1 k 2 − x 2 + i | sin π(x − k)| · 1 − 2 · . x(x + k) |x − k| k

Next, note that ˜ 2 ) 2 (x 2 − n2 )2 + n2 n4 + 1 G(x (x 2 − k 2 )2 + k 2 = · . G(x 2 ) (x 2 − n2 )2 + 1 n4 + n2 (x 2 − k 2 )2 + 1 n∈N

Combining this, we conclude that 2 |x 2 − k 2 + ki| G ˜ x k −1 x −1 , x(x + k)

x > 1.

2 . Then E is of order 1/2 and finite type, we have |E (x)| 1, ˜ Set E0 (z) = (z + i)(G(z)) 0 0 1/2 x > 1, and log |E(x)| |x| , x → −∞. Therefore 1 ∈ Assoc(H(E0 )). Let E0 = A0 − iB0 . Changing slightly the function E0 we may assume that A0 (0) = 1, B0 (0) = 0. Then by [8, Theorems 27, 28], there exist real entire functions C0 , D0 , with D0 (0) = 1, C0 (0) = 0, such that for the matrix A0 B0 W := C0 D 0


2627

the space K(W ) exists. Thus also the space K(W˜ ) exists, where W˜ :=

D0 C0

B0 A0

.

Let H(E1 ) be an arbitrary de Branges space, and set E = A − iB, where (A, B) := (A1 , B1 )W˜ . Define

f+ f−

:=

B0 (z) W˜ (z)J − J 1 z , = A (z)−1 0 0 z z

J=

0 1

−1 . 0

Then ff+− ∈ K(W ) and so f+ A1 + f− B1 ∈ H(E) L. However, this function is majorized by mE1 on (0, ∞). Thus RmE1 |(0,∞) (H(E)) H(E1 ). 5.11. Example. There exist spaces H = H(E), L = H(E1 ), with functions E, E1 , of arbitrarily small order such that L ∈ Sub H and, for D = R + iy0 , R∇L |D (H) = L. Thus, in the statement of Theorem 5.9 we cannot replace RmE1 |D (H) by R∇L |D (H). To show this, let α > 1, let tn = |n|α , n ∈ Z \ {0}, and let μn = |n|2α−2 . Put q(z) =

μn

n∈Z\{0}

1 1 . − tn − z tn

The series converges since μn n

tn2

=

|n|2α−2 n

|n|2α

=

1 < ∞. n2 n

There exist real entire functions A1 and B1 such that q = B1 /A1 and H(E1 ) exists. We show that for L = H(E1 ) and y0 > 0 we have A1 (x + iy0 ) ∇L (x + iy0 ),

x ∈ R.

(5.3)

Put Θ = E1−1 E1# . Then (5.3) is equivalent to 1 + Θ(x + iy0 )2 1 − Θ(x + iy0 )2 ,

x ∈ R.

Also, q = i 1−Θ 1+Θ and so Im q(x + iy0 ) =

|n|2α−2 1 − |Θ(x + iy0 )|2 = y0 . 2 2 α 2 |1 + Θ(x + iy0 )| n (x − |n| ) + y0

(5.4)

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It is easy to see that Im q(x + iy0 ) 1, which implies (5.4). Indeed, let x ∈ [k α , (k + 1)α ], k ∈ N. Then |x − k α | k α−1 with constants independent of k. Hence n

|n|2α−2 k 2α−2 > 1. (x − |n|α )2 + y02 (x − k α )2 + y02

5.12. Remark. We return to the comment made in Remark 4.6. Seeking an example that the growth assumption in Theorem 4.2 is necessary, the first idea would be to proceed in the same way as in Example 5.10. But this is not possible. The reason is that there exists no function E0 ∈ HB, such that 1 ∈ Assoc(H(E0 )) and E0 = A0 − iB0 is bounded on D. Indeed, if E0 would 1 ∈ H 2 , which implies that |E0 (x + i)| |x + i|−1 , x ∈ R. Also have these properties, then (z+i)E 0 since E0 is not a polynomial, log |E0 (iy)| > N log y, y → ∞, for any fixed N > 0. Applying the Poisson formula in the angle {Re z > 0, Im z > 1} to log |E0 | (“Two Constant Theorem”) we conclude that E0 is unbounded on D. Acknowledgments We would like to thank Alexei Poltoratski who suggested the use of weak type estimates for the proof of Theorem A.1. The first author was partially supported by the grants MK-5027.2008.1 and NSH-2409.2008.1. Appendix A. Estimates of inner functions on horizontal rays In this appendix we prove a theorem about asymptotic behaviour of inner functions along horizontal rays. It is well known that for any ray D := eiπβ [0, ∞) with 0 < β < 1, the estimate 2iα e − Θ(z) |z + i|−1 , z ∈ D, is equivalent to e2iα − Θ ∈ H(E). If Θ = E −1 E # is a meromorphic inner function the latter condition means that eiα E − e−iα E # ∈ H(E). For the de Branges space setting, see [8, Theorem 22], the case of general inner functions is discussed, e.g., in [1]. We show that an analogous and even stronger statement is true for the rays iy0 + [0, ∞), y0 > 0. Each inner function Θ generates a model subspaces KΘ = H 2 ΘH 2 of H 2 . A.1. Theorem. Let Θ be an inner function in C+ and let y0 > 0. Assume that there exists a function f ∈ KΘ and a positive constant C, such that f (x + iy0 ) + 1 − Θ(x + iy0 )

C , |x + iy0 |

x > 0.

(A.1)

Then 1 − Θ ∈ KΘ . For the proof of this result we will combine weak type estimates for the Hilbert transform and properties of the Clark measures. We will throughout this appendix keep the following notation: (i) The Lebesgue measure on R is denoted by m. Moreover, Π denotes the Poisson measure on R, that is dΠ(t) = (1 + t 2 )−1 dm(t).


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(ii) The set of all functions q which are defined and analytic in C+ and have nonnegative real part throughout this half-plane is denoted by C. (iii) Recall that a function q belongs to C if and only if it has an integral representation of the form 1 i t q(z) = −ipz + i Im q(i) + dμ(t), (A.2) + π z − t 1 + t2 R

where p 0,

μ is a positive Borel measure, R

dμ(t) < ∞. 1 + t2

The data p and μ in this representation are uniquely determined by the function q (see, e.g., [25, 5.3, 5.4]). Note that, if the function q has a continuous extension to the closed half-plane C+ ∪ R, then the measure μ is absolutely continuous with respect to m and dμ(t) = Re q(t) dm(t). (iv) Two subclasses of C are defined as

1 C1 := q ∈ C: p = lim Re q(iy) = 0 , y→+∞ y C0 := q ∈ C: the limit lim yq(iy) exists . y→+∞

Recall that q ∈ C0 if and only if in (A.2) we have dμ(t) < ∞,

p = 0,

Im q(i) = −

R

1 π

R

t dμ(t), 1 + t2

i.e., if and only if q can be represented in the form q(z) =

i π

R

dμ(t) z−t

with a finite positive Borel measure μ, see, e.g., [13, Theorem 6.4]. In this case we have dμ(t) = π lim yq(iy). y→∞

R

Weak type estimates enter the discussion in the form Lemma A.2 below, and can be used to conclude that 1 − Θ ∈ KΘ , cf. Lemma A.3.

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A.2. Lemma. Let y0 > 0 be given. (i) Whenever q ∈ C1 , we have lim a · Π x ∈ R: q(x + iy0 ) > a = 0.

a→+∞

(ii) There exists a positive constant A, such that a · m x ∈ R: q(x + iy0 ) > a A · lim yq(iy), y→+∞

a > 0, q ∈ C0 .

Proof. Let q ∈ C, and consider the function Q(z) := q(z + iy0 ),

z ∈ C+ ∪ R.

Then Q is continuous in C+ ∪ R and belongs to C. In particular, y0 Re Q(x) = py0 + π

R

dμ(t) , (t − x)2 + y02

and it is easy to see that Re Q(x) ∈ L1 (Π). Moreover, Q ∈ C1 (or Q ∈ C0 ) if and only if q has the respective property. For the proof of (i), assume that q ∈ C1 . Then we have Im Q(x) − Im Q(i) = lim Im Q(x + iy) − Im Q(i) y0

1 y0 π

= lim

R

x−t t Re Q(t) dt. + (x − t)2 + y 2 1 + t 2

By Kolmogorov’s Theorem on the harmonic conjugate, to be more specific by [19, p. 65, Corollary], we have lim a · Π x ∈ R: Im Q(x) − Im Q(i) > a = 0.

a→+∞

Since also a · Π x ∈ R: Re Q(x) > a

a→+∞

χ{Re Q>a} Re Q dΠ −−−−−→ 0, R

the desired limit relation follows. For the proof of (ii), assume that q ∈ C0 . Then we have Re Q(x) ∈ L1 (m), and Q(z) =

i π

R

Re Q(t) dm(t), z−t

z ∈ C+ .


2631

This shows that Im Q(x) is the standard Hilbert transform of Re Q(x). Thus, by [26, V, Lemma 2.8], we have the weak type estimate e m x ∈ R: Im Q(x) > a a

R

e Re Q(t) dm(t) = π lim yq(iy), a y→+∞

where e is the Euler number. Since 1 m x ∈ R: Re Q(x) > a a

χ{Re Q>a} Re Q dm R

π a

lim yq(iy),

y→+∞

√ we obtain the desired estimate, e.g., with the constant A := π 2(1 + e).

2

A.3. Lemma. Let Θ be an inner function in C+ , and let y0 > 0. Assume that there exist positive constants c, c , and r0 , such that m x ∈ [r, 2r]: 1 − Θ(x + iy0 )

c |x + iy0 |

c r,

r r0 .

(A.3)

Then 1 − Θ ∈ KΘ . Proof. Consider the function q(z) :=

1 + Θ(z) , 1 − Θ(z)

z ∈ C+ .

For r > 0 set Mr := x ∈ [r, 2r]: 1 − Θ(x + iy0 )

c . |x + iy0 |

Then, by our hypothesis (A.3), we have m(Mr ) c r, r r0 . Assume that a > 1 and let x ∈ Mr with r > ca. Then q(x + iy0 ) = 1 + Θ(x + iy0 ) 2 − |1 − Θ(x + iy0 )| 1 − Θ(x + iy ) |1 − Θ(x + iy0 )| 0

2|x + iy0 | − 1 2a − 1 > a, c

since x r ca. Thus, we have Mr ⊆ x ∈ R: q(x + iy0 ) > a , It follows that for a > 1, r ca,

a > 1, r ca.

2632


Π x ∈ R: q(x + iy0 ) > a Π(Mr ) =

Mr

dm(t) 1 + t2

1 c r m(M ) . r 1 + 4r 2 1 + 4r 2

(A.4)

If a rc0 , then r := ca r0 , and we may use this particular value of r in (A.4). It follows that, for a > max(1, r0 /c), Π x ∈ R: q(x + iy0 ) > a

c ca d , 2 2 a 1 + 4c a

where the constant d depends only on c and c . The function q is analytic in C+ and has nonnegative real part throughout this half-plane. By Lemma A.2(i), it cannot belong to the subclass C1 , i.e., we have lim

y→+∞

1 q(iy) > 0. y

However, this property of q is, e.g., by the discussion in [1], equivalent to 1 − Θ belonging to KΘ . 2 Proof of Theorem A.1. Assume on the contrary that 1 − Θ ∈ / KΘ . Our aim is to show that, under the assumptions of the theorem, the relation (A.3) holds for some appropriate values of c, c and r0 . Once this has been achieved, Lemma A.3 implies that 1 − Θ ∈ KΘ , and we have derived a contradiction. The function 1 − Θ does not belong to KΘ if and only if q = 1+Θ 1−Θ is in C1 , that is, p = 0 in (A.2). Thus, 1 i t 1 + Θ(z) dμ(t). = i Im q(i) + + 1 − Θ(z) π z − t 1 + t2 R

The measure μ, called the Clark measure, has many important properties (see, e.g., [22, Vol. 2, Part D, Chapter 4]). In particular, it was shown in [23] that each function f ∈ KΘ has radial boundary values μ-a.e. and the restriction operator f → f |supp(μ) is a unitary operator from KΘ onto L2 (μ). Note that Θ = 1 μ-a.e. on supp(μ). For z ∈ C+ denote by kz the reproducing kernel of KΘ , kz (ζ ) =

i 1 − Θ(z)Θ(ζ ) · . 2π ζ −z

Then, for f ∈ KΘ and z ∈ C+ , we have f (z) = (f, kz )L2 (μ) = since Θ = 1 μ-a.e.

1 − Θ(z) 2πi

R

f (t) dμ(t), t −z


2633

Now let f ∈ KΘ be a function as in the hypothesis of Theorem A.1. Note that, for each M > 0, there exists a positive constant CM such that f (t) CM , x ∈ R. (A.5) dμ(t) |x + iy | t − x − iy0 0 [−M,M]

Let > 0 be fixed. We have f ∈ L2 (μ) and (|t| + 1)−1 ∈ L2 (μ) and so (|t| + 1)−1 f ∈ Using (A.5), we obtain that there exists M > 0 and C > 0 such that the function (μ := 1 μ| R\[−M ,M ] ) 2π

L1 (μ).

f (z) :=

1 − Θ(z) i

R

f (t) dμ (t), t −z

(A.6)

satisfies (1)

R

|f (t)| |t|

dμ (t) < ,

(2) |f (x + iy0 ) + 1 − Θ(x + iy0 )|

C |x+iy0 | ,

x > 0.

For the time being, let > 0 be arbitrary; we will make a particular choice later. The representation (A.6) of the function f may be rewritten as f (t) f (t) dμ (t) 1 − Θ(z) . dμ (t) + z · f (z) = i t t t −z R R

(A.7)

=:γ (z)

Let u(t) = Re f (t) t and let u+ = max{u, 0}, u− = u+ − u. Set 1 q(z) := i

u+ (t) dμ (t). t −z

R

(A.8)

Then q ∈ C0 and lim yq(iy) =

y→+∞

u+ (t) dμ (t) .

Using Lemma A.2(ii), we obtain A m x ∈ R: q(x + iy0 ) > a , a

a > 0.

The same argument applies when we take u− , as well as v+ and v− for v(t) = Im f (t) t , instead of u+ in the definition (A.8) of q. Altogether, we conclude that

f (t) dμ (t) 16A >a · , a > 0. m x ∈ R: t t −z a R

2634


Let r > 0 and x ∈ [r, 2r] be given, then γ (x + iy0 )

f (t) dμ (t) |f (t)| . dμ (t) +(2r + y0 ) · |t| t t −z R R

It follows that, for any r > 0 and a > 0, m x ∈ [r, 2r]: γ (x + iy0 ) > + (2r + y0 )a

f (t) dμ (t) 16A >a · , m x ∈ [r, 2r]: t t −z a R

and hence 16A m x ∈ [r, 2r]: γ (x + iy0 ) + (2r + y0 )a r − . a Assume that r y0 , and use this inequality for the particular value a := that

√

r

of a. Then it follows

√ √ m x ∈ [r, 2r]: γ (x + iy0 ) + 3 r(1 − 16A ).

(A.9)

√ At this point we make a particular choice of , namely, we take > 0 so small that + 3 √ and 16A 12 . Then (A.9) gives

1 1 r, m x ∈ [r, 2r]: γ (x + iy0 ) 2 2

1 2

r y0 .

However, if x ∈ [r, 2r] is such that |γ (x +iy0 )| 12 , then by the hypothesis (A.1) and the relation (A.7) we obtain that 1 − Θ(x + iy0 ) = |f (x + iy0 ) + 1 − Θ(x + iy0 )| 2C . |1 − iγ (x + iy0 )| |x + iy0 | We conclude that, for r y0 ,

2C |x + iy0 |

1 1 r, m x ∈ [r, 2r]: γ (x + iy0 ) 2 2

m x ∈ [r, 2r]: 1 − Θ(x + iy0 )

i.e. (A.3) holds.

2


2635

Appendix B. Summary of results Let H = H(E) be a de Branges space and let L = H(E1 ) ∈ Sub H. a. Necessary conditions for L = Rm (H) D

Condition on L

w∈R\D

dL (w) = dH (w)

Im z ψ(Re z), z ∈ D, ψ positive, even, increasing on [0, ∞), ∞ 2 −1 0 (t + 1) ψ(t) dt < ∞

mtH L = 0

b. Sufficient conditions for L = Rm (H) D

Representation of L

Assumption on L

Assumption on H

R ∪ i[0, ∞) R ∪ eiπβ [0, ∞), β ∈ (0, 1)

L = R∇L |D (H) L = RmE |D (H)

– –

– –

i[h, ∞), h > 0 eiπβ [h, ∞), h > 0, β ∈ (0, 12 ) ∪ ( 12 , 1)

L = R∇L |D (H) L = R∇L |D (H)

dL = dH dL = dH

– order (2 − 2β)−1 zero type

R

L = RmE |D (H) 1 L ⊆ R∇L |D (H) ⊆ L˘

mtH L = 0

–

mtH L = 0, 0

1

L = R∇L |D (H)

dL = dH

< ∞, ∀K: K = K ˘ supR ϕE

L = RmE |D (H) 1

dL = dH , mtH L = 0 dL = dH , mtH L = 0

–

dL = dH

order 12 zero type

L ⊆ R∇L |D (H) ⊆ L˘ iy0 + [h, ∞), h ∈ R, y0 0

1

L = RmE |D (H) 1

–

References [1] A.D. Baranov, Isometric embeddings of the spaces Kθ in the upper half-plane, J. Math. Sci. 105 (5) (2001) 2319– 2329. [2] A.D. Baranov, Polynomials in the de Branges spaces of entire functions, Ark. Mat. 44 (2006) 16–38. [3] A. Baranov, H. Woracek, Subspaces of de Branges spaces generated by majorants, Canad. J. Math. 61 (3) (2009) 503–517. [4] A. Baranov, H. Woracek, Majorization in de Branges spaces II. Banach spaces generated by majorants, submitted for publication, preprint available online as ASC Preprint Series 13/2009, http://asc.tuwien.ac.at. [5] A. Beurling, P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962) 291– 309. [6] A. Borichev, M. Sodin, Weighted polynomial approximation and the Hamburger moment problem, in: Complex Analysis and Differential Equations, Proceedings of the Marcus Wallenberg Symposium in Honor of Matts Essén, Uppsala University, 1998. [7] L. de Branges, Some Hilbert spaces of entire functions, Proc. Amer. Math. Soc. 10 (5) (1959) 840–846. [8] L. de Branges, Hilbert Spaces of Entire Functions, Prentice Hall, London, 1968. [9] H. Dym, H. McKean, Gaussian Processes, Function Theory, and the Inverse Spectral Problem, Academic Press, New York, 1976. [10] J.B. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.

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[11] I. Gohberg, M.G. Kre˘ın, Theory and Applications of Volterra Operators in Hilbert Space, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1970. [12] L. Golinskii, I. Mikhailova, Hilbert spaces of entire functions as a J -theory subject, Oper. Theory Adv. Appl. 95 (1997) 205–251. [13] M.L. Gorbachuk, V.I. Gorbachuk, M.G. Krein’s Lectures on Entire Operators, Oper. Theory Adv. Appl., vol. 97, Birkhäuser Verlag, Basel, 1997. [14] S. Hassi, H.S.V. de Snoo, H. Winkler, Boundary-value problems for two-dimensional canonical systems, Integral Equations Operator Theory 36 (4) (2000) 445–479. [15] V.P. Havin, B. Jöricke, The Uncertainty Principle in Harmonic Analysis, Springer-Verlag, Berlin, 1994. [16] V.P. Havin, J. Mashreghi, Admissible majorants for model subspaces of H 2 . Part I: slow winding of the generating inner function, Canad. J. Math. 55 (6) (2003) 1231–1263. [17] V.P. Havin, J. Mashreghi, Admissible majorants for model subspaces of H 2 . Part II: fast winding of the generating inner function, Canad. J. Math. 55 (6) (2003) 1264–1301. [18] M. Kaltenbäck, H. Woracek, De Branges spaces of exponential type: General theory of growth, Acta Sci. Math. (Szeged) 71 (1/2) (2005) 231–284. [19] P. Koosis, The Logarithmic Integral I, Cambridge Stud. Adv. Math., vol. 12, Cambridge University Press, 1988. [20] P. Koosis, The Logarithmic Integral II, Cambridge Stud. Adv. Math., vol. 21, Cambridge University Press, 1992. [21] N. Nikolski, Treatise on the Shift Operator, Springer-Verlag, Berlin, Heidelberg, 1986. [22] N. Nikolski, Operators, Functions, and Systems: An Easy Reading, vols. 1–2, Math. Surveys Monogr., vols. 92–93, Amer. Math. Soc., Providence, RI, 2002. [23] A.G. Poltoratski, On the boundary behavior of pseudocontinuable functions, St. Petersburg Math. J. 5 (2) (1994) 389–406. [24] C. Remling, Schrödinger operators and de Branges spaces, J. Funct. Anal. 196 (2) (2002) 323–394. [25] M. Rosenblum, J. Rovnyak, Topics in Hardy Classes and Univalent Functions, Birkhäuser Verlag, Basel, 1994. [26] E. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, New Jersey, 1971.


A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions ✩ Pekka Koskela a,∗ , Dachun Yang b , Yuan Zhou b,a a University of Jyväskylä, Department of Mathematics and Statistics, PO Box 35 (MaD),

Fin-40014 University of Jyväskylä, Finland b School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems,

Ministry of Education, Beijing 100875, People’s Republic of China Received 9 July 2009; accepted 4 November 2009

Communicated by N. Kalton

Abstract In this paper, we establish the equivalence between the Hajłasz–Sobolev spaces or classical Triebel– Lizorkin spaces and a class of grand Triebel–Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p ∈ (n/(n + 1), ∞), we give a new characterization of the Hajłasz–Sobolev spaces M˙ 1,p (Rn ) via a grand Littlewood–Paley function. © 2009 Elsevier Inc. All rights reserved. Keywords: Sobolev spaces; Triebel–Lizorkin space; Calderón reproducing formula

1. Introduction Recently, analogs of the theory of first order Sobolev spaces on doubling metric spaces have been established both based on upper gradients [23,9,32] and on pointwise inequalities [18]. For surveys on this see [19,20,24]. These different approaches result in the same function class ✩ Dachun Yang was supported by the National Natural Science Foundation (Grant No. 10871025) of China. Pekka Koskela and Yuan Zhou were supported by the Academy of Finland grant 120972. * Corresponding author. E-mail addresses: [email protected] (P. Koskela), [email protected] (D. Yang), [email protected] (Y. Zhou).


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P. Koskela et al. / Journal of Functional Analysis 258 (2010) 2637–2661

if the underlying space supports a suitable Poincaré inequality [25]. In this paper we further investigate the spaces introduced by Hajłasz [18] (also see [39]) that are defined via pointwise inequalities. Definition 1.1. Let (X , d) be a metric space equipped with a regular Borel measure μ such that all balls defined by d have finite and positive measures. Let p ∈ (0, ∞) and s ∈ (0, 1]. The homogeneous fractional Hajłasz–Sobolev space M˙ s,p (X ) is the set of all measurable functions p f ∈ Lloc (X ) for which there exists a non-negative function g ∈ Lp (X ) and a set E ⊂ X of measure zero such that f (x) − f (y) d(x, y) s g(x) + g(y)

(1.1)

for all x, y ∈ X \ E. Denote by D(f ) the class of all nonnegative Borel functions g satisfying (1.1). Moreover, define f M˙ s,p (X ) ≡ infg∈D(f ) {gLp (X ) }, where the infimum is taken over all functions g as above. In the Euclidean setting, M˙ 1,p coincides with the usual homogeneous first order Sobolev space W˙ 1,p , [18], provided 1 < p < ∞. For p ∈ (n/(n + 1), 1], it was very recently proved [30] that this pointwise definition yields the corresponding Hardy–Sobolev space. Consequently, 1 (Rn ), for n/(n + 1) < p < ∞, where F˙ 1 (Rn ) refers to a homogeneous M˙ 1,p (Rn ) = F˙p,2 p,q Triebel–Lizorkin space (see Theorem 5.2.3/1 in [34] and [30]). In the fractional order, s ∈ (0, 1), s (Rn ), provided 1 < p < ∞. Notice the jump in case it was shown in [39] that M˙ s,p (Rn ) = F˙p,∞ the index q when s crosses 1 and that the result in the fractional order case does not allow for values of p below 1. We will next introduce a class of grand Triebel–Lizorkin spaces that allow us to characterize conveniently the fractional Hajłasz–Sobolev spaces for n/(n + s) < p < ∞. The definition is based on grand Littlewood–Paley functions and we later extend it to the metric space setting, establishing an analogous characterization. Let Z+ ≡ N ∪ {0}. Let S(Rn ) be the Schwartz space, namely, the space of rapidly decreasing functions endowed with a family of seminorms { · Sk,m (Rn ) }k,m∈Z+ , where for any k ∈ Z+ and m ∈ (0, ∞), we set ϕSk,m (Rn ) ≡

sup

m sup 1 + |x| ∂ α ϕ(x).

α∈Zn+ , |α|k x∈Rn

Here we recall that for any α ≡ (α1 , . . . , αn ) ∈ Zn+ , |α| = α1 + · · · + αn and ∂ α ≡ ( ∂x∂ 1 )α1 · · · ( ∂x∂ n )αn . It is known that S(Rn ) forms a locally convex topology vector space. Denote by S (Rn ) the dual space of S(Rn ) endowed with the weak∗ -topology. Moreover, for each N ∈ Z+ , denote by SN (Rn ) the space of all functions f ∈ S(Rn ) satisfying that Rn x α f (x) dx = 0 for all α ∈ Zn+ with |α| N . For the convenience, we also write S−1 (Rn ) ≡ S(Rn ). For any ϕ ∈ S(Rn ), t > 0 and x ∈ Rn , set ϕt (x) ≡ t −n ϕ(t −1 x). For each N ∈ Z+ ∪ {−1}, m ∈ (0, ∞) and ∈ Z+ , our class of test functions is AN,m ≡ φ ∈ SN Rn : φSN++1,m (Rn ) 1 .

(1.2)


2639

Definition 1.2. Let s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. Let A be a class of test functions as s (Rn ) is defined as the collection in (1.2). The homogeneous grand Triebel–Lizorkin space AF˙p,q of all f ∈ S (Rn ) such that when q ∈ (0, ∞),

1/q

ksq q

f AF˙ s (Rn ) ≡

2 sup |φ2−k ∗ f |

p,q k∈Z

φ∈A

< ∞,

Lp (Rn )

and when q = ∞,

ks sup ≡ 2 sup |φ ∗ f | f AF˙p,∞ s −k

n 2 (R ) k∈Z

φ∈A

Lp (Rn )

< ∞.

s (Rn ) as A n ˙s For A ≡ AN,m , we also write AF˙p,q N,m Fp,q (R ). Moreover, if N ∈ Z+ and f AF˙ s (Rn ) = 0, then it is easy to see that f ∈ PN , where PN is the space of polynomials p,q s (Rn )/P is a quasi-Banach space. As with degree no more than N . So the quotient space AF˙p,q N s n s (Rn ), is simply referred to usual, an element [f ] = f + PN ∈ AF˙p,q (R )/PN with f ∈ AF˙p,q s (Rn )/P as AF˙ s (Rn ). by f . By abuse of the notation, we always write the space AF˙p,q N p,q The grand Triebel–Lizorkin spaces are closely connected with Hajłasz–Sobolev spaces and (consequently) with the classical Triebel–Lizorkin spaces.

Theorem 1.1. Let s ∈ (0, 1] and p ∈ (n/(n + s), ∞). If A = A0,m with ∈ Z+ and m ∈ s (Rn ) with equivalent norms. (n + 1, ∞), then M˙ s,p (Rn ) = AF˙p,∞ To prove Theorem 1.1, for any f ∈ Lp (Rn ), we introduce a special g ∈ D(f ) via a variant of the grand maximal function; see (2.1) below. When s = 1, comparing this with the proof of Theorem 1 of [30], we see that the gradient on f appearing there is transferred to the vanishing moments of the test functions and the size conditions of the test functions and their first-order derivatives (see A) here. We point out that the choice of the set A is very subtle. This is the key point which allows us to extend Theorem 1.1 to certain metric measure spaces. Moreover, to prove Theorem 1.1, an imbedding theorem established by Hajłasz [19] is also employed. Theorem 1.1 also has a higher-order version. Definition 1.3. Let p ∈ (0, ∞) and s ∈ (k, k + 1] with k ∈ N. The homogeneous Hajłasz– p Sobolev space M˙ s,p (Rn ) is defined to be the set of all measurable functions f ∈ Lloc (Rn ) such that for all α ∈ Zn+ with |α| = k, ∂ α f ∈ M˙ s−k,p (Rn ), and normed by f M˙ s,p (Rn ) ≡ α |α|=k ∂ f M˙ s−k,p (Rn ) . Corollary 1.1. Let N ∈ Z+ , s ∈ (N, N + 1] and p ∈ (n/(n + N − s), ∞). If A = AN,m with ∈ Z+ and m ∈ (n + N + 2, ∞) when s = N + 1 or m ∈ (n + N + 1, ∞) when s ∈ (N, N + 1), s (Rn ) with equivalent norms. then M˙ s,p (Rn ) = AF˙p,∞ The essential point in the proof of Corollary 1.1 is to establish a lifting property for s (Rn ) via Theorem 1.1. This is done with the aid of auxiliary lemmas (see Lemmas 2.4 AF˙p,∞ and 2.5 below), where in Lemma 2.4, we decompose a test function in SN (Rn ) into a sum of test

2640


functions in Sk (Rn ) with subtle controls on their semi-norms for all −1 k N − 1. The decomposition of a test function in S0 (Rn ) into functions in S(Rn ) already plays a key role in [30]. The proof of Corollary 1.1 also uses Theorem 1.2 below. Now we recall the definition of Triebel–Lizorkin spaces on Rn . Definition 1.4. Let s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. Let ϕ ∈ S(Rn ) satisfy that ϕ (ξ ) constant > 0 if 3/5 |ξ | 5/3. supp ϕ ⊂ ξ ∈ Rn : 1/2 |ξ | 2 and

(1.3)

s (Rn ) is defined as the collection of all f ∈ S (Rn ) The homogeneous Triebel–Lizorkin space F˙p,q such that

1/q

ksq q

2 |ϕ ∗ f | max{s, J − n − s}

and m > max{J, n + N + 1},

(1.4)

s (Rn ) = F˙ s (Rn ) with equivalent norms. then AF˙p,q p,q

To prove Theorem 1.2, we use the Calderón reproducing formula in [31,12] and the boundedness of almost diagonal operators on the sequence spaces corresponding to the Triebel–Lizorkin spaces. The almost diagonal operators were introduced by Frazier and Jawerth [13] and proved to be a very powerful tool therein (see also [8]). It is perhaps worthwhile to point out that the proof of Theorem 1.1 does not rely on Theorem 1.2. 1 (Rn ) when p ∈ (n/(n + 1), ∞) by [30], AF˙ 0 (Rn ) = Lp (Rn ) Recall that M˙ 1,p (Rn ) = F˙p,2 p,∞ 0 (Rn ) = H p (Rn ) when p ∈ (n/(n + 1), 1], where A = A when p ∈ (1, ∞) and AF˙p,∞ −1,m with 1 and m ∈ (n+1, ∞), and H p (Rn ) is the classical real Hardy space (see [33,16]). Combining these facts with Theorem 1.1, we have the following result. Corollary 1.2. (i) If p ∈ (n/(n + 1), ∞) and A = A0,m with ∈ Z+ and m ∈ (n + 1, ∞), then 1 (Rn ) = M ˙ 1,p (Rn ) = F˙ 1 (Rn ) with equivalent norms. AF˙p,∞ p,2 (ii) If s ∈ (0, 1), p ∈ (n/(n + s), ∞) and A = A0,m with ∈ Z+ and m ∈ (n + 1, ∞), then s (Rn ) = M s (Rn ) with equivalent norms. ˙ s,p (Rn ) = F˙p,∞ AF˙p,∞ 0 (Rn ) = (iii) Let A ≡ A−1,m with 1 and m ∈ (n + 1, ∞). If p ∈ (n/(n + 1), 1], then AF˙p,∞ 0 p n n 0 n H (R ) = F˙p,2 (R ) with equivalent norms; if p ∈ (1, ∞), then AF˙p,∞ (R ) = Lp (Rn ) = 0 (Rn ) with equivalent norms. F˙p,2


2641

Moreover, for N ∈ Z+ , s ∈ (N, N + 1] and p ∈ (n/(n + N − s), ∞), by Corollary 1.1, Theorem 1.2 and the lifting property of homogeneous Triebel–Lizorkin spaces, we have that s (Rn ) with equivalent norms when s ∈ (N, N + 1), and M ˙ N +1,p (Rn ) = M˙ s,p (Rn ) = F˙p,∞ N +1 n F˙p,2 (R ) with equivalent norms. Remark 1.1. (i) In a sense, Corollary 1.2(i) gives a grand maximal characterizations of Hardy– 1 (Rn ) with p ∈ (n/(n + 1), 1], where H ˙ 1,p (Rn ) is defined as Sobolev spaces H˙ 1,p (Rn ) = F˙p,2 n p n the space of all f ∈ S (R ) such that ∇f ∈ H (R ). We point out the advantage of this grand maximal characterization is that it only depends on the first-order derivatives of test functions, which can be replaced by Lipschitz regularity (see Definition 1.5). In fact, our approach transfers the derivatives on f to vanishing moments, size conditions and Lipschitz regularity of test functions. This is a key observation, which allows us to extend this characterization to certain metric measure spaces without any differential structure; see Theorems 1.3 and 1.4 below. (ii) We point out that Auchser, Russ and Tchamitchian [3] characterized the Hardy–Sobolev 1 (Rn ) via a maximal function which is obtained by transferring the gradient on f to a space F˙p,2 size condition on the divergence of the vectors formed by certain test functions; see Theorem 6 of [3]. However, this characterization still depends on the derivatives. (iii) We also point out that Cho [11] characterized Hardy–Sobolev spaces H˙ k,p (Rn ) = k F˙p,2 (Rn ) with k ∈ N via a nontangential maximal function by transferring the derivatives on the distribution to a fixed specially chosen Schwartz function; see Theorem I of [11]. a continuous version of the grand Littlewood–Paley function (iv) We finally remark2 that 1/2 with a different choice of A was used by Wilson [38] to solve ( k∈Z supφ∈A |φ2−k ∗ f | ) a conjecture of R. Fefferman and E.M. Stein on the weighted boundedness of the classical Littlewood–Paley S-function. Finally, we discuss the metric space setting. Let (X , d, μ) be a metric measure space. For any x ∈ X and r > 0, let B(x, r) ≡ {y ∈ X : d(x, y) < r}. Recall that (X , d, μ) is called an RD-space if there exist constants 0 < C1 1 C2 and 0 < κ n such that for all x ∈ X , 0 < r < 2 diam(X ) and 1 λ < 2 diam(X )/r, C1 λκ μ B(x, r) μ B(x, λr) C2 λn μ B(x, r) ,

(1.5)

where and in what follows, diam X ≡ supx,y∈X d(x, y); see [21]. We point out that (1.5) implies the doubling property, there exists a constant C0 ∈ [1, ∞) such that for all x ∈ X and r > 0, μ(B(x, 2r)) C0 μ(B(x, r)), and the reverse doubling property: there exists a constant a ∈ (1, ∞) such that for all x ∈ X and 0 < r < diam X /a, μ(B(x, ar)) 2μ(B(x, r)). For more equivalent characterizations of RD-spaces and the fact that each connected doubling space is an RD-space, see [40]. In what follows, we always assume that (X , d, μ) is an RD-space. We also assume that ˚ 2), μ(X ) = ∞ in this section and in Section 4. In the remaining part of this section, let G(1,

−k ˚ ˚ G(x, 2 , 1, 2), (G(1, 2)) and (G0 (β, γ )) be as in Section 4.

2642


Definition 1.5. Let s ∈ (0, 1], p ∈ (0, ∞) and q ∈ (0, ∞]. Let A := {Ak (x)}x∈X , k∈Z and ˚ 2), φ ˚ −k Ak (x) = {φ ∈ G(1, G (x,2 ,1,2) 1} for all x ∈ X . The homogeneous grand Triebel– s (X ) is defined to be the set of all f ∈ (G(1, 2)) that satisfy Lizorkin space AF˙p,q

∞ 1/q

q

f AF˙ s (X ) ≡

2ksq sup f, φ

p,q

φ∈Ak (·)

n + 1 n + s and φ

2−k

Rn

φ(x) dx = 0, it follows that for all k ∈ Z and x ∈ X ,

∗ f (x) = φ2−k (x − y) f (y) − X

∞

f (z) dz dy

– B(x,2−k )

2

–

−(m−n)i

i=0

f (y) −

B(x,2−k+i )

f (z) dz dy

– B(x,2−k )

(n+s)/n 2−ks M g n/(n+s) (x) ,

(2.3)

which together with the Lp(n+s)/n -boundedness of M implies that if p ∈ (n/(n + s), ∞), then

(n+s)/n

p sup 2ks φ2−k ∗ (f ) p M g n/(n+s) gLp (X ) .

sup L (X ) L (X )

k∈Z φ∈A (Rn ) 0,m

Moreover, without loss of generality, we may assume that M(g n/(n+s) )(0) < ∞. Then for any ψ ∈ S(Rn ), by an argument similar to that of (2.3), we have that f (x)ψ(x) dx ψ 1 L (X ) X

– f (z) dz + B(0,1)

ψL1 (X )

f (x) −

Rn

– f (z) dzψ(x) dx B(0,1)

(n+s)/n – f (z) dz + ψS0,m (Rn ) M g n/(n+s) (0)

B(0,1)

C(f )ψS0,m (Rn ) , s (Rn ) and f which implies that f ∈ S (Rn ). Thus, f ∈ AF˙p,∞ s AF˙p,∞ (Rn ) f M˙ s,p (Rn ) , which completes the proof of Theorem 1.1. 2

To prove Theorem 1.2, we need the following estimate. Lemma 2.3. Let N ∈ Z+ ∪ {−1} and m ∈ (n + N + 1, ∞). Then there exists a positive constant C such that for all x ∈ Rn and i, j ∈ Z with i j , φ ∈ SN (Rn ) and ψ ∈ S(Rn ), φ

2−i

−m ∗ ψ2−j (x) CφSN+1,m (Rn ) ψS0,m (Rn ) 2−(i−j )(N +1) 2j n 1 + 2j |x| .

Proof. Without loss of generality, we may assume that φSN+1,m (Rn ) = ψSN+1,m (Rn ) = 1. For simplicity, we only consider the case N 0. If j = 0, then by φ ∈ SN (Rn ) and the Taylor formula, we have


2647

1 α α y ∂ ψ(x) dy ∗ ψ(x) = φ2−i (y) ψ(x − y) − α!

φ

2−i

Rn

|α|N

|y|N +1 ∂ α ψ(x − θy) dy

φ

2−i (y)

|α|=N +1

|y|(1+|x|)/2

+

φ

2−i (y)ψ(x

− y) dy

|y|>(1+|x|)/2

+

|y||α| ∂ α ψ(x) dy ≡ I1 + I2 + I3 ,

φ

2−i (y)

|α|N

|y|>(1+|x|)/2

where θ ∈ [0, 1]. If |y| (1 + |x|)/2, then for any θ ∈ [0, 1], we have that 1 + |x| 1 + |x − θy| + |y| and hence, 1 + |x| 2(1 + |x − θy|). By this and m ∈ (n + N + 1, ∞), we obtain I1 |y|(1+|x|)/2

2−i(N +1)

2in |y|N +1 1 dy (1 + 2i |y|)m (1 + |x − θy|)m

1 (1 + |x|)m

−m 2in |2i y|N +1 dy 2−i(N +1) 1 + |x| . i m (1 + 2 |y|)

Rn

For I2 and I3 , we also have I2 |y|>(1+|x|)/2

−m 2in 1 dy 2−i(N +1) 1 + |x| (1 + 2i |y|)m (1 + |x − y|)m

and I3 |y|>(1+|x|)/2

−m 2in 1 |y|N dy 2−i(N +1) 1 + |x| . i m m (1 + 2 |y|) (1 + |x|)

For j = 0, we obtain φ

2−i

−m ∗ ψ(x) = 2j n φ2−(i−j ) ∗ ψ 2j x 2−|i−j |(N +1) 2j n 1 + 2j |x| ,

which completes the proof of Lemma 2.3.

2

Now we turn to the proof of Theorem 1.2. Proof of Theorem 1.2. Let A = AN,m with ∈ Z+ , N and m satisfying (1.4). Obviously, s (Rn ) is continuously imbedded into F˙ s (Rn ). We now prove that if f ∈ F˙ s (Rn ), then AF˙p,q p,q p,q s (Rn ) and f f ∈ AF˙p,q n f ˙ s n . This proof is similar to the proof that the defi˙s AFp,q (R )

Fp,q (R )

s (Rn ) is independent of the choice of ϕ satisfying (1.3), but a bit more complicated. nition of F˙p,q

2648


In fact, we need to use the boundedness of almost diagonal operators in sequences spaces. For reader’s convenience, we sketch the argument. there exists a function ψ ∈ S(Rn ) satisfying the same conditions as ϕ such that Recall that −k (2−k ξ ) = 1 for all ξ ∈ Rn \ {0}; see [14, Lemma 6.9]. Then the Calderón reproϕ (2 ξ )ψ k∈Z ducing formula says that for all f ∈ S (Rn ), there exist polynomials Pf and {Pi }i∈N depending on f such that for all x ∈ Rn , ∞ ϕ2−j ∗ ψ2−j ∗ f (x) + Pi (x) , (2.4) f (x) + Pf (x) = lim i→−∞

j =i

s (Rn ), it is where the series converges in S (Rn ); see, for example, [31,12]. When f ∈ F˙p,q known that the degrees of the polynomials {Pi }i∈N here are no more than s − n/p; see [13, pp. 153–155], and also [31, p. 53] and [34, pp. 17–18]. Recall that α for α ∈ R denotes the maximal integer no more than α. Moreover, as shown in [13, pp. 153–155], f + Pf is the canonical representative of f in the sense that if ϕ (i) , ψ (i) satisfy (1.3) and (1) (2) n (i) −k (i) −k k∈Z ϕ (2 ξ )ψ (2 ξ ) = 1 for all ξ ∈ R \ {0} for i = 1, 2, then Pf − Pf is a poly(i)

nomial of degree no more than s − n/p, where Pf is as in (2.4) corresponding to ϕ (i) , ψ (i) for i = 1, 2. So in this sense, we identify f with f≡ f + Pf . Let ϕ (x) = ϕ(−x) for all x ∈ Rn . Denote by Q the collection of the dyadic cubes on Rn . For any dyadic cube Q ≡ 2−j k + 2−j [0, 1]n ∈ Q with certain k ∈ Zn , we set xQ ≡ 2−j k, denote by (Q) ≡ 2−j the side length of Q and write ϕQ (x) ≡ 2j n/2 ϕ(2j x − k) = 2−j n/2 ϕ2−j (x − xQ ) for all x ∈ Rn . It is known that for all x ∈ Rn , f, ϕQ ψQ (x) (2.5) ϕ2−j ∗ ψ2−j ∗ f (x) = (Q)=2−j

in S (Rn ) and pointwise; see [12,14] and also [8, Lemma 2.8]. Notice also that N + 1 > s implies s (Rn ), φ ∈ S (Rn ) with N s − n/p, i ∈ Z and that N s − n/p. Then for all f ∈ F˙p,q N n x ∈ R , by (2.4) and (2.5), we have f∗ φ2−i (x) =

f, ϕQ ψQ ∗ φ2−i (x) =

Q∈Q

tQ ψQ ∗ φ2−i (x),

Q∈Q

ϕQ , and by [13, Theorem 2.2] or [14, Theorem 6.16], where tQ = f, f F˙ s

p,q

(Rn ) ∼ {tQ }Q∈Q f˙s

p,q

−s/n−1/2 q 1/q

|Q| ≡ |t |χ Q Q

(Rn )

Lp (Rn )

Q∈Q

Moreover, by Lemma 2.3, for all R ∈ Q with (R) = 2−i and x ∈ R, we have −|i−j |(N +1) (i∧j )n −j n/2 f∗ φ −i (x) 2 2 2 2 j ∈Z

(Q)=2−j

j ∈Z (Q)=2−j

2−|i−j |(n/2+N +1)

|tQ | (1 + 2i∧j |x − xQ |)m

|tQ | |R|−1/2 . (1 + 2i∧j |xR − xQ |)m

.

(2.6)


2649

For R, Q ∈ Q with (R) = 2−i and (Q) = 2−j , setting −m , aRQ = 2−|i−j |(n/2+N +1) 1 + 2i∧j |xR − xQ | by (1.4), we have −J −

n+ −n |xR − xQ | (R) s (R) 2 (Q) J + 2 1+ min , aRQ (Q) max{(R), (Q)} (Q) (R) s (Rn ), which is for certain > 0. Thus {aRQ }R,Q∈Q forms an almost diagonal operator on f˙p,q s n known to be bounded on f˙p,q (R ); see [13, Theorem 3.3] and also [14, Theorem 6.20]. Therefore, by (2.6), we have

q 1/q

−s/n−1/2

|R| aRQ tQ χR fAF˙ s (Rn )

p,q

R∈Q

fp,q (Rn )

which completes the proof of Theorem 1.2.

Lp (Rn )

Q∈Q

{tQ }Q∈Q ˙s

∼ f F˙ s

p,q (R

n)

,

2

s (Rn ), which heavily To prove Corollary 1.1, we need to establish a lifting property of AF˙p,q depends on the following result.

Lemma 2.4. Let N ∈ Z+ , ϕ ∈ SN (Rn ) and 1 k N + 1. Then there exist functions {ϕα }α∈Zn+ ,|α|=k ⊂ SN −k (Rn ) such that ϕ = |α|=k ∂ α ϕα ; moreover, for any ∈ Z+ , there exists a positive constant C, depending on N, k, and m, but not on ϕ and ϕα , such that ϕα SN++1,m−kn (Rn ) CϕSN++1,m (Rn ) . (2.7) |α|=k

Proof. We begin by proving Lemma 2.4 for k = 1. We point out that when N = 0, this proof is essentially given by [30, Lemma 6] and [1, Lemma 3.29] except for checking the estimate (2.7). Now assume N 0. We decompose ϕ by using the idea appearing in the proof of [30, Lemma 6] and then verify (2.7). x Let ϕ ∈ SN (Rn ). We apply induction on n. For n = 1, set ψ(x) ≡ −∞ ϕ(y) dy for all x ∈ R. d Then ϕ(x) = dx ψ(x) for x ∈ R. Moreover, for any 0 j N − 1, by integration by parts, we 1 j +1 dx = 0, which means ψ ∈ S have R ψ(x)x j dx = − j +1 N −1 (R). Moreover, for all R ϕ(x)x x ∈ R, since ϕ ∈ SN (R), ψ(x) ϕS

∞ N++1,m (R)

|x|

−(m−1) 1 dy ϕSN++1,m (R) 1 + |x| , (1 + |y|)m

and for all 1 j N + + 1, j j −1 d d −(m−1) . dx j ψ(x) = dx j −1 ϕ(x) ϕSN++1,m (R) 1 + |x| Thus Lemma 2.4 holds for n = 1.

2650


Suppose that Lemma 2.4 holds true for a fixed n 1. Let ϕ ∈ SN (Rn+1 ). Without loss of generality, we may assume that ϕSN++1,m (Rn+1 ) = 1. For any x ∈ Rn+1 , we write x = (x , xn+1 ) and define h(x ) ≡ R ϕ(x , u) du, where x = (x1 , . . . , xn ) ∈ Rn . Then h ∈ SN (Rn ). Moreover, for all x ∈ Rn and α ∈ Zn+ with |α | N + + 1, we have α ∂ h x

R

1 (|1 + |x | + |u|)m

du

1 , (|1 + |x |)m−1

which implies that hSN++1,m−1 (Rn ) 1. By induction hypothesis, we write h(x ) = n ∂ n i=1 ∂xi hi (x ) with hi ∈ SN −1 (R ) and hi SN++1,m−n−1 (Rn ) 1 for i = 1, . . . , n. Let a ∈ For all x ∈ Rn+1 , set ϕn+1 (x) ≡ be fixed and R a(u) du = 1. S(R) xn+1 −∞ [ϕ(x , u) − a(u)h(x )] du and ϕi (x) ≡ a(xn+1 )hi (x ) with i = 1, . . . , n. Then ϕi ∈ SN −1 (Rn+1 ) for i = 1, . . . , n. For any N − 1 and |α| with α = (α , αn+1 ) ∈ Zn+1 + , by integration by parts again, we have

xn+1 α αn+1 ϕn+1 (x)x dx = ϕ x , u x xn+1 du dxn+1 dx α

Rn R −∞

Rn+1

=−

1 αn+1 + 1

α αn+1 +1 ϕ(x) x xn+1 dx = 0.

Rn+1

For any α ∈ Zn+1 + , for i = 1, . . . , n, we have α ∂ ϕi (x)

−(m−n−1) hi SN++1,m−n−1 (Rn ) , m−n−1 1 + |x| m−n−1 1 + |x | (1 + |xn+1 |)

with |α| N + + 1, if which implies that ϕi SN++1,m−n−1 (Rn+1 ) 1. For any α ∈ Zn+1 + α αn+1 = 0, then by hSN++1,m−1 (Rn ) 1, we have that |∂ ϕn+1 (x)| (1 + |x|)−(m−n−1) for all x ∈ Rn ; if αn+1 = 0, then by R [ψ(x , u) − a(u)h(x )] du = 0, we have that for all x ∈ Rn , α ∂ ϕn+1 (x)

∞

|xn+1 |

1 1 du + m (1 + |x | + u) (1 + |x |)m−n

−(m−n−1) ϕSN++1,m (Rn+1 ) 1 + x .

∞ |xn+1 |

1 du (1 + |u|)m

Thus, ϕn+1 SN++1,m−n−1 (Rn+1 ) 1, which completes the proof of Lemma 2.4.

2

N +1 (Rn ) = Lemma 2.5. For any N , ∈ Z+ and m, m ∈ (n + N + 2, ∞), A0N,m F˙p,∞ N +1 (Rn ) with equivalent norms. AN,m F˙p,∞ N +1 (Rn ), then f ∈ A0 ˙ N +1 n Proof. It suffices to prove that if f ∈ AN,m F˙p,∞ N,m Fp,∞ (R ) and f A0 F˙ N+1 (Rn ) f A F˙ N+1 (Rn ) . Without loss of generality, we may assume that m m . N,m p,∞

N,m p,∞


2651

To this end, fix ψ ∈ S(Rn ) such that Rn ψ(x) dx = 1. Obviously, for any α ∈ Zn+ with |α| = N + 1, if β ∈ Zn+ , |β| N + 1 and β = α, then Rn ∂ α ψ(x)x β dx = 0; if α = β, then α α N +1 . For any φ ∈ A0 Rn ∂ ψ(x)x dx = (−1) N,m , let

φ = φ − (−1)N +1

φ(x)x α dx ∂ α ψ.

(2.8)

|α|=N +1 Rn

Then φ ∈ SN +1 (Rn ). Moreover, for |α| = N + 1, since φ ∈ A0N,m with m ∈ (n + N + 2, ∞), we have

φ(x)x α dx

Rn

Rn

|x|N +1 dx 1, (1 + |x|)m

which implies that φ is a fixed constant multiple of an element of A0N +1,m . Notice that {∂ α ψ}|α|=N +1 are also fixed constant multiples of elements of AN,m . Then, by (2.8), we have sup φ2−k ∗ f (x) sup φ2−k ∗ f (x) + φ∈AN,m

φ∈A0N,m

φ

sup φ∈A0N+1,m

2−k

∗ f (x),

which implies that f A0 F˙ N+1 (Rn ) f A F˙ N+1 (Rn ) + f A0 F˙ N+1 (Rn ) . By TheoN,m p,∞ N,m p,∞ N+1,m p,∞ rem 1.2 together with m ∈ (n + N + 2, ∞), we have that f A0 F˙ N+1 (Rn ) ∼ f F˙ N+1 (Rn ) N+1,m p,∞

f A

p,∞

, which yields that f A0 F˙ N+1 (Rn ) f A F˙ N+1 (Rn ) . This finishes the N,m p,∞ N,m p,∞ proof of Lemma 2.5. 2 ˙ N+1 n N,m Fp,∞ (R )

Proof of Corollary 1.1. First, let f ∈ M˙ s,p (Rn ). Then by Theorem 1.1, for any ∈ Z+ and m ∈ s−N (Rn ). ((n + 2)N + 1, ∞), and all α ∈ Zn+ with |α| = N , ∂ α f ∈ M˙ s−N,p (Rn ) = A0,m−nN F˙p,∞ Moreover, for any φ ∈ AN,m , by Lemma 2.4, there exist {φα }|α|=N and a positive constant C independent of φ such that { C1 φα }|α|=N ⊂ A0,m−nN and φ = |α|=N ∂ α φα . This implies that for all x ∈ Rn , ∂ α φα 2−k ∗ f (x) = 2kN (−1)N φ2−k ∗ f (x) = (φα )2−k ∗ ∂ α f (x), |α|=N

|α|=N

and thus, sup φ2−k ∗ f (x) 2kN sup

sup

|α|=N φ∈A 0,m−nN

φ∈AN,m

φ −k ∗ ∂ α f (x). 2

s−N (Rn ) for all α ∈ Zn with |α| = N together with TheoFrom this and ∂ α f ∈ A0,m−nN F˙p,∞ + s (Rn ) and rem 1.1, it follows that f ∈ AN,m F˙p,∞

f A

n ˙s N,m Fp,∞ (R )

∂ α f

|α|=N

s−N A0,m−nN F˙p,∞ (Rn )

∼

∂ α f

|α|=N

M˙ s−N,p (Rn )

∼ f M˙ s,p (Rn ) .

2652


s (Rn ). Let N and m ∈ (n + N + 1, ∞). Observe that On the other hand, let f ∈ AN,m F˙p,∞ −N n for any φ ∈ A0,m and α ∈ Z+ with |α| = N , ∂ α φ ∈ AN,m . Thus for all k ∈ Z, sup φ2−k ∗ ∂ α f sup 2kN φ2−k ∗ (f ), φ∈AN,m

φ∈A−N 0,m

−N ˙ s−N which implies that {∂ α f }|α|=N ⊂ A0,m Fp,∞ (Rn ) = M˙ s−N,p (Rn ) and thus, f ∈ M s,p (Rn ) and

∂ α f ˙ s−N,p n ∼

∂ α f −N ˙ s−N n f ˙ s f M˙ s,p (Rn ) ∼ Fp,∞ (Rn ) . A M F (R ) A (R ) |α|=N

0,m

|α|=N

p,∞

N,m

Finally, combining the above results with Lemma 2.5 and Theorem 1.2, for all ∈ Z+ and m ∈ (n + N + 2, ∞) when s = N + 1 or m ∈ (n + N + 1, ∞) when s ∈ (N, N + 1), we have that s (Rn ). This finishes the proof of Corollary 1.1. 2 M˙ s,p (Rn ) = A F˙p,∞ N,m

3. Inhomogeneous versions of Theorems 1.1 and 1.2 We first recall the definitions of inhomogeneous Triebel–Lizorkin spaces; see [34]. Definition 3.1. Let s ∈ R, p ∈ (0, ∞) and q ∈ (0, ∞]. Let ϕ ∈ S(Rn ) satisfy (1.3) and Φ ∈ S(Rn ) ⊂ B(0, 2) and |Φ(ξ )| constant > 0 for all |ξ | 5/3. The inhomogeneous be such that supp Φ s (Rn ) is defined as the collection of all f ∈ S (Rn ) such that Triebel–Lizorkin space Fp,q

∞ 1/q

ksq q s (Rn ) ≡ Φ ∗ f Lp (Rn ) +

f Fp,q 2 |ϕ2−k ∗ f |

k=1

0, set V (x, y) ≡ μ(B(x, d(x, y))) and Vr (x) ≡ μ(B(x, r)). It is easy to see that V (x, y) ∼ V (y, x) for all x, y ∈ X . Definition 4.1. Let x1 ∈ X , r ∈ (0, ∞), β ∈ (0, 1] and γ ∈ (0, ∞). A function ϕ on X is said to be in the space G(x1 , r, β, γ ) if there exists a nonnegative constant C such that γ 1 r (i) |ϕ(x)| C Vr (x1 )+V for all x ∈ X ; (x1 ,x) r+d(x1 ,x) d(x,y) β γ r 1 (ii) |ϕ(x) − ϕ(y)| C r+d(x1 ,x) Vr (x1 )+V for all x, y ∈ X satisfying that (x1 ,x) r+d(x1 ,x) d(x, y) (r + d(x1 , x))/2. Moreover, for any ϕ ∈ G(x1 , r, β, γ ), its norm is defined by ϕG (x1 ,r,β,γ ) ≡ inf{C: (i) and (ii) hold}. Throughout the whole paper, we fix x1 ∈ X and letG(β, γ ) ≡ G(x1 , 1, β, γ ). Then G(β, γ ) is ˚ γ ) = {f ∈ G(β, γ ): a Banach space. We also let G(β, X f (x) dμ(x) = 0}. Denote by (G(β, γ )) ˚ γ )) the dual spaces of G(β, γ ) and G(β, ˚ γ ), respectively. Obviously, (G(β, ˚ γ )) = and (G(β, (G(β, γ )) /C. For any given ∈ (0, 1], let G0 (β, γ ) be the completion of the set G( , ) in the space G(β, γ ) when β, γ ∈ (0, ]. Obviously, G0 ( , ) = G( , ). If ϕ ∈ G0 (β, γ ), define ϕG0 (β,γ ) ≡ ϕG (β,γ ) . Obviously, G0 (β, γ ) is a Banach space. The space G˚0 (β, γ ) is defined to be the com˚ ) in G(β, ˚ γ ) when β, γ ∈ (0, ]. Let (G (β, γ )) and (G (β, γ )) pletion of the space G( , 0 0 be the dual space of G0 (β, γ ) and G0 (β, γ ), respectively. Also we have that (G˚0 (β, γ )) = (G0 (β, γ )) /C. Remark 4.1. Because (G˚0 (β, γ )) = (G0 (β, γ )) /C, if we replace (G˚0 (β, γ )) with (G0 (β, γ )) /C or (G0 (β, γ )) in Definition 1.6, then we obtain a new Triebel–Lizorkin space which, modulo constants, is equivalent to the original Triebel–Lizorkin space. So we can replace (G˚0 (β, γ )) with (G0 (β, γ )) /C or (G0 (β, γ )) in the Definition 1.6 if need be, in what follows.


2655

Now we recall the notion of approximations of the identity on RD-spaces, which were first introduced in [21]. Definition 4.2. Let 1 ∈ (0, 1]. A sequence {Sk }k∈Z of bounded linear integral operators on L2 (X ) is called an approximation of the identity of order 1 (for short, 1 -AOTI) with bounded support, if there exist constants C3 , C4 > 0 such that for all k ∈ Z and all x, x , y and y ∈ X , Sk (x, y), the integral kernel of Sk is a measurable function from X × X into C satisfying 1 ; (i) Sk (x, y) = 0 if d(x, y) > C4 2−k and |Sk (x, y)| C3 V −k (x)+V −k (y) 2

2

1 for d(x, x ) max{C4 , 1}21−k ; (ii) |Sk (x, y) − Sk (x , y)| C3 2k 1 [d(x, x )] 1 V −k (x)+V −k (y) 2

2

(iii) Property (ii) holds with x and y interchanged; 1

1

)] [d(y,y )] (iv) |[Sk (x, y) − Sk (x, y )] − [Sk (x , y) − Sk (x , y )]| C3 22k 1 [d(x,x V −k (x)+V −k (y) for d(x, x ) 2

max{C4 , 1}21−k and d(y, y ) max{C4 , 1}21−k ; (v) X Sk (x, y) dμ(y) = 1 = X Sk (x, y) dμ(x).

2

It was proved in [21, Theorem 2.6] that there always exists a 1-AOTI with bounded support on an RD-space. To prove Theorem 1.3, we need a Sobolev embedding theorem, which for s = 1 is due to Hajłasz [19, Theorem 8.7], and for s ∈ (0, 1) can be proved by a slight modification of the proof of [19, Theorem 8.7]. We omit the details. Lemma 4.1. Let s ∈ (0, 1], p ∈ (0, n/s) and p ∗ = np/(n − sp). Then there exists a positive ∗ constant C such that for all u ∈ M˙ s,p (X ), g ∈ D(u) and all balls B0 with radius r0 , u ∈ Lp (B0 ) and

1/p∗

1/p p∗ s p inf – |u − c| dμ Cr0 . – g dμ

c∈R

B0

2B0

By Lemma 4.1, we have the following version of Lemma 2.2. Lemma 4.2. Let s ∈ (0, 1], p ∈ [n/(n + s), n/s) and p∗ = np/(n − sp). Then for each u ∈ M˙ s,p (X ), there exists constant C such that u − C ∈ Lp∗ (X ) and u − CLp∗ (X ) Cu M˙ s,p (X ) , is a positive constant independent of u and C. where C With the aid of Lemmas 4.1 and 4.2, we can prove Theorem 1.3 by following the ideas used in the proof of Theorem 1.1. For reader’s convenience, we sketch the argument. s (X ), then f ∈ M ˙ s,p (X ) and Proof of Theorem 1.3. We first prove that if f ∈ AF˙p,∞ f M˙ s,p (X ) f AF˙p,∞ s (X ) . Let {Sk }k∈Z be a 1-AOTI with bounded support. If f is a locally integrable function, using Sk (f ) to replace ϕ2−k ∗ f and following the procedure as in the proof of Theorem 1.1, we know that f ∈ M˙ s,p (X ) and

g(·) = sup

sup 2ks f, φ ∈ Lp (X )

k∈Z φ∈Ak (·)

2656


˙s and f M˙ s,p (X ) f AF˙p,∞ s (X ) . If f ∈ AFp,∞ (X ) is only known to be an element in (G(1, 2)) , we may also identify f with a locally integrable function f in (G(1, 2)) and fM˙ s,p (X ) f AF˙p,∞ s (X ) by using Lemma 4.2 and an argument used in that of Theorem 1.1. In this sense, we have that f ∈ M˙ s,p (X ) and f M˙ s,p (X ) f AM˙ s,p (X ) . Conversely, let f ∈ M˙ s,p (X ). Choose g ∈ D(f ) such that gLp (X ) 2f M˙ s,p (X ) . Then for all x ∈ X , k ∈ Z and φ ∈ Ak (x), similarly to the proof of (2.3) and using Lemma 4.1, we have that

f, φ 2−ks M g n/(n+s) (x) (n+s)/n , which together with the Lp(n+s)/n (X )-boundedness of M implies that f AF˙ s (X ) gLp (X ) p,q for all p ∈ (n/(n + s), ∞). Moreover, without loss of generality, we may assume that M(g n/(n+s) )(x1 ) < ∞. Then for all ψ ∈ G(1, 2), letting σ := X ψ(y) dμ(y), by Lemma 4.2 and an argument similar to the proof of (2.3), we have that f ∈ L1loc (X ) and f (x)ψ(x) dμ(x) = f (x) ψ(x) − σ S0 (x1 , x) dμ(x) + |σ |S0 (f )(x1 ) X

X

ψL1 (X )

–

f (z) dμ(z) + ψG (1,2) M g n/(n+s) 1+s/n (x1 )

B(0,2C4 )

C(f )ψG (1,2) , s (X ) and f which implies that f ∈ (G(1, 2)) . Thus f ∈ AF˙p,∞ s AF˙p,∞ (X ) f M˙ s,p (X ) , which completes the proof of Theorem 1.3. 2

Remark 4.2. By the above proof, if we replace the space G(1, 2) of test functions by G(β, γ ) with β ∈ [s, 1] and γ ∈ (s, ∞) in Definition 1.5, then Theorem 1.3 still holds true. Thus the s (X ) is independent of the choice of the space of test functions definition of the space AF˙p,∞ G(β, γ ) with β ∈ [s, 1] and γ ∈ (s, ∞). To prove Theorem 1.4, we need the following homogeneous Calderón reproducing formula established in [21]. We first recall the following construction given by Christ in [10], which provides an analogue of the set of Euclidean dyadic cubes on spaces of homogeneous type. Lemma 4.3. Let X be a space of homogeneous type. Then there exists a collection {Qkα ⊂ X : k ∈ Z, α ∈ Ik } of open subsets, where Ik is some index set, and constants δ ∈ (0, 1) and C5 , C6 > 0 such that (i) (ii) (iii) (iv) (v)

μ(X \ α Qkα ) = 0 for each fixed k and Qkα ∩ Qkβ = ∅ if α = β; for any α, β, k, with k, either Qβ ⊂ Qkα or Qβ ∩ Qkα = ∅; for each (k, α) and each < k, there exists a unique β such that Qkα ⊂ Qβ ; diam(Qkα ) C5 δ k ; each Qkα contains some ball B(zαk , C6 δ k ), where zαk ∈ X .


2657

In fact, we can think of Qkα as being a dyadic cube with diameter roughly δ k and centered at zαk . In what follows, to simplify our presentation, we always suppose that δ = 1/2; otherwise, we need to replace 2−k in the definition of approximations to the identity by δ k and some other changes are also necessary; see [21] for more details. In the following, for k ∈ Z and τ ∈ Ik , we denote by Qk,ν τ , ν = 1, 2, . . . , N (k, τ ), the set of k+j k k all cubes Qτ ⊂ Qτ , where Qτ is the dyadic cube as in Lemma 4.3 and j is a fixed positive large integer such that 2−j C5 < 1/3. Denote by zτk,ν the “center” of Qk,ν τ as in Lemma 4.3 and by yτk,ν a point in Qk,ν . τ Lemma 4.4. Let ∈ (0, 1) and {Sk }k∈Z be a 1-AOTI with bounded support. For k ∈ Z, set k }k∈Z of linear Dk := Sk − Sk−1 . Then, for any fixed j ∈ N large enough, there exists a family {D k,ν k,ν operators such that for any fixed yτ ∈ Qτ with k ∈ Z, τ ∈ Ik and ν = 1, . . . , N (k, τ ), x ∈ X , and all f ∈ (G˚0 (β, γ )) with β, γ ∈ (0, ), f (x) =

(k,τ ) ∞ N k x, yτk,ν Dk (f ) yτk,ν , D μ Qk,ν τ k=−∞ τ ∈Ik

ν=1

where the series converge in (G˚0 (β, γ )) . Moreover, for any ∈ ( , 1), there exists a positive k (x, y), of the operators D k constant C, depending on , such that the kernels, denoted by D satisfy 1 k (x, y)| C (i) for all x, y ∈ X , |D V −k (x)+V (x,y) 2

2−k , 2−k +d(x,y)

(ii) for all x, x , y ∈ X with d(x, x ) (2−k + d(x, y))/2, D k x , y C k (x, y) − D

d(x, x ) 2−k + d(x, y)

2−k 1 , V2−k (x) + V (x, y) 2−k + d(x, y)

k (x, y) dμ(y) = 0 = (iii) for all k ∈ Z, X D X Dk (x, y) dμ(x). s (X ), by Remark 4.2 and the fact that AF˙ s (X ) ⊂ Proof of Theorem 1.4. If f ∈ AF˙p,q p,q s (X ), we know that f ∈ (G(β, γ )) with β ∈ (s, 1) and γ ∈ (s, ∞) and thus, f ∈ F˙ s (X ) AF˙p,∞ p,q s (X ). By Lemma 4.4, for all and f F˙ s (X ) f AF˙ s (X ) . Conversely, assume that f ∈ F˙p,q p,q p,q x ∈ X , ∈ Z and φ ∈ A (x), we have

f, φ =

(k,τ ) ∞ N k,ν k z, yτk,ν φ(z) dμ(z), D μ Qk,ν (f ) y D k τ τ k=−∞ τ ∈Ik

ν=1

X

where we fix yτk,ν ∈ Qk,ν τ such that Dk (f ) y k,ν 2 inf Dk (f )(z). τ z∈Qk,ν τ

(4.1)

2658


k depends on the choice of yτk,ν and thus on f , but they do have uniform estimates Recall that D as in Lemma 4.4, which is enough for us. In fact, by these estimates and φ ∈ G˚0 (β, γ ), we further know that for any fixed β ∈ (s, β) and γ ∈ (s, γ ) satisfying (1.6), D k z, yτk,ν φ(z) dμ(z) 2−|k−|β X

1

γ

2−(k∧)

V2−(k∧) (x) + V (x, yτk,ν ) 2−(k∧) + d(x, yτk,ν )

;

see [21] for a detailed proof. Thus, choosing an r ∈ (n/(n + [β ∧ γ ]), min{p, q}), by (4.1), we have γ N (k,τ ) ∞ k,ν 2−(k∧) μ(Qk,ν τ )|Dk (f )(yτ )| f, φ 2−|k−|β k,ν −(k∧) + d(x, y k,ν ) τ k=−∞ τ ∈Ik ν=1 V2−(k∧) (x) + V (x, yτ ) 2

∞

2

−|k−|β

2[(k∧)−k]n(1−1/r) M

∞

τ

τ ∈Ik

k=−∞

(k,τ ) N Dk (f ) y k,ν r χ

Qk,ν τ

1/r

(x)

ν=1

r 1/r 2−|k−|β 2[(k∧)−k]n(1−1/r) M Dk (f ) (x) .

k=−∞

This implies that

∞ ∞

(−k)sq f AF˙ s (X )

2 2−|k−|β 2[(k∧)−k]n(1−1/r) p,q

k=−∞

=−∞

r 1/r × M 2ksr Dk (f )

q 1/q Lp (X ) .

Applying the Hölder inequality when q > 1 and the inequality that ( k |ak |)q k |ak |q when q ∈ (0, 1] for all {ak }k∈Z ⊂ C, and using the vector-valued inequality of the Hardy-Littlewood maximal operator (see [17]), we then have

∞ 1/q

r q/r

f AF˙ s (X )

M 2ksr Dk (f )

p,q

Lp (X )

k=−∞

This finishes the proof of Theorem 1.4.

f F˙ s (X ) . p,q

2

5. Inhomogeneous versions of Theorems 1.3 and 1.4 We consider both cases μ(X ) < ∞ and μ(X ) = ∞ at the same time. We next recall the notions of inhomogeneous Besov and Triebel–Lizorkin spaces from [21]. We call {Sk }k∈N to be an inhomogeneous approximation of the identity of order with bounded support if their kernels satisfy (i) through (v) of Definition 4.1.


2659

Definition 5.1. Let , s, p, q, β, γ be as in Definition 1.6. Let {Sk }k∈N be an inhomogeneous approximation of the identity of order with bounded support. For k ∈ N, set Dk ≡ Sk − Sk−1 . Let {Q0,ν τ : τ ∈ I0 , ν = 1, . . . , N (0, τ )} with a fixed large j ∈ N be dyadic cubes as in Section 4. s (X ) is defined to be the set of all Let s ∈ (0, ). The inhomogeneous Triebel–Lizorkin space Fp,q

f ∈ (G0 (β, γ )) that satisfy s (X ) ≡ f Fp,q

(0,τ ) N p mQ0,ν S0 (f ) μ Q0,ν τ

1/p

τ

τ ∈I0

ν=1

∞ 1/q

q

ksq Dk (f ) +

2 Lp (X ) < ∞

k=1

with the usual modification made when q = ∞. s (X ) is independent of the choices of , β, γ and the As shown in [40], the definition of Fp,q inhomogeneous approximation of the identity.

Definition 5.2. Let s ∈ (0, 1], p ∈ (0, ∞) and q ∈ (0, ∞]. Let A :≡ {Ak (x)}k∈Z+ , x∈X with A0 (x) = {φ ∈ G(1, 2), φG (x,1,1,2) 1} and for k ∈ N, Ak (x) := φ ∈ G(1, 2), φG˚ (x,2−k ,1,2) 1 . s (X ) is defined to be the set of all f ∈ The inhomogeneous grand Triebel–Lizorkin space AF˙p,q (G(1, 2)) that satisfy

∞ 1/q

q

s (X ) ≡

f AFp,q 2ksq sup f, φ

φ∈Ak (·) k=0

0,

is zero. Proof. The following inclusions are immediate from the definition of operator H : 1 1 D(H ) ⊂ H 1,2 Rd ∩ D V+2 ∩ D V−2 , D(H ) ⊂ D(Hmax ),

(5)

2666

D. Kinzebulatov, L. Shartser / Journal of Functional Analysis 258 (2010) 2662–2681

where D(Hmax ) := f ∈ H: f ∈ D Rd ∩ L1loc Rd , Vf ∈ L1loc Rd , −f + Vf ∈ H . Therefore, D(H ) ⊂ YVweak and if u ∈ D(H ) is a solution to (5), then |u| = (V − λ)u

a.e. in Rd .

By Kato’s theorem [8] u has compact support. Now Theorem 3 follows from Theorem 1.

2

3. Historical context 1) D. Jerison and C. Keing [6] and E.M. Stein [21] proved the validity of the SUC property d

d

,∞

2 2 (Ω) and Lloc (Ω) (weak type d/2 Lorentz space), respectively. for potentials from classes Lloc Below · p,∞ denotes weak type p Lorentz norm. One has

d

2 Lloc (Ω)

d Fβ,loc ,

(6)

d Fβ,loc .

(7)

β>0 d

,∞

2 Lloc (Ω)

β>0

The first inclusion follows straightforwardly from the Sobolev embedding theorem. For the following proof of the second inclusion let us note first that 1B(x

0 ,ρ)

|W |

d−1 4

(−)−

d−1 2

|W |

d−1 4

d−1 d−1 2 1B(x0 ,ρ) 2 →2 = 1B(x0 ,ρ) |V | 4 (−)− 4 2 →2 .

Next, if V ∈ Ld/2,∞ , then 1B(x

0 ,ρ)

|V |

d−1 4

d−1 (−)− 4 2 →2

2d −1 π d2 c 1 2

( d2 )c d 2

d−1

1B(x0 ,ρ) V d 4 ,

(8)

2 ,∞

which is a special case of Strichartz inequality with sharp constants, proved in [13]. Inclusion (7) follows. To see that the latter inclusion is strict we introduce a family of potentials V (x) :=

C(1B(1+δ) (x) − 1B(1−δ) (x)) (|x| − 1)

2 d−1

(− ln ||x| − 1|)b

,

where b >

2 , 0 < δ < 1. d −1 d−1

d−1

d A straightforward computation shows that V ∈ Fβ,loc , as well as V ∈ Lloc2 (Ω) \ Lloc2 d 2 ,∞

any ε > 0, so that V ∈ / Lloc (Ω).

(9)

+ε

(Ω) for

D. Kinzebulatov, L. Shartser / Journal of Functional Analysis 258 (2010) 2662–2681 d

2667

,∞

2 The result in [21] can be formulated as follows. Suppose that d 3 and V ∈ Lloc (Ω). There exists a sufficiently small constant β such that if

sup lim 1B(x0 ,ρ) V d ,∞ β,

x0 ∈Ω ρ→0

2

2,p¯

2d then (1) has the SUC property in YV := Hloc (Ω), where p¯ := d+2 . (It is known that the assumption of β being sufficiently small cannot be omitted, see [12].) In view of (6), (7), the results in [21] and in [6] follow from Theorem 2 provided that we 1 show |V | 2 u ∈ X2 . Indeed, let Lq,p be the (q, p) Lorentz space (see [22]). By Sobolev em2,p¯ q, ¯ p¯ 2d [22]. Hence, by bedding theorem for Lorentz spaces Hloc (Ω) → Lloc (Ω) with q¯ := d−2 q, ¯ p¯

1

d/2,∞

Hölder inequality in Lorentz spaces |V | 2 u ∈ X2 whenever u ∈ Lloc (Ω) and V ∈ Lloc . Also, 2,p¯ 1,p¯ 2,p¯ Hloc (Ω) → Hloc (Ω), so Hloc (Ω) ⊂ YVstr , as required. 2) E.T. Sawyer [18] proved uniqueness of continuation for the case d = 3 and potential V from the local Kato-class

Kβ,loc := W ∈ L1loc (Ω): sup lim sup (−)−1 1BK (x0 ,ρ) |W | ∞ β , K ρ→0 x0 ∈K

where K is a compact subset of Ω. It is easy to see that Kβ,loc Fβ,loc . To see that the latter inclusion is strict consider, for instance, potential

Vβ (x) := βv0 ,

v0 :=

d −2 2

2

|x|−2 .

By Hardy’s inequality, Vβ ∈ Fβ,loc . At the same time, (−)−1 v0 1B(ρ) ∞ = ∞ for all ρ > 0, hence Vβ ∈ / Kβ,loc for all β = 0. The next statement is essentially due to E.T. Sawyer [18]. Theorem 4. Let d = 3. There exists a constant β < 1 such that if V ∈ Kβ,loc then (1) has the WUC property in YVK := {f ∈ X1 : f ∈ X1 , Vf ∈ X1 }. The proof of Theorem 4 is provided in Section 5. Despite the embedding Kβ,loc → Fβ,loc , Theorem 1 does not imply Theorem 4. The reason is simple: YVK ⊂ YVweak . 3) S. Chanillo and E.T. Sawyer showed in [3] the validity of the SUC property for (1) in YV = 2,2 Hloc (Ω) (d 3) for potentials V locally small in Campanato–Morrey class M p (p > d−1 2 ), M p := W ∈ Lp : W M p :=

sup x∈Ω, r>0

r

2− pd

1B(x,r) W p < ∞ .

2668


Note that for p >

d−1 2 p

Mloc

d Fβ,loc

β>0

(see [3,4,10]). To see that the above inclusion is strict one may consider, for instance, potential defined in (9). p 2,2 It is easy to see, using Hölder inequality, that if u ∈ Hloc (Ω) and V ∈ Mloc (p > d−1 2 ), then 1

2,2 |V | 2 u ∈ X2 , i.e., u ∈ YVweak . However, the assumption ‘u ∈ Hloc ’ is in general too restrictive for application of this result to the problem of absence of positive eigenvalues (see Remark 1).

Remark 1. Below we make several comments about H 2,q -properties of the eigenfunctions of the self-adjoint Schrödinger operator H = (− V+ ) (−V− ), V = V+ − V− , defined by (2) in the assumption that condition V− β(H0 V+ ) + cβ ,

β < 1, cβ < ∞,

(10)

is satisfied. (Note that (10) implies condition (3). We say that (10) is satisfied with β = 0 if (10) holds for any β > 0 arbitrarily close to 0, for an appropriate cβ < ∞.) Let u ∈ D(H ) and H u = μu. Then e−tH u = e−tμ u, As is shown in [14], for every 2 r < that

2d √1 d−2 1− 1−β

t > 0. there exists a constant c = c(r, β) > 0 such

−tH d 1 1 e f r ct − 2 ( 2 − r ) f 2 ,

(11) p

where f ∈ L2 = L2 (Rd ). Let us now consider several possible Lp and Lp,∞ (as well as Lloc and p,∞ Lloc ) conditions on potential V . p (A) Suppose in addition to (10) that V ∈ Lloc for some 1 p < d2 . Then by Hölder inequality q q and (11) V u ∈ Lloc and, due to inclusion D(H ) ⊂ D(Hmax ), u ∈ Lloc for any q such that √ 1 1 d −21− 1−β > + . q p d 2 The latter implies that q < p¯ in general, i.e., when β in (10) is close to 1. Hence, in general 2,p¯ 2,2 ’) is too restrictive for applications to the the assumption ‘u ∈ Hloc ’ (and, moreover, ‘u ∈ Hloc problem of absence of positive eigenvalues even under additional hypothesis of the type V ∈ p p d d−1 d Lcom , d−1 2 p < 2 or V ∈ Mcom , 2 p < 2 (cf. [3,17]). d p ∞ (B1) If V = V1 + V2 ∈ L + L , p > 2 , then (10) holds with β = 0 and u ∈ L∞ , therefore, 1

|V | 2 u ∈ L2 (cf. D(H ) and YVweak ). 2,p It follows that u ∈ C 0,α for any α ∈ (0, 1 − d2 ]. Therefore, u ∈ Hloc and, in particular, for d 4, u ∈ H 2,2 .


2669 2,p

p

(B2) Assume in addition to (10) that V ∈ Lloc , p > d2 , and β = 0. Then u ∈ Hloc , p > d2 . 1

Using Hölder inequality, one immediately obtains |V | 2 u ∈ X2 . 2,p¯ 2,2 2d / Hloc , but of course u ∈ Hloc , p¯ = d+2 (< p, cf. If d = 3, and p > d2 is close to d2 , then u ∈ remark in [1, p. 166]). d (B3) If V = V1 + V2 ∈ L 2 + L∞ , then (10) is satisfied with β = 0 and u ∈ 2r

cβ , β

one immediately obtains that (λ + Hˆ p¯ )−1 : Lp¯ → L2 , i.e., any eigenfunction of operator Hˆ p¯ belongs to L2 . Furthermore, an analogous representation for (λ + H )−1 yields the identity (λ + H )−1 f = (λ + Hˆ p¯ )−1 f,

f ∈ L2 ∩ Lp¯ .

Therefore, any eigenfunction of Hˆ p¯ is an eigenfunction of H (cf. [21]). The converse statement is valid, e.g., for eigenfunctions having compact support. d

,∞

2,q

2 If V ∈ Lloc and (12) holds, then u ∈ Hloc 0 for some q0 > p. ¯ Indeed, we have V ∈ Lrloc for q0 d 2d p for a certain q0 > p, ¯ any r < 2 , and so by (11) u ∈ L for some p > d−2 . Thus, V u ∈ Lloc

2,q

1

hence u ∈ Hloc 0 and, therefore, |V | 2 u ∈ X2 . The latter confirms that the result in [21] applies to the problem of absence of positive eigenvalues (cf. D(H )).

2670


4. Proofs of Theorems 1 and 2 Let us introduce some notations. In what follows, we omit index K in BK (x0 , ρ), and write simply B(x0 , ρ). Let W ∈ X d−1 , x0 ∈ Ω, ρ > 0, d 3, define 2

d−1 d−1 d−1 τ (W, x0 , ρ) := 1B(x0 ,ρ) |W | 4 (−)− 2 |W | 4 1B(x0 ,ρ) 2 →2 .

(13)

d Note that if V is a potential from Fβ,loc , and V1 := |V | + 1, then

τ (V1 , x0 , ρ) τ (V , x0 , ρ) + ε(ρ),

(14)

where ε(ρ) → 0 as ρ → 0. Let 1B(ρ\a) be the characteristic function of set B(0, ρ) \ B(0, a), where 0 < a < ρ. We define integral operator z − 2z (−) (−)− 2 N (x, y)f (y) dy, 0 Re(z) d − 1 f (x) := N Rd z

whose kernel [(−)− 2 ]N (x, y) is defined by subtracting Taylor polynomial of degree N − 1 at x = 0 of function x → |x − y|z−d , N −1 (x · ∇)k − 2z z−d z−d (−) |0 − y| (x, y) := cz |x − y| − , N k! k=0

where (x · ∇)k :=

∂k k! α |α|=k α1 !...αd ! x ∂x α1 ...∂x αn n

is the multinomial expansion of (x · ∇)k . Define,

1

further,

z z (−)− 2 N,t := ϕt (−)− 2 N ϕt−1 , where ϕt (x) := |x|−t . 4.1. Proof of Theorem 1 Our proof is based on the inequalities of Proposition 1 and Lemma 1. Proposition 1. If τ (V , 0, ρ) < ∞, then there exists a constant C = C(ρ, δ, d) > 0 such that 1B(ρ\a) |V | 12 (−)−1

N,Ndδ

2 1 |V | 2 1B(ρ\a) 2 →2 Cτ (V , 0, ρ) d−1 ,

for all positive integers N , where 0 < δ < 1/2 and

d −3 d δ Nd := N + −δ . 2 d −1


2671

Lemma 1. There exists a constant C = C(d) such that

N (−)−1 (x, y) CN d−3 |x| (−)−1 (x, y) N |y| for all x, y ∈ Rd and all positive integers N . Lemma 1 is a simple consequence of Lemma 3 below for γ = 0. Lemmas 2 and 3 are required for analytic interpolation procedure used in the proof of Proposition 1 when d 4. Lemma 2. (See [6].) There exist constants C2 = C2 (ρ1 , ρ2 , δ, d) and c2 = c2 (ρ1 , ρ2 , δ, d) > 0 such that 1B(ρ \a) (−)−iγ 1 C2 ec2 |γ | , 1 N,N + d −δ B(ρ2 \a) 2 →2 2

where 0 < δ < 1/2, for all γ ∈ R and all positive integers N . Lemma 3. There exist constants C1 = C1 (d) and c1 = c1 (d) > 0 such that

N C1 ec1 γ 2 |x| (−)− d−1 (−)− d−1 2 (1+iγ ) 2 (x, y) (x, y) N |y| for all x, y ∈ Rd , all γ ∈ R and all positive integers N . We prove Lemma 3 at the end of this section. Proof of Proposition 1. If d = 3, then Proposition 1 follows immediately from Lemma 1. Suppose that d 4. Consider the operator-valued function F (z) := 1B(ρ\a) |V |

d−1 4 z

d−1 ϕN +( d −δ)(1−z) (−)− 2 z N ϕ −1

N +( d2 −δ)(1−z)

2

|V |

d−1 4 z

1B(ρ\a)

defined on the strip {z ∈ C: 0 Re(z) 1} and acting on L2 . By Lemma 2, F (iγ ) C2 ec2 |γ | , γ ∈ R, 2 →2 and by Lemma 3, F (1 + iγ )

2 →2

2

τ (V , 0, ρ)C1 ec1 γ ,

γ ∈ R.

Together with obvious observations about analyticity of F this implies that F satisfies all con2 ) : L2 → L2 is bounded, which ditions of Stein’s interpolation theorem. In particular, F ( d−1 completes the proof. 2 Proof of Theorem 1. Let u ∈ YVweak . Without loss of generality we may assume u ≡ 0 on B(0, a) for a > 0 sufficiently small, such that there exists ρ > a with the properties ρ < 1 and ¯ 3ρ) ⊂ Ω. In order to prove that u vanishes on Ω it suffices to show that u ≡ 0 on B(0, ρ) B(0, for any such ρ.

2672


Let η ∈ C0∞ (Ω) be such that 0 η 1, η ≡ 1 on B(0, 2ρ), η ≡ 0 on Ω \ B(0, 3ρ), |∇η| ρc , |η| p
0, c > 0. Here PN −1 (t, θ ) denotes the Taylor polynomial of degree N − 1 at point z = 0 of function z = teiθ → |1 − z|−1 . Similarly to [18], via summation of geometric series we obtain a representation PN −1 (t, θ ) =

N −1

γ

am (θ )t m ,

m=0

where γ am (θ ) :=

−1 + 2 l

k+l=m

iγ 2

− 12 + k

iγ 2

ei(k−l)θ .


2675

Note that − 1 − 1 2 2 =1 l k

0 (0) = am

k+l=m

since ∞

0 am (0)t m

−1

= (1 − t)

=

m=0

∞

t m.

m=0

0 (0) = 1 yield Now estimate (16) and identity am − 1 − 1 γ 2 2 am (θ ) 2 2 2cγ = e2cγ . l k e k+l=m

We have to distinguish between four cases t 2, 1 < t < 2, 0 t 12 and 12 < t < 1. Below we consider only the cases t 2 and 1 < t < 2 (proofs in two other cases are similar). If t 2, then −1 N γ am (θ )t m e2cγ 2 t N 3 e2cγ 2 t N 1 − teiθ −1 PN −1 (t, θ ) 2 m=0

since 1 32 t|1 − teiθ |−1 . Hence, using ||1 − teiθ |−1−iγ | t N |1 − teiθ |−1 , it follows 1 − teiθ −1−iγ − PN −1 (t, θ ) t N 1 − teiθ −1 + 3 e2cγ 2 t N 1 − teiθ −1 2 2cγ 2 N iθ −1 Ce t 1 − te for an appropriate C > 0, as required. If 1 < t < 2, then, after two summations by parts, we derive PN −1 (t, θ ) =

N −3 l=0

+

− 12 + S l

N −2

S

l=0

+

N −1 k=0

iγ 2

k=0

− 12 + l

− 12 + k

Dl (¯z)

N −l−3

iγ 2

iγ 2

− 12 + S k

iγ 2

Dk (z)

− 12 + iγ2 D (¯z)DN −l−2 (z) N −l−2 l

− 12 + iγ2 zk DN −1−k (z) = J1 + J2 + J3 , N −k−1

where δ δ δ S := − , k k k+1

Dk (z) :=

k j =0

zj .

2676


We use estimate 1 − + S 2 k

iγ 2

1 − + = 2 k

iγ 2

− 12 + iγ2 1+k

C(k + 1)− 12 ecγ 2 2

to obtain, following an argument in [18], that each Ji (i = 1, 2, 3) is majorized by Cecγ t N |1 − teiθ |−1 for some C > 0. Since ||1 − teiθ |−1−iγ | t N |1 − teiθ |−1 , Lemma 3 follows. 2 4.2. Proof of Theorem 2 Choose Ψj ∈ C ∞ (Ω) in such a way that 0 Ψj 1, Ψj (x) = 1 for |x| > j2 , Ψj (x) = 0 for

|x| < j1 , |∇Ψj (x)| c j , |Ψj (x)| c j 2 .

Proposition 2. Let τ (V , 0, ρ) < ∞. There exists a constant C = C(ρ, δ, d) > 0 such that for all positive integers N and j 2 1 1B(ρ) Ψj |V | 12 (−)−1 |V | 2 Ψj 1B(ρ) 2 →2 Cτ (V , 0, ρ) d−1 , N,Ndδ 2 1 1B(ρ) Ψj |V | 12 (−)−1 |V | 2 1B(3ρ\ρ) 2 →2 Cτ (V , 0, 3ρ) d−1 , N,Ndδ 1 1B(ρ) Ψj |V | 12 (−)−1 1 Cτ (V , 0, ρ) d−1 , N,Ndδ B( j2 \ j1 ) p →2 1 1B(ρ) Ψj |V | 12 (−)−1 1 Cτ (V , 0, 3ρ) d−1 , N,N δ B(3ρ\2ρ) p →2

(E1) (E2) (E3) (E4)

d

where p =

2d d+2 .

We prove Proposition 2 at the end of this section. Proof of Theorem 2. We use the same notations as in the proof of Theorem 1. Suppose that u ∈ YVstr satisfies (1) and vanishes to an infinite order at 0 ∈ Ω. We wish to obtain an estimate of the form ϕ δ 1B(ρ) Nd u C. (17) ϕN δ (ρ) 2 d

Then, letting N → ∞, we would derive the required identity: u ≡ 0 in B(0, ρ). The same argument as in the proof of Theorem 1 leads us to an identity uηj = (−)−1 (−uηj ),

ηj = ηΨj ,

which, in turn, implies 1

1B(ρ) Ψj V12 ϕN δ u d

1 1 1 −ηj u = 1B(ρ) Ψj V12 (−)−1 N,N δ V12 ϕN δ + 1B(ρ) Ψj V12 (−)−1 N,N δ ϕN δ Ej (u). 1 d d d d V12


2677

Here 0 < δ < 1/2 is fixed, 2/j ρ, uηj = ηj u + Ej (u) and Ej (u) := 2∇ηj ∇u + (ηj )u. Letting I to denote the left hand side of the previous identity, and, respectively, I1 and I2 the two summands of the right hand side, we rewrite the latter as I = I1 + I2 . 1

1,p

Note that I ∈ L2 , since Hloc (Ω) ⊂ X2 by Sobolev embedding theorem, and |V | 2 u ∈ X2 by the definition of YVstr . c , where Next, we expand I1 as a sum I11 + I11 1 1 −Ψj u I11 := 1B(ρ) Ψj V12 (−)−1 N,N δ V12 1B(ρ) ϕN δ 1 d d V12

and 1 1 −ηu c I11 := 1B(ρ) Ψj V12 (−)−1 N,N δ V12 1cB(ρ) ϕN δ . 1 d d 2 V1

Proposition 2 and inequalities (E1) and (E2) imply the required estimates: 2

I11 2 Cτ (V1 , 0, ρ) d−1 I 2 and c 2 I Cϕ δ (ρ)τ (V1 , 0, 3ρ) d−1 1B(3ρ) |V | 12 u . 11 2 N 2 d

Finally, we represent I2 as a sum I21 + I22 , where 1 (1) I21 := 1B(ρ) Ψj V12 (−)−1 N,N δ 1B( 2 \ 1 ) ϕN δ Ej (u) d

j

j

d

and 1 (2) I22 := 1B(ρ) Ψj V12 (−)−1 N,N δ 1B(3ρ\2ρ) ϕN δ Ej (u). d

d

Here Ej(1) (u) := −2∇Ψj ∇u − (Ψj )u,

E (2) (u) := −2∇η∇u − (η)u.

In order to derive an estimate on I21 2 , we expand I21 = I21 + I21 ,

where

2678

D. Kinzebulatov, L. Shartser / Journal of Functional Analysis 258 (2010) 2662–2681 1 I21 := 1B(ρ) Ψj V12 (−)−1 N,N δ 1B( 2 \ 1 ) ϕN δ (−Ψj )u, j

d

d

j

:= 1B(ρ) Ψj V1 (−)−1 N,N δ 1B( 2 \ 1 ) ϕN δ (−2∇η∇u). I21 1 2

j

d

d

j

presents no problem: by (E3), 1) Term I21 1 I 1B(ρ) Ψj V 2 (−)−1 1 1 ϕ (Ψj )u 2 21 2 1 N,N δ B( 2 \ 1 ) p →2 B( 2 \ 1 ) N δ

Cτ (V1 , 0, ρ)

1 d−1

1

j

d

j

j

B( 2 \ 1 ) ϕN δ (Ψj )u 2 , j

d

j

d

j

where 1 2 1 ϕ δ (Ψj )u Cj Ndδ +2 1 2 u2 → 0 as j → ∞, B( \ ) N B( ) 2 j

d

j

j

by the definition of the SUC property. , we once again use inequality (E3): 2) In order to derive an estimate on I21 1 I 1B(ρ) Ψj V 2 (−)−1 1 1 ϕ ∇Ψj ∇up 21 2 1 N,N δ B( 2 \ 1 ) p →2 B( 2 ) N δ j

d

Cτ (V1 , 0, ρ) where p :=

2d d+2 .

1 d−1

j

˜ 1B( 2 ) ϕN δ ∇Ψj ∇up Cj j

d

j

Ndδ +1

d

1B( 2 ) ∇up , j

We must estimate 1B( 2 ) ∇u2 by 1B( 4 ) u2 in order to apply the SUC propj

j

erty. For this purpose, we make use of the following well-known interpolation inequality d d d+6 1B( 2 ) ∇up Cj p C j 2 −1 1B( 4 ) u2 + j 2 1B( 4 ) ur , j

j

j

2d where r := d+4 (see [15]). Using differential inequality (1), we reduce the problem to the probμ lem of finding an estimate on 1B( 4 ) V ur in terms of 1B( 4 ) u2 , μ > 0. By Hölder inequality, j

j

d−1 1 2 1− 2 d 1B( 4 ) u2 d , 1B( 4 ) V ur 1B( 4 ) |V | 2 u 2d 1B( 4 ) V d−1 j

j

j

2

j

as required. As the last step of the proof, we use inequality (E4) to derive an estimate on term I22 : (2) 1 I22 2 Cτ (V1 , 0, 3ρ) d−1 ϕN δ (ρ) Ej (u) p . d

This estimate and the estimates obtained above imply (17).

2

Proof of Proposition 2. Estimates (E1) and (E2) follow straightforwardly from Proposition 1. In order to prove estimate (E3), we introduce the following interpolation function: F1 (z) := 1B(ρ) Ψj |V |

d−1 4 z

d−1 ϕN +( d −δ)(1−z) (−)− 2 z N ϕ −1 2

1 2 1 , N +( d2 −δ)(1−z) B( j \ j )

0 Re(z) 1.


2679

According to Lemma 2, F1 (iγ )2 →2 C1 ec1 |γ | for appropriate C1 , c1 > 0. Further, according to Lemma 3, F1 (1 + iγ )

2d 2d−1 →2

d−1 d−1 2 C2 ec2 γ 1B(ρ) |V | 4 (−)− 2

2d 2d−1 →2

d−1 d−1 d−1 2 C2 ec2 γ 1B(ρ) |V | 4 (−)− 4 2 →2 (−)− 4 1 d−1 2 C2 ec2 γ τ (V , x0 , ρ) 2 (−)− 4

for appropriate C2 , c2 > 0, where, clearly, (−)− terpolation theorem,

2 F1 d −1

p →2

d−1 4

2d 2d−1 →2

2d 2d−1 →2

2d 2d−1 →2

< ∞. Therefore, by Stein’s in-

1

Cτ (V , x0 , ρ) d−1 .

The latter inequality implies (E3). The proof of estimate (E4) is similar: it suffices to consider interpolation function F2 (z) := 1B(ρ) Ψj |V | for 0 Re(z) 1.

d−1 4 z

d−1 ϕN +( d −δ)(1−z) (−)− 2 z N ϕ −1

1 N +( d2 −δ)(1−z) B(3ρ\2ρ)

2

2

5. Proof of Theorem 4 Proof of Theorem 4. Let u ∈ YVK . Suppose that u ≡ 0 in some neighborhood of 0. Assume ¯ 2ρ) ⊂ Ω, and let η ∈ C ∞ (Ω) be such that η ≡ 1 that ρ > 0 is sufficiently small, so that B(0, on B(0, ρ), η ≡ 0 on Ω \ B(0, 2ρ). We may assume, without loss of generality, that V 1. The standard limiting argument implies the following identity: 1B(ρ) u = 1B(ρ) (−)−1 N (−uη ). Therefore, we can write 1B(ρ) ϕN V u

−1 −1 c = 1B(ρ) ϕN V (−)−1 N ϕN 1B(ρ) ϕN (−u) + 1B(ρ) ϕN V (−)−1 N ϕN 1B(ρ) ϕN (−uη ),

or, letting K to denote the left hand side and, respectively, K1 and K2 the two summands of the right hand side of the last equality, we rewrite the latter as K = K1 + K2 . Note that K ∈ L1 (Rd ), as follows from definition of space YVK . Lemma 1 implies that 1B(ρ) ϕN V (−)−1 ϕ −1 f C 1B(ρ) V (−)−1 f Cβf 1 , N N 1 1

2680


for all f ∈ L1 (Ω), which implies an estimate on K1 : K1 1 CβK1 . In order to estimate K2 , we first note that 1cB(ρ) (−uη ) = 1B(2ρ\ρ) (−uη ). According to Lemma 1 there exists a constant Cˆ > 0 such that 1B(2ρ) ϕN V (−)−1 ϕ −1 ˆ C. N N 1 →1 Hence, ˆ −N uη 1 . K2 1 Cˆ 1B(2ρ\ρ) ϕN (−uη ) 1 Cρ Let us choose β > 0 such that Cβ < 1. Then the estimates above imply ˆ (1 − Cβ) 1B(ρ) ρ N ϕN u 1 (1 − Cβ) ρ N K 1 ρ N K2 1 Cu η 1 . Letting N → ∞, we obtain u ≡ 0 in B(0, ρ).

2

Acknowledgments We are grateful to Yu.A. Semenov for introducing us to the subject of unique continuation and close guidance throughout our work on this article, and to Pierre Milman for his supervision and, in particular, help in communicating our results here. References [1] W.O. Amrein, A.M. Bertier, V. Georgescu, Lp -inequalities for the Laplacian unique continuation, Ann. Inst. Fourier (Grenoble) 31 (1981). [2] T. Carleman, Sur un probleme d’unicite pour les systemes d’equations aux derives partielles a deux variables independantes, Ark. Mat. B 26 (1939) 1–9. [3] S. Chanillo, E.T. Sawyer, Unique continuation for + V and Fefferman–Phong class, Trans. Amer. Math. Soc. 318 (1990) 275–300. [4] C. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. 9 (1983) 129–206. [5] R. Froese, I. Herbst, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, L2 -exponential lower bounds to solutions of the Schrödinger equations, Commun. Math. Phys. 87 (1982) 265–286. [6] D. Jerison, C.E. Kenig, Unique continuation absence of positive eigenvalues for Laplace operator, Ann. of Math. 121 (1985) 463–494. [7] T. Kato, Notes on some inequalities for linear operators, Math. Ann. 4 (1952) 208–212. [8] T. Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Commun. Pure Appl. Math. 12 (1959) 403–425. [9] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, 1966. [10] R. Kerman, E.T. Sawyer, The trace inequality eigenvalue estimates for Schrödinger operators, Ann. Inst. Fourier (Grenoble) 36 (1986) 207–228. [11] D. Kinzebulatov, L. Shartser, Towards an optimal result on unique continuation for solutions of Schrödinger inequality, C. R. Math. Acad. Sci. Soc. R. Can. 30 (2009) 106–114. [12] H. Koch, D. Tataru, Sharp counterexamples in unique continuation for second order elliptic equations, J. Reine Angew. Math. 542 (2002) 133–146. 1/2 [13] V.F. Kovalenko, M.A. Perelmuter, Yu.A. Semenov, Schrödinger operators with LW (R l )-potentials, J. Math. Phys. 22 (1981) 1033–1044. [14] V. Liskevich, Yu.A. Semenov, Some problems on Markov semigroups, Adv. Partial Differ. Equ. 11 (1996) 163–217.


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[15] V. Maz’ya, Sobolev Spaces, Springer-Verlag, 1985. [16] M. Reed, B. Simon, Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness, Academic Press, 1975. [17] A. Ruiz, L. Vega, Unique continuation for Schrödinger operators with potentials in Morrey spaces, Publ. Mat. 35 (1991) 291–298. [18] E.T. Sawyer, Unique continuation for Schrödinger operators in dimensions three or less, Ann. Inst. Fourier (Grenoble) 34 (1984) 189–200. [19] M. Schechter, B. Simon, Unique continuation for Schrödinger operators with unbounded potentials, J. Math. Anal. Appl. 77 (1980) 482–492. [20] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [21] E.M. Stein, Appendix to “Unique continuation”, Ann. of Math. 121 (1985) 488–494. [22] E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. [23] T.H. Wolff, Unique continuation for |u| V |∇u| and related problems, Rev. Mat. Iberoamericana 6 (1990) 155– 200.


A joint similarity problem for n-tuples of operators on vector-valued Bergman spaces Olivia Constantin Faculty of Mathematics, University of Vienna, Norbergstrasse 15, 1090 Vienna, Austria Received 17 July 2009; accepted 19 October 2009 Available online 27 October 2009 Communicated by J. Bourgain

Abstract We investigate n-tuples of commuting Foias–Williams/Peller type operators acting on vector-valued weighted Bergman spaces. We prove that a commuting n-tuple of such operators is jointly (completely) polynomially bounded if and only if it is similar to an n-tuple of contractions, if and only if each of the n operators is polynomially bounded. © 2009 Elsevier Inc. All rights reserved. Keywords: Hankel operators; Vector-valued Bergman spaces; Joint polynomial boundedness; Joint similarity to an n-tuple of contractions

1. Introduction Given a separable Hilbert space H, denote by B(H) the set of bounded linear operators acting on H. An operator A ∈ B(H) is called similar to a contraction if it can be written as A = V −1 SV , where S, V ∈ B(H) with V invertible and S 1. By von Neumann’s inequality [8] it follows that an operator A ∈ B(H) that is similar to a contraction is polynomially bounded (see Preliminaries for the relevant definitions). Whether the converse holds, i.e. if a polynomially bounded operator is similar to a contraction was a famous long-standing problem in operator theory formulated by Halmos [7] in 1970 and solved by Pisier [14] in 1997. Attempting to find a counterexample, Peller [12] considered the following type of operators (sometimes called Foias–Williams/Peller type operators or Foguel–Hankel operators) E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.012

O. Constantin / Journal of Functional Analysis 258 (2010) 2682–2694

RT =

S∗ 0

ΓT S

2683

(1.1)

.

Here RT is acting on the direct sum H 2 ⊕ H 2 , where H 2 is the usual scalar-valued Hardy space, S is the shift operator on H 2 , and ΓT is the Hankel operator with (analytic) symbol T . Results by Peller [12], Bourgain [4], Aleksandrov and Peller [1] show that this operator is polynomially bounded if and only if it is similar to a contraction, and hence it does not provide a counterexample to the Halmos problem. Subsequently, Pisier [14] had the crucial idea to consider an operator of type (1.1), but acting on the direct sum of Hardy spaces with values in an infinite-dimensional Hilbert space. This way, he provided an example of a polynomially bounded operator that is not similar to a contraction and the conjecture was solved in the negative. The corresponding problem for operators of type (1.1) acting on scalar-valued Bergman spaces was investigated by Ferguson and Petrović in [6]. They prove that polynomial boundedness and similarity to a contraction are equivalent for RT in this context. Surprisingly, these results continue to hold for Bergman spaces with values in an infinite-dimensional Hilbert space (see [2]), illustrating a completely different picture than the one provided by Pisier for vectorvalued Hardy spaces. Subsequent to Pisier’s solution to the similarity problem, questions concerning n-tuples of (completely) polynomially bounded operators naturally arised. Petrović [13] showed that there exist commuting operators A1 , A2 such that each of them is polynomially bounded, but the product A1 A2 is not polynomially bounded, and hence the pair (A1 , A2 ) is not jointly polynomially bounded (see Preliminaries for the definitions). Moreover, Pisier [15] constructed an example of two commuting operators A1 , A2 , each of which is similar to a contraction, but the pair (A1 , A2 ) is not jointly polynomially bounded. The joint similarity problem for pairs of commuting operators of type (1.1) acting on Bergman spaces was considered in [5,6]. Ferguson and Petrović [6] proved that, in the context of scalar standard-weighted Bergman spaces, a commuting pair of operators (RT1 , RT2 ) is jointly similar to a pair of contractions if and only if the pair is jointly polynomially bounded, if and only if each of RT1 , RT2 is polynomially bounded. These results continue to hold in the vector-valued case (see [5]), that is, for commuting pairs (RT1 , RT2 ) acting on standard-weighted Bergman spaces with values in an infinite-dimensional Hilbert space. There is a fundamental difference between the cases n 2 and n > 2 in the study of the above mentioned properties for general n-tuples of operators. For n > 2 a number of new difficulties appear. For example, the existence of commuting unitary dilations for 3 commuting contractions is not always guaranteed (see [9]) and von Neumann’s inequality does not generalize to n ( 3) commuting contractions (see [17]). Moreover, for n > 2 the concept of joint complete polynomial boundedness of an n-tuple is stronger than the one of joint similarity to an n-tuple of contractions, and it is actually equivalent to joint similarity to an n-tuple of contractions possessing commuting unitary dilations on a common Hilbert space. However, a clear picture can be provided for n-tuples of Foguel–Hankel operators (RT1 , . . . , RTn ) acting on standard weighted vector-valued Bergman spaces. In the present paper we prove that for this n-tuple the concepts of joint complete polynomial boundness, joint polynomial boundedness, joint similarity to an n-tuple of contractions are all equivalent to the condition: RTi is polynomially bounded for all i ∈ {1, . . . , n}. In terms of the symbols Ti , this can be expressed as sup 1 − |z|2 Ti (z) < ∞, z∈D

1 i n.

2684


Finally, we show how the above considerations can be used in the study of more general commuting n-tuples of operators (RX1 , . . . , RXn ), where RX i =

S∗ 0

Xi S

1 i n,

,

and the Xi ’s are bounded linear operators acting on vector-valued Bergman spaces. It is somewhat surprising that in spite of the above mentioned difficulties it is possible to extend the results for pairs from [5] to n-tuples of operators. The reason for this is that in our present approach, which is also more direct, we can take advantage of some structural properties of the vectorvalued Bergman space. 2. Preliminaries We start by presenting some definitions concerning the operators and the spaces that will be used in our further considerations. For a separable Hilbert space H, we let B(H) denote the bounded linear operators on H. Given A1 , . . . , An ∈ B(H), the n-tuple (A1 , . . . , An ) is called jointly similar to an n-tuple of contractions if there exist an invertible operator V ∈ B(H), and S1 , . . . , Sn ∈ B(H) with S1 , . . . , Sn 1, such that Ai = V −1 Si V , 1 i n. Let d be a positive integer and denote by Dd the polydisc in Cd . A commuting d-tuple of operators (A1 , A2 , . . . , Ad ), with Ai ∈ B(H) (1 i d), is said to be jointly polynomially bounded if there exists a positive constant k such that the following inequality holds p(A1 , A2 , . . . , Ad ) k sup p(z),

(2.2)

z∈Dd

for any analytic polynomial of d variables p. Note that this condition is equivalent to the boundedness of the representation π : A(Dd ) → B(H), with π(zi ) = Ai (1 i d), where A(Dd ) is the polydisc algebra. A commuting d-tuple (A1 , A2 , . . . , Ad ) is said to be jointly completely polynomially bounded if there exists a constant k > 0 such that

pij (A1 , A2 , . . . , Ad )xi , yj 1i,j n

kP Mn (A(Dd ))

n

1/2 xi

i=1

2

n

1/2 yj

2

,

(2.3)

j =1

whenever P = (pij )1i,j n is an n × n matrix of analytic polynomials of d variables, n = 1, 2, . . . , and {xi }ni=1 , {yj }nj=1 are vectors in H. Here P Mn (A(Dd )) =

sup (z1 ,...,zd )∈Dd

pij (z1 , . . . , zd )

Mn

.

Denoting by P (A1 , A2 , . . . , Ad ) the n×n matrix acting on n H, whose entries pij (A1 , . . . ,Ad ) are analytic polynomials of A1 , . . . , Ad , we can rewrite (2.3) in the form


P (A1 , . . . , Ad ) kP

2685

Mn (A(Dd )) ,

and hence (2.3) is equivalent to the complete boundedness of the map π defined above. For d = 1 in the previous definitions, we simply say that A1 is polynomially bounded, respectively completely polynomially bounded. The following result due to Paulsen provides a necessary and sufficient condition for a commuting d-tuple of operators to be jointly completely polynomially bounded. Theorem 2.1. (See [10,11].) A commuting d-tuple of operators (A1 , . . . , Ad ) on a Hilbert space H satisfies (2.3) if and only if there is an invertible operator V on H such that the commuting d-tuple (V −1 A1 V , . . . , V −1 Ad V ) satisfies (2.3) with constant k = 1. We would like to point out that in case d = 1, 2 one can combine the above theorem with the existence of a unitary dilation for a contraction, respectively the existence of commuting unitary dilations for a pair of commuting contractions (see [3]), to deduce that (joint) complete polynomial boundedness is equivalent to (joint) similarity to a (pair of) contraction(s). For d 3 it follows by Theorem 2.1 that joint complete polynomial boundedness is equivalent to joint similarity to an n-tuple of contractions possessing an n-tuple of commuting unitary dilations (see [16]). Recall that n commuting contractions S1 , . . . , Sn ∈ B(H) possess an n-tuple commuting unitary dilations if there exists a Hilbert space K ⊇ H and commuting unitary operators U1 , . . . , Un ∈ B(K) with PH U1k1 . . . Unkn |H = S1k1 . . . Snkn ,

k1 , . . . , kn ∈ N.

Let dA denote the normalized area measure on the unit disc and for α > −1 let dμα (z) = (α + 1)(1 − |z|2 )α dA(z). We consider the standard weighted vector-valued Bergman spaces L2,α a (H) which consist of analytic functions x : D → H with x =

1 2 x(z)2 dμα (z) < ∞. H

(2.4)

D

We write L2,α a for the scalar space, i.e. when H = C. For an analytic operator-valued function T : D → B(H), the (little) Hankel operator ΓT is defined by means of the Hankel form

ΓT x, y = lim

r→1

T (rz)x(r z¯ ), y(rz) dA(z),

D

where x, y are H-valued analytic functions in a disk of radius strictly larger than 1 (as it is wellknown these functions form a dense subset in L2,α a (H)). It turns out (see [2]) that ΓT extends to a bounded linear operator on L2,α a (H) if and only if sup 1 − |z|2 T (z) < ∞. z∈D

2686


In the proof of our main result we shall make use of the next lemma, together with the wellknown formulas (2.6)–(2.7) included below for the sake of completeness. Lemma 2.1. (See [2].) Let γ 0 and let T : D → B(H) be an analytic operator-valued function satisfying supz∈D (1 − |z|2 )γ T (z) < ∞. Then the following equality holds

γ T (z)x(¯z), y(z) 1 − |z|2 dμα (z)

D

=

1 α+γ +1

γ +1 y(z) dμα (z), T (z)x(¯z), 1 − |z|2 z

(2.5)

D

for any x, y ∈ L2,α a (H) with y(0) = 0. For A ∈ B(Cn , H), the space of bounded linear operators from Cn to H, we denote by AB2 its Hilbert–Schmidt norm, that is AB2 =

n

1/2 Aek

2

,

k=1

where {ek }nk=1 is some orthonormal basis of Cn . On B(Cn ) we consider the usual operator norm and also the trace norm defined by Atr = tr |A| ,

A ∈ B Cn ,

where |A| = (A∗ A)1/2 . As is well known, the following inequalities hold. For A, B ∈ B(Cn ), we have tr(AB) A · Btr ,

(2.6)

and for T ∈ B(H), and X, Y ∈ B(Cn , H), we have ∗ Y T X XB Y B T . 2 2 tr

(2.7)

This last inequality is usually stated for operators acting on the same space, but the more general version stated above is a consequence of the classical one. In fact, if M is an n-dimensional subspace of H containing the range of X and U : M → Cn denotes a unitary operator, then ∗ Y T X = Y ∗ T U −1 U X Y ∗ T U −1 U XB 2 tr tr B2 ∗ ∗ XB2 Y T U −1 B = XB2 U −1 T ∗ Y B 2 2 −1 ∗ ∗ XB2 Y B2 U T XB2 Y B2 T .


2687

3. Main results Let us first consider the d-tuple of operators (RT1 , RT2 , . . . , RTd ), with RTi =

Mz∗ 0

ΓTi Mz

,

1 i d,

2,α acting on the direct sum L2,α a (H) ⊕ La (H) where Mz denotes the operator of multiplication by 2,α z on La (H) and ΓTi , 1 i d are Hankel operators. Since any Hankel operator ΓT satisfies ΓT Mz = Mz∗ ΓT , the operators RTi , 1 i d, commute. These operators are convenient for our purposes as the action of an analytic polynomial of d variables on the d-tuple (RT1 , RT2 , . . . , RTd ) is easily computed. It is straightforward to show by induction that, if p is such a polynomial, then

p(RT1 , RT2 , . . . , RTd ) =

T1 ,T2 ,...,Td (p) , p(Mz , . . . , Mz )

p(Mz∗ , . . . , Mz∗ ) 0

where T1 ,T2 ,...,Td (p) =

d

ΓTi (∂zi p)(Mz , . . . , Mz ).

i=1

Given an n × n matrix of analytic polynomials of d variables P = (pij ), by a change of basis (the so-called canonical shuffle) one obtains ∗ ∗ P (RT , . . . , RT ) = (pij (Mz , . . . , Mz )) d 1 0

(T1 ,...,Td (pij )) . (pij (Mz , . . . , Mz ))

(3.8)

Also, if P = (pij )1i,j n is such a matrix of analytic polynomials, we shall denote by P # the #) matrix with entries (pij 1i,j n , where # pij (z1 , z2 , . . . , zd ) = pij (¯z1 , z¯ 2 , . . . , z¯ d ),

1 i, j n.

Theorem 3.1. Let Ti : D → B(H) be holomorphic operator-valued functions with supz∈D (1 − |z|2 )Ti (z) < ∞, 1 i d. Then the following are equivalent: The operators RTi , 1 i d, are polynomially bounded; The d-tuple (RT1 , . . . , RTd ) is jointly polynomially bounded; The d-tuple (RT1 , . . . , RTd ) is jointly completely polynomially bounded; The d-tuple (RT1 , . . . , RTd ) is jointly similar to a d-tuple of contractions; The d-tuple (RT1 , . . . , RTd ) is jointly similar to a d-tuple of contractions possessing a dtuple of commuting unitary dilations; (vi) supz∈D (1 − |z|2 )Ti (z) < ∞, 1 i d.

(i) (ii) (iii) (iv) (v)

Proof. The implications (iii) ⇒ (ii) ⇒ (i) and (iv) ⇒ (i) are obvious. The equivalence of (i) and (vi) was proved in [2]. The implication (iii) ⇒ (v) follows from Theorem 2.1, and (v) ⇒ (iv) is obvious. Hence it is enough to prove (vi) ⇒ (iii). Suppose condition (vi) holds. Throughout the

2688


proof C > 0 stands for a generic constant. Since Mz is a contraction, the d-tuples (Mz∗ , . . . , Mz∗ ) and (Mz , . . . , Mz ) are jointly completely polynomially bounded (being one variable objects). In view of (3.8) we deduce that the d-tuple (RT1 , . . . , RTd ) is jointly completely polynomially bounded if and only if the map T1 ,...,Td is completely bounded from the polydisc algebra A(Dd ) to B(L2,α a (H)). Let us show that T1 ,...,Td is completely bounded. For n ∈ N, let P = (pij )1i,j n be a matrix of analytic polynomials of d variables. Denote by ∂zk P the matrix with entries (∂zk pij )1i,j n , 1 k d. We shall first assume that z12 (∂zk P ) (z, . . . , z) is a matrix of analytic polynomials of one variable for any k ∈ {1, . . . , d}. The map T1 ,...,Td is completely bounded if d

ΓTk (∂zk pij )(Mz , . . . , Mz )xi , yj 1i,j n k=1

CP Mn (A)

n i=1

1 2

xi

2

n

1 2

yj

2

(3.9)

j =1

for xi , yj ∈ L2,α a (H), n 1. Note that (vi) implies supz∈D Tk (z) < ∞, and supz∈D (1 − |z|2 )Tk (z) < ∞. Hence, for every 1 k d, we can apply Lemma 2.1 twice to obtain

ΓTk (∂zk pij )(Mz , . . . , Mz )xi , yj

1i,j n

=

Tk (z)xi (¯z), yj (z) (∂zk pij )(¯z, . . . , z¯ ) dμα (z)

1i,j n D

1 = α+1

1i,j n D

1 = (α + 1)(α + 2)

(∂zk pij )(¯z, . . . , z¯ ) 1 − |z|2 dμα (z) Tk (z)xi (¯z), yj (z) z¯

1i,j n D

2

(∂zk pij )(¯z, . . . , z¯ ) Tk (z)xi (¯z), yj (z) 1 − |z|2 dμα (z). 2 z¯

For each z ∈ D we denote by Tk (z) the n × n matrix with entries xy

1 − |z|2 Tk (z)xi (¯z), yj (z) i,j and let 2 (∂zk p ij )# (z, . . . , z) k (z) = (1 − |z| ) (∂zk P )# (z, . . . , z) = 1 − |z|2 P . z2 z2 1i,j n Then k (z) = P

z, . . . , z¯ ) 2 (∂zk pij )(¯ 1 − |z| . z¯ 2 1i,j n


2689

With these notations we have

2

(∂zk pij )(¯z, . . . , z¯ ) Tk (z)xi (¯z), yj (z) 1 − |z|2 dμα (z) 2 z¯

1i,j n D

=

k (z)Txy (z)t dμα (z). tr P k

D

k (z) and T (z) as operators acting on Cn . Using (2.6) we get We regard P k xy

xy tr P k (z)Txy (z)t dμα (z) P k (z) Tk (z)tr dμα (z) k D

D

=

xy P k (z) Tk (z)tr dμα (z).

(3.10)

D

For any a, b ∈ Cn with a = b = 1, the function Dd (z1 , . . . , zd ) → P # (z1 , . . . , zd )a, b is an analytic polynomial of d variables. Hence, by the Schwarz Lemma, we deduce

(∂z P )# (z1 , . . . , zd )a, b 1 − |zk |2 = ∂z P # (z1 , . . . , zd )a, b 1 − |zk |2 k k

C sup P # (z1 , . . . , zd )a, b zk ∈D

C

sup

#

P (z1 , . . . , zd )a, b

(z1 ,...,zd )∈Dd

CP Mn (A) ,

z ∈ D.

Now first put z1 = · · · = zd = z in the above relation, and then use the maximum modulus principle to deduce (1 − |z|2 )

# (∂zk P ) (z, . . . , z)a, b CP Mn (A) , z2

z ∈ D.

Since a, b were arbitrarily chosen such that a = b = 1, we obtain P k (z) CP M (A) , n

z ∈ D, 1 k d.

From this estimate and relation (3.10) we obtain xy tr P k (z)Txy (z)t dμα (z) CP M (A) Tk (z)tr dμα (z). n k D

D

xy Regard Tk (z) as an element of B(Cn ). For a, b ∈ Cn , we have

(3.11)

2690


1 − |z|2 Tk (z)xi (¯z), yj (z) ai b¯j

xy Tk (z)a, b =

1i,j n

=

1 − |z|2 Tk (z) ai xi (¯z) , bj yj (z) 1j n

1in

= 1 − |z|2 Tk (z)Xa, Y b ,

(3.12)

where X, Y : Cn → H are the linear operators defined on the standard basis of Cn , {ei }ni=1 , by X(ei ) = xi (¯z),

Y (ej ) = yj (z),

1 i, j n.

Then from (3.12) we get xy

Tk (z)a, b = 1 − |z|2 Tk (z)Xa, Y b = Y ∗ 1 − |z|2 Tk (z) Xa, b , where Y ∗ : H → C is the adjoint of Y . Now use (2.7) to obtain xy Tk (z)tr = Y ∗ 1 − |z|2 Tk (z) X tr XB2 Y B2 1 − |z|2 Tk (z). With these estimates we get from (3.11) and the Cauchy–Schwarz inequality tr P k (z)Txy (z)t dμα (z) k D

CP Mn (A)

XB2 Y B2 1 − |z|2 Tk (z) dμα (z)

D

1 1 2 2 xi 2 yj 2 C sup 1 − |z|2 Tk (z) P Mn (A)

z∈D

M (A) Xn Y n . = CP n Hence, for any k ∈ {1, . . . , d} we have

ΓTk (∂zk pij )(Mz , . . . , Mz )xi , yj CP Mn (A) Xn Y n . 1i,j n

Thus (3.9) follows. If z12 (∂zk P )(z, . . . , z) is not a matrix of analytic polynomials of one variable for some k ∈ {1, . . . , d}, we apply the above procedure to the matrix Q(z1 , . . . , zd ) = P (z1 , . . . , zd ) − L(z1 , . . . , zd ), where


L(z1 , . . . , zd ) =

β

2691

β

∂zβ11 . . . ∂zβdd P (0, . . . , 0)z1 1 . . . zd d ,

β1 +···+βd 2

with β1 , . . . , βd ∈ N. Doing this we get T

1 ,...,Td

(qij ) ij CQMn (A) .

Taking into account β ∂ 1 . . . ∂ βd P (0, . . . , 0) CP M (A) , z1 zd n

β1 + · · · + βd 2,

(3.13)

we deduce QMn (A) CP Mn (A) . Note also that d (l ) ΓTk (∂zk lij )(Mz , . . . , Mz ) ij CP Mn (A) , 1 ,...,Td ij ij

T

k=1

where the last inequality follows by (3.13). So by the above we obtain T ,...,T (pij ) C T ,...,T (qij ) + T ,...,T (lij ) d d d 1 1 1 ij ij ij C QMn (A) + P Mn (A) CP Mn (A) , and the proof is complete.

2

Remark 3.1. It was shown in [2] that RT is similar to a contraction, if and only if RT is polynomially bounded, if and only if RT is power bounded (i.e. supn1 RTn < ∞). In view of this result, the statements (i)–(vi) in Theorem 3.1 are actually equivalent to the statement: RT1 , . . . , RTd are power bounded. Let us consider the more general d-tuple of operators (RX1 , . . . , RXd ), where RXi =

Mz∗ 0

Xi Mz

,

1 i d,

(3.14)

and Xi ∈ B(L2,α a (H)). Note that the commutativity of RXi and RXj , 1 i, j d, is equivalent to (Xi − Xj )Mz = Mz∗ (Xi − Xj ), that is Xi − Xj = ΓTij is a Hankel operator. If p is an analytic polynomial of d variables note that p(RX1 , . . . , RXd ) =

p(Mz∗ , . . . , Mz∗ ) 0

δ(X1 ,...,Xd ) (p) , p(Mz , . . . , Mz )

where the map p → δ(X1 ,...,Xd ) (p) ∈ B(L2,α a (H)) is a derivation, that is, for any p, q, we have

2692


δ(pq) = p Mz∗ , . . . , Mz∗ δ(q) + δ(p)q(Mz , . . . , Mz ). Since Mz is a contraction, the d-tuple (RX1 , . . . , RXd ) is jointly (completely) polynomially bounded if and only if δ extends to a (completely) bounded map from A(Dd ) to B(L2,α a (H)). The next result shows that the study of the joint (complete) polynomial boundedness for the dtuple (RX1 , . . . , RXd ) reduces to investigating the same property for simpler objects. Its proof is similar to the proofs presented in [5,6] for pairs of such operators. Proposition 3.1. Assume X1 , . . . , Xd ∈ B(L2,α a (H)) are such that RX1 , . . . , RXd commute. Then δ(X1 ,...,Xd ) extends to a (completely) bounded map on A(Dd ) if and only if the derivations δ(X1 −Xi ,0,...,0) , δ(0,X2 −Xi ,0,...,0) , . . . , δ(0,...,0,Xd −Xi ) and δ(Xi ,Xi ,...,Xi ) are (completely) bounded on A(Dd ), for some i ∈ {1, . . . , d}. Moreover δ(X1 ,...,Xd ) (f ) = δ(Xi ,Xi ,...,Xi ) (f ) + δ(X1 −Xi ,0,...,0) (f ) + δ(0,X2 −Xi ,0,...,0) (f ) + · · · + δ(0,...,0,Xd −Xi ) (f ),

(3.15)

for all f ∈ A(Dd ). Proof. For any j ∈ {1, . . . , d} the operators RXi , RXj commute, and therefore Xj − Xi is a Hankel operator that we denote by ΓTj . We shall prove that (3.15) holds for all analytic polynomials j

j

j

p(z1 , . . . , zd ). To this end, it is enough to show that (3.15) holds for all monomials z11 z22 . . . zdd , for all integers j1 , . . . , jd 1. By straight-forward induction we obtain j j 1 −1 2 −1 j1 ∗ k ∗ j1 +k jd j1 +···+jd −k−1 j +···+jd −k−1 Mz X 1 Mz Mz + X 2 Mz 2 δ(X1 ,...,Xd ) z1 . . . zd = k=0

+ ··· +

k=0 j d −1

∗ j1 +···+jd−1 +k j −k−1 Mz X d Mz d .

k=0

Replacing Xj by Xi + ΓTj , 1 j d, in the above relation, we obtain j j j j δ(X1 ,...,Xd ) z11 . . . zdd = δ(Xi ,Xi ,...,Xi ) z11 . . . zdd + δ(ΓT1 ,0,...,0) (f ) j j j j + δ(0,ΓT2 ,0,...,0) z11 . . . zdd + · · · + δ(0,...,0,ΓTd ) z11 . . . zdd , and hence (3.15) holds for all polynomials p(z1 , . . . , zd ). Since (3.15) holds on a dense subset of A(Dd ), it is clear that if δ(ΓT1 ,0,...,0) , δ(0,ΓT2 ,0,...,0) , . . . , δ(0,...,0,ΓTd ) and δ(Xi ,Xi ,...,Xi ) are (completely) bounded on A(Dd ), then δ(X1 ,...,Xd ) is (completely) bounded on A(Dd ). On the other hand, if we suppose δ(X1 ,...,Xd ) is (completely) bounded on A(Dd ), then the d-tuple (RX1 , . . . , RXd ) is jointly (completely) polynomially bounded. In particular RXi is (completely) polynomially bounded and each pair (RXi , RXj ), 1 j d, is jointly (completely) polynomially bounded. Then the d-tuple (RXi , . . . , RXi ) is jointly (completely) polynomially bounded, and hence the map δ(Xi ,...,Xi ) is (completely) bounded on A(Dd ). Moreover, by Proposition 3.1 in [5] we have


2693

δ(Xi ,Xj ) = δ(Xi ,Xi ) + δ(0,Xj −Xi ) = δ(Xi ,Xi ) + δ(0,ΓTj ) . As the pair (RXi , RXj ) is jointly (completely) polynomially bounded, we deduce that δ(Xi ,Xj ) is (completely) bounded. Then the last equality above shows that δ(0,ΓTj ) is (completely) bounded, and by Theorem 3.1 we obtain sup 1 − |z|2 Tj (z) < ∞,

1 j d.

z∈D

Again, by Theorem 3.1 we obtain that δ(ΓT1 ,0,...,0) , δ(0,ΓT2 ,0,...,0) , . . . , δ(0,...,0,ΓTd ) are completely bounded (the notation T1 ,...,Td used in Theorem 3.1 corresponds to δ(ΓT1 ,...,ΓTd ) ). 2 The next corollary is an immediate consequence of Theorem 3.1 and Proposition 3.1. Corollary 3.1. Let RX1 , RX2 , . . . , RXd be commuting operators as in (3.14). Assume that for some i ∈ {1, . . . , d} we have sup 1 − |z|2 Tj (z) < ∞,

1 j d,

(3.16)

z∈D

where Tj is the holomorphic symbol of the Hankel operator Xj − Xi = ΓTj . Then the following are equivalent: (i) RXi is similar to a contraction. (ii) The d-tuple (RX1 , . . . , RXd ) is jointly completely polynomially bounded. Proof. We clearly have (ii) ⇒ (i). Suppose (i) holds. Then the d-tuple (RXi , . . . , RXi ) is jointly completely polynomially bounded, being a one-variable object, and therefore δXi ,...,Xi is completely bounded on A(Dd ). In view of our assumption on the Tj ’s, by Theorem 3.1 we deduce that δ(ΓT1 ,0,...,0) , . . . , δ(0,...,0,ΓTd ) are completely bounded. Thus δ(X1 ,...,Xd ) is completely bounded on A(Dd ) by Proposition 3.1. 2 In particular, we have Corollary 3.2. Let RX1 , RX2 , . . . , RXd be commuting operators as in (3.14). Assume that for some fixed i ∈ {1, . . . , d} and for each j ∈ {1, . . . , d}, the pair (RXi , RXj ) is jointly polynomially bounded. Then the following are equivalent: (i) RXi is similar to a contraction. (ii) The d-tuple (RX1 , . . . , RXd ) is jointly completely polynomially bounded. Proof. Examining the proof of Proposition 3.1, we notice that if the pair (RXi , RXj ), 1 j d, is jointly polynomially bounded, then condition (3.16) in Corollary 3.1 is satisfied. 2 We would like to conclude with the following remark. If (RX1 , RX2 , . . . , RXd ) is a commuting d-tuple given by (3.14), which is jointly similar to a d-tuple of contractions, then, by the existence of a unitary dilation for a commuting pair of contractions we deduce that (RXi , RXj ) is

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jointly polynomially bounded for 1 i, j d. Hence the hypotheses of Corollary 3.2 are satisfied, and we deduce that (RX1 , RX2 , . . . , RXd ) is jointly completely polynomially bounded. Thus the concepts of joint complete polynomial boundedness and joint similarity to a d-tuple of contractions are equivalent for commuting d-tuples (RX1 , RX2 , . . . , RXd ) acting on vector-valued Bergman spaces. Acknowledgments The author is very grateful to Catalin Badea for useful comments and suggestions. Constructive criticism by the referee has the led to an improvement of the exposition. References [1] A.B. Aleksandrov, V.V. Peller, Hankel operators and similarity to a contraction, Int. Math. Res. Not. 6 (1996) 263– 275. [2] A. Aleman, O. Constantin, Hankel operators on Bergman spaces and similarity to contractions, Int. Math. Res. Not. 35 (2004) 1785–1801. [3] T. Ando, On a pair of commutative contractions, Acta Sci. Math. (Szeged) 24 (1963) 88–90. [4] J. Bourgain, On the similarity problem for polynomially bounded operators on Hilbert space, Israel J. Math. 54 (1986) 227–241. [5] O. Constantin, F. Jaëck, A joint similarity problem on vector-valued Bergman spaces, J. Funct. Anal. 256 (2009) 2768–2779. [6] S.H. Ferguson, S. Petrović, The joint similarity problem for weighted Bergman shifts, Proc. Edinb. Math. Soc. (2) 45 (1) (2002) 117–139. [7] P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. (N.S.) 76 (1970) 887–933. [8] J. von Neumann, Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, Math. Nachr. 4 (1951) 258–281. [9] S. Parrott, Unitary dilations for commuting contractions, Pacific J. Math. 34 (1970) 481–490. [10] V.I. Paulsen, Every completely polynomially bounded operator is similar to a contraction, J. Funct. Anal. 55 (1984) 1–17. [11] V.I. Paulsen, Completely bounded homomorphisms of operator algebras, Proc. Amer. Math. Soc. 92 (1984) 225– 228. [12] V.V. Peller, Estimates of functions of Hilbert space operators, similarity to a contraction and related function algebras, in: Lecture Notes in Math., vol. 1043, Springer-Verlag, Berlin, 1984, pp. 199–204. [13] S. Petrović, Polynomially unbounded product of two polynomially bounded operators, Integral Equations Operator Theory 27 (1997) 473–477. [14] G. Pisier, A polynomially bounded operator on Hilbert space which is not similar to a contraction, J. Amer. Math. Soc. 10 (1997) 351–369. [15] G. Pisier, Joint similarity problems and the generation of operator algebras with bounded length, Integral Equations Operator Theory 31 (1998) 353–370. [16] G. Pisier, Similarity Problems and Completely Bounded Maps, Lecture Notes in Math., vol. 1618, Springer-Verlag, Berlin, 2001. [17] N. Varopoulos, On an inequality of von Neumann and an application of the metric theory of tensor products to operators theory, J. Funct. Anal. 16 (1974) 83–100.


Vanishing of second cohomology for tensor products of type II1 von Neumann algebras Florin Pop a , Roger R. Smith b,∗ a Department of Mathematics and Computer Science, Wagner College, Staten Island, NY 10301, United States b Department of Mathematics, Texas A&M University, College Station, TX 77843, United States

Received 18 July 2009; accepted 13 January 2010 Available online 27 January 2010 Communicated by S. Vaes

Abstract We show that the second cohomology group H 2 (M ⊗ N, M ⊗ N ) is always zero for arbitrary type II1 von Neumann algebras M and N . © 2010 Elsevier Inc. All rights reserved. Keywords: von Neumann algebra; Hochschild cohomology; Tensor product

1. Introduction The theory of bounded Hochschild cohomology for von Neumann algebras was initiated by Johnson, Kadison and Ringrose in a series of papers [13,16,17], which laid the foundation for subsequent developments. These were a natural outgrowth of the theorem of Kadison [15] and Sakai [27] which established that every derivation δ : M → M on a von Neumann algebra M is inner; H 1 (M, M) = 0 in cohomological terminology. While cohomology groups can be defined for general M-bimodules (see Section 2 for definitions), this derivation result ensured special significance for M as a bimodule over itself. When M is represented on a Hilbert space H , then B(H ) is also an important M-bimodule, but here the known results are less definitive. For example, it is not known whether every derivation δ : M → B(H ) is inner, a problem known to be equivalent to the similarity problem [18]. In [17], it was shown the H n (M, M) = 0 for all n 1 * Corresponding author.

E-mail addresses: [email protected] (F. Pop), [email protected] (R.R. Smith). 0022-1236/$ – see front matter © 2010 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2010.01.013

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F. Pop, R.R. Smith / Journal of Functional Analysis 258 (2010) 2695–2707

when M is an injective von Neumann algebra, a class which includes the type I algebras. They conjectured that this should be true for all von Neumann algebras, now known as the Kadison– Ringrose conjecture. The purpose of this paper is to verify this for the second cohomology of tensor products of type II1 von Neumann algebras. The study of this conjecture reduces to four cases in parallel with the type decomposition of Murray and von Neumann. Three of these are solved. As noted above, the type I case was determined at the outset of the theory, while the types II∞ and III cases were solved by Christensen and Sinclair [9] after they had developed the theory of completely bounded multilinear maps [8] and applied it, jointly with Effros [5], to cohomology into B(H ). They showed that n (M, M) = 0, n 1, for all von Neumann algebras, where the subscript indicates that all releHcb vant multilinear maps are required to be completely bounded. Since then, all progress has hinged on reducing a given cocycle to one which is completely bounded and then quoting their result. In this paper we follow a different path, although complete boundedness will play an important role. The one remaining open case is that of type II1 von Neumann algebras. There are several positive results for special classes: the McDuff factors [9], those factors with Cartan subalgebras [22,6,29,1], and those with property Γ [9,4,7]. While tensor products form a large class of type II1 von Neumann algebras, the prime factors fall outside our scope. The best known examples are the free group factors, shown to be prime by Ge [12], and these do not lie in any of the classes already mentioned, so nothing is known about their cohomology. Section 2 gives a brief review of definitions and some results that we will need subsequently. The heart of the paper is Section 3 where we prove that H 2 (M ⊗ N, M ⊗ N ) = 0 for separable type II1 von Neumann algebras. This restriction is made in order to be able to choose certain special hyperfinite subalgebras that are only available in this setting. The proof proceeds through a sequence of lemmas which reduce a given cocycle to one with extra features, after which we can exhibit it as a coboundary. In this process, particular use is made of complete boundedness and the basic construction [14] for containments of type II1 algebras. Section 4 handles the general case by deducing it from the separable situation using several known techniques to be found in [28, §6.5]. However, Lemma 4.1 appears to be new. For general background on cohomology we refer to the survey article [26] and the monograph [28]. The theory of complete boundedness is covered in several books [11,19,20], while [31] contains an introduction to the basic construction algebra. 2. Preliminaries and notation Since this paper is only concerned with second cohomology, we will only give the definitions at this level, referring to [28] for the general case. Let A be a C ∗ -algebra with an A-bimodule V . For n = 1, 2, 3, Ln (A, V ) denotes the space of bounded n-linear maps from A × · · · × A into V , while L0 (A, V ) is defined to be V . For v ∈ V and φn ∈ Ln (A, V ), n = 1, 2, the coboundary map ∂ : Ln (A, V ) → Ln+1 (A, V ) is defined as follows: ∂v(a) = va − av,

a ∈ A,

∂φ1 (a1 , a2 ) = a1 φ1 (a2 ) − φ1 (a1 a2 ) + φ1 (a1 )a2 ,

(2.1) ai ∈ A,

(2.2)

∂φ2 (a1 , a2 , a3 ) = a1 φ2 (a2 , a3 ) − φ2 (a1 a2 , a3 ) + φ2 (a1 , a2 a3 ) − φ2 (a1 , a2 )a3 ,

ai ∈ A.

(2.3)


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An algebraic calculation gives ∂∂ = 0. Cocycles are those maps φ for which ∂φ = 0, while coboundaries are maps of the form ∂ξ . The nth cohomology group H n (A, V ) is then the quotient space of n-cocycles modulo n-coboundaries. In particular, H 1 (A, V ) is the space of derivations modulo inner derivations. Since we plan to prove that certain second cohomology groups are zero, this amounts to showing that each 2-cocycle is a 2-coboundary. There is a considerable theory of cohomology, much of which is summarized in [28]. We use this monograph as our standard reference, but include some results below which are not to be found there. The first two of these concern complete boundedness, the second of which is a small extension of the factor case of [9] (the results of this paper appear in [28]). Lemma 2.1. Let M ⊆ B(H ) and S ⊆ B(K) be type II1 von Neumann algebras with S hyperfinite. If φ : M ⊗ S → B(H ⊗2 K) is bounded, normal and (I ⊗ S)-modular, then φ is completely bounded. Proof. We regard M and S as both represented on B(H ⊗2 K). The (I ⊗ S)-modularity implies that the restriction ψ of φ to M ⊗ I maps into S , so φ maps the minimal tensor product M ⊗min S into C ∗ (S , S). Now S contains arbitrarily large matrix subfactors Mn , n 1, and φ|M⊗Mn can be regarded as the composition of ψ ⊗ idn : M ⊗ Mn → S ⊗ Mn with a ∗isomorphism πn : S ⊗ Mn → C ∗ (S , Mn ). The uniform bound φ on each of these restrictions then shows that ψ is completely bounded. Hyperfiniteness of S gives a ∗-homomorphism ρ : S ⊗min S → C ∗ (S , S) defined on elementary tensors by s ⊗ s → s s [10, Proposition 4.5], so φ|M⊗min S is the composition ρ ◦ (ψ ⊗ idS ), showing complete boundedness on M ⊗min S. The same conclusion on M ⊗ S now follows from normality of φ and the Kaplansky density theorem applied to M ⊗min S ⊗ Mn ⊆ M ⊗ S ⊗ Mn . 2 The proof that we have given of this result relies on normality and hyperfiniteness, and it would be interesting to know if it holds without these restrictions. The next result is known for factors [9], but does not appear to be in the literature in the generality that we require. The theory of multimodular maps plays a significant role in the study of cohomology, so we recall the definition for linear and bilinear maps. Let A ⊆ B be C ∗ -algebras. A map ψ : B → B is A-modular if a1 ψ(b)a2 = ψ(a1 ba2 ),

a1 , a2 ∈ A, b ∈ B.

(2.4)

A bilinear map φ : B × B → B is A-multimodular if a1 φ(b1 a2 , b2 )a3 = φ(a1 b1 , a2 b2 a3 ),

ai ∈ A, bi ∈ B.

(2.5)

Lemma 2.2. Let M and S be type II1 von Neumann algebras with S hyperfinite, and let Q ⊆ M be a hyperfinite von Neumann subalgebra with Q ∩ M = Z(M). Let φ : (M ⊗ S) × (M ⊗ S) → M ⊗ S be a bounded separately normal bilinear map which is Q ⊗ S-multimodular. Then φ is completely bounded. Proof. The hypothesis of multimodularity allows us to apply Lemma 5.4.5(ii) of [28] which yields the inequality

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n n 1/2 n 1/2 ∗ ∗ φ(xi , yi ) 2 φ xi xi yi yi i=1

i=1

(2.6)

i=1

for arbitrary finite sets of elements xi , yi ∈ M ⊗ S, 1 i n. If φn denotes the nth amplification of φ to M ⊗ S ⊗ Mn , then (2.6) says that φn (R, C) 2 φ R C

(2.7)

for operators R and C in the respective row and column spaces Rown (M ⊗ S) and Coln (M ⊗ S). Now S contains arbitrarily large matrix subfactors and so has no finite dimensional representations. From [21, Proposition 3.4], S norms M ⊗ S. Thus, for each pair X, Y ∈ Mn (M ⊗ S), φn (X, Y ) = sup Rφn (X, Y )C : R ∈ Rown (I ⊗ S), C ∈ Coln (I ⊗ S), R , C 1 = sup φn (RX, Y C): R ∈ Rown (I ⊗ S), C ∈ Coln (I ⊗ S), R , C 1 2 φ X Y ,

(2.8)

where the second equality uses (I ⊗ S)-modularity and the final inequality is (2.7) applied to the row RX and the column Y C. Since n was arbitrary in (2.8), complete boundedness of φ is established by this inequality. 2 In [25], it was shown that, for von Neumann algebras M ⊆ B(H ), every derivation δ : M → B(H ) is automatically bounded and ultraweakly continuous. We will require two further facts about derivations which we quote from the work of Christensen in the next two lemmas. In the first one, our statement is extracted from the proof of [4 ⇒ 2] in the referenced theorem. Lemma 2.3. (See Theorem 3.1 in [3].) Each completely bounded derivation δ : M → B(H ) is inner and is implemented by an operator in B(H ). Lemma 2.4. (See special case of Theorem 5.1 in [2].) If M ⊆ N is an inclusion of finite von Neumann algebras, then each derivation δ : M → N is inner and is implemented by an element of N . 3. Separable algebras In this section we will prove the vanishing of second cohomology for tensor products of type II1 von Neumann algebras under the additional hypothesis that each algebra is separable. In this context, separability of a von Neumann algebra means the existence of a countable ultraweakly dense subset or, equivalently, a faithful normal representation on a separable Hilbert space. If M is a separable type II1 factor, then it was shown in [23] that M has a maximal abelian subalgebra (masa) A and a hyperfinite subfactor R such that A ⊆ R ⊆ M and R ∩ M = C1. This was generalized to separable type II1 von Neumann algebras with the modifications that R is now a hyperfinite von Neumann subalgebra and that R ∩ M is now the center Z(M). This may be found in [24, proof of 3.3] with a complete proof in [30]. Separability is essential for these results and this is the reason for restricting to separable algebras in this section. Throughout we assume that M and N are separable type II1 von Neumann algebras with respective centers Z(M) and Z(N ). We fix choices of masas A and B and hyperfinite type II1 subalgebras R and S so that


A ⊆ R ⊆ M,

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B ⊆ S ⊆ N,

(3.1)

S ∩ N = Z(N ).

(3.2)

and R ∩ M = Z(M),

We also note the trivial fact that centers are always contained in masas. We wish to consider a bounded 2-cocycle φ : (M ⊗ N ) × (M ⊗ N ) → M ⊗ N and show that it is a coboundary. The general reduction results of [28, Chapter 3] allow us to impose the following extra conditions on φ: (C1) φ is separately normal in each variable; (C2) φ(x, y) = 0 whenever x or y lies in R ⊗ S; (C3) φ is R ⊗ S-multimodular. The latter condition is a consequence of (C2), from [28, Lemma 3.2.1], so (C2) is a slightly stronger requirement. We begin by making a further reduction. Lemma 3.1. Let φ be a 2-cocycle on M ⊗ N which satisfies conditions (C1)–(C3). Then φ is equivalent to a 2-cocycle ψ on M ⊗ N satisfying (C1)–(C3) and the additional condition ψ(m1 ⊗ I, m2 ⊗ I ) = ψ(I ⊗ n1 , I ⊗ n2 ) = 0

(3.3)

for m1 , m2 ∈ M, n1 , n2 ∈ N . Proof. Multimodularity with respect to I ⊗ S shows that (I ⊗ s)φ(m1 ⊗ I, m2 ⊗ I ) = φ(m1 ⊗ s, m2 ⊗ I ) = φ(m1 ⊗ I, m2 ⊗ s) = φ(m1 ⊗ I, m2 ⊗ I )(I ⊗ s),

m1 , m2 ∈ M, s ∈ S,

(3.4)

from which it follows that φ(m1 ⊗ I, m2 ⊗ I ) ∈ (I ⊗ S) ∩ (M ⊗ N ) = M ⊗ Z(N ) for all m1 , m2 ∈ M. Note that Z(N) ⊆ B ⊆ S, and so M ⊗ Z(N ) ⊆ M ⊗ S. Since φ(m1 ⊗ s1 , m2 ⊗ s2 ) = φ(m1 ⊗ I, m2 ⊗ I )(I ⊗ s1 s2 )

(3.5)

for m1 , m2 ∈ M, s1 , s2 ∈ S, we conclude that φ maps (M ⊗ S) × (M ⊗ S) to M ⊗ S. Thus the restriction of φ to M ⊗ S is completely bounded by Lemma 2.2. It follows from [9] that there is a normal (R ⊗ S)-modular map α : M ⊗ S → M ⊗ S such that φ|M ⊗ S = ∂α, and a similar argument gives a normal (R ⊗ S)-modular map β : R ⊗ N → R ⊗ N such that φ|R ⊗ N = ∂β. Using the normal conditional expectations EM ⊗ S and ER ⊗ N of M ⊗ N onto M ⊗ S and R ⊗ N ˜ β˜ : M ⊗ N → M ⊗ N by respectively, we now extend α and β to (R ⊗ S)-modular maps α, α˜ = α ◦ EM ⊗ S ,

β˜ = β ◦ ER ⊗ N .

(3.6)

˜ a 2-cocycle equivalent to φ. We verify the desired properties Now define ψ = φ − ∂ α˜ − ∂ β, for ψ . Separate normality is clear from the choices of α and β, so ψ satisfies (C1).

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Since φ satisfies (C2), we have φ(I, I ) = 0. Then the cocycle identity I α(I ) − α(I ) + α(I )I = φ(I, I ) = 0

(3.7)

gives α(I ) = 0, and modularity then implies that α| ˜ R ⊗ S = α|R ⊗ S = 0, with a similar result ˜ A straightforward calculation then shows that ∂ α(x, ˜ y) are both 0 whenever for β. ˜ y) and ∂ β(x, at least one of x and y lies in R ⊗ S. Thus ψ satisfies (C2) and hence (C3). It remains to show that (3.3) is satisfied. We consider only the relation ψ(m1 ⊗ I, m2 ⊗ I ) = 0 for m1 , m2 ∈ M, since the argument for the second is identical. For m ∈ M, ˜ ⊗ I ) = β ER (m) ⊗ I = 0, β(m

(3.8)

˜ M⊗I = 0. Consequently ψ|M⊗I = φ|M⊗I − ∂ α| since β vanishes on R ⊗ S, and thus ∂ β| ˜ M⊗I , and we determine the latter term. For m ∈ M, α(m ˜ ⊗ I ) = α m ⊗ ES (I ) = α(m ⊗ I ),

(3.9)

so ∂ α(m ˜ 1 ⊗ I, m2 ⊗ I ) = ∂α(m1 ⊗ I, m2 ⊗ I ) = φ(m1 ⊗ I, m2 ⊗ I ), since φ = ∂α on M ⊗ S. This shows that (3.3) holds.

m1 , m2 ∈ M,

(3.10)

2

In light of this lemma, we may henceforth assume that the 2-cocycle φ on M ⊗ N not only satisfies (C1)–(C3) but also condition (3.3). We will need to make use of the basic construction for an inclusion P ⊆ Q of finite von Neumann algebras, where Q has a specified normal faithful trace τ . Then Q acts on the Hilbert space L2 (Q, τ ), which we abbreviate to L2 (Q), and its commutant is J QJ , where J is the canonical conjugation. We will use J for all such conjugations, which should be clear from the context. The Hilbert space projection of L2 (Q) onto L2 (P ) is denoted by ep , and the basic construction Q, ep is the von Neumann algebra generated by Q and ep . Since Q, ep = J P J [14], it is clear that Q, ep is hyperfinite precisely when P has this property. For the inclusions A ⊗ B ⊆ R ⊗ S ⊆ M ⊗ N , we obtain an inclusion M ⊗ N, eR ⊗ S ⊆ M ⊗ N, eA ⊗ B of hyperfinite von Neumann algebras. Since J (A ⊗ B)J is a masa in J (M ⊗ N)J , the general theory of extended cobounding [17] or [28], allows us to find a bounded (R ⊗ S)-modular map λ : M ⊗ N → M ⊗ N, eA ⊗ B so that φ = ∂λ. Hyperfiniteness gives a conditional expectation E : M ⊗ N, eA ⊗ B → M ⊗ N, eR ⊗ S , and the (R ⊗ S)-modular composition γ = E ◦ λ : M ⊗ N → M ⊗ N, eR ⊗ S also has the property that φ = ∂γ . Moreover, the results of [13] allow us to further assume that γ is normal. We now introduce three auxiliary linear maps. At the outset these are not obviously bounded, and so can only be defined on the algebraic tensor product M ⊗ N . We define f, g : M ⊗ N → M ⊗ N, eR ⊗ S and h : M ⊗ N → M ⊗ N on elementary tensors m ⊗ n ∈ M ⊗ N by


2701

f (m ⊗ n) = φ(m ⊗ I, I ⊗ n) + γ (m ⊗ n),

(3.11)

g(m ⊗ n) = φ(I ⊗ n, m ⊗ I ) + γ (m ⊗ n),

(3.12)

h(m ⊗ n) = g(m ⊗ n) − f (m ⊗ n) = φ(I ⊗ n, m ⊗ I ) − φ(m ⊗ I, I ⊗ n).

(3.13)

The next lemma lists some basic properties of these maps. Lemma 3.2. The following properties hold: (i) The restrictions γ |M ⊗ S and γ |R ⊗ N are completely bounded derivations, spatially implemented by elements of M ⊗ N, eR ⊗ S . (ii) The restrictions f |M⊗I , f |I ⊗N , g|M⊗I , and g|I ⊗N are equal to the respective restrictions of γ to these subalgebras, and are all bounded derivations spatially implemented by elements of M ⊗ N, eR ⊗ S . (iii) The restrictions h|M⊗I and h|I ⊗N are both 0. Proof. (i) We consider only γ |M ⊗ S , the other case being similar. Since φ|M⊗I = 0, from (3.3), and φ = ∂γ , we see that γ |M⊗I is a derivation. The (R ⊗ S)-modularity then implies that γ |M⊗S is a derivation, with the same conclusion for γ |M ⊗ S by normality of γ . Since γ is, in particular, (I ⊗ S)-modular, complete boundedness of γ |M ⊗ S follows from Lemma 2.1. Thus γ |M ⊗ S is implemented by an operator t ∈ B(L2 (M ⊗ N )), from Lemma 2.3. By hyperfiniteness of M ⊗ N, eR ⊗ S , there is a conditional expectation E of B(L2 (M ⊗ N )) onto this subalgebra, and so γ |M ⊗ S is also implemented by E(t) ∈ M ⊗ N, eR ⊗ S . (ii) From (3.11) and (C2), f (m ⊗ I ) = φ(m ⊗ I, I ⊗ I ) + γ (m ⊗ I ) = γ (m ⊗ I ),

m ∈ M,

(3.14)

so f |M⊗I = γ |M⊗I is a derivation on M ⊗ I spatially implemented by an element of M ⊗ N, eR ⊗ S from (i). The other three restrictions are handled similarly. (iii) From (ii) h|M⊗I = g|M⊗I − f |M⊗I = γ |M⊗I − γ |M⊗I = 0, with a similar result for h|I ⊗N .

2

Proposition 3.3. The map f of (3.11) is a derivation on M ⊗ N . Proof. For m ∈ M and n ∈ N , Lemma 3.2(ii) gives (m ⊗ I )f (I ⊗ n) + f (m ⊗ I )(I ⊗ n) = (m ⊗ I )γ (I ⊗ n) + γ (m ⊗ I )(I ⊗ n) = (m ⊗ I )γ (I ⊗ n) − γ (m ⊗ n) + γ (m ⊗ I )(I ⊗ n) + γ (m ⊗ n)

(3.15)

2702


= φ(m ⊗ I, I ⊗ n) + γ (m ⊗ n) = f (m ⊗ n),

(3.16)

using φ = ∂γ . A similar calculation leads to (I ⊗ n)g(m ⊗ I ) + g(I ⊗ n)(m ⊗ I ) = g(m ⊗ n).

(3.17)

We now use (3.16) and (3.17) to calculate ∂f on pairs of elementary tensors, noting that f is a derivation on M ⊗ I and I ⊗ N . For m1 , m2 ∈ M, n1 , n2 ∈ N , ∂f (m1 ⊗ n1 , m2 ⊗ n2 ) = (m1 ⊗ n1 )f (m2 ⊗ n2 ) − f (m1 m2 ⊗ n1 n2 ) + f (m1 ⊗ n1 )(m2 ⊗ n2 ) = (m1 ⊗ n1 ) (m2 ⊗ I )f (I ⊗ n2 ) + f (m2 ⊗ I )(I ⊗ n2 ) − (m1 m2 ⊗ I )f (I ⊗ n1 n2 ) + f (m1 m2 ⊗ I )(I ⊗ n1 n2 ) + (m1 ⊗ I )f (I ⊗ n1 ) + f (m1 ⊗ I )(I ⊗ n1 ) (m2 ⊗ n2 ) = (m1 m2 ⊗ n1 )f (I ⊗ n2 ) + (m1 ⊗ n1 )f (m2 ⊗ I )(I ⊗ n2 ) − (m1 m2 ⊗ I ) (I ⊗ n1 )f (I ⊗ n2 ) + f (I ⊗ n1 )(I ⊗ n2 ) − (m1 ⊗ I )f (m2 ⊗ I ) + f (m1 ⊗ I )(m2 ⊗ I ) (I ⊗ n1 n2 ) + (m1 ⊗ I )f (I ⊗ n1 )(m2 ⊗ n2 ) + f (m1 ⊗ I )(m2 ⊗ n1 n2 ) = (m1 m2 ⊗ n1 )f (I ⊗ n2 ) + (m1 ⊗ n1 )f (m2 ⊗ I )(I ⊗ n2 ) − (m1 m2 ⊗ n1 )f (I ⊗ n2 ) − (m1 m2 ⊗ I )f (I ⊗ n1 )(I ⊗ n2 ) − (m1 ⊗ I )f (m2 ⊗ I )(I ⊗ n1 n2 ) − f (m1 ⊗ I )(m2 ⊗ n1 n2 ) + (m1 ⊗ I )f (I ⊗ n1 )(m2 ⊗ n2 ) + f (m1 ⊗ I )(m2 ⊗ n1 n2 ) = (m1 ⊗ I ) (I ⊗ n1 )f (m2 ⊗ I ) + f (I ⊗ n1 )(m2 ⊗ I ) (I ⊗ n2 ) − (m1 ⊗ I ) (m2 ⊗ I )f (I ⊗ n1 ) + f (m2 ⊗ I )(I ⊗ n1 ) (I ⊗ n2 ).

(3.18)

Recalling that f , g and γ agree on M ⊗ I and I ⊗ N , while φ = ∂γ , (3.18) becomes ∂f (m1 ⊗ n1 , m2 ⊗ n2 ) = (m1 ⊗ I ) (I ⊗ n1 )γ (m2 ⊗ I ) + γ (I ⊗ n1 )(m2 ⊗ I ) (I ⊗ n2 ) − (m1 ⊗ I ) (m2 ⊗ I )γ (I ⊗ n1 ) + γ (m2 ⊗ I )(I ⊗ n1 ) (I ⊗ n2 ) = (m1 ⊗ I ) φ(I ⊗ n1 , m2 ⊗ I ) + γ (m2 ⊗ n1 ) (I ⊗ n2 ) − (m1 ⊗ I ) φ(m2 ⊗ I, I ⊗ n1 ) + γ (m2 ⊗ n1 ) (I ⊗ n2 ) = (m1 ⊗ I ) g(m2 ⊗ n1 ) − f (m2 ⊗ n1 ) (I ⊗ n2 ) = (m1 ⊗ I )h(m2 ⊗ n1 )(I ⊗ n2 ).

(3.19)


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Here we have used the relations (3.11)–(3.13). Now define F = ∂f . Then (3.19) is F (m1 ⊗ n1 , m2 ⊗ n2 ) = (m1 ⊗ I )h(m2 ⊗ n1 )(I ⊗ n2 ).

(3.20)

The identity ∂F = 0 for the triple (I ⊗ n1 , m2 ⊗ I, m3 ⊗ I ) yields (m2 ⊗ n1 )h(m3 ⊗ I ) − (m2 ⊗ I )h(m3 ⊗ n1 ) + h(m2 m3 ⊗ n1 ) − h(m2 ⊗ n1 )(m3 ⊗ I ) = 0, (3.21) and so h(m2 m3 ⊗ n1 ) = (m2 ⊗ I )h(m3 ⊗ n1 ) + h(m2 ⊗ n1 )(m3 ⊗ I )

(3.22)

since h|M⊗I = 0. It follows from (3.22) that, for each fixed n1 ∈ N , the map δ(m ⊗ I ) = h(m ⊗ n1 ), m ∈ M, defines a derivation of M ⊗ I into M ⊗ N . Since M ⊗ N is finite, δ is implemented by an element a ∈ M ⊗ N by Lemma 2.4. For r ∈ R, δ(r ⊗ I ) = h(r ⊗ n1 ) = φ(I ⊗ n1 , r ⊗ I ) − φ(r ⊗ I, I ⊗ n1 ) = 0,

(3.23)

from (3.13) and (C2). Thus a ∈ (R ⊗ I ) ∩ (M ⊗ N ) = Z(M) ⊗ N , so a commutes with M ⊗ I . We conclude that h(m ⊗ n1 ) = 0 for m ∈ M. Since n1 ∈ N was arbitrary, h = 0 and, from (3.19), ∂f = 0. This shows that f is a derivation on the algebraic tensor product M ⊗ N . 2 Proposition 3.4. There exists a bounded normal map ξ : M ⊗ N → M ⊗ N such that ξ(m ⊗ n) = φ(m ⊗ I, I ⊗ n),

m ∈ M, n ∈ N.

(3.24)

Proof. From Proposition 3.3, f is a derivation on M ⊗ N with values in M ⊗ N, eR ⊗ S = M, eR ⊗N, eS . By Lemma 3.2(ii), f |M⊗I is a completely bounded derivation implemented by an element t ∈ M, eR ⊗N, eS . Define a derivation δ : M ⊗ N → M, eR ⊗N, eS by δ(m ⊗ n) = f (m ⊗ n) − t (m ⊗ n) − (m ⊗ n)t ,

m ∈ M, n ∈ N.

(3.25)

Then δ|M⊗I = 0 from (3.25), so δ is (M ⊗ I )-modular. From Lemma 3.2(ii), f |1⊗N is a derivation implemented by an element of M, eR ⊗N, eS , so from (3.25) there is an element b in this algebra such that δ(I ⊗ n) = b(I ⊗ n) − (I ⊗ n)b,

n ∈ N.

(3.26)

The (M ⊗ I )-modularity of δ shows that (m ⊗ I )δ(I ⊗ n) = δ(m ⊗ n) = δ(I ⊗ n)(m ⊗ I ),

m ∈ M, n ∈ N,

(3.27)

and we conclude that the range of δ|I ⊗N lies in (M ⊗ I ) ∩ M, eR ⊗N, eS . This algebra is (M ∩ M, eR ) ⊗N, eS , equal to (J MJ ∩ (J RJ ) ) ⊗N, eS , and in turn equal to (J Z(M)J ) ⊗N, eS . The latter algebra is hyperfinite, so if we take a conditional expectation

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onto it and apply this to (3.26), then we conclude that the element b of (3.26) may be assumed to lie in (J Z(M)J ) ⊗N, eS . Then b commutes with M ⊗ I , so δ(m ⊗ n) = (m ⊗ I )δ(I ⊗ n) = (m ⊗ I ) b(I ⊗ n) − (I ⊗ n)b = b(m ⊗ n) − (m ⊗ n)b,

m ∈ M, n ∈ N.

(3.28)

Thus δ has a unique bounded normal extension to M ⊗ N , and (3.25) shows that the same is then true for f . Since ξ = f − γ on M ⊗ N from (3.11), and γ is already bounded and normal on M ⊗ N , this gives a bounded normal extension of ξ to M ⊗ N . 2 Remark 3.5. Eq. (3.25) shows that the derivation f on M ⊗ N has a unique normal extension to M ⊗ N . Taking ultraweak limits in the equation f (xy) = xf (y) + f (x)y,

x, y ∈ M ⊗ N,

(3.29)

shows that this extension is also a derivation on M ⊗ N . We now come to the main result of this section. Theorem 3.6. Let M and N be separable type II1 von Neumann algebras. Then H 2 (M ⊗ N, M ⊗ N ) = 0.

(3.30)

Proof. We have already reduced consideration of a general cocycle φ to one which satisfies (C1)–(C3) and (3.3). With the previously established notation, Proposition 3.4 and Remark 3.5 show that ξ and f have bounded normal extensions from M ⊗ N to M ⊗ N . Using the same letters for the extensions, we see that ξ maps M ⊗ N to itself, while f is a derivation on M ⊗ N from Remark 3.5. Thus φ = ∂γ = ∂f − ∂ξ = ∂(−ξ )

(3.31)

on (M ⊗ N ) × (M ⊗ N ). This shows that φ is a coboundary with respect to the bounded linear map −ξ , proving the result. 2 Remark 3.7. We will require one more piece of information about maps ξ on M ⊗ N for which φ = ∂ξ , namely that they can be chosen so that ξ C φ for an absolute constant C. The argument is already essentially in [28, Lemma 6.5.1], so we only sketch it here. If no such C existed, then it would be possible to find separable type II1 algebras Mn and Nn for n 1, and cocycles φn on Mn ⊗ Nn of norm 1 so

that any ξn satisfying

φ∞n = ∂ξn necessarily had norm at least n. Form separable algebras M = ∞ n=1 Mn and N = n=1 Nn and define a cocycle φ on M ⊗ N by φ(mi ⊗ nj , m ˜ k ⊗ n˜ ) = φi (mi ⊗ ni , m ˜ i ⊗ n˜ i ) when i = j = k = , and 0 otherwise. By Theorem 3.6 there exists a bounded map ξ on M ⊗ N so that φ = ∂ξ (which can be assumed to be Z(M ⊗ N )-modular), but this would then contradict the lower bounds on ξn by restricting ξ to the component algebras.


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4. The general case The techniques of Section 3 relied heavily on the existence of hyperfinite subalgebras whose relative commutants are the center, and these are only guaranteed to exist in the separable case. We will use Theorem 3.6 and Remark 3.7 to derive the general result, but we require some preliminary lemmas. A complication for a general type II1 von Neumann algebra M is that it need not have a faithful normal trace. However, a standard maximality argument gives a family of central projections pλ with sum I such that each Mpλ has such a trace. Until we reach Theorem 4.5, we restrict attention to those algebras which do have faithful normal traces. Lemma 4.1. Let M and N be type II1 von Neumann algebras with faithful normal unital traces τM and τN respectively, and let Q ⊆ M ⊗ N be a separable von Neumann subalgebra. Then there exist separable type II1 von Neumann subalgebras M0 ⊆ M and N0 ⊆ N such that Q ⊆ M0 ⊗ N 0 . Proof. We may certainly assume that Q contains arbitrarily large matrix subalgebras of M ⊗ I and I ⊗ N , and this will guarantee that the M0 and N0 that we construct are type II1 . Let τ = τM ⊗ τN be a faithful normal unital trace on M ⊗ N , and fix a countable ultraweakly dense sequence {qn }∞ n=1 in the unit ball of Q. The ultrastrong and · 2 -norm topologies are equivalent on the unit ball of M ⊗ N , and so the qn ’s are the · 2 -limits of sequences from M ⊗ N , each element of which is a finite sum of elementary tensors. Let M0 (respectively N0 ) be the von Neumann algebra generated by the first (respectively second) entries in all of these elementary tensors. Each is separable. Then L2 (Q) ⊆ L2 (M0 ⊗ N0 ) and so Q ⊆ M0 ⊗ N0 by considering the conditional expectation of M ⊗ N onto M0 ⊗ N0 , which also defines the Hilbert space projection of L2 (M ⊗ N ) onto L2 (M0 ⊗ N0 ) (see the proof of [1, Lemma 2.2]). 2 Lemma 4.2. Let M and N be type II1 von Neumann algebras with faithful normal unital traces τM and τN respectively. Let φ be a separately normal bounded bilinear map from (M ⊗ N) × (M ⊗ N ) to M ⊗ N . Given a finite set F ⊆ M ⊗ N , there exist separable type II1 von Neumann subalgebras MF ⊆ M and NF ⊆ N such that F ⊆ MF ⊗ NF and φ maps (MF ⊗ NF ) × (MF ⊗ NF ) to (MF ⊗ NF ). Proof. We apply Lemma 4.1 repeatedly. Let Q0 be the von Neumann generated by F and choose separable von Neumann algebras M0 ⊆ M and N0 ⊆ N so that Q0 ⊆ M0 ⊗ N0 . Then let Q1 be the von Neumann algebra generated by M0 ⊗ N0 and the range of φ|M0 ⊗ N0 ; the separate normality of φ ensures that Q1 is separable. Now choose separable von Neumann algebras so that Q1 ⊆ M1 ⊗ N1 . By construction, φ maps (M0 ⊗ N0 ) × (M0 ⊗ N0 ) into M1 ⊗ N1 . Continuing in this way, we obtain an ascending sequence {Mi ⊗ Ni }∞ i=0 of separable von Neumann algebras so that φ maps (Mi ⊗ Ni ) × (Mi ⊗ Ni ) into M ⊗ N i+1 . Define MF and NF as the respec∞ i+1 tive ultraweak closures of ∞ M and N . Then separate normality shows that φ maps i=0 i i=0 i (MF ⊗ NF ) × (MF ⊗ NF ) into MF ⊗ NF as required. 2 The next result is a special case of the subsequent main result.

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Proposition 4.3. Let M and N be type II1 von Neumann algebras with faithful normal unital traces. Then H 2 (M ⊗ N, M ⊗ N ) = 0.

(4.1)

Proof. Theorem 3.3.1 of [28] allows us to restrict attention to a separately normal 2-cocycle φ on M ⊗ N . For each finite subset F of M ⊗ N , let MF and NF be the separable von Neumann subalgebras constructed in Lemma 4.2, so that φ maps (MF ⊗ NF ) × (MF ⊗ NF ) to MF ⊗ NF . Let φF be the restriction of φ to this subalgebra. By Theorem 3.6, there is a bounded linear map ξF : MF ⊗ NF → MF ⊗ NF so that φF = ∂ξF , and Remark 3.7 allows us to assume a uniform bound on ξF independent of F . The construction of a bounded map ξ : M ⊗ N → M ⊗ N such that φ = ∂ξ now follows the proof of [28, Theorem 6.5.3]. 2 Remark 4.4. An examination of the proof of [28, Theorem 6.5.3] combined with Remark 3.7 shows the existence of an absolute constant K so that, under the hypotheses of Proposition 4.3, to each 2-cocycle φ on M ⊗ N there corresponds a bounded map ξ on M ⊗ N satisfying φ = ∂ξ and ξ K φ . The final step is to remove the hypothesis of faithful traces from Proposition 4.3. Theorem 4.5. Let M and N be type II1 von Neumann algebras. Then H 2 (M ⊗ N, M ⊗ N ) = 0.

(4.2)

Proof. As noted earlier, there are orthogonal sets of central projections pλ ∈ Z(M) and qμ ∈ Z(N ), each summing to I , so that Mpλ and N qμ have faithful normal unital traces. Given a separately normal 2-cocycle φ on M ⊗ N , [28, Theorem 3.2.7] allows us to assume that it is Z(M ⊗ N)-multimodular. Thus the restriction φλ,μ of φ to Mpλ ⊗ N qμ maps back to this algebra. By Proposition 4.3 and Remark 4.4, there are maps ξλ,μ : Mpλ ⊗ N qμ → Mpλ ⊗ N qμ so that φλ,μ = ∂ξλ,μ with a uniform bound on ξλ,μ . This allows us to define a bounded map

ξ : M ⊗ N → M ⊗ N by ξ = λ,μ ξλ,μ , and then φ = ∂ξ . 2 References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

J. Cameron, Hochschild Cohomology of II1 factors with Cartan masas, Proc. Edinb. Math. Soc. 52 (2009) 287–295. E. Christensen, Extension of derivations, J. Funct. Anal. 27 (1978) 234–247. E. Christensen, Extensions of derivations II, Math. Scand. 50 (1982) 111–122. E. Christensen, Finite von Neumann algebra factors with property Γ , J. Funct. Anal. 186 (2001) 366–380. E. Christensen, E.G. Effros, A.M. Sinclair, Completely bounded multilinear maps and C ∗ -algebraic cohomology, Invent. Math. 90 (1987) 279–296. E. Christensen, F. Pop, A.M. Sinclair, R.R. Smith, On the cohomology groups of certain finite von Neumann algebras, Math. Ann. 307 (1997) 71–92. E. Christensen, F. Pop, A.M. Sinclair, R.R. Smith, Hochschild cohomology of factors with property Γ , Ann. of Math. (2) 158 (2003) 635–659. E. Christensen, A.M. Sinclair, Representations of completely bounded multilinear operators, J. Funct. Anal. 72 (1987) 151–181. E. Christensen, A.M. Sinclair, On the Hochschild cohomology for von Neumann algebras, unpublished manuscript. E.G. Effros, E.C. Lance, Tensor products of operator algebras, Adv. Math. 25 (1977) 1–34. E.G. Effros, Z.-J. Ruan, Operator Spaces, London Math. Soc. Monogr. Ser., vol. 23, The Clarendon Press/Oxford Univ. Press, New York, 2000.


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[12] L. Ge, Applications of free entropy to finite von Neumann algebras. II, Ann. of Math. (2) 147 (1998) 143–157. [13] B.E. Johnson, R.V. Kadison, J.R. Ringrose, Cohomology of operator algebras. III. Reduction to normal cohomology, Bull. Soc. Math. France 100 (1972) 73–96. [14] V.F.R. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. [15] R.V. Kadison, Derivations of operator algebras, Ann. of Math. (2) 83 (1966) 280–293. [16] R.V. Kadison, J.R. Ringrose, Cohomology of operator algebras I. Type I von Neumann algebras, Acta Math. 126 (1971) 227–243. [17] R.V. Kadison, J.R. Ringrose, Cohomology of operator algebras II. Extended cobounding and the hyperfinite case, Ark. Mat. 9 (1971) 55–63. [18] E. Kirchberg, The derivation problem and the similarity problem are equivalent, J. Operator Theory 36 (1996) 59–62. [19] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge Univ. Press, Cambridge, UK, 2002. [20] G. Pisier, Introduction to Operator Space Theory, London Math. Soc. Lecture Note Ser., vol. 294, Cambridge Univ. Press, Cambridge, UK, 2003. [21] F. Pop, A.M. Sinclair, R.R. Smith, Norming C ∗ -algebras by C ∗ -subalgebras, J. Funct. Anal. 175 (2000) 168–196. [22] F. Pop, R.R. Smith, Cohomology for certain finite factors, Bull. London Math. Soc. 26 (1994) 303–308. [23] S. Popa, On a problem of R.V. Kadison on maximal abelian ∗-subalgebras in factors, Invent. Math. 65 (1981) 269–281. [24] S. Popa, Hyperfinite subalgebras normalized by a given automorphism and related problems, in: Operator Algebras and Their Connections with Topology and Ergodic Theory, Bu¸steni, 1983, in: Lecture Notes in Math., vol. 1132, Springer, Berlin, 1985, pp. 421–433. [25] J.R. Ringrose, Automatic continuity of derivations of operator algebras, J. London Math. Soc. (2) 5 (1972) 432–438. [26] J.R. Ringrose, Cohomology of operator algebras, in: Lectures on Operator Algebras, Tulane Univ. Ring and Operator Theory Year, 1970–1971, vol. II, Dedicated to the Memory of David M. Topping, in: Lecture Notes in Math., vol. 247, Springer, Berlin, 1972, pp. 355–434. [27] S. Sakai, Derivations of W ∗ -algebras, Ann. of Math. (2) 83 (1966) 273–279. [28] A.M. Sinclair, R.R. Smith, Hochschild Cohomology of von Neumann Algebras, London Math. Soc. Lecture Note Ser., vol. 203, Cambridge Univ. Press, Cambridge, UK, 1995. [29] A.M. Sinclair, R.R. Smith, Hochschild cohomology for von Neumann algebras with Cartan subalgebras, Amer. J. Math. 120 (1998) 1043–1057. [30] A.M. Sinclair, R.R. Smith, Cartan subalgebras of finite von Neumann algebras, Math. Scand. 85 (1999) 105–120. [31] A.M. Sinclair, R.R. Smith, Finite von Neumann Algebras and Masas, London Math. Soc. Lecture Note Ser., vol. 351, Cambridge Univ. Press, Cambridge, UK, 2008.


Malnormal subgroups of lattices and the Pukánszky invariant in group factors Guyan Robertson a,∗ , Tim Steger b a School of Mathematics and Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, England, UK b Istituto Di Matematica e Fisica, Università degli Studi di Sassari, Via Vienna 2, 07100 Sassari, Italy

Received 22 July 2009; accepted 28 August 2009 Available online 10 September 2009 Communicated by D. Voiculescu

Abstract Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A. This implies that the II1 factor VN(Γ ) contains a masa A with Pukánszky invariant {∞}. © 2009 Published by Elsevier Inc. Keywords: von Neumann algebra; Semisimple group

1. Introduction A subgroup Γ0 of a group Γ is malnormal if xΓ0 x −1 ∩Γ0 = {1} for all x ∈ Γ −Γ0 . An abelian malnormal subgroup is necessarily maximal abelian. The main result of this article is Theorem 1.1, which rests upon work of Prasad and Rapinchuk [11]. Theorem 1.1. Let G be a connected semisimple real algebraic group and let d be the R-rank of G. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Then Γ contains a malnormal abelian subgroup A ∼ = Zd . * Corresponding author.

E-mail addresses: [email protected] (G. Robertson), [email protected] (T. Steger). 0022-1236/$ – see front matter © 2009 Published by Elsevier Inc. doi:10.1016/j.jfa.2009.08.016

G. Robertson, T. Steger / Journal of Functional Analysis 258 (2010) 2708–2713

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Theorem 1.1 will be applied to the group factor VN(Γ ). Recall that if Γ is a group, then the von Neumann algebra VN(Γ ) is the convolution algebra VN(Γ ) = f ∈ 2 (Γ ): f 2 (Γ ) ⊆ 2 (Γ ) . It is well known that if Γ is an infinite conjugacy class group then VN(Γ ) is a factor of type II1 . This is true if Γ is a lattice in a semisimple Lie group [4, Lemma 3.3.1]. If Γ0 is a subgroup of Γ , then VN(Γ0 ) embeds naturally as a subalgebra of VN(Γ ) via f → f¯, where f (x) ¯ f (x) = 0

if x ∈ Γ0 , otherwise.

This article is concerned with examples where Γ0 = A is an abelian subgroup of Γ and A = VN(A) is a maximal abelian -subalgebra (masa) of M = VN(Γ ). Recall that A is the von Neumann subalgebra of B(2 (Γ )) defined by the left convolution operators λ(f ) : φ → f φ where f ∈ 2 (A) and f 2 (Γ ) ⊆ 2 (Γ ). The algebra A also acts on 2 (Γ ) by right convolution ρ(f ) : φ → φ f. Let Aopp be the von Neumann subalgebra of B(2 (Γ )) defined by this right action of A. Let B be the von Neumann subalgebra of B(2 (Γ )) generated by A ∪ Aopp and let p denote the orthogonal projection of 2 (Γ ) onto the closed subspace generated by A. Then p is in the centre of the commutant B , and B p is abelian. The von Neumann algebra B (1 − p) is of type I and may therefore be expressed as a direct sum Bn1 ⊕ Bn2 ⊕ · · · of algebras Bni of type Ini , where 1 n1 < n2 < · · · ∞. The Pukánszky invariant [15, Chapter 7] is the set {n1 , n2 , . . .}. It is an isomorphism invariant of the pair (A, M). It has been shown [8, Corollary 3.3] that each nonempty subset S of the natural numbers containing 1 can be realized as the Pukánszky invariant of some masa in the hyperfinite II1 factor R. This was extended [3,14] to subsets S containing ∞ for R and for the free group factor. It was later extended [16] to arbitrary subsets S ⊂ N ∪ {∞} for R (and for certain other McDuff factors). It is known that every factor of type II1 contains a singular masa [9]. S. Popa [10, Remark 3.4] showed that if the Pukánszky invariant of (A, M) does not contain 1, then A is a singular masa in M. K. Dykema [2] has shown (using Voiculescu’s free entropy dimension) that the Pukánszky invariant of any masa in the free group factor must either contain ∞ or be unbounded. This means that it is not possible for any singleton other than {∞} to be a possible Pukánszky invariant occurring in every II1 factor. Jolissaint [5] has shown that if F0 is the cyclic subgroup generated by the first generator of Thompson’s group F then VN(F0 ) has Pukánszky invariant {∞}. A natural question arises. • Does every II1 factor M contain a masa A with Pukánszky invariant {∞}? This article uses Theorem 1.1 to provide an affirmative answer for M = VN(Γ ), where Γ is a torsion-free uniform lattice subgroup of a connected semisimple real algebraic group G without compact factors. If G has R-rank 2, then VN(Γ ) has Kazhdan’s property T . This is the first result on possible values of the Pukánszky invariant in an II1 factor with property T .

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Many thanks are due to G. Prasad and Y. Shalom for their assistance with the proofs of Theorem 1.1 and Theorem 3.2, respectively. S. White provided valuable background information. 2. Malnormal abelian subgroups of lattices This section is devoted to the proof of Theorem 1.1. Let G be a connected semisimple real algebraic group and let d be the R-rank of G. Assume that G(R) has no compact factors and let Γ be a torsion-free uniform lattice subgroup of G(R). Since Γ is finitely generated, Γ < G(K) for some finitely generated subfield K of R. By the Borel Density Theorem [6, Chapter II, Corollary 4.4], Γ is Zariski dense in G(R). Therefore, according to Theorems 1 and 2 of [11], there exists a maximal abelian torus subgroup T of G with the following properties: (1) The R-rank of T is d. (2) A = T (R) ∩ Γ is a uniform lattice in T (R). (3) T has no proper algebraic subgroups defined over K. Moreover, T is the K-Zariski closure of a single R-regular element x0 in A [11]. In fact there are many such elements x0 [11, Remark 2]. We claim that A is a malnormal subgroup of Γ . To this end, fix an arbitrary element x ∈ Γ − A. We must show that xAx −1 ∩ A = {1}. Let Ts (respectively Ta ) be the maximal R-split (respectively R-anisotropic) subtorus of T . Then T = Ts · Ta (an almost direct product) [1, Proposition 8.15] and Ts is a maximal R-split torus in G [11, Remark 1]. Thus T (R) = S ·C, where S = Ts (R) ∼ = Rd and C = Ta (R) ∼ = (R/Z)r , where r is the dimension of Ta . Since Γ is torsion free and discrete, A is a uniform lattice in S. In particular, A ∼ = Zd . Since T is the K-Zariski closure of A, it follows that T is defined over K. Since x ∈ Γ ⊆ G(K), xT x −1 is also defined over K. According to condition (3), there are only two possibilities: • T ∩ xT x −1 = {1}; • T = xT x −1 . In the first case we also have T (K) ∩ xT (K)x −1 = {1}, and a fortiori A ∩ xAx −1 = {1}, as required. To show that the second case does not occur, assume that T = xT x −1 . This implies that T (R) is stable under conjugation by x. Also Γ is stable under conjugation by x. Therefore xAx −1 = A, since A = Γ ∩ T (R). There are two possibilities to consider for the action αx : a → xax −1 on A. (a) αx fixes only the trivial element of A; (b) αx fixes some nontrivial element of A. Since conjugation by x stabilizes T , it also stabilizes Ts and Ta separately [1, Proposition 8.15(3)]. Thus xSx −1 = S. The symmetric space of G(R) is X = G(R)/K, where K is a maximal compact subgroup of G(R). The group Γ acts freely on X, since it is torsion free. Since Ts is a maximal R-split torus of G, there is a unique flat F in X such that SF = F and S acts simply transitively on F [7, Lemma 5.1]. Now xF is another such flat, since SxF = xSx −1 xF = xSF = xF. Hence xF = F .


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The action of x on F is by some rigid motion and the action of A on F is by translations. No nontrivial element in Γ can act trivially on F , so we can calculate the conjugation by x of any element y ∈ A by considering the actions of these elements on F . The two cases above correspond to: (a) x acts on F by a rigid motion whose linear part has trivial 1-eigenspace; (b) x acts on F by a rigid motion whose linear part has nontrivial 1-eigenspace. In case (a), x necessarily has a fixed point in F . Therefore x = 1, since Γ acts freely on X. Consider case (b). The algebraic subgroup ZG (x) ∩ T , which consists of the elements commuting with x, is defined over K. In case (b), ZG (x) ∩ T contains nontrivial elements of A. That is, it has nontrivial K-points. Hence, it must be nontrivial (as an algebraic group). By condition (3) it must be all of T . Hence x commutes with every element of T . Therefore the algebraic closure of {x} ∪ T over K is commutative. However T is a maximal abelian subgroup over K, and so x ∈ T (K). Therefore x ∈ A, contrary to assumption. This completes the proof of Theorem 1.1. 3. The Pukánszky invariant The following result was proved in [13, Proposition 3.6] and later extended in [14, Theorem 4.1]. Proposition 3.1. Suppose that A is an abelian subgroup of a countable group Γ such that A = VN(A) is a masa of VN(Γ ). If A is malnormal in Γ then the Pukánszky invariant of A contains precisely one element n = #(A\Γ /A − {A}). In view of Theorem 1.1, the next result is enough to provide examples of masas with Pukánszky invariant {∞}. Theorem 3.2. Let G be a connected semisimple real algebraic group. Assume that G(R) has no compact factors. Let Γ be a torsion free uniform lattice subgroup of G(R) and let A < Γ be an abelian subgroup. Then #(A\Γ /A) = ∞. Proof. Suppose that #(A\Γ /A) < ∞. Then Γ =

AxA

x∈F

where F ⊂ Γ is finite. Taking Zariski closures, it follows from the Borel Density Theorem that G(R) = Γ¯ =

x∈F

AxA.

(1)

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¯ A¯ is locally closed in the Zariski topology, since it is an orbit of A¯ acting on For each x ∈ F , Ax ¯ A¯ = U ∩ E where U is Zariski-open and E is G(R)/A¯ [17, Corollary 3.1.5]. This means that Ax Zariski closed. Therefore ¯ A) ¯ ∪ G(R) \ U . AxA = U ∩ E ⊆ E ⊆ (U ∩ E) ∪ G(R) \ U = (Ax Since U ⊆ G(R) is Zariski open (and G(R) is Zariski connected), G(R)\U has measure zero, relative to Haar measure μ on G(R) [6, Chapter I, Proposition 2.5.3]. Therefore, ¯ A). ¯ μ(AxA) = μ(Ax

(2)

¯ A) ¯ = 0. Each element of A is semisimple, since each element of Γ Now we show that μ(Ax is [7, Section 11]. Therefore each element of the Zariski closure A¯ is also semisimple; in other words, A¯ is a torus subgroup. The dimension of A¯ as a Lie group is no larger than the absolute rank d of G (the rank over C of the Lie algebra of G). The number of positive roots of the compexified Lie algebra of G is at least d , and the total number of roots is at least 2d . Thus the total dimension of the root spaces is at least 2d . This means that d is at most one third of the dimension of G. Thus the dimension of A¯ × A¯ is at most two thirds the dimension of G. The map (a1 , a2 ) → a1 xa2 from A¯ × A¯ to G(R) is C ∞ (in fact polynomial). Therefore, by the above ¯ A) ¯ = 0. However, dimension count, its image has measure zero. It follows from (2) that μ(Ax this contradicts (1), thereby proving the result. 2 An immediate consequence of Theorems 1.1 and 3.2 is Corollary 3.3. Let G be a connected semisimple real algebraic group such that G(R) has no compact factors. Let Γ be a uniform lattice subgroup of G(R). Then there exists an abelian subgroup A < Γ such that VN(A) is a masa of VN(Γ ) with Pukánszky invariant {∞}. Proof. This follows immediately from Proposition 3.1.

2

Remark 3.4. A similar result was obtained by geometrical methods in [12, Theorem 4.6], if Γ is the fundamental group of a compact locally symmetric space M of constant negative curvature and A is generated by the homotopy class of a simple closed geodesic in M. References [1] A. Borel, Linear Algebraic Groups, 2nd edition, Springer-Verlag, New York, 1991. [2] K. Dykema, Two applications of free entropy, Math. Ann. 308 (1997) 547–558. [3] K.J. Dykema, A.M. Sinclair, R.R. Smith, Values of the Pukánszky invariant in free group factors and the hyperfinite factor, J. Funct. Anal. 240 (2006) 373–398. [4] F. Goodman, P. de la Harpe, V. Jones, Coxeter Graphs and Towers of Algebras, Springer-Verlag, New York, 1989. [5] P. Jolissaint, Operator algebras related to Thompson’s group F , J. Aust. Math. Soc. 79 (2005) 231–241. [6] G.A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991. [7] G.D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Princeton University Press, New Jersey, 1973. [8] C. Neshveyev, E. Størmer, Ergodic theory and maximal abelian subalgebras of the hyperfinite factor, J. Funct. Anal. 195 (2002) 239–261. [9] S. Popa, Singular maximal abelian ∗-subalgebras in continuous von Neumann algebras, J. Funct. Anal. 50 (1983) 151–166. [10] S. Popa, Notes on Cartan subalgebras in type II 1 factors, Math. Scand. 57 (1985) 171–188.


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[11] G. Prasad, A.S. Rapinchuk, Existence of irreducible R-regular elements in Zariski-dense subgroups, Math. Res. Lett. 10 (2003) 21–32. [12] G. Robertson, Abelian subalgebras of von Neumann algebras from flat tori in locally symmetric spaces, J. Funct. Anal. 230 (2006) 419–431. [13] G. Robertson, T. Steger, Maximal abelian subalgebras of the group factor of an A˜ 2 group, J. Operator Theory 36 (1996) 317–334. [14] A.M. Sinclair, R.R. Smith, The Pukánszky invariant for masas in group von Neumann factors, Illinois J. Math. 49 (2003) 325–343. [15] A.M. Sinclair, R.R. Smith, Finite von Neumann Algebras and Masas, Cambridge University Press, Cambridge, 2008. [16] S. White, Values of the Pukánszky invariant in McDuff factors, J. Funct. Anal. 254 (2008) 612–631. [17] R.L. Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, Boston, 1985.


Direct limits, multiresolution analyses, and wavelets ✩ Lawrence W. Baggett a , Nadia S. Larsen b , Judith A. Packer a , Iain Raeburn c,∗ , Arlan Ramsay a a Department of Mathematics, University of Colorado, Boulder, CO 80309, USA b Mathematics Institute, University of Oslo, Blindern, NO-0316 Oslo, Norway c School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Australia

Received 22 July 2009; accepted 28 August 2009 Available online 16 September 2009 Communicated by D. Voiculescu

Abstract A multiresolution analysis for a Hilbert space realizes the Hilbert space as the direct limit of an increasing sequence of closed subspaces. In a previous paper, we showed how, conversely, direct limits could be used to construct Hilbert spaces which have multiresolution analyses with desired properties. In this paper, we use direct limits, and in particular the universal property which characterizes them, to construct wavelet bases in a variety of concrete Hilbert spaces of functions. Our results apply to the classical situation involving dilation matrices on L2 (Rn ), the wavelets on fractals studied by Dutkay and Jorgensen, and Hilbert spaces of functions on solenoids. © 2009 Elsevier Inc. All rights reserved. Keywords: Wavelet; Direct limit; Multiresolution

0. Introduction Suppose that H is a Hilbert space equipped with a unitary operator D, which we think of as a dilation, and a unitary representation T : Γ → U (H ) of an abelian group, which we think ✩

This research was supported by the Australian Research Council, the National Science Foundation (through grant DMS-0701913), the Research Council of Norway, and the University of Oslo. * Corresponding author. E-mail addresses: [email protected] (L.W. Baggett), [email protected] (N.S. Larsen), [email protected] (J.A. Packer), [email protected] (I. Raeburn), [email protected] (A. Ramsay). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.08.011

L.W. Baggett et al. / Journal of Functional Analysis 258 (2010) 2714–2738

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of as a group of translations. A multiresolution analysis (MRA) for (H, D, T ) consists of an increasing sequence of closed subspaces Vn , whose union is dense, whose intersection is {0}, and which satisfy D(Vn ) = Vn+1 , together with a scaling vector φ ∈ V0 whose translates Tγ φ form an orthonormal basis for V0 ; in a generalized multiresolution analysis (GMRA), the existence of the scaling vector is relaxed to the requirement that V0 is T -invariant. MRAs and GMRAs play an important role in the construction of wavelets: a wavelet is a vector ψ whose translates form an orthonormal basis for W0 := V1 V0 , and then {D j Tγ ψ: j ∈ Z, γ ∈ Γ } is an orthonormal basis for H . A famous theorem of Mallat [15] gives a procedure for constructing wavelets in the Hilbert space L2 (R), starting from a quadrature mirror filter, which is a function m : T → C satisfying |m(z)|2 + |m(−z)|2 = 2, and proceeding through an MRA for the usual dilation operator and integer translations. Baggett, Courter, Merrill, Packer and Jorgensen have generalized Mallat’s construction to GMRAs [1,2]. Writing a Hilbert space H as an increasing union of closed subspaces Vn amounts to realizing H as a Hilbert-space direct limit lim Vn . In [14], Larsen and Raeburn constructed MRAs for −→

L2 (R) by constructing a direct system based on a single isometry Sm on L2 (T) associated to a quadrature mirror filter m, and using the universal property of the direct-limit construction to identify the direct limit lim(L2 (T), Sm ) with L2 (R). This yielded a new proof of Mallat’s −→ theorem. Subsequently the present authors used a similar construction to settle a question about multiplicity functions of generalized multiresolution analyses [3]. Here we will show that the universal properties of direct limits provide useful insight in a variety of situations involving wavelets and their generalizations. Our techniques provide efficient proofs of known results concerning classical wavelets and the wavelets on fractals studied by Dutkay and Jorgensen [9]. We also obtain some interesting new results. We provide, building on our previous work in [3], easily verified and very general criteria which imply that the isometries Sm associated to filters are pure isometries (see Theorem 3.1). We use our direct-limit approach, and in particular the uniqueness of such limits, to settle a question of Ionescu and Muhly [12] about the support of measures in realizations of MRAs in L2 -spaces on solenoids. We begin with a short section in which we recall general results on direct limits and MRAs from [3], and indicate what extra information is needed to yield wavelet bases associated with these MRAs. In an attempt to emphasize how general our approach is, we will work whenever possible with an abstract translation group Γ , and for most purposes this poses no extra difficulty. In Section 2, we discuss the filters from which we build MRAs and the filter banks from which we build wavelet bases. One key hypothesis in our general theory says that the isometry Sm associated to a filter is a pure isometry, in the sense that its Wold decomposition has no unitary summand, and we prove our new criterion for pureness in Section 3. In Section 4 we prove our main theorem on identifying direct limits, and illustrate its usefulness by applying it in the classical situation of a low-pass filter associated to dilation by an expansive integer matrix on Rn . In the next two sections, we give several other applications of this theorem. The first involves the wavelets on fractals studied by Dutkay and Jorgensen. Starting with a filter which is definitely not low-pass, we run our direct-limit construction, and identify the direct limit as a Hilbert space of functions on a “filled-in Cantor set” constructed in [9]. Second, under a nonsingularity hypothesis on the filter m, we realize our direct limits as spaces of functions on solenoids. This realization applies to both the classical case and the fractal case, and in both cases comparing the solenoidal realization with the original gives interesting information: in the fractal case, we recover Dutkay’s Fourier transform from [8], and in the classical case, we deduce that the measure defining the L2 -space on the solenoid is supported on a

2716


“winding line”, thereby confirming a conjecture of Ionescu and Muhly [12]. In the final section, we show that our methods can be used to obtain (a slight variation of) a theorem of Jorgensen on wavelet representations of the Baumslag–Solitar group [13]. Notation and standing assumptions We consider an additive countable abelian group Γ and its compact dual group Γ . We write f (k) dk for the integral of f with respect to normalized Haar measure on Γ . Γ Throughout the paper, we consider an injective endomorphism α of Γ such that α(Γ ) has finite index N in Γ and n0 α n (Γ ) = {0}. We write α ∗ for the endomorphism ω → ω ◦ α of Γ; observe that α ∗ is surjective, that |ker α ∗ | = N , and that n0 ker α ∗n is dense in Γ. The example to bear in mind is the endomorphism of Γ = Z defined by α(n) = N n, when α ∗ is the endomorphism z → zN of T. To simplify formulas, we sometimes write (K, β) for (Γ, α ∗ ). 1. Wavelet bases in direct limits Suppose that S is an isometry on a Hilbert space H , and let (H∞ , Un ) be the Hilbertspace direct limit of the direct system (Hn , Tn ) in which each (Hn , Tn ) = (H, S). We proved in [3, Theorem 5] that there is a unitary operator S∞ on H∞ characterized by S∞ Un = Un S = Un−1 , and that the subspaces Vn of H∞ defined by Vn :=

Un (H ) |n| S∞ (V0 )

if n 0, if n < 0

(1.1)

satisfy Vn ⊂ Vn+1 , n∈Z Vn = H∞ and S∞ (Vn+1 ) = Vn . In addition, we have n∈Z Vn = {0} if and only if S is a pure isometry, in which case the subspaces Wn := Vn+1 Vn give an orthogonal decomposition H∞ = n∈Z Wn . Now suppose that μ : Γ → U (H ) is a unitary representation such that Sμγ = μα(γ ) S for γ ∈ Γ . Then we proved in [3, Theorem 5(d)] that there is a representation μ∞ of Γ on H∞ characterized by μ∞ (γ )Un = Un μα n (γ ) ; we then have S∞ μ∞ (γ ) = μ∞ (α(γ ))S∞ , and the triple −1 ) is a generalized multiresolution analysis (GMRA) for H if and only if S is a ({Vn }, μ∞ , S∞ ∞ pure isometry. At this point, we ask what extra input we need to ensure that this GMRA is associated to a wavelet or multiwavelet basis for H∞ . Proposition 1.1. Suppose that S is a pure isometry on H . Suppose there are a Hilbert space L, a unitary representation ρ : Γ → U (L), an orthonormal set B in L such that {ργ l: l ∈ B, γ ∈ Γ } is an orthonormal basis for L, and a unitary isomorphism S1 of L onto (SH )⊥ such that S1 ργ = μα(γ ) S1 . Then −j S∞ μ∞ (γ )ψ: j ∈ Z, γ ∈ Γ, ψ ∈ U1 S1 (B)

(1.2)

is an orthonormal basis for H∞ . Proof. We know that U1 is an isomorphism of H onto V1 , and U1 (SH ) = U0 H = V0 , so U1 is an isomorphism of (SH )⊥ onto W0 := V1 V0 . Thus {U1 S1 ργ l: l ∈ B} is an orthonormal basis −j for W0 . Now S∞ maps W0 onto Wj , and hence


−j S∞ U1 S1 ργ l: j ∈ Z, γ ∈ Γ, l ∈ B

2717

(1.3)

is an orthonormal basis for H∞ . But

U1 S1 ργ = U1 μ α(γ ) S1 = μ∞ (γ )U1 S1 , so (1.3) is the desired orthonormal basis (1.2).

2

2. Filters and isometries In this section we will only use the dual endomorphism α ∗ , so we simplify notation by writing (K, β) for (Γ, α ∗ ). Recall that β is surjective and N := |ker β| is finite. A filter for β is a Borel function m : K → C such that

m(ak) 2 = N

for almost all k ∈ K.

(2.1)

a∈ker β

A filter bank for β consists of Borel functions ma : K → C parametrized by a ∈ ker β such that

ma (dk)mb (dk) = δa,b N

for almost all k ∈ K;

(2.2)

d∈ker β

Eq. (2.2) says that the matrix (N −1/2 ma (dk))a,d is unitary for almost all k; in particular, each ma is a filter in its own right. Examples 2.1. (a) In the classical situation, we have Γ = Z, K = T, β(z) = z2 and N = 2, and in this case we recover the usual notions of conjugate mirror filter and filter bank with perfect reconstruction. More generally, we could take for β the endomorphism of Tn induced by an integer matrix B: β(e2πix ) = e2πiBx for x ∈ Rn , in which case N = |det B|. such (b) To get a filter for a more general β ∈ End K, choose characters γ0 , . . . , γN −1 in K that (ker β)∧ = {γj |ker β : 0 j N − 1}. Then for every unit vector c = (cj ) in CN , m(k) := N −1 1/2 cj γj (k) defines a filter m for β. To see this we just need to recall that the characters j =0 N form an orthonormal basis for 2 ((ker β)∧ ), and compute: −1 N

m(ak) 2 = N ci γi (ak)cj γj (ak) a∈ker β i,j =0

a∈ker β

=

N −1

N ci γi (k)cj γj (k)

i,j =0

=

N −1

2

N|cj |2 γj (k) ,

j =0

which is N because γj (k) ∈ T and c is a unit vector.

a∈ker β

γi (a)γj (a)

2718


(c) To construct filter banks, we generalize method from [11]. Choose an orthonormal basis −1 a1/2 N ca,j γj (k). Then, as in the previous calculation, ca = (ca,j ) for CN , and take ma (k) = N j =0

ma (dk)mb (dk) =

N −1

N ca,i γi (k)cb,j γj (k) γi (a)γj (a) = N (ca | cb ).

i,j =0

d∈ker β

a∈ker β

The next result is well known in special cases (see [6], for example). Proposition 2.2. (a) If m is a filter for β, then the formula (Sm f )(k) = m(k)f (β(k)) defines an isometry Sm on L2 (K). (b) If {ma : a ∈ ker β} is a filter bank for β, then {Sma : a ∈ ker β} satisfies the Cuntz relation

∗ Sm a Sm = 1. a

a∈ker β

Part (a) implies that for every filter m we can run the argument of Section 1 with S = Sm ; if pure, we obtain a GMRA for the direct L2 (K)∞ . Part (b) implies that for every a, Sm is limit 2 S1 := b∈ker β, b =a Smb is an isometry of b =a L (K) onto

2 ⊥

⊥ ∗ Sma L2 (K) L (K) = Sm a Sm = a

2 ∗ L (K) ; Sm b Sm b

b∈ker β, b =a

thus, when a filter m is a member of a filter bank, we can use Proposition 1.1 to generate a multiwavelet basis for L2 (K)∞ . To prove Proposition 2.2, we need an elementary lemma. Notice that our countability hypoth implies that there is always a Borel section c for the surjection β : K → K. esis on Γ = K Lemma 2.3. Suppose that c : K → K is a Borel map such that β(c(k)) = k for all k ∈ K. Then for every continuous function f on K we have (a) K f (β(k)) dk = K f (k) dk, and (b) K f (k) dk = K N −1 ( a∈ker β f (ac(k))) dk. Proof. For (a), we define I (f ) := K f (β(k)) dk. Since β is surjective, it follows easily from the translation invariance of Haar measure on K that I is also a translation-invariant integral on K; since I (1) = 1, it must be the Haar integral, and (a) follows. For (b), we use (a) to simplify the right-hand side: K

N −1

f ac(k) dk = N −1 f ac(k) dk

a∈ker β

a∈ker β K

=

a∈ker β K

N −1 f β ac(k) dk


=

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N −1 f (k) dk,

a∈ker β K

which since N = |ker β| gives (b). Proof of Proposition 2.2. To Lemma 2.3:

Sm f 2 =

2

see that Sm is an isometry, we compute using part (b) of

m(k)f β(k) 2 dk

K

= K

=

N −1

(2.3)

m ac(k) f β ac(k) 2 dk

(2.4)

a∈ker β

2 −1

f (k) 2 dk, m ac(k) N a∈ker β

K

which by the filter equation (2.1) is precisely f 2 . For (b), we use Lemma 2.3(b) again to check that

∗ ma dc(k) f dc(k) = N −1 ma (l)f (l), Sma f (k) = N −1 d∈ker β

β(l)=k

compute

∗ Sm a Sm f (k) = ma (k)N −1 a

ma (l)f (l) = ma (k)N −1

ma (dk)f (dk),

d∈ker β

β(l)=β(k)

and add to get

∗ −1 Sma Sma f (k) = N ma (k)ma (dk) f (dk). a∈ker β

d∈ker β

a∈ker β

Now the term in brackets is the inner product of two columns of the unitary matrix (ma (dk))a,d , and hence vanishes unless d = 1, in which case we are left with N −1 Nf (k). 2 3. When Sm is a pure isometry A crucial hypothesis in the general theory of Section 1 is that the isometry S is pure. Our next theorem gives easily verifiable criteria which imply that an isometry of the form Sm is pure. We stress that this is not an elementary fact: the proof uses results from [3] which rely on the reverse martingale convergence theorem. Theorem 3.1. Suppose that B is a Borel subset of Γ and m : Γ → C is a Borel function such that

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m(ζ ) 2 = N χB (ω)

for almost all ω ∈ Γ,

(3.1)

α ∗ (ζ )=ω

and define Sm : L2 (B) → L2 (B) by (Sm f )(ω) = m(ω)f (α ∗ (ω)). If either (a) Γ\B has positive Haar measure, or (b) |m(ω)| = 1 on a set of positive measure, then Sm is a pure isometry. Proof. In the language of [3], the hypothesis on m says that “m is a filter relative to the multiplicity function χB : Γ → {0, 1} and the endomorphism β := α ∗ ”. We are not assuming that m is a low-pass filter, but that hypothesis is not used in the proof of [3, Theorem 8] until after Proposition 12. So we know from [3, §4] that Sm is an isometry. We will assume that Sm is not pure, to prove that neither (a) nor (b) holds. Saying that Sm is not pure means that and aim n L2 (B) is non-zero, and hence that there exists a unit vector f in R . ProposiS R∞ := ∞ ∞ n=0 m ∗n f satisfy tion 12 of [3] implies that the functions fn := Sm

fn β n (ω) → 1 as n → ∞ for almost all ω ∈ Γ.

(3.2)

We claim that |m(ω)| 1 for almost all ω. To establish this claim, we again suppose not, so that there exists > 0 and a Borel set C of positive (Haar) measure such that |m(ω)| 1 − for ω ∈ C. Let δ > 0. Then we can deduce from (3.2) and Egorov’s theorem that there exist a Borel set E ⊂ C and M ∈ N such that E has positive measure and nM

and ω ∈ E

⇒

1 − δ < fn β n (ω) < 1 + δ.

Lemma 2.3 implies that β is measure-preserving, so the Poincaré recurrence theorem (as in [17, Theorem 2.3.2]) implies that there is a Borel set E ⊂ E such that zero ∞E \ E has measure n and {n ∈ N: β (ω) ∈ E } is infinite for every ω ∈ E . Writing E = n=M {ω ∈ E : β n (ω) ∈ E } implies that there exists n M such that F := {ω ∈ E : β n (ω) ∈ E } has positive measure. In particular, for ω ∈ F , β n (ω) belongs to C, and

1 − δ fn β n (ω) = (Sm fn+1 ) β n (ω)

= m β n (ω) fn+1 β (n+1) (ω) (1 − )(1 + δ). Since this is true for every δ > 0, we can let δ → 0+ and deduce that 1 1 − , which is a contradiction. Thus |m(ω)| 1 for almost all ω, and the left-hand side of the filter equation (3.1) is N for almost all ω. Since the right-hand side of is N , both sides must equal N , which implies that χB (ω) = 1 and |m(ω)| = 1 for almost all ω, so that neither (a) nor (b) holds, as required. 2 Remark 3.2. When B = Γ = T, this follows from Theorem 3.1 of [6]. That theorem also asserts that when |m| ≡ 1, the space R∞ is spanned by a single function ξ : T → T, and that m then has


2721

the form m(z) = λξ(z)ξ(zN ) for some λ ∈ T. These extra assertions also extend to the general case. nf To see this, we again consider a unit vector f in R∞ , and deduce from the equations f = Sm n and |m| ≡ 1 that

n−1

f (ω) = m β k (ω) fn β n (ω) = fn β n (ω) .

k=0

Thus |f (ω)| = |f (ωζ )| for ω and every ζ ∈ ker β n . Since the right-regular representa almost all n tion ρ is continuous and n1 ker β is dense in Γ, this implies that ρζ (|f |) = |f | for all ζ ∈ Γ. The Fourier transform |f |∧ then satisfies ζ (γ )|f |∧ (γ ) = |f |∧ (γ ) for all ζ ∈ Γ and all γ ∈ Γ , so |f |∧ (γ ) = 0 for γ = 0, and |f | is constant. So |f | is constant for every f ∈ R∞ . This implies that R∞ is one-dimensional: if f, g ∈ R∞ are non-zero, then 2 Re f g = |f + g|2 − |f |2 − |g|2 and 2 Im f g = |f + ig|2 − |f |2 − |g|2 are constant, so f g is constant and f = (f g)g/|g|2 is a constant multiple of g. If we choose a spanning element ξ which is a unit vector, so that |ξ | ≡ 1, then Sm ξ is also a unit vector in R∞ . Thus there exists λ ∈ T such that Sm ξ = λξ , which says that m(ω)ξ(β(ω)) = λξ(ω) for almost all ω. 4. Identifying the direct limit The universal property of the direct limit implies that, to identify H∞ with a given space K, we only need to find isometries Rn : H → K such that Rn+1 S = Rn and ∞ n=0 Rn H is dense in K. In [14], for example, we applied this strategy to identify L2 (T)∞ with L2 (R) when S is the isometry Sm associated to a quadrature mirror filter on T. If we have a candidate for the unitary S∞ , it is even easier. Theorem 4.1. Suppose that μ : Γ → U (H ) is a unitary representation, and S is an isometry on H such that Sμγ = μα(γ ) S for γ ∈ Γ . Suppose that λ : Γ → U (K) is a unitary representation and D is a unitary operator on K such that Dλγ D ∗ = λα(γ ) for γ ∈ Γ . If there is an isometry R : H → K such that (a) RS = DR, and (b) Rμγ = λγ R for γ ∈ Γ , −n then there is an isomorphism R∞ of H∞ onto the subspace ∞ n=0 D R(H ) of K such that ∗ ∗ −n R∞ S∞ R∞ = D and R∞ μ∞ R∞ = λ. The subspaces D R(H ) form a GMRA of R∞ (H∞ ) relative to D and λ if and only if S is a pure isometry. Proof. We define Rn : H → K by Rn = D −n R. Then each Rn is an isometry, and from (a) we have

Rn+1 S = D −(n+1) RS = D −n D −1 (DR) = D −n R = Rn . Thus the Rn induce an isometry R∞ of H∞ into K, and this is a unitary isomorphism onto the −n subspace ∞ n=0 D R(H ) of K. For each n 1 we have

2722


R∞ S∞ Un = R∞ Un S = Rn S = Rn−1 = DD −n R = DRn = DR∞ Un , so R∞ intertwines S∞ and D. For γ ∈ Γ and n 0, we have R∞ μ∞ (γ )Un = R∞ Un μα n (γ ) = Rn μα n (γ ) = D −n Rμα n (γ ) = D −n λα n (γ ) R = λγ D −n R = λγ Rn = λγ R∞ Un , ∗ = λ . The last assertion holds because the subspaces V and this implies that R∞ μ∞ (γ )R∞ γ n defined by (1.1) are a GMRA for H∞ if and only if S is pure. 2

To construct the isometry R when S is the isometry Sm associated to a filter m, we use a scaling function φ for the filter. We illustrate how this works by applying Theorem 4.1 in the classical situation of a dilation by an integer matrix on Rn , thereby showing that the approach taken in [14] also covers this situation. Example 4.2 (Classical wavelets). Let A ∈ GLn (Z) be an integer matrix such that every eigenvalue λ has |λ| > 1, and define α ∈ End Zn by α(k) = Ak (using multi-index notation). Note that N := |Zn /AZn | = |det A|. The dual endomorphism α ∗ of Tn is given on e2πix := t (e2πix1 , . . . , e2πixn ) by α ∗ (e2πix ) = e2πiA x . Suppose that m : Tn → C is a filter which is low1/2 pass, in the sense that m(1) = N , and is Lipschitz near 1; suppose also that m is non-vanishing on a suitably large neighbourhood of 1 (this is Cohen’s condition; see [18, Theorem 1.9], for example). Theorem 3.1 implies that Sm is a pure isometry. Under our hypotheses on m the infinite product1 φ(x) =

∞

t −n N −1/2 m e2πi(A ) x

(4.1)

n=1

converges pointwise almost everywhere for x ∈ Rn and in L2 (Rn ) to a unit vector φ ∈ L2 (Rn ); the limit φ is continuous near 0, satisfies φ(0) = 1,

N 1/2 φ At x = m e2πix φ(x),

(4.2)

φ(x + k) 2 = 1

(4.3)

and

k∈Zn

for almost all x ∈ Rn . 1 The assertions in this sentence are all well known (see [18], for example), but it is hard to point to an efficient derivation. They can, however, be deduced from the more general results in [2, Proposition 3.1] and [1, Lemma 3.3]; there we need to take the multiplicity function to be identically 1 on Tn , so that the matrix H consists of the single function denoted here by m, and observe that in this case the functions M˜ n and M n in [1, §3] coincide.


2723

We now define R : L2 (Tn ) → L2 (Rn ) by

(Rf )(x) = f e2πix φ(x). With B =

n

j =1 [0, 1),

Rn is the disjoint union of the sets B + k for k ∈ Zn , and

Rf 2 =

f e2πix φ(x + k) 2 dx k∈Zn B

=

2πix 2

f e

φ(x + k) 2 dx k∈Zn

B

= f 2 by (4.3). Thus R is an isometry. With (Dg)(x) := N 1/2 g(At x), the scaling equation (4.2) gives

t t

(RSm f )(x) = m e2πix f e2πiA x φ(x) = N 1/2 f e2πiA x φ At x = (DRf )(x), and with μ : Zn → U (L2 (Tn )) defined by (μk f )(z) = zk f (z) and λ : Zn → U (L2 (Rn )) by (λk f )(x) = e2πix·k g(x), we can easily check that Rμk = λk R. Thus Theorem 4.1 implies that 2 n −j 2 n there is an isomorphism R∞ of L2 (Tn )∞ onto the subspace ∞ j =0 D R(L (T )) of L (R ) which intertwines (S∞ , μ∞ ) and (D, λ). Since R is an isometry, the functions ek φ : x → e2πik·x φ(x) form an orthonormal basis for V0 := R(L2 (Tn )), and hence the functions D −j (ek φ) n form an orthonormal basis for Vj := D −j R(L2 (T )). Thus we can 2run nthe standard argument (as on page 212 of [1], for example) to see that Vj is dense in L (R ). We deduce that the subspaces {Vj } form a multiresolution analysis for L2 (Rn ). Now suppose that m1 := m is part of a filter bank {mw : w ∈ ker α ∗ } parametrized by ker α ∗ = w ∈ Tn : w = e2πix for some x ∈ Rn such that At x ∈ Zn . (It is known that for every filter m there is always a filter bank containing m [5, p. 494], but our construction depends on fixing one.) Since {Smw : w ∈ ker α ∗ } is a Cuntz family, S1 :=

Sm w :

w =1

L2 T n → L2 T n

(4.4)

w =1

is an isometry with range (Sm L2 (Tn ))⊥ . Thus we can apply Proposition 1.1 with S1 given by (4.4). Note that D −1 R is an isomorphism of (Sm L2 (Tn ))⊥ onto W0 := V1 V0 . Let 1w denote (Tn ), so that the functions {x → e2πik·x 1w : w ∈ the constant function 1 in the wth copy of L2 ∗ ker α , w = 1} form an orthonormal basis for w =1 L2 (Tn ), and set

−1 t −1

x . ψw (x) := D −1 RS1 1w (x) = N −1/2 mw e2πi(A ) x φ At Proposition 1.1 implies that the functions

j t j ψw,j,k (x) := N j/2 e2πik·(A ) x ψw At x

2724


form an orthonormal basis for L2 (Rn ), and the inverse Fourier transforms {ψˇ w : w ∈ ker α ∗ , w = 1} form a multi-wavelet for L2 (Rn ). Example 4.3. Consider the multiplicity function χB : T → {0, 1} associated to the interval (− 13 , 13 ] (or rather to the set B := {e2πix : x ∈ (− 13 , 13 ]}). We can check that the function m : e2πix → 21/2 χ(− 1 , 1 ] (x) satisfies the generalized filter equation (3.1) with N = 2, and hence 6 6

Theorem 3.1 implies that Sm : L2 (B) → L2 (B) is a pure isometry. The function φ := χ(− 1 , 1 ] 3 3

satisfies the scaling equation 21/2 φ(2x) = m(e2πix )φ(x), so in parallel with the classical case we define R : L2 (B) → L2 (R) by

(Rf )(x) = f e2πix χ(− 1 , 1 ] (x). 3 3

Calculations show that the usual dilation operator defined by (Dξ )(x) = 21/2 ξ(2x) satisfies DR = RSm , and that R intertwines the representations μ and λ of Z defined by (μn f )(z) = zn f (z) and (λn ξ )(x) = e2πinx ξ(x). The range of R is the subspace L2 (− 13 , 13 ] of L2 (R) conn n sisting of functions which vanish for |x| > 13 , and D −n (L2 (− 13 , 13 ]) = L2 (− 23 , 23 ], so the ∞ dominated convergence theorem implies that n=0 D −n R(L2 (B)) is dense in L2 (R). Thus Theorem 4.1 implies that the subspaces D −j R(L2 (B)) form a GMRA for L2 (R). Since the functions en : x → e2πinx form an orthonormal basis for L2 (− 12 , 12 ], and since multiplication by φ = χ(− 1 , 1 ] is the orthogonal projection on L2 (− 13 , 13 ], the functions λn φ 3 3

form a Parseval frame for RL2 (B) = L2 (− 13 , 13 ]. The inverse Fourier transform of λn φ is the ˇ − n), and hence we have just shown that the inverse Fourier transforms Vj := translate φ(· (D −j R(L2 (B)))∨ form a frame multiresolution analysis in the sense of [4] — indeed, we have just recovered Example 4.10(a) of [4]. 5. Wavelets associated to the Cantor set The characteristic function χC of the middle-third Cantor set in [0, 1] satisfies

χC 3−1 x = χC (x) + χC (x − 2)

for all x ∈ R.

(5.1)

Dutkay and Jorgensen observed in [9] that this is formally similar to saying that χC satisfies a scaling equation involving the dilation (Df )(x) = f (3−1 x) and two translations. The righthand side can be viewed as convolution with the measure δ0 + δ2 , which is the inverse Fourier transform of 1 + z2 ∈ L2 (T). So one is led to view 1 + z2 as a filter, and consider the associated isometry on L2 (T). We consider the function m : T → C defined by m(z) = 2−1/2 (1 + z2 ); the normalising factor of 2−1/2 ensures that m satisfies

m(z) 2 + m(ωz) 2 + m ω2 z 2 = 3,

(5.2)

where ω := e2πi/3 is a cube root of unity, so that m is a filter for multiplication by 3. Notice that m is not low-pass: it satisfies m(1) = 21/2 rather than m(1) = 31/2 . A key point established in [9] is that when we mimic the classical construction of wavelets on R using this filter, we wind up in a Hilbert space of functions determined by a measure which is supported on a set of Lebesgue


2725

measure 0. Our goal in this section is to show that our recognition theorem also applies in this situation. Theorem 3.1 implies that the operator on L2 (T) defined by (Sm f )(z) = m(z)f (z3 ) is a pure isometry. With α ∈ End Z defined by α(n) = 3n and μ : Z → U (L2 (T)) given by (μn f )(z) = zn f (z), we have Sm μn = μ3n Sm = μα(n) Sm . We want to identify the direct limit (L2 (T)∞ , S∞ , μ∞ ) using φ := χC as scaling function. When we normalize m by multiplying by 2−1/2 , we need to multiply both sides of the scaling equation (5.1) by 2−1/2 , and hence the appropriate dilation operator is given by (Df )(x) = 2−1/2 f (3−1 x). Following [9], we define R :=

3−n (C + k): k, n ∈ Z ,

and let ν denote the Borel measure on R which has ν(C) = 1, is invariant for the action of Z by translation on R, and satisfies

−1 f (x) dν(x) = 2 (5.3) f 3−1 x dν(x) for every f ∈ L1 (R, ν). (See [9, Proposition 2.4].) Thus D is a unitary operator on L2 (R, ν), and the scaling function χC is a unit vector. We define λ : Z → U (L2 (R, ν)) by (λn f )(x) = f (x − n). A straightforward calculation shows that Dλn = λ3n D, so that Dλn D ∗ = λ3n . Proposition 5.1. The direct limit (L2 (T)∞ , S∞ , μ∞ ) is isomorphic to (L2 (R, ν), D, λ). The subspaces Vn = span D −n λk (χC ): k ∈ Z form an MRA for L2 (R, ν), and {λk (χC ): k ∈ Z} is an orthonormal basis for V0 . To apply Theorem 4.1, we need an isometry R : L2 (T) → L2 (R, ν). This one looks a little different to those in the previous section because the scaling equation in the form (5.1) involves a convolution rather than a pointwise multiplication in the Fourier domain. Lemma 5.2. For n ∈ Z, let en denote the function z → zn . Then there is an isometry R of L2 (T) into L2 (R, ν) such that Ren = λn χC = χC+n for n ∈ Z. Proof. Since {en : n ∈ Z} is an orthonormal basis for L2 (T), it suffices for us to check that the elements λn χC = χC+n form an orthonormal set in L2 (R, ν). Since singleton sets have νmeasure zero, we can delete 1 from C without changing the element χC of L2 (R, ν); now the sets C + n are disjoint, so the functions are mutually orthogonal, and since ν(C + n) = ν(C) = 1, each χC+n is a unit vector. 2 To get surjectivity of our isomorphism R∞ , we need the following lemma.2 2 This result is stated as Proposition 2.8(iii) in [9], but there seems to be a gap in the proof. This was observed and fixed independently by Sam Webster and Kathy Merrill. The proof of Lemma 5.3 is similar to the proof in Sam’s honours thesis (University of Newcastle, 2006); Kathy’s argument is generalized in [7].

2726


Lemma 5.3. The functions χ3−n (C+k) = 2−n/2 D −n λk (χC ): n, k ∈ Z span a dense subspace of L2 (R, ν). −n Proof. Since R = ∞ n=0 3 ( k∈Z (C +k)) is an increasing union of almost disjoint unions, two applications of the dominated convergence theorem show that it suffices to approximate functions f with support in 3−N (C + K) for fixed N 0 and K ∈ Z. Then λ−K D N f has support in C. We now consider the sets 3−n (C + k) which are contained in C. For each n 0, there are exactly 2n such sets, and they are disjoint; each 3−n (C + k) = 3−(n+1) (C + 3k) ∪ 3−(n+1) (C + 3k + 2). Thus two such sets are either disjoint or one is contained in the other, and A := span χ3−n (C+k) : n 0, k ∈ Z, and 3−n (C + k) ⊂ C is a ∗-subalgebra of C(C); since A contains the characteristic functions of arbitrarily small sets, it separates points of C, and hence by the Stone–Weierstrass theorem is uniformly dense in C(C). Since ν is inner regular and C has finite measure, the restriction of ν to C is a regular Borel measure, and C(C) is dense in L2 (C, ν). Thus we can find a function g in span{χ3−n (C+k) : n, k ∈ Z} = span D −n λk (χC ): n, k ∈ Z such that λ−K D N f − g is small. Since λK and D −N are unitary, f − D −N λK g is also small. But

D −N λK D −n λk (χC ) = D −(N +n) λ3n K+k (χC ), so D −N λK g has the required form.

2

Proof of Proposition 5.1. We next check that RSm = DR (equation (a) of Theorem 4.1). For each n ∈ Z, we have

(DRen )(x) = (DχC+n )(x) = 2−1/2 χC+n 3−1 x = 2−1/2 χC 3−1 (x − 3n) , which in view of the scaling equation (5.1) gives

(DRen )(x) = 2−1/2 χC (x − 3n) + χC (x − 3n − 2) = R 2−1/2 (e3n + e3n+2 ) (x). Since

(Sm en )(z) = 2−1/2 1 + z2 en z3 = 2−1/2 1 + z2 z3n = 2−1/2 (e3n + e3n+2 )(z), we deduce that RSm and DR agree on the basis elements en , and hence are equal.


2727

To check the hypothesis (b) of Theorem 4.1, observe that μn ek = ek+n . Thus for n, k ∈ Z we have (Rμn )ek = Ren+k = χC+n+k = λn (χC+k ) = (λn R)ek . Now Theorem 4.1 gives an isometry R∞ of (L2 (T)∞ , S∞ , μ∞ ) into (L2 (R, ν), D, λ). Since the range of R contains the vectors λn (χC ), it follows from Lemma 5.3 that n0 D −n (R(L2 (T))) is dense in (L2 (R, ν), D, λ), and the result follows. 2 To get a wavelet basis for L2 (R, ν), we observe that m0 = m and m1 (z) = z, m2 (z) = form a filter bank: with ω = exp(2πi/3), the matrix

2−1/2 (1 − z2 )

3

−1/2

m0 (z) m0 (ωz) m0 (ω2 z)

m1 (z) m1 (ωz) m1 (ω2 z)

m2 (z) m2 (ωz) m2 (ω2 z)

is unitary for every z ∈ T. Proposition 2.2 implies that the operators Ti := Tmi on L2 (T) form a Cuntz family with T0 = Sm , and Sm is pure by Theorem 3.1. Thus the operator S1 : L2 (T) ⊕ L2 (T) → L2 (T) defined by S1 (f, g) = T1 f + T2 g is a unitary isomorphism of L := L2 (T) ⊕ L2 (T) onto the complement (Sm (L2 (T)))⊥ , and the hypotheses of Proposition 1.1 are satisfied with B = {(1, 0), (0, 1)} and ρ = μ ⊕ μ. We deduce that the set

U1 S1 (1, 0), U1 S1 (0, 1) = {U1 T1 1, U1 T2 1} = {U1 m1 , U1 m2 }

generates a wavelet basis

−j

S∞ μ∞ (k)U1 mi : j ∈ Z, k ∈ Z, i = 1, 2

for L2 (T)∞ . Applying the isomorphism R∞ gives an orthonormal basis

D −j λk R∞ U1 mi : j ∈ Z, k ∈ Z, i = 1, 2

for L2 (R, ν). Let

ψi (x) = R∞ (U1 mi )(x) = (R1 mi )(x) = D −1 Rmi (x) = 21/2 (Rmi )(3x); in terms of the basis en for L2 (T) used to define R in Lemma 5.2, we have m1 = e1 and m2 = 2−1/2 (e0 − e2 ), so ψ1 (x) = 21/2 χC+1 (3x) = 21/2 χ3−1 (C+1) (x), and

ψ2 (x) = 21/2 2−1/2 χC − 2−1/2 χC+2 (3x) = χ3−1 C − χ3−1 (C+2) (x). Thus we recover the following theorem of Dutkay and Jorgensen [9]:

2728


Theorem 5.4. Let ψ1 = 21/2 χ3−1 (C+1) and ψ2 = χ3−1 C − χ3−1 (C+2) . Then

ψi,j,k (x) = 2j/2 ψi 3j x − k : i = 1, 2, j ∈ Z, k ∈ Z

is an orthonormal basis for L2 (R, ν). Example 5.5. More generally, one can form a one-parameter family of multi-wavelets corresponding to dilation and translation on the filled-out Cantor set R. For r satisfying |r| 2−1/2 set m0 (z) = 2−1/2 (1 + z2 ), as above, and take 1/2 1/2 2

+ 21/2 rz + 1 − 2r 2 /2 z , m1,r (z) := − 1 − 2r 2 /2 1/2

m2,r (z) := r + 1 − 2r 2 z − rz2 . The remarks made in Example 2.1(c) imply that {m0 , m1,r , m2,r } is a filter bank, and the above argument shows that the pair 1/2 1/2

ψ1,r := − 1 − 2r 2 χ3−1 C + 2rχ3−1 (C+1) + 1 − 2r 2 χ3−1 (C+2) ,

1/2

ψ2,r := 21/2 rχ3−1 C + 1 − 2r 2 χ3−1 (C+1) − rχ3−1 (C+2) is a multi-wavelet for dilation by 3 on L2 (R, ν); to recover Theorem 5.4, take r = 2−1/2 . There is also a version of Theorem 5.4 which starts from the characteristic function of the Sierpinski gasket (see [7]). 6. Functions on solenoids Suppose that m : Γ → C is a filter for α ∗ ∈ End Γ. Then the representation μ : Γ → U (L2 (Γ)) defined by (μγ f )(ζ ) = ζ (γ )f (ζ ) satisfies Sm μγ = μα(γ ) Sm . Thus the direct limit construction of Section 1 gives a direct limit (L2 (Γ)∞ , Un ) together with a dilation S∞ and a representation μ∞ of Γ on L2 (Γ)∞ such that S∞ Un = Un Sm and S∞ μ∞ (γ ) = μ∞ (α(γ ))S∞ . We want to identify this direct limit with an L2 -space of functions on the solenoid Sα ∗ := lim(Γ, α ∗ ); ←− this is motivated by previous work of Jorgensen [13] and Dutkay [8, §5.2], where Γ = Z, α is multiplication by N , and Sα ∗ is the usual solenoid SN := lim(T, z → zN ). Then, as applications ←− of our result, we will rederive a theorem of Dutkay on a “Fourier transform” for the Cantor set, and settle a question of Ionescu and Muhly about the support of the measure on the solenoid when m is a low-pass filter. To define the L2 -space on the solenoid, we need some background material on measures on solenoids. The first lemma is a modern formulation of a classical result (see, for example, [16, Proposition 27.8]). Lemma 6.1. Suppose that rn : Tn+1 → Tn is an inverse system of compact spaces with each rn surjective, and μn is a family of measures on Tn such that μ0 is a probability measure and

(f ◦ rn ) dμn+1 =

f dμn

for f ∈ C(Tn ).

(6.1)


2729

Let T∞ = lim(Tn , rn ), and denote the canonical map from T∞ to Tn by πn . Then there is a unique ←− probability measure μ on T∞ such that

(f ◦ πn ) dμ =

f dμn

for f ∈ C(Tn ).

∗ Proof. Since each rn is surjective, ∞so is ∗each πn , and the map πn : f → f ◦ πn of C(Tn ) into C(T∞ ) is isometric. The subset n=0 πn (C(Tn )) of C(T∞ ) is a unital ∗-subalgebra of C(T∞ ) which separates points of T∞ , and hence by the Stone–Weierstrass theorem is dense in C(T∞ ). Construct a functional φ on the dense subset πn∗ (C(Tn )) of C(T∞ ) by φ(πn∗ (f )) = f dμn for f ∈ C(Tn ); Eq. (6.1) implies that φ is well-defined. Taking f = 1 in (6.1) shows that each μn is a probability measure; since the maps πn∗ are isometric, this implies that φ is a positive functional with norm 1. Thus φ extends to a positive functional of norm 1 on C(T∞ ), and the Riesz theorem gives us the measure μ. The uniqueness follows from density of ∞ representation ∗ n=0 πn (C(Tn )). 2

Now we return to our specific situation, where we again write (K, β) for (Γ, α ∗ ). Proposition 6.2. Denote by πn the canonical map of Sβ := lim(K, β) onto the nth copy of K. ←−

There is a unique probability measure3 τ on Sβ such that for every f ∈ C(K),

(f ◦ πn ) dτ =

Sβ

n−1

2 j

m β (k) f (k) dk.

(6.3)

j =0

K

3 When K = T and β(z) = zN , this is same as the measure constructed by Dutkay in [8, Proposition 4.2(i)]. In our notation, his defining property is

(f ◦ πn ) dτ = SN

T

1 Nn

n−1 j 2 N

m w dz, f (w)

(6.2)

j =0

n

w N =z

and his uniqueness statement is [8, Proposition 4.2(ii)]. To see that our defining property is equivalent, notice that for any g ∈ L∞ (T) and any p ∈ N, we have T

1 p

w p =z

1 p−1 1 2π i(j +x)/p dx g(w) dz = g e p 0

=

j =0

(j +1)/p p−1

g e2π it dt

j =0 j/p

=

g(z) dz. T

2730


For the proof we need the following lemma, which follows from part (b) of Lemma 2.3 by essentially the same calculation which proves that Sm is an isometry (see (2.3)). Lemma 6.3. For every g ∈ L∞ (K) we have

2

g β(k) m(k) dk =

K

g(k) dk. K

Proof of Proposition 6.2. We take τ0 to be normalized Haar measure, and define measures τn for n 1 by

f dτn =

n−1

2 j

m β (k) f (k) dk

for f ∈ C(K).

(6.4)

j =0

K

To verify the consistency condition (6.1), let f ∈ C(K). Then

(f ◦ rn ) dτn+1 =

n

j 2

m β (k) m(k) 2 dk. f β(k)

K

(6.5)

j =1

Now Lemma 6.3 implies that the right-hand side of (6.5) is

f (k)

K

n

j −1 2

m β (k) dk = f dτn . j =1

Thus the measures τn satisfy the hypotheses of Lemma 6.1, and the result follows from that lemma. 2 We now want to identify the direct limit (L2 (K)∞ , Un ) with (L2 (Sβ , τ ), πn∗ ). For this to be useful, we need to know what the isomorphism does to the dilation S∞ and the translations μ∞ (γ ). To describe the dilation on L2 (Sβ , τ ) we need the shift h : Sβ → Sβ characterized by πn (h(ζ )) = πn−1 (ζ ); if we realise elements of the inverse limit as sequences ζ = {ζn : n 0} satisfying β(ζn+1 ) = ζn , then h(ζ0 , ζ1 , . . .) = (β(ζ0 ), ζ0 , ζ1 , . . .). Theorem 6.4. Suppose that m : Γ → C is a filter for α ∗ ∈ End Γ such that m−1 (0) has Haarmeasure zero. Let τ be the measure on Sα ∗ described in Proposition 6.2. Then there is an isomorphism V∞ of L2 (Sα ∗ , τ ) onto the direct limit L2 (Γ)∞ = lim(L2 (Γ), Sm ) such that −→

∗j (a) V∞ (g ◦ πn ) = Un (g( n−1 j =0 (m ◦ α ))); ∗ (b) (V∞ S∞ V∞ f )(ζ ) = m(π0 (ζ ))f (h(ζ )); and ∗ μ (γ )V f )(ζ ) = π (ζ )(γ )f (ζ ). (c) (V∞ ∞ ∞ 0 We have chosen to look for an isomorphism from L2 (Sα ∗ , τ ) to L2 (Γ)∞ because this will be more convenient in the applications. However, this choice means that we cannot simply apply


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Theorem 4.1 to find the desired isomorphism. So we need to find different ways of exploiting the universal property of the direct limit. Proof. Again we write (K, β) for (Γ, α ∗ ). We begin by showing that the direct limit system defining L2 (K)∞ , in which each Hilbert space is L2 (K), is isomorphic to one in which the nth Hilbert space is L2 (K, τn ) (where τn is the measure defined in (6.4)). We define 2 2 T n : L (K, τn ) → L (K, τn+1 ) by Tn f = f ◦ β; the consistency condition (f ◦ rn ) dτn+1 = f dτn (checked in the proof of Proposition 6.2) says that Tn is an isometry. With V0 = 1 and n−1

Vn f := m ◦ β j f, j =0

we have the following commutative diagram of isometries: L2 (K)

Sm

V0

L2 (K)

Sm

V1

T0

L2 (K)

L2 (K, τ1 )

L2 (K)

Sm

···

T2

···

V2

T1

L2 (K, τ2 )

Since the filter m is non-zero except on a set of measure zero, each Vn is surjective, and the Vn form an isomorphism of the direct systems. To identify the direct limit of the new system, we consider the maps Rn : f → f ◦πn ; Eq. (6.3) implies that Rn is an isometry of L2 (K, τn ) into L2 (Sβ , τ ), and the formula β ◦ πn+1 = πn implies that we have a commutative diagram T0

L2 (K)

R0

L2 (K, τ1 )

R1

T1

L2 (K, τ2 )

T2

···

R2

L2 (Sβ , τ ) Since the functions of the form f ◦ πn span a dense subspace of C(Sβ ) and hence also of L2 (Sβ , τ ), the isometries Rn induce an isomorphism of the direct limit onto L2 (Sβ , τ ). Alternatively, we can say that (L2 (Sβ , τ ), Rn ) is a direct limit for the system. Since isomorphic direct systems have isomorphic direct limits, we deduce that there is an isomorphism V∞ of L2 (Sβ , τ ) onto L2 (K)∞ such that V∞ Rn = Un Vn , which is equation (a). It is enough to verify formulas (b) and (c) for f of the form f = Rn g = g ◦ πn . For (b), we have

∗ ∗ ∗ S∞ V∞ Rn = V∞ S∞ Un Vn = V∞ Un (Sm Vn ) = Rn Vn∗ Vn+1 Tn . V∞

2732


To compute the latter, we let g ∈ C(K) and ζ ∈ Sβ . Then

Rn Vn∗ Vn+1 Tn g (ζ ) = Vn∗ Vn+1 Tn g πn (ζ ) n−1 n

−1

j j = m β πn (ζ ) m β πn (ζ ) g β πn (ζ ) j =0

= m β n πn (ζ ) g πn−1 (ζ )

= m π0 (ζ ) (Rn g) h(ζ ) ,

j =0

and (b) follows. For (c), we begin by expanding ∗ ∗ ∗ V∞ μ∞ (γ )V∞ Rn = V∞ μ∞ (γ )Un Vn = V∞ Un μα n (γ ) Vn .

Now we observe that both μα n (γ ) and Vn are multiplication operators, and hence commute (formally at least: strictly speaking, the two μα n (γ ) act on different spaces). Thus ∗ ∗ μ∞ (γ )V∞ Rn = V∞ Un Vn μα n (γ ) = Rn μα n (γ ) . V∞

For g ∈ C(K) and ζ ∈ Sβ , we have

(Rn μα n (γ ) g)(ζ ) = (μα n (γ ) g) πn (ζ ) = πn (ζ ) α n (γ ) g πn (ζ )

= β n πn (ζ ) (γ )(Rn g)(ζ ) = π0 (ζ )(γ )(Rn g)(ζ ), which gives (c).

2

6.1. Dutkay’s Fourier transform for R As a first application of Theorem 6.4, we apply it with Γ = Z, α(j ) = 3j and m(z) = 2−1/2 (1+z2 ). The resulting isometry Sm on L2 (T) is the same one we considered in Section 5, so Theorem 6.4 gives an alternative realization of the direct limit L2 (T)∞ as a space of functions on the solenoid S3 . Combining this isomorphism with that of Proposition 5.1 gives an isomorphism of L2 (S3 , τ ) onto L2 (R, ν). The inverse of this isomorphism is Dutkay’s “Fourier transform for R”, as established in [8, Corollary 5.8]. Corollary 6.5. Consider the filter m(z) = 2−1/2 (1+z2 ) for dilation by 3, and let (L2(R, ν), D, λ) be as in Section 5. Let τ be the measure on the solenoid S3 = lim(T, z → z3 ) described in ←−

Proposition 6.2. Then there is an isomorphism F of L2 (R, ν) onto L2 (S3 , τ ) such that (a) (F DF ∗ f )(ζ ) = m(π0 (ζ ))f (h(ζ )), (b) (F λk F ∗ f )(ζ ) = π0 (ζ )k f (ζ ), and (c) F (χC ) = 1.


2733

Proof. The composition of the isomorphism V∞ : L2 (S3 , τ ) → L2 (T)∞ of Theorem 6.4 with the isomorphism R∞ : L2 (T)∞ → L2 (R, ν) constructed in the proof of Proposition 5.1 is an isomorphism of L2 (S3 , τ ) onto L2 (R, ν); we take F := (R∞ ◦ V∞ )∗ . Then (a) and (b) follow from the properties of R∞ and V∞ . For (c), we compute

R∞ V∞ (1) = R∞ V∞ (1 ◦ π0 ) = R∞ U0 (1) = R0 (1) = χC . 2 Dutkay’s proof of Corollary 6.5 uses a uniqueness theorem for a family of “wavelet representations” of the Baumslag–Solitar group Z[N −1 ] Z due to Jorgensen [13, Theorem 2.4]. In the next section we show that Jorgensen’s theorem also follows easily from our Theorem 4.1. Corollary 6.5 and Theorem 5.4 imply that the functions ψˆ1 = 21/2 F (χ3−1 (C+1) )

and ψˆ2 = F (χ3−1 C − χ3−1 (C+2) )

generate a wavelet basis for L2 (S3 , τ ) with respect to the dilation described in (a) and the translation described in (b). 6.2. The winding line When m : T → C is a low-pass filter for dilation by N and m−1 (0) has measure zero, we can identify the direct limit lim(L2 (T), Sm ) with either L2 (R) (as in Example 4.2) or L2 (SN , τ ) −→

(using Theorem 6.4). Combining these two results gives an isomorphism R∞ ◦ V∞ of L2 (SN , τ ) onto L2 (R), from which we will obtain a completely different description of the measure τ as Lebesgue measure on a “winding line” obtained from an embedding of R in the solenoid. We begin by deriving a formula for R∞ ◦ V∞ on functions of the form g ◦ πn . We resume the notation of Example 4.2, and define DN : L2 (R) → L2 (R) by (DN f )(t) = N 1/2 f (N t). Then part (a) of Theorem 6.4 gives n−1 j (x) m zN R∞ ◦ V∞ (g ◦ πn )(x) = R∞ ◦ Un z → g(z) j =0

n−1 j −n (x) m zN = DN R z → g(z)

j =0

−n

2πiN −n x n−1 −n/2 2πiN −n+j x φ N x , g e m e =N j =0

and n applications of the scaling identity (4.2) imply that

−n R∞ ◦ V∞ (g ◦ πn )(x) = g e2πiN x φ(x). So we introduce the function w : R → SN which is uniquely characterized by

−n πn w(x) = e2πiN x for x ∈ R and n 0; this is the “winding line” referred to above.

(6.6)

2734


Theorem 6.6. Suppose that m : T → C is a low-pass filter for dilation by N which is Lipschitz near 1, which satisfies Cohen’s condition, and for which m−1 (0) has measure zero. Let φ ∈ L2 (R) be the associated scaling function satisfying (4.1), (4.2) and (4.3). Let w : R → SN be the function satisfying (6.6). Then the measure τ of Proposition 6.2 satisfies

f dτ = SN

2

f w(x) φ(x) dx

for f ∈ C(SN ),

(6.7)

R

and the formula (Tf )(x) := f (w(x))φ(x) defines a unitary isomorphism T of L2 (SN , τ ) onto ∗ S V )T ∗ = D and T (V ∗ μ (k)V )T ∗ is multiplication by e2πikx . L2 (R) such that T (V∞ ∞ ∞ N ∞ ∞ ∞ Proof. We fix g ∈ C(T), n 0, and compute:

2

(g ◦ πn ) w(x) φ(x) dx =

R

2

−n g e2πiN x φ(x) dx

R

= R

=

2 g e2πis N n φ N n s ds

2

2πis n−1 2πiN j s 2

m e φ(s) ds g e

R

=

1

2πis n−1 j s 2 2πiN

φ(s + k) 2 ds

m e g e j =0

k∈Z 0

n−1 j 2

m zN dz = g(z) T

=

(using (4.2))

j =0

(using (4.3))

j =0

(g ◦ πn ) dτ

(by (6.3)).

We can now deduce (6.7) from the uniqueness in Proposition 6.2. Eq. (6.7) implies that T is an isometry of L2 (SN , τ ) into L2 (R); surjectivity will be easy after we have the other properties of T. For the last two assertions, we let f ∈ L2 (SN , τ ). First, we use part (b) of Theorem 6.4 to see that

∗

∗

T V∞ S∞ V∞ f (x) = V∞ S∞ V∞ f w(x) φ(x)

= m π0 w(x) f h w(x) φ(x)

= m e2πix f w(N x) φ(x), which by the scaling equation is N 1/2 φ(N x)f (w(N x)) = (DN Tf )(x). Next, we use part (c) of Theorem 6.4 to see that


2735

∗

∗

T V∞ μ∞ (k)V∞ f (x) = V∞ μ∞ (k)V∞ f w(x) φ(x)

k

= π0 w(x) f w(x) φ(x) = e2πikx (Tf )(x). We still have to prove that T is surjective. For f ∈ L2 (T), we have T (f ◦ π0 )(x) = f (e2πix )φ(x), so the range of T contains the subspace V0 = span x → e2πikx φ(x): k ∈ Z in the usual multiresolution analysis {Vj } for L2 (R) associated to the low-pass filter m for dila∗ S V ) = D T implies that the range tion by N (as in Example 4.2). Since the formula T (V∞ ∞ ∞ N of T is closed under dilation, the range of T is a closed subspace containing j Vj , and hence must be all of L2 (R). 2 Remark 6.7. Ionescu and Muhly [12] have also recognised that the direct limit L2 (T)∞ can be realised as both L2 (R) and L2 (SN , τ ), and conjectured that the measure τ is supported on the winding line and is absolutely continuous with respect to the measure pulled over from Lebesgue measure on R (see the second last paragraph of [12]). The formula (6.7) confirms this conjecture, and also identifies the Radon–Nikodym derivative in terms of the scaling function φ. Remark 6.8. Theorem 6.6 holds without significant change for any dilation matrix A and lowpass filter m : Tn → C satisfying the hypotheses of Example 4.2. In this case A : Rn → Rn induces an endomorphism α of Tn = Rn /Zn , and the theorem gives an embedding w of Rn round the solenoid SA := lim(Tn , α) which carries the measure |φ(x)|2 dx into τ . ←−

7. Uniqueness of the wavelet representation We let (Γ∞ , ιn ) denote the direct limit lim(Γ, α), and write α∞ for the automorphism of Γ∞ −→

characterized by α∞ ◦ ιn = ιn ◦ α. We identify Γ with the subgroup ι0 (Γ ) of Γ∞ , so that α = α∞ |Γ . The semidirect product BS(Γ, α) := Γ∞ α∞ Z is known as the Baumslag–Solitar group of α (see, for example, [10]). Unitary representations W : BS(Γ, α) → U (H ) are determined by a unitary representation T = W |Γ and a unitary operator U = W(0,1) satisfying U Tγ = Tα(γ ) U ; −n T U n+j . Associated to the unitary representation T is a −n we recover W as W(α∞ γ (γ ),j ) = U representation πW : C(Γ) → B(H ) which takes the functions γ : ω → ω(γ ) to the operators Tγ ; the pair (πW , U ) is then covariant in the sense that U πW (f )U ∗ = πW (f ◦ α ∗ ). Now suppose that m is a filter for α ∗ and h : Γ → [0, ∞) is an integrable function such that 1 N

m(aω) 2 h(ω) = h α ∗ (ω)

for almost all ω ∈ Γ.

a∈ker α ∗

In this section we suppose that m is a continuous function (but see Remark 7.3 below). Following [13], we say that a unitary representation W of BS(Γ, α) on H is a wavelet representation for m with correlation function h if there is a cyclic vector φ ∈ H such that

2736


(WR1) U φ = πW (m)φ, and (WR2) (Tγ φ | φ) = Γ ω(γ )h(ω) dω for every γ ∈ Γ ; we then call φ a scaling element for W . Notice that if h = 1, then (WR2) says that the set {Tγ φ: γ ∈ Γ } is orthonormal, so in general the correlation function is a measure of the extent to which this set is not orthonormal. Example 7.1. We define a measure σ on Γ by f dσ = Γ f (ω)h(ω) dω, and then a routine calculation, as in [13, Lemma 3.2], shows that the operator Sm is isometric on L2 (Γ, σ ). Apγf plying the construction of Section 1 to Sm and the representation μ defined by μγ : f → gives a direct limit (L2 (Γ, σ )∞ , Un ), a unitary dilation S∞ of Sm , and a representation μ∞ of Γ on L2 (Γ, σ )∞ such that S∞ Un = Un Sm and S∞ μ∞ (γ ) = μ∞ (α(γ ))S∞ . This last identity says that (S∞ , μ∞ ) determines a unitary representation W of the Baumslag–Solitar group BS(Γ, α) on L2 (Γ, σ )∞ , which we claim is a wavelet representation for m and h. γ span a dense subset of First note that the elements μ∞ (γ )U0 (1) = U0 (μγ (1)) = U0 −n maps the range of U onto the range of U , it follows that the eleU0 (L2 (Γ, σ )). Since S∞ 0 n −n μ (γ )U (1) = W −n 2 ments S∞ ∞ 0 (α∞ (γ ),n) U0 (1) span a dense subspace of L (Γ , σ )∞ , and hence φ := U0 (1) is cyclic. To verify (WR1), notice that both sides are continuous in m, and so it γ . Then suffices to consider m = γ ∈Γ aγ πW (m)U0 (1) =

aγ μ∞ (γ )U0 (1) =

γ ∈Γ

aγ U0 ( γ ) = U0

γ ∈Γ

aγ γ

γ ∈Γ

= U0 (m) = U0 Sm (1) = S∞ U0 (1). For (WR2), we compute

γ )U0 (1) | U0 (1) = μ∞ (γ )U0 (1) | U0 (1) = U0 ( γ ) | U0 (1) = ( γ | 1), πW ( which is the right-hand side of (WR2). In the previous example, we have basically summarized the discussion in [13, pp. 15–20] under slightly different hypotheses (see Remark 7.3). The next result is the analogue of uniqueness in [13, Theorem 2.4], and our proof differs from the original in its use of the universal property via Theorem 4.1. Proposition 7.2 (Jorgensen). Suppose that W : BS(Γ, α) → U (H ) is a wavelet representation for m with correlation function h and scaling element φ. Then there is an isomorphism X of L2 (Γ, σ )∞ onto H such that (a) W(γ ,0) = Xμ∞ (γ )X ∗ for γ ∈ Γ , (b) W(0,1) = XS∞ X ∗ , and (c) XU0 (1) = φ. Proof. We aim to apply Theorem 4.1 with λγ = W(γ ,0) and D = W(0,1) . We define R : C(Γ) → H by Rf = πW (f )φ, and claim that R extends to an isometry on L2 (Γ, σ ). Since σ is a regular Borel measure, C(Γ) is dense in L2 (Γ, σ ), and it suffices to check that Rf 2 = f 2 for f


2737

of the form f = cγ γ . This follows from a straightforward calculation using the equality in (WR2) above. The relation Dλγ D ∗ = λα(γ ) is the covariance relation which characterizes the representations of BS(Γ, α). The covariance of (πW , D) = (πW , W(0,1) ) implies that

(RSm )f = πW m(f ◦ α ∗ ) φ = πW (f ◦ α ∗ )πW (m)φ = πW (f ◦ α ∗ )Dφ = DπW (f )φ = (DR)f, γ f we have and hence RSm = DR. Since μγ (f ) is the pointwise product

(Rμγ )f = R( γ f ) = πW ( γ f )φ = πW ( γ ) πW (f )φ = W(γ ,0) (Rf ) = (λγ R)f, and Rμγ = λγ R. So Theorem 4.1 gives an isomorphism R∞ of L2 (Γ, σ )∞ onto the closure of ∞ −n 2 γ ), and every D n λγ (φ) = n=0 D R(L (Γ , σ )). The range of R contains every λγ (φ) = R( R(Sm γ ) with n > 0, so the cyclicity of φ implies that R∞ is surjective. Properties (a) and (b) of X := R∞ follow from the properties of R∞ in Theorem 4.1. For (c), notice that XU0 (1) = R∞ U0 (1) = R(1) = φ, as required. 2 Remark 7.3. When Γ = Z and α(j ) = Nj , we recover a characterization of the wavelet representations of the classical Baumslag–Solitar group Z[N −1 ] Z. This is slightly different from Theorem 2.4 of [13], since we have assumed that m is continuous. The result in [13] applies to Borel filters m, but requires an extra hypothesis on the representation W which ensures that the representation πW of C(T) extends to a normal representation of L∞ (T), so that one can make sense of πW (m) in such a way that the covariance of (πW , U ) is preserved. It is not immediately obvious that when m(z) = 2−1/2 (1 + z2 ), the representation W of Z[3−1 ] Z on L2 (R, ν) constructed in Section 5 satisfies this normality hypothesis, so the above version of [13, Theorem 2.4] may be better suited to the application in [8, §5.2]. 8. Conclusions We have tackled a variety of problems associated with multiresolution analyses and wavelets using a systematic approach based on direct limits of Hilbert spaces and their universal properties. Previous authors have observed the connection with direct limits (often referring to them as “inductive limits”, and often referring to the process of turning an isometry into a unitary as “dilation”); the innovation in our approach lies in the systematic use of the universal property to identify a particular direct limit with a concrete Hilbert space of functions, such as L2 (R) or L2 (SN ). This approach does not eliminate the need for analytic arguments, but it does seem to help identify exactly what analysis is needed: in each situation we have considered, we have quickly been able to identify the ingredients necessary to make our approach work. References [1] L.W. Baggett, J.E. Courter, K.D. Merrill, The construction of wavelets from generalized conjugate mirror filters in L2 (Rn ), Appl. Comput. Harmon. Anal. 13 (2002) 201–223. [2] L.W. Baggett, P.E.T. Jorgensen, K.D. Merrill, J.A. Packer, Construction of Parseval wavelets from redundant filter systems, J. Math. Phys. 46 (2005) 1–28, #083502.

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[3] L.W. Baggett, N.S. Larsen, K.D. Merrill, J.A. Packer, I. Raeburn, Generalized multiresolution analyses with given multiplicity functions, J. Fourier Anal. Appl. (2008), published online in May, arXiv:0710.2071. [4] J.J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal. 5 (1998) 389–427. [5] M. Bownik, The construction of r-regular wavelets for arbitrary dilations, J. Fourier Anal. Appl. 7 (2001) 489–506. [6] O. Bratteli, P.E.T. Jorgensen, Isometries, shifts, Cuntz algebras and multiresolution analyses of scale N , Integral Equations Operator Theory 28 (1997) 382–443. [7] J. D’Andrea, K.D. Merrill, J.A. Packer, Fractal wavelets of Dutkay–Jorgensen type for the Sierpinski gasket space, in: Frames and Operator Theory in Analysis and Signal Processing, in: Contemp. Math., vol. 451, Amer. Math. Soc., Providence, 2008, pp. 69–88. [8] D.E. Dutkay, Low-pass filters and representations of the Baumslag–Solitar group, Trans. Amer. Math. Soc. 358 (2006) 5271–5291. [9] D.E. Dutkay, P.E.T. Jorgensen, Wavelets on fractals, Rev. Mat. Iberoamericana 22 (2006) 131–180. [10] D.E. Dutkay, P.E.T. Jorgensen, A duality approach to representations of Baumslag–Solitar groups, in: Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey, in: Contemp. Math., vol. 449, Amer. Math. Soc., Providence, 2008, pp. 99–127. [11] R.A. Gopinath, C.S. Burrus, Wavelet transforms and filter banks, in: C.K. Chui (Ed.), Wavelets: A Tutorial in Theory and Applications, Academic Press, San Diego, 1992, pp. 603–654. [12] M. Ionescu, P.S. Muhly, Groupoid methods in wavelet analysis, in: Group Representations, Ergodic Theory, and Mathematical Physics: A Tribute to George W. Mackey, in: Contemp. Math., vol. 449, Amer. Math. Soc., Providence, 2008, pp. 193–208. [13] P.E.T. Jorgensen, Ruelle operators: Functions which are harmonic with respect to a transfer operator, Mem. Amer. Math. Soc. 152 (720) (2001). [14] N.S. Larsen, I. Raeburn, From filters to wavelets via direct limits, in: Operator Theory, Operator Algebras and Applications, in: Contemp. Math., vol. 414, Amer. Math. Soc., Providence, 2006, pp. 35–40. [15] S.G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. 315 (1989) 69–87. [16] K.R. Parthasarathy, Introduction to Probability and Measure, Macmillan, Delhi, 1977, Springer-Verlag, New York, 1978. [17] K. Petersen, Ergodic Theory, Cambridge Univ. Press, Cambridge, 1983. [18] R.S. Strichartz, Construction of orthonormal wavelets, in: J.J. Benedetto, M.W. Frazier (Eds.), Wavelets: Mathematics and Applications, CRC press, Boca Raton, 1994, pp. 23–50.


Spectral inequalities for non-selfadjoint elliptic operators and application to the null-controllability of parabolic systems M. Léautaud a,b,∗ a Université Pierre et Marie Curie Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France b Laboratoire POEMS, INRIA Paris-Rocquencourt/ENSTA, CNRS UMR 2706, France

Received 23 July 2009; accepted 20 October 2009

Communicated by J.-M. Coron

Abstract We consider elliptic operators A on a bounded domain, that are compact perturbations of a selfadjoint operator. We first recall some spectral properties of such operators: localization of the spectrum and resolvent estimates. We then derive a spectral inequality that measures the norm of finite sums of root vectors of A through an observation, with an exponential cost. Following the strategy of Lebeau and Robbiano (1995) [25], we deduce the construction of a control for the non-selfadjoint parabolic problem ∂t u + Au = Bg. In particular, the L2 norm of the control that achieves the extinction of the lower modes of A is estimated. Examples and applications are provided for systems of weakly coupled parabolic equations and for the measurement of the level sets of finite sums of root functions of A. © 2009 Elsevier Inc. All rights reserved. Keywords: Non-selfadjoint elliptic operators; Spectral theory; Parabolic systems; Controllability

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2740 Spectral theory of perturbated selfadjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2744

* Address for correspondence: Université Pierre et Marie Curie Paris 6, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France. E-mail address: [email protected].


2740

M. Léautaud / Journal of Functional Analysis 258 (2010) 2739–2778

3. Spectral inequality for perturbated selfadjoint elliptic operators . . . . . . . 4. From the spectral inequality to a parabolic control . . . . . . . . . . . . . . . . 5. Application to the controllability of parabolic coupled systems . . . . . . . 6. Application to the controllability of a fractional order parabolic equation 7. Application to level sets of sums of root functions . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2748 2752 2763 2769 2772 2773 2773 2777

1. Introduction 1.1. Results in an abstract setting Various methods have been developed to prove the null-controllability of parabolic equations ∂t u + Au = Bg, (1) u|t=0 = u0 where A is of elliptic type, and B is a bounded operator (for instance a distributed control). A large number of such results are based on the seminal papers of either G. Lebeau and L. Robbiano [25] or A. Fursikov and O.Yu. Imanuvilov [11]. The first approach has been used for the treatment of self-adjoint operators A. The second approach also permits to address nonselfadjoint operators and semilinear equations. Each method has advantages. We shall focus on those of the Lebeau–Robbiano method. First, this method only relies on elliptic Carleman estimates that are simpler to produce than their parabolic counterparts as needed for the Fursikov–Imanuvilov approach. Second, it enlights some fundamental properties of the elliptic operator A through a spectral inequality. This type of spectral inequality, further investigated in [26] and [20], has a large field of applications, like the measurement of the level sets of sums of root functions (see [20] or Section 7). For control issues, it yields the cost of the nullcontrollability of the low frequencies of the elliptic operator A. Finally, based on this fine spectral knowledge of A, this approach gives an explicit iterative construction of the control function, using both the partial controllability result and the natural parabolic dissipation. In this paper, we extend the method of [25] for the abstract parabolic system (1) to cases in which A is a non-selfadjoint elliptic operator. More precisely, we treat the case where A is a small perturbation of a selfadjoint operator, under certain spectral assumptions. We suppose A = A0 + A1 where A1 is a relatively compact perturbation of an elliptic selfadjoint operator A0 . We first obtain spectral inequalities of the following type. Denoting by Πα the projector on the spectral subspaces of A associated to eigenvalues with real part less than α, we have, for some θ 1/2 θ Πα wH CeDα B ∗ Πα w Y ,

w ∈ H.

(2)

Here, B ∗ ∈ L(H ; Y ) denotes a bounded observation operator and the state space H and the observation space Y are Hilbert spaces. Typically, B ∗ = 1ω is a distributed observation. This spectral inequality yields the cost of the measurement of some finite sums of root-vectors of A through the partial observation B ∗ , in terms of the largest eigenvalue of A in the considered


2741

spectral subspace. The difficulty here is not in the elliptic estimates since Carleman inequalities remains unchanged under the addition of lower order terms, but in the spectral theory of the nonselfadjoint operator A. This motivates the exposition of the spectral theory of such operators in Section 2, following [5]. Inequalities of the type of (2) were first established in [25] for the Laplace operator under a relatively different form. In fact, the inequality proved in [25] reads Πα w+ 2H + Πα w− 2H √ √ √ 2 CeD α ϕB ∗ et A Πα w+ + e−t A Πα w− L2 (0,T ;Y ) ,

w+ , w− ∈ H,

(3)

where ϕ(t) is a smooth cut-off function. Here, in the non-selfadjoint case, we prove a spectral inequality of the form √ √ θ Πα wH CeDα ϕB ∗ et A Πα + e−t A Πα w L2 (0,T ;Y ) ,

w ∈ H.

(4)

We also prove that such an inequality is sufficient to deduce controllability results (in the case θ < 1). Spectral inequality of the form (2) first appeared in [26] and [20] for the Laplace operator with θ = 1/2. Comments can be made about the two different forms of spectral inequalities, (2) on the one hand, and (3), (4) on the other hand, that involves an integration with respect to an additional variable. Both types can be proved with interpolation inequalities (like (12) or (14) below), themselves following from local Carleman inequalities. The interpolation inequality (12) used to prove (3) exhibits a spacialy distributed observation, whereas the interpolation inequality (14), used to prove (2) is characterized by a boundary observation at time t = 0. This latter form is more convenient to use for control results (see Section 6). Yet, for systems, the derivation of such an interpolation inequality with boundary observation at time t = 0 is open (see Section 5). In Section 4, we deduce from the spectral inequality (4) the construction of a control function for the parabolic problem (1). Following [25] and an idea that goes back to the work of D.L. Russell [31], the spectral inequality yields the controllability of the parabolic system on the θ related finite dimensional spectral subspace Πα H with a control cost of the form CeCα . In the case θ < 1, we can then construct a control to the full parabolic equation (1). We improve the method of [25] from a spectral point of view. The proof of the controllability in [25] relies on the Hilbert basis property of the eigenfunctions of the Laplace operator. Here we only use some resolvent estimate away from the spectrum. No (Hilbert or Riesz) basis property is required in the construction. The main results of this article can be summarized as follows • Starting from an interpolation inequality of the Lebeau–Robbiano type (12) (resp. (14)), the spectral inequality (4) (resp. (2)) holds (see Theorem 3.2, resp. 3.3). • The spectral inequality (4) (resp. (2)) implies that the control cost needed to drive any initial θ datum to (I − Πα )H is bounded by CeCα (see Theorem 4.10, resp. Proposition 6.2). • In the case θ < 1, the non-selfadjoint parabolic system (1) is null-controllable in any positive time (see Theorem 4.13). The question of the optimality of the power θ in these spectral inequalities remains open. For an elliptic operator A, in the results we obtain, the power θ increases linearly with the space

2742


dimension n: θ ≈ max{1/2; n/2 − cst}. In the selfadjoint case however, the spectral inequalities (2) and (4) always hold with θ = 1/2, and the controllability result always follows. The power 1/2 is known to be optimal in this case (see [20] or [24]). Moreover, thanks to global parabolic inequalities, the controllability result of Theorem 4.13 is in general known to be true, independently on θ (see [11]). We are thus led to consider that the spectral inequalities should be true with θ = 1/2: the power θ we obtain does not seem to be natural but only technical. Since θ ≈ max{1/2; n/2 − cst}, our controllability results, obtained for θ < 1, are limited to small dimensions n. This problem arises in all the applications we give. In fact, in each of them, the power θ is the limiting point of the theory. The origin of the dimension-depending term n/2 − cst in θ cannot be found in the elliptic estimates or in the control theory described above, but only in the resolvent estimates we use (see Section 2). If one wants to improve our results, one has to improve the resolvent estimates of [5] (maybe taking into account that A is a differential operator). On the contrary, if one wants to prove the optimality of the power θ , one needs to produce lower bounds for the resolvent of A. In any case, it does not seem to be an easy task at all. Remark 1.1. Note that we can replace A by A + λ in (1) without changing the controllability properties of the parabolic problem. More precisely, suppose that the problem

∂t v + (A + λ)v = Bf, v|t=0 = u0

is null-controllable in time T > 0 by a control function f . Then, the control function g = eλt f drives the solution of Problem (1) to zero in time T and has the same regularity as f . Hence, in the sequel, we shall consider that the operator A is sufficiently positive without any loss of generality. Remark 1.2. In the sequel, C will denote a generic constant, whose value may change from line to line. 1.2. Some applications We give several applications of such a spectral approach in Sections 5, 6 and 7. This work has first been motivated by the problem of Section 5, i.e. the null-controllability in any time T > 0 of parabolic systems of the type ⎧ ∂ u + P1 u1 + au1 + bu2 = 0 ⎪ ⎨ t 1 ∂t u2 + P2 u2 + cu1 + du2 = 1ω g ⎪ u1|t=0 = u01 , u2|t=0 = u02 ⎩ u1 = u2 = 0

in (0, T ) × Ω, in (0, T ) × Ω, in Ω, on (0, T ) × ∂Ω,

(5)

where Pi = − div(ci ∇·), i = 1, 2, are selfadjoint elliptic operators, ω is a non-empty subset of the bounded open set Ω ⊂ Rn , and a, b, c, d are bounded functions of x ∈ Ω and the coupling factor b does not vanish in an open subset O ⊂ Ω. Such parabolic systems have already been investigated. The first result was stated in [32] in the context of insensitizing control (thus, one of the equations is backward in time). In this work, a positive answer to the null-controllability problem (5) is given in the case a = c = d = 0, b = 1O and O ∩ ω = ∅. The control problem (5)


2743

was then solved in [1] and [12] in the case O ∩ ω = ∅. In all these references, the authors used global parabolic Carleman estimates to obtain an observability estimate. They can thus treat timedependant coefficients and semilinear reaction–diffusion problems. To the author’s knowledge, the null-controllability in the case O ∩ ω = ∅ remains an open problem. A recent step in this direction has been achieved in [23], proving approximate controllability by a spectral method in the case a = c = d = 0, b = 1O without any condition on O ∩ ω. The field of coupled systems of d parabolic equations, d 2, has also been investigated in [2] where the authors prove a Kalman-type rank condition in the case where the coefficients are constant and in [13] where the authors prove the controllability of a cascade system with nonvanishing variable coupling coefficients by one control force. Here, we obtain a spectral inequality of the type (4) with θ = max{1/2; n/2 − 1} for the operator

A=

P1 + a c

b P2 + d

with the measurement on only one component. As opposed to the results of [1] and [12], we can provide an estimation on the control cost of low modes for system (5). The null-controllability of (5) follows from the method described above in the case n 3 (corresponding to θ < 1 in the spectral inequality). In Section 6, we address the fractional power controllability problem

∂t u + Aν u = Bg, u|t=0 = u0 ∈ H,

that has been solved in the selfadjoint case in [29] and [28]. We prove the null-controllability in any time T > 0 when ν > θ . The case ν θ is open. In particular when ν = 1, it allows us to give an estimation on the control cost of low modes for the following heat equation with a transport term

∂t u − u + b · ∇u + cu = 1ω g u|t=0 = u0 u=0

in (0, T ) × Ω, in Ω, on (0, T ) × ∂Ω.

(6)

In the case n 2 (corresponding to θ < 1 in the spectral inequality), this estimate is sufficient to recover the null-controllability of (6). In Section 7, we are not concerned with controllability issues, but with the measurement of the (n − 1)-dimensional Hausdorff measure of the level sets of sums of root functions of A = − + b · ∇ + c on the n-dimensional compact manifold Ω. Yet, the technique and the estimates used here are the same as in the other sections. We obtain the following estimate, for sums of root functions associated with eigenvalues of real part lower than α, i.e., ϕ ∈ Πα L2 (Ω), we have Hn−1 {ϕ = K} C1 α θ + C2 ,

1 n−1 . θ = max ; 2 2

This type of inequality has already been proved in the selfadjoint case for the Laplace operator. In [7] and [8], H. Donnelly and C. Fefferman proved, for ϕ an eigenfunction associated with the

2744


eigenvalue α, the estimate Hn−1 ({ϕ = K}) Cα 1/2 . This was generalized to sums of eigenfunctions associated with eigenvalues lower than α in [20]. We generalize this last result for operators that are lower order perturbations of the Laplace operator. Here, however, θ is in general strictly greater than 1/2. 2. Spectral theory of perturbated selfadjoint operators To prepare the following sections, we shall first recall a theorem about the spectral theory of some operators close to being selfadjoint. This result can be presented with numerous variations and refinements. This subject has been a well-developed research field since the 60 s. The first step of the theory is the Keldysh theorem on the completeness of the root vectors of a weakly perturbed compact selfadjoint operator (see [14] for more precisions). The main result here, Theorem 2.5, was first proved by A.S. Markus and by D.S. Katsnel’son [21]. A simplified proof was given later by A.S. Markus and V.I. Matstaev with Matsaev’s method of “artificial lacuna”. This proof is presented in the book [27, Chapter 1] by A.S. Markus in the more general case of weak perturbations of certain normal operators. For later use in the following sections, we sharpen some of the results as given by M.S. Agranovich [5], following the steps of his proof. Let H be a separable Hilbert space, (·,·)H be the scalar product on H and .H be the associated norm. The set of bounded linear operators on H is denoted by L(H ) and is equiped with the subordinated norm · L(H ) . As usual, for a given operator A on H , we denote by D(A) its domain, by ρ(A) its resolvent set, i.e., the subset of C where the resolvent RA (z) = (z − A)−1 is defined and bounded, and by Sp(A) = C \ ρ(A) its spectrum. Proposition 2.1. Let A = A0 + A1 be an operator on H . Assume that (a) Re(Au, u)H 0 for all u ∈ D(A), (b) A0 : D(A0 ) ⊂ H → H is selfadjoint, positive, with a dense domain and compact resolvent, (c) A1 : D(A1 ) ⊂ H → H is q-subordinate to A0 , i.e., there exist k0 > 0 and q ∈ [0, 1) such 1/2 1/2 2q 2−2q that for every u ∈ D(A0 ), |(A1 u, u)H | k20 A0 uH uH . We then have, (i) D(A) = D(A0 ) ⊂ D(A1 ) ⊂ H and A has a dense domain and a compact resolvent, q (ii) Sp(A) ⊂ Ok0 = {z ∈ C, Re(z) 0, | Im(z)| < k0 |z|q } q 2 2 (iii) RA (z)L(H ) d(z,Sp(A d(z,R for z ∈ C \ O2k0 , where d(z, S) denotes the euclidian +) 0 )) distance from z to the subset S of C. Remark 2.2. Note that the subordination assumption (b) can be found under the different forms: q

q

q

• D(A0 ) ⊂ D(A1 ), and A1 uH CA0 uH for u ∈ D(A0 ) in [4]; q 1−q • D(A0 ) ⊂ D(A1 ), and A1 uH CA0 uH uH for u ∈ D(A0 ) in [27, Chapter 1]; −q • A1 A0 L(H ) = C < +∞ in [9]. Replacing assumption (b) by one of these assertions does not change the conclusions of Proposition 2.1 and Theorem 2.5.


2745

Fig. 1. Complex contours around the spectrum of A.

Remark 2.3. Point (i) implies that Sp(A) contains only isolated eingenvalues, with finite multiplicity and without any accumulation point. Furthermore, for every λ ∈ Sp(A), the sequence of iterated null-spaces Nk = N ((A − λ)k ) is stationary. For what follows, we shall need the spectrum of A to be located in a “parabolic neighborhood” of the real positive axis. We note that there exist λ0 > 0, K0 > 0 such that q q q Sp(A + λ0 ) ⊂ Ok0 + λ0 ⊂ O2k0 + λ0 ⊂ PK0 = z ∈ C, Re(z) 0, | Im(z)| < K0 Re(z)q , (7) see Fig. 1. We now fix λ0 and K0 satisfying this assumption. Referring to Remark 1.1, we shall work with the operator A + λ0 , yet writing A for simplicity. Definition 2.4. Let (αk )k∈N ⊂ R+ be an increasing sequence, 0 < α0 < α1 < · · · < αk < · · · tending to +∞, and such that αk ∈ / Re(Sp(A)), for every k ∈ N. Then, we set Ik = {z ∈ C, Re(z) = q αk , | Im(z)| K0 αk } and by γk , we denote the positively oriented contour delimited by the verq tical line segments Ik on the right and Ik−1 on the left and by the parabola ∂PK0 on the upper and the lower side (see Fig. 1). We also define the associated spectral projectors Pk =

1 2iπ

RA (z) dz. γk

Note that the spectral projector Pk is a projector on the characteristic subspaces of A associated with the eigenvalues that are inside γk . The projectors satisfy the identity Pk Pj = δj k Pk . Moreover, thanks to Remark 2.3, the projectors Pk , k ∈ N, have finite rank. We can now state the main spectral result, that can (at least partially) be found under different forms in [4], [27, Chapter 1], [5,9].

2746


Theorem 2.5. Let 0 < λ1 λ2 · · · λk · · · be the spectrum of the selfadjoint operator A0 . We assume the additional condition that there exists p > 0 such that lim supj →∞ λj j −p > 0. Then, setting β = max{0, p −1 − (1 − q)}, there exists a sequence (αk )k∈N ⊂ R+ as in Definition 2.4 such that for some C > 0 RA (z)

β

L(H )

eCαk ,

k ∈ N, z ∈ Ik .

(8)

Remark 2.6. Note that Proposition 2.1 point (iii) implies that RA (z)L(H ) q

z ∈ γk ∩ ∂PK0 . Thus the resolvent estimate (8), RA (z)L(H ) e

β Cαk

2 q K0 αk

2 q K0 α0

for

holds for all z ∈ γk .

Remark 2.7. Comments can be made about this theorem and its proof: • The idea of the proof of Theorem 2.5 is to find uniform gaps around vertical lines Re(z) = αk in the spectrum of A, to be sure that the resolvent RA is well-defined in these regions. To find such gaps, one proves the existence of sufficiently large gaps in the spectrum of A0 and places αk in these zones. The results presented in [4], [27, Chapter 1], [5,9] are in fact a bit stronger than Theorem 2.5 since they contain not only the resolvent estimate (8) but also some basis properties. • In the case p(1 − q) 1 one can prove a Riesz basis property for the subspaces Pk H , i.e. one can write H = k∈N Pk H and there exists c > 0 such that for all u ∈ H , c−1 u2H 2 2 k∈N Pk uH cuH . • In the case p(1 − q) < 1 one can prove a weaker basis property (so-called Abel basis) for the subspaces Pk H . For α ∈ R+ we define the spectral projector on the characteristic subspace of H associated with the eigenvalues of real part less than max{αk ; αk α}: Πα =

αk α

Pk =

1 2iπ

RA (z) dz = Γα

1 2iπ

RA (z) dz,

αk α γk

where Γα is the positively oriented contour delimited by the vertical line segments Re(z) = q max{αk ; αk α} on the right and Re(z) = α0 > 0 on the left and by the parabola PK0 on the upper and the lower side. On each finite dimensional subspace Pk H (or equivalently Πα H ) we have a holomorphic calculus for A (e.g. see [22]); given a holomorphic function f , we have 1 f (A)Pk = f (APk ) = 2iπ

f (z)RA (z) dz ∈ L(Pk H ). γk

In the subsequent sections, we shall consider the adjoint problem of the abstract parabolic system (1), involving A∗ = A0 + A∗1 . The spectral theory of A and A∗ and their respective functional calculus are connected by the following proposition (see [22]).


2747

Proposition 2.8. Let f be a holomorphic function, γ a positively oriented contour in C \ Sp(A). f (z)RA (z) dz and γ the positively oriented complex We denote fˇ : z → f (¯z), f γ (A) = 1 2iπ

γ

conjugate contour of γ . Then, γ is a contour in C \ Sp(A∗ ) and (f γ (A))∗ = fˇγ (A∗ ). With this new notation, f γk (A) = f (A)Pk = f (APk ). Noting that with the choices made above, γk = γk , Γα = Γα , we obtain ∗ Pk A∗ = 1γk A∗ = 1γk (A) = Pk (A)∗ = Pk∗ and Πα (A∗ ) = (Πα (A))∗ = Πα∗ . More generally, if fˇ = f , we have f γk (A∗ ) = (f γk (A))∗ and f Γα (A∗ ) = (f Γα (A))∗ . √ √ Example 2.9. For t ∈ R, the functions f (z) = etz or f (z) = et z (taking for z the principal value of the square root of z ∈ C) or f (z) = R ψ(t)e−itz dt (ψ being a real function) fulfill the property fˇ = f . For all these functions, we have f γk (A∗ ) = (f γk (A))∗ , f Γα (A∗ ) = (f Γα (A))∗ . This will be used in the following sections.

Remark 2.10. Note that RA∗ (z)L(H ) = RA (¯z)L(H ) . As a consequence, any sequence satisfying (8) for A also satisfies (8) for A∗ . In the course of the construction of a control function that we present below, we shall need a precise asymptotics of the increasing sequence (αk )k∈N , that is not given in [5]. We first remark that if (αk )k∈N is a sequence satisfying properties (i)–(iii) of Theorem 2.5, then every subsequence of (αk )k∈N also satisfies these properties. We shall thus seek a minimal growth for the asymptotics of (αk )k∈N . For μ ∈ R, we set N (μ) = #{k ∈ N, λk ∈ Sp(A0 ), λk μ}. Here, we prove the following proposition. 1

Proposition 2.11. If the eigenvalues of A0 satisfy the following asymptotics: N (μ) = m0 μ p + 1

o(μ p ), as μ → +∞, then, for all δ > 1 we can choose the sequence (αn )n∈N such that there exists N ∈ N and for every n N , we have δ n−1 αn δ n . First note that the assumption we make here for the asymptotics of the eigenvalues is stronger than that made in Theorem 2.5 for it implies λk ∼k→∞ Ck p . To prove Proposition 2.11, we briefly recall how the sequence (αk )k∈N is built in [5]. Every q q αn is in the interval [μn − hμn , μn + hμn ], where (μn )n∈N is a sequence increasing to infinity q q β such that #{k ∈ N, λk ∈ Sp(A0 ), λk ∈ [μn − hμn , μn + hμn ]} Cμn . We thus need to show the existence of such a sequence (μn )n∈N , having a precise asymptotics as n goes to infinity. This is the aim of Lemma 2.12 below, replacing [5, Lemma 7]. With such a result, for all fixed δ > 1 we can build (αn )n∈N such that there exists N ∈ N and for every n N , αn satisfies δ n−1 αn δ n . We can then follow the proof of [5] to finish that of Proposition 2.11, estimating the resolvent on q q vertical lines Re(z) = C ∈ [μn − hμn , μn + hμn ]. Lemma 2.12. We set r = p −1 and recall that β = max{0, r − 1 + q}. Assume that as μ → +∞. N (μ) = m0 μr + o μr

(9)

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Then, for every h > 0, δ > 1, there exist C > 0 and N ∈ N such that for every n N , there exists q q q q β μn such that [μn − hμn , μn + hμn ] ⊂ [δ n−1 , δ n ] and N (μn + hμn ) − N (μn − hμn ) Cμn . Proof. We proceed by contradiction. Let h > 0, δ > 0, and suppose that ∀C > 0, ∀N ∈ N, ∃n0 N + 1, ∀μ satisfying μ − hμq , μ + hμq ⊂ δ n0 −1 , δ n0 , we have N μ + hμq − N μ − hμq > Cμβ . (10) We choose 0 < ε < m0 . From the asymptotics (9) of N , there exists N0 ∈ N such that for every n N0 , (m0 − ε)δ nr N δ n (m0 + ε)δ nr . 2hδ β+1 δ−1 (m0 + ε + 1) and N ∈ N such that 2hδ β+1 nβ 0. Such a N exists δ−1 (m0 + ε + 1)δ

Let us fix C − ε)δ nr

(11)

N N0 , and for all n N + 1,

(m0 − since β r + q − 1 < r. Then for these C and N , assumption (10) gives the existence of n0 N + 1 N0 + 1 such that for every μ q β satisfying [μ − hμq , μ + hμq ] ⊂ [δ n0 −1 , δ n0 ], we have N (μ + hμq ) − N (μ − hμ . We n) >nCμ −1 0 −δ 0 disdenote by x the floor function. In the interval [δ n0 −1 , δ n0 ], there are at least δ 2hδ n0 q joint intervals of the type [μ − hμq , μ + hμq ], each containing more than Cδ (n0 −1)β eigenvalues of A0 . Hence, n0

δ − δ n0 −1 (n0 −1)β δ − 1 n0 (1−q) Cδ δ N δ n0 − N δ n0 −1 Cδ (n0 −1)β − 1 . 2hδ n0 q 2hδ Then, taking into account the lower bound on C and the asymptotics (11) for n = n0 − 1 N N0 , we obtain 2hδ β+1 N δ n0 (m0 + ε + 1)δ n0 β+n0 (1−q) − (m0 + ε + 1)δ (n0 −1)β + (m0 − ε)δ r(n0 −1) δ−1 (m0 + ε + 1)δ n0 (β+1−q) , since n0 N + 1. The asymptotics (11) for n0 gives N (δ n0 ) (m0 + ε)δ n0 r (m0 + ε)δ n0 (β+1−q) since β r − 1 + q. This yields a contradiction and concludes the proof of the lemma. 2 3. Spectral inequality for perturbated selfadjoint elliptic operators In this section, we prove in an abstract setting some spectral inequalities where the norm of a finite sum of root vectors of A is bounded by a partial measurement of these root vectors. For the proof, we assume that some interpolation inequality holds. This inequality will be proved in the case of different elliptic operators in Section 5. Such interpolation inequality were used in [25] and [26] to achieve a spectral inequality of the type we prove here. Note that this type of spectral inequality can however be obtained by other means (e.g. doubling properties [3], or directly from global Carleman estimates [6]).


2749

Here, we suppose that A = A0 + A1 : D(A) ⊂ H −→ H satisfies some of the spectral properties of Section 2, i.e. (a)–(c) of Proposition 2.1 and the resolvent estimate (8) of Theorem 2.5. We define the following Sobolev spaces based on the selfadjoint operator A0 . Definition 3.1. For s ∈ N and τ1 < τ2 , we set Hs (τ1 , τ2 ) =

s

n/2 , H s−n τ1 , τ2 ; D A0

n=0

which is a Hilbert space with the natural norm vHs (τ1 ,τ2 ) =

1/2

m n/2 2 ∂ A v 2 t

L (τ1 ,τ2 ;H )

0

≈

s n=0

n+ms

1/2 v2

n/2

H s−n (τ1 ,τ2 ;D (A0 ))

.

Note that H0 (τ1 , τ2 ) = L2 (τ1 , τ2 ; H ). Let Y be another Hilbert space, B ∗ ∈ L(H, Y ) be a bounded operator. Let T0 be a positive number, ϕ ∈ C0∞ (0, T0 ; C), ϕ = 0 and θ = max{1/2, p −1 − (1 − q)} = max{1/2, β}. Theorem 3.2. Suppose that there exist C > 0, ζ ∈ (0, T0 /2) and ν ∈ (0, 1] such that for every v ∈ H2 (0, T0 ) vH1 (ζ,T0 −ζ ) C v1−ν H1 (0,T

0)

∗ ϕB v

L2 (0,T0 ;Y )

ν + −∂t2 + A∗ v H0 (0,T ) . 0

(12)

Then, there exist positive constants C, D such that for every positive α, for all w ∈ Πα∗ H , √ √ ∗ θ ∗ wH CeDα ϕB ∗ et A + e−t A w L2 (0,T

0 ;Y )

(13)

.

In other situations, we can prove another interpolation inequality with an observation at the boundary t = 0. In this case, we obtain a simpler spectral inequality, involving no time integration in the observation term. Theorem 3.3. Suppose that there exist C > 0, ζ ∈ (0, T0 /2) and ν ∈ (0, 1] such that for every v ∈ H2 (0, T0 ) satisfying v(0) = 0, we have vH1 (ζ,T0 −ζ ) C v1−ν H1 (0,T

0)

∗ ν B ∂t v(0) + −∂ 2 + A∗ v 0 t Y H (0,T ) . 0

(14)

Then, there exist positive constants C, D such that for every positive α, for all w ∈ Πα∗ H , θ wH CeDα B ∗ w Y .

(15)

The estimation of the constant in the inequality in terms of the maximal eigenvalue in the finite sum is the key point in the control applications below.

2750


Proof of Theorem 3.2. For w ∈ Πα∗ H , we set √ √ ∗ 1 ∗ v(t) = et A + e−t A w = 2iπ

√ t √z e + e−t z RA∗ (z)w dz.

Γα

We have v ∈ H 2 (0, T0 ; H ) ∩ H 1 (0, T0 ; D(A)) ⊂ H2 (0, T0 ) as D(A) = D(A0 ). We first notice that (−∂t2 + A∗ )v = 0. Second, we have to estimate every single term of 1/2 2 v2H1 (0,T ) = v2L2 (0,T ;H ) + ∂t v2L2 (0,T ;H ) + A0 v L2 (0,T ;H ) , 0 0 0 0 √ t √A∗ ∗ −t A ∗ Πα L(H ) wH +e vH e √ 1 t √z −t z e RA∗ (z) dz +e wH 2iπ L(H ) Γα

√ √ C meas (Γα ) sup RA∗ (z)L(H ) sup et z + e−t z wH

Cαe

z∈Γα √ C(α β +t α)

z∈Γα

wH β

since meas (Γα ) Cα, RA∗ (z)L(H ) eCα from estimate (8) of Theorem 2.5, Remark 2.10 and, √ √ t √z e + e−t z 2et Re( z) ,

√ √ √ with Re( z) |z| C α for z ∈ Γα . θ Thus, for some C > 0, vL2 (0,T0 ;H ) CeCα wH . Concerning the second term in vH1 (0,T0 ) , we have √ √ √ ∗ ∗ ∂t vL2 (0,T0 ;H ) = A∗ et A − e−t A w L2 (0,T

0 ;H )

,

θ

and similar computations show that ∂t vL2 (0,T0 ;H ) CeCα wH . In order to estimate the third term in vH1 (0,T0 ) , we use assumption (a) of Proposition 2.1, observing that for all u ∈ H , 1/2 2 A u = (A0 u, u)H = A∗ u, u − A∗ u, u 1 0 H H H k0 1/2 2q 2−2q A∗ u, u H + A0 uH uH 2 ∗ k0 q1 1/2 2 − 1 qε A0 uH + (1 − q)ε 1−q u2H A u, u H + 2 for every positive ε and every q ∈ (0, 1), thanks to Young’s inequality. We then choose ε such that

1 γ q 2 qε

1 2

and we obtain for every u ∈ H ,

1/2 2 A u C A∗ u, u + u2 C A∗ u uH + u2 . H H 0 H H H


2751

The same estimate is obvious in the case q = 0. Similar computations as those performed above θ 1/2 yield A0 vL2 (0,T0 ;H ) CeCα wH , and we have finally w ∈ Πα∗ H.

θ

vH1 (0,T0 ) CeCα wH ,

We√now produce a lower bound for the left-hand side of (12). We first notice that the operator √ ∗ ∗ t A −t A +e )Πα∗ is an isomorphism on Πα∗ H and we compute an upper bound for its inverse: (e √ √ 1 t √A∗ −1 t √z ∗ −t z −1 e ∗ e + e−t A Πα∗ L(H ) = + e R (z) dz A 2iπ L(H ) Γα

√ √ β −1 C meas (Γα )eCα sup et z + e−t z z∈Γα

CαeC(α

β +t

√

α)

sup z∈Γα

1

√ |e2t z

+ 1|

.

Then, we have √ 2t √z √ √ e + 1 e2t Re( z) − 1 2t Re( z) 2t α0 ,

(16)

√ √ since Re( z) α0 > 0 on Γα . Hence, for some constant C > 0, √ t √A∗ −1 ∗ e + e−t A Π ∗

L(H )

α

C C(α β +t √α) e . t

Concerning the left-hand side of (12), we thus have the following lower bound: T0 −ζ

vH1 (ζ,T −ζ ) v2L2 (ζ,T −ζ ;H ) 0 0 2

√ t √A∗ −1 −2 ∗ e + e−t A Π ∗

α

2 L(H ) dtwH

ζ

Ce−2Cα

β

T0 −ζ

t 2 e−2tC

√

α

dtw2H

ζ

Ce

−2Cα β 2

T0 −ζ

ζ

e−2tC

√

α

dtw2H

ζ

Ce

−2Cα β 2 e

ζ

√ −T0 C α

√ α

Cζ 2 (T0 − 2ζ )e−C(α

√ sinh C α(T0 − 2ζ ) w2H

β +√α)

w2H Ce−Cα w2H . θ

The interpolation inequality (12) then gives √ 1−ν ∗ t √A∗ θ θ ∗ ν ϕB e Ce−Cα wH CeCα wH + e−t A w L2 (0,T

0 ;Y )

.

2752


1−ν Dividing both side by wH , we finally obtain the existence of positive constants C, D such that for every positive α, for all w ∈ Πα∗ H , √ √ ∗ θ ∗ wH CeDα ϕB ∗ et A + e−t A w L2 (0,T

0 ;Y )

2

.

Proof of Theorem 3.3. This proof follows the same; for w ∈ Πα∗ H , we take −1/2 √ 1 v(t) = A∗ sinh t A∗ w = 2iπ

√ sinh(t z) RA∗ (z)w dz, √ z

Γα

√

√

∗

∗

instead of v(t) = (et A + e−t A )w, see [20,26]. It satisfies (−∂t2 + A∗ )v = 0, v(0) = 0 θ and ∂t v(0) = w. We also have v ∈ H2 (0, T0 ) and vH1 (0,T0 ) CeCα wH as above. Only the lower bound for vH1 (ζ,T0 −ζ ) has to be proved. To begin with, the operator √ (A∗ )−1/2 sinh(t A∗ )Πα∗ is an isomorphism on Πα∗ H and we compute an upper bound for its inverse: √

1 sinh(t z) −1 RA∗ (z) dz √ L(H ) = 2iπ z L(H )

∗ −1/2 √ −1 A sinh t A∗ Π ∗ α

Γα

Cαe

Cα β

√ z sup √ sinh(t z)

z∈Γα β

Cα 3/2 eCα sup z∈Γα

Cα 3/2 eC(α

|et

β +t √α)

√ z

sup z∈Γα

√

1 − e−t

√

z|

1

√ |e2t z

− 1|

.

√

Then, |e2t z − 1| e2t Re( z) − 1, and (16) gives a lower bound for vH1 (ζ,T0 −ζ ) vL2 (ζ,T0 −ζ ;H ) as in the preceding proof. The conclusion follows as in the proof of Theorem 3.2. 2 4. From the spectral inequality to a parabolic control In this section we construct a control for the parabolic abstract problem (1). We follow the method introduced by G. Lebeau and L. Robbiano in [25]. The non-selfadjoint nature of the problem requires however modifications in their approach. Let H and Y be two Hilbert spaces, H standing for the state space and Y the control space. We suppose that B ∈ L(Y, H ) is a bounded control operator and A is an unbounded operator A : D(A) ⊂ H −→ H that satisfies all the spectral properties of Section 2, i.e. (a)–(c) (and thus also (i)–(iii)) of Proposition 2.1, the resolvent estimate (8) of Theorem 2.5, and the asymptotics given by Proposition 2.11. In particular, the properties (a)–(c) of Proposition 2.1 imply that −A generates a C 0 -semigroup of contraction on H . If we take u0 in H , problem (1) is then well-posed in H . Let T0 be a positive number, ϕ ∈ C0∞ (0, T0 ; C), ϕ = 0, and B ∗ ∈ L(H, Y ) the adjoint operator of B, i.e., such that (By, h)H = (B ∗ h, y)Y for every y ∈ Y , h ∈ H . We assume that the result of


2753

Theorem 3.2, (i.e. the spectral inequality (13)) holds. An example will be given in Section 5. We shall first interpret this spectral inequality (13) of Theorem 3.2 as an observability estimate for an elliptic evolution problem. Remark 4.1. Note that if we suppose the spectral inequality (15) of Theorem 3.3 instead of (13), the construction of the control function follows that of [28] or [24] and is much simpler. In fact, the spectral inequality (15) directly yields an observability inequality for the partial problem (27) and implies an analogous of Theorem 4.9. This proof can be found in Section 6, taking ν = 1. Section 4.4 then ends the proof of the null-controllability. Note that in this case, there is no restriction on the subordination number q (in Proposition 4.5, we require q < 3/4) since there is no need of the regularisation with a Gevrey function. 4.1. Elliptic controllability on Πα H with initial datum in Pk H From the spectral inequality (13), we deduce a controllability result for a family of (finite dimensional) elliptic evolution problems. Let first Gα be the following gramian operator T0 √ √ √ 2 √ ∗ t A ∗ Gα = e + e−t A Πα B ϕ(t) B ∗ et A + e−t A Πα∗ dt. 0

Lemma 4.2. The operator Gα in an isomorphism from Πα∗ H onto Πα H . We denote by Gα−1 the inverse of Gα on Πα H . Then, there exists D0 > 0 and for every s ∈ N there exists Cs such that for every T0 > 0, α > 0, k ∈ N∗ satisfying αk α, for every vk ∈ Pk H , the function √ 2 √ ∗ ∗ hk (t) = ϕ(t) B ∗ et A + e−t A Gα−1 vk

(17)

satisfies (i) hk ∈ C0∞ (0, T0 ; Y ); s θ (ii) ∂ts hk L∞ (0,T0 ;Y ) Cs α 2 eD0 α vk H ; √ √ ∗ ∗ (iii) for all w ∈ Πα∗ H , (vk , w)H = (Bhk , (et A + e−t A )w)L2 (0,T0 ;H ) . Remark 4.3. Here, we could actually take the “initial datum” v in Πα H and the result and its proof remain the same. We have chosen to take v in Pk H since it is the precise result we use in Proposition 4.5, in particular to prove the estimate (iii) of Proposition 4.5, which in turn is a key point in the proof of Theorem 4.10 below. Proof. We first observe that the spectral inequality (13) implies that Gα in an isomorphism from Πα∗ H onto Πα H . Then, we note that point (ii) implies (i), and is itself a direct consequence of expression (17) θ where B ∗ is bounded and Gα−1 L(H ) CeD0 α from the spectral inequality (13). Finally, we check that (iii) holds. For w ∈ Πα∗ H , we compute

2754

M. Léautaud / Journal of Functional Analysis 258 (2010) 2739–2778 √ √ ∗ ∗ Bhk , et A + e−t A w L2 (0,T ;H ) 0 √ √ 2 ∗ t √A∗ √ ∗ ∗ −t A∗ Gα−1 vk , et A + e−t A w L2 (0,T = B ϕ(t) B e +e

0 ;H )

T0 √ √ 2 ∗ t √A∗ t √A ∗ A −t A −t ∗ −1 e Πα B ϕ(t) B e Πα dt Gα vk , w = +e +e 0

!"

#

Gα

= (vk , w)H .

H

2

Remark 4.4. Lemma 4.2 corresponds to a null-controllability property on [0, T0 ] in Πα H for the elliptic control problem with initial condition in Pk H ⎧ −∂ 2 u + Au = Πα Bhk , ⎪ ⎪ ⎨ t u|t=0 = 0, u = v ∈ Pk H, ∂ ⎪ ⎪ ⎩ t |t=0 u|t=T0 = ∂t u|t=T0 = 0, whose dynamics remains in Πα H for every t ∈ [0, T0 ]. The adjoint problem is the following, well-posed in Πα∗ H : ⎧ 2 ∗ ⎨ −∂t z + A z = 0, z = w ∈ Πα∗ H, ⎩ |t=T0 ∂t z|t=T0 = 0. The estimation of the “cost” of the control hk is the key point for the following parabolic partial controllability properties. 4.2. Parabolic controllability on Πα H with initial datum in Pk H From the elliptic controllability result of Lemma 4.2, we now deduce a corresponding parabolic controllability result. The main tools here are the transformation introduced in [31] and a Paley–Wiener-type theorem. Proposition 4.5. We suppose that the subordination number satisfies q < 34 and we fix γ > max{ 2(1−q) 3−4q , 1}. Then, for all T > 0 there exists D1 > 0 and for every s ∈ N there exists Cs such that for every 0 < T < T , α > 0, k ∈ N∗ satisfying αk α, for every uk,0 ∈ Pk H , there exists a control function Gk such that: (i) Gk ∈ C0∞ (0, T ; Y );

γ

(ii) ∂ts Gk L∞ (0,T ;Y ) Cs T −2γ s exp(D1 (α θ + T1γ + (T αk ) 2γ −1 ) − T αk−1 )uk,0 H ; ∗ ∗ (iii) for all w ∈ Πα∗ H , −(uk,0 , e−T A w)H = (BGk , e−(T −t)A w)L2 (0,T ;H ) . Remark 4.6. For technical requirements, that can be found in the proof of Lemma A.1 in Appendix A.1, we have assumed that the subordination q of A1 to A0 is less than 34 here. This will be the case in all the applications we present in Section 5.


2755

Remark 4.7. The new variable T here is the time in which we want to control the full equation (29), appearing in Section 4.4. Proof. From Lemma 4.2, we have for any positive T0 (that will be fixed equal to 1 below): for all vk ∈ Pk H , there exists hk (that we know with precision) such that for all w ∈ Πα∗ H , √ √ ∗ ∗ (vk , w)H = vk , Pk∗ w H = Bhk , et A + e−t A Pk∗ w L2 (0,T

0 ;H )

T0 √ t √A −t A e Pk Bhk (t) dt, w = +e H

0

T0

=

1 2iπ

=

√ t √z −t z e RA (z) dz Bhk (t) dt, w +e H

γk

0

1 2iπ

T0 √ √ t z −t z e hk (t) dt dz, w . RA (z)B +e

γk

H

0

We introduce the Fourier–Laplace transform of a function: ˆ f (z) = f (t)e−itz dt, z ∈ C. R

If f ∈ C0∞ (0, T0 ; Y ), then fˆ is an entire function with values in Y . We write fˆ ∈ H(C; Y ). Recalling that hk ∈ C0∞ (0, T0 ; Y ), we have hˆ k ∈ H(C; Y ) and

(vk , w)H =

1 2iπ

√ √ RA (z)B hˆ k (i z) + hˆ k (−i z) dz, w

.

(18)

H

γk

From Lemma 4.2 (ii), taken with s = 0, we obtain hˆ k (z)Y C0 eD0 α eT0 | Im(z)| vk H . Following [31] and [25], we set θ

Qk −iz2 = hˆ k (iz) + hˆ k (−iz), and note that Qk (z) is an entire function with respect to z. We now have for every vk ∈ Pk H , the existence of Qk ∈ H(C; Y ) such that for all w ∈ Πα∗ H ,

(vk , w)H =

1 2iπ

RA (z)BQk (−iz) dz, w

,

(19)

H

γk

with Qk satisfying θ

Qk (z)Y C0 eD0 α eT0

√

|z|

vk H .

(20)

2756


The goal is now to see Qk as the Fourier–Laplace transform of a regular function with compact support in (0, T ). However, (a Hilbert-valued version of) the Paley–Wiener theorem [18, Theorem 15.1.5] indicates that the inverse Fourier transform of Qk is only in the dual space (G2 (R; Y )) of the space of Gevrey functions of order 2. With the convolution by a function ˆ we can now regularize the inverse Fourier transe ∈ Gσ , σ ∈ (1, 2), i.e. by multiplying Qk by e, form of Qk . We fix T0 = 1 and set σ = 2 − γ1 ∈ (1, 2). Since we have required that q < 34 and γ > max{ 2(1−q) 3−4q , 1}, the Gevrey index σ = 2 −

1 γ

satisfies σ >

1 2(1−q)

what is equivalent to

1 < 1 − 2σ

q . Under these conditions, Lemma A.1 of Appendix A.1 gives the existence of a Gevrey function e ∈ Gσ satisfying (the constants ci are positive) ⎧ supp(e) = [0, 1] and 0 < e(t) 1 for all t ∈ (0, 1), ⎪ ⎪ 1 ⎨ e(z) ˆ c1 e−c2 |z| σ if Im(z) 0, (21) ⎪ 1 ⎪ q ⎩ e(z) σ −c |z| 1−q q 4 c3 e ˆ in − iPK T 1−q = −i z ∈ C, Re(z) 0, Im(z) < K0 T Re(z) . 0

q

q

The parabola PK T 1−q is chosen here so that for every k ∈ N, for every z ∈ γk ⊂ PK0 (defined 0 q in (7)), we have T z ∈ PK T 1−q and the lower bound of (21) holds. 0 We set, for T < T , ˆ z)Qk (z) ∈ H(C; Y ), gˆ k (z) = e(T and because of (20) and the first two points of (21), gˆ k (z) satisfies the following estimates √ gˆ k (z) C0 eT Im(z) eD0 α θ e |z| vk H Y

gˆ k (z) C0 c1 e Y

1 −c2 |T z| σ

θ

e D0 α e

√

|z|

for every z ∈ C,

vk H

if Im(z) 0.

(22) (23)

From (22), (23) and the Paley–Wiener-type theorem given in Proposition A.3 in Ap pendix A.2, there exists gk ∈ C0∞ (0, T ; Y ) such that gˆ k (z) = R gk (t)e−izt dt for every z ∈ C. 1 iτ t dτ . From (23), g satisfies the estiThe function gk is then given by gk (t) = 2π k R gˆ k (τ )e mates 1 √ s C0 c 1 D 0 α θ s −c2 |T τ | σ ∂ g k ∞ e |τ | e e |τ | dτ vk H , s ∈ N. t L (0,T ;Y ) 2π R

The Laplace method [10] applied to this integral (only dependent on s and γ ) finally implies that there exists C > 0 and for every s ∈ N there exists Cs > 0 such that for every T > 0, α > 0, k ∈ N∗ such that αk α, for every vk ∈ Pk H , s ∂ gk t

L∞ (0,T ;Y )

C

Cs eD0 α T −2γ s e T γ vk H . θ

(24)

Let us now properly construct the control function that satisfies the three assertions of the proposition. We first note that for T < T , the operators e(−iT ˆ A) and e−T A are two isomorq −T ˆ z) and e z do not vanish in PK0 . Given phisms of Pk H since the holomorphic functions e(−iT


2757

uk,0 ∈ Pk H , we set vk = −[e(−iT ˆ A)]−1 e−T A uk,0 ∈ Pk H and gk the associated control function given as preceding. We set Gk (t) = gk (T − t), which satisfies point (i) of the proposition. From (24), we obtain s −1 C θ ∂ Gk ∞ ˆ A) Pk L(H ) e−T A Pk L(H ) uk,0 H . (25) Cs eD0 α T −2γ s e T γ e(−iT t L (0,T ;Y ) We can estimate 1 1 R = (z) dz A 2iπ L(H ) e(−iT ˆ z) L(H )

−1 e(−iT ˆ A) Pk

γk

1 1 meas(γk ) sup sup RA (z)L(H ) 2π | e(−iT ˆ z)| z∈γk z∈γk

1 sup RA (z)L(H ) |e(−iT ˆ z)| z∈γk

Cαk sup

z∈γk

1 σ

β

C αk sup c3−1 ec4 |T z| eCαk z∈γk

1

C ec$4 (|T αk | σ +α ) , θ

where we have used the third property of the Gevrey function e given in (21) and the resolvent estimate (8) of Theorem 2.5 on γk . A similar estimate for e−T A Pk L(H ) gives −T A e Pk

Cαk e−T αk−1 +Cα . θ

L(H )

We finally obtain from (25) that for all T > 0 there exists D1 > 0 and for every s ∈ N there exists Cs > 0 such that for every 0 < T < T , α > 0, k ∈ N∗ such that αk α, for every uk,0 ∈ Pk H , s ∂ Gk t

L∞ (0,T ;Y )

1 σ

D1

Cs eD1 α T −2γ s e T γ eD1 |T αk | e−T αk−1 uk,0 H . θ

1 σ

Point (ii) of the proposition is thus proved recalling that (BGk , e

−(T −t)A∗

w)L2 (0,T ;H ) , with w

∈ Πα∗ H .

=

γ 2γ −1 .

(26)

To prove (iii), we compute

We have

∗ ∗ BGk , e−(T −t)A w L2 (0,T ;H ) = Bgk , e−tA w L2 (0,T ;H ) = e−tA Bgk , w L2 (0,T ;H ) T

=

1 2iπ

=

1 2iπ

RA (z) dz Bgk (t) dt, w

H

T

RA (z)B γk

As supp(gk ) ⊂ (0, T ), we obtain

e

−tz

γk

0

0

e−tz gk (t) dt dz, w

. H

2758


∗ BGk , e−(T −t)A w L2 (0,T ;H ) =

=

1 2iπ

RA (z)B gˆ k (−iz) dz, w

H

γk

1 2iπ

RA (z)B e(−iT ˆ z)Qk (−iz) dz, w H

γk

1 = e(−iT ˆ A) RA (z)BQk (−iz) dz, w 2iπ H γk

because the holomorphic calculus gives for φ ∈ H(C; C) and ψ ∈ H(C; H )

1 2iπ

φ(z)RA (z) dz

1 2iπ

γk

1 RA (z)ψ(z) dz = φ(z)RA (z)ψ(z) dz. 2iπ

γk

γk

From (19), we then have

BGk , e

−(T −t)A∗

w L2 (0,T ;H )

=

1 2iπ

RA (z)BQk (−iz) dz, eˆ −iT A∗ w

γk

H

= vk , eˆ −iT A∗ w H as e(−iT ˆ A∗ )w ∈ Πα∗ H . Recalling that vk = −[e(−iT ˆ A)]−1 e−T A uk,0 , we have obtained that the ∗ function Gk constructed here satisfies for all w ∈ Πα H −1 ∗ BGk , e−(T −t)A w L2 (0,T ;H ) = − e(−iT ˆ A) e−T A uk,0 , eˆ −iT A∗ w H = − e−T A uk,0 , w H , for every uk,0 ∈ Pk H . Point (iii) is thus proved. This concludes the proof of Proposition 4.5.

2

Remark 4.8. Proposition 4.5 is a null-controllability property on [0, T ] in the finite dimensional space Πα H for the parabolic control problem: ⎧ ⎨ ∂t u + Au = Πα BGk , u|t=0 = uk,0 ∈ Pk H, ⎩u |t=T = 0. This also means that for every initial datum u0,k ∈ Pk H and T > 0, there exists Gk ∈ C0∞ (0, T ; Y ) such that the solution of

satisfies Πα u(T ) = 0.

∂t u + Au = BGk , u|t=0 = uk,0 ∈ Pk H,


2759

4.3. Parabolic controllability on Πα H We shall now combine the controllability results for initial datum in Pk H , αk α to obtain a null-controllability result for an initial datum in Πα H , i.e., for the problem ⎧ ⎨ ∂t u + Au = Πα BGα , u|t=0 = u0 ∈ Πα H (27) ⎩u |t=T = 0. The norm of the control function Gα will be estimated to prepare for the next section, where a control for an arbitrary initial condition in H is constructed. Theorem 4.9. For the control problem (27), for every positive T , for every γ sufficiently large, there exists a control function Gα ∈ C0∞ (0, T ; Y ) driving u0 to 0 in time T with a cost given by Gα L∞ (0,T ;Y ) C exp(D(α θ + T1γ ))u0 H . Actually, we prove the following more precise result. Theorem 4.10. Let q < 34 and γ > max{ 2(1−q) 3−4q , 1}. For all T > 0, there exists D > 0 and for every s ∈ N there exists Cs such that for every 0 < T < T , α > 0, for every u0 ∈ Πα H , there exists a control function Gα such that: (i) Gα ∈ C0∞ (0, T ; Y ); (ii) ∂ts Gα L∞ (0,T ;Y ) Cs T −2γ s exp(D(α θ + T1γ ))u0 H ; ∗ ∗ (iii) for all w ∈ Πα∗ H , −(u0 , e−T A w)H = (BGα , e−(T −t)A w)L2 (0,T ;H ) . Proof. We write u0 = αk α Pk u0 ∈ Πα H , with Pk u0 ∈ Pk H and Proposition 4.5 gives for every k the existence of a control function Gk satisfying: ⎧ ∗ ∗ ⎨ for all w ∈ Πα∗ H, − Pk u0 , e−T A w = BGk , e−(T −t)A w 2 ; H L (0,T ;H ) (28) γ s ⎩ ∂ Gk ∞ Cs T −2γ s exp D1 (α θ + T1γ + (T αk ) 2γ −1 ) − T αk−1 Pk u0 H t L (0,T ;Y ) We set Gα =

Gk ∈ C0∞ (0, T ; Y )

αk α

and (i) is clear as the sum is finite. To prove (iii), given w ∈ Πα∗ H , we simply compute ∗ ∗ ∗ − u0 , e−T A w H = BGk , e−(T −t)A w L2 (0,T ;H ) − Pk u0 , e−T A w H = αk α

αk α

∗ = BGα , e−(T −t)A w L2 (0,T ;H ) .

We now prove point (ii). Here, we use the asymptotic estimation for the sequence (αk )k∈N given in Proposition 2.11. Let δ > 1, there exists N ∈ N such that δ k−1 αk δ k if k N . We then have

2760


s ∂ Gα

L∞ (0,T ;Y )

t

∂ s Gk

L∞ (0,T ;Y )

t

αk α

Cs T −2γ s eD1 (α

θ+ 1 Tγ

)

eD1 (T αk )

γ 2γ −1 −T α k−1

Pk u0 H

αk α

Cs T −2γ s eD1 (α

θ+ 1 Tγ

)

u0 H

eD1 (T δ

γ k ) 2γ −1 −T δ k−2

.

αk α

It remains to estimate the sum. Recalling that 2γγ−1 < 1, the function x → eD1 x bounded on R+ by a constant κ = κ(γ , δ). We thus have

γ

e

D1 (T δ k ) 2γ −1 −T δ k−2

Nκ +

αk α

kN αk α

γ 2γ −1 −δ −2 x

is

ln(α) . κ κ N + ln(δ)

Finally, changing the constants Cs , we conclude that s ∂ Gα t

L∞ (0,T ;Y )

Cs T

−2γ s

1 θ exp D α + γ u0 H . T

2

Remark 4.11. Here, we have constructed a control in C0∞ (0, T ; Y ). In the case Y = H and p for every p ∈ N, B ∗ ∈ we are able to construct with the same techniques a control %L(D(A )), ∞ function in C0 (0, T ; p∈N D(Ap )), following [25]. 4.4. Decay property for the semigroup and construction of the final control We shall now conclude the proof of the main controllability theorem. We consider the full controllability problem: given T > 0, we construct a control function g such that the solution of the problem

∂t u + Au = Bg, u|t=0 = u0 ∈ H,

(29)

satisfies u(T ) = 0 in H . The proof uses both the partial control result of Theorem 4.9 and the decay rate of the semigroup generated by −A, once restricted to (I − Πα )H . We first prove an estimate of this decay rate. We denote by (SA (t))t∈R+∗ the C 0 -semigroup of contraction generated by −A. $ ∈ N such that for every k N $, t Proposition 4.12. There exist C > 0 and N Cαkθ −tαk SA (t)(I − Πα ) . k L(H ) Ce

1 αk ,


2761

Proof. From [30, Theorem 1.7.7] we first write the semigroup generated by −A(I − Παk ) as an q integral over the infinite positively oriented contour ∂PK0 SA (t)(I − Παk ) = SA(I −Παk ) (t) =

1 2iπ

e−tz RA(I −Παk ) (z) dz.

q

∂ PK

0

− We set Λk = {z ∈ PK0 , Re(z) αk }, ∂Λ+ k = {z ∈ ∂Λk , Re(z) > αk , Im(z) 0} and ∂Λk = + − {z ∈ ∂Λk , Re(z) > αk , Im(z) 0} so that ∂Λk = ∂Λk ∪ Ik ∪ ∂Λk and is a positively oriented contour. q Since RA(I −Παk ) is holomorphic in C \ Λk , we may shift the path of integration from ∂PK0 to ∂Λk without changing the value of the integral. Hence, q

1 SA (t)(I − Παk ) = 2iπ

e−tz RA (z) dz

∂Λk

1 = 2iπ

e−tz RA (z) dz +

∂Λ+ k

1 2iπ

e−tz RA (z) dz +

1 2iπ

Ik

e−tz RA (z) dz,

∂Λ− k

where −tz e R (z) dz A ∂Λ+ k

∞ L(H )

Ce−tx dx C

e−tαk , t

αk

since RA (z) is uniformly bounded on ∂Λ0 from Proposition 2.1 point (iii). The same estimate holds for the integral over ∂Λ− k . Finally estimate (8) of Theorem 2.5 gives e−tz RA (z) dz Ik

Cαk eCαk −tαk . q

L(H )

θ

Then, for some C > 0, we obtain the following estimation of the semigroup

e−tαk Cαkθ −tαk SA (t)(I − Πα ) , C e + k L(H ) t $ ∈ N such that for k N $, Thus, taking N

1 αk

e−Cαk , we finally obtain θ

Cαkθ −tαk SA (t)(I − Πα ) , k L(H ) Ce and the proposition is proved.

t > 0.

$, t 1 , kN αk

2

For the sake of completeness, we now construct the control function for the parabolic problem (29), following [25]. The decay rate proved in Proposition 4.12 shows that we have now to

2762


restrict ourselves to the case θ < 1, i.e. p −1 + q < 2. We recall that N is an integer such that for all n N , δ n−1 αn δ n (see Proposition 2.11). Theorem 4.13. Suppose that q < 34 and θ < 1. Then, for every T > 0, u0 ∈ H , there exists a control function g ∈ C0∞ (0, T ; Y ) such that the solution u of the problem (29) satisfies u(T ) = 0. Proof. We first fix ρ and γ > max{ 2(1−q) , 1} such that 0 < ρ < min{ γ1 , 1−θ θ }. We set 3−4q −jρθ T = Kδ , K being such that 2T = T . We divide the time interval [0, T ] = j j ∈N j [a , a ], with (a ) such that a = 0 and a = a + 2T . Hence, lim a j j ∈N 0 j +1 j j j →∞ j = T . j ∈N j j +1 For uj ∈ Παj H , we define by Gαj (uj , Tj ) the control function given by Theorem 4.10 that drives uj to zero in time Tj , which in particular satisfies the estimate (ii) of this theorem. $} (so that δ j −1 αj δ j for j J0 and the We set J0 an integer such that J0 max{N, N decay rate of Proposition 4.12 holds) and Tj δ −(j −1) αj−1 for every j J0 . We now define the control function g: • if t aJ0 , we set g = 0, and u(t) = SA (t)u0 ; • if j J0 , t ∈ (aj , aj + Tj ], we set g = Gαj (Παj u(aj ), Tj ), and t u(t) = SA (t − aj )u(aj ) +

SA (t − s)Bg(s) ds; aj

• if j J0 , t ∈ (aj + Tj , aj +1 ], we set g = 0, and u(t) = SA (t − aj − Tj )u(aj + Tj ). We recall that SA (t)L(H ) 1 because we have required A to be positive. During the first phase 0 t aJ0 , we simply have u(aJ0 )H u0 H . The choice of the control function during the second phase implies for j J0 , Παj u(aj + Tj ) = 0 and u(aj + Tj ) C exp Cδ j θ + Cδ j γρθ u(aj ) exp C δ j θ u(aj ) , H H H as γρ < 1. Finally, during the third phase, the decay rate of the semigroup is given for j J0 by Proposition 4.12 and we then have u(aj +1 ) C exp Cδ j θ − Tj δ j −1 u(aj + Tj ) H H j θ j (1−ρθ) u(aj + Tj )H . exp C δ − Kδ Combining the estimations given on the three phases, we obtain for j J0 j u(aj +1 ) exp C δ kθ − Kδ k(1−ρθ) u0 H . H k=J0


Because of our choice ρ <

1−θ θ ,

j

2763

we have 1 − ρθ > θ , and for some c > 0,

kθ

C δ

− Kδ

k(1−ρθ)

c exp −cδ j (1−ρθ) ,

k=J0

and thus for every j J0 , u(aj +1 ) c exp −cδ j (1−ρθ) u0 H . H

(30)

From Theorem 4.10 point (ii) and estimate (30), we have

1 −2γ s θ u(aj ) H sup Cs Tj exp D αj + γ L∞ (0,T ;Y ) Tj j J0 sup Cs δ 2jρθγ s exp D δ j θ + Kδ jργ θ − cδ (j −1)(1−ρθ) u0 H < +∞,

s ∂ g t

j J0

following the same estimations as above. Thus, g ∈ C0∞ (0, T ; Y ). This implies in particular that the solution u of (29) is continuous with values in H . Hence, from (30) we directly obtain u(T ) = lim u(aj +1 ) = 0, H H j →∞

and u(T ) = 0 in H .

2

5. Application to the controllability of parabolic coupled systems In this section, we apply the abstract results proved in the previous sections to second order elliptic operators and to the controllability of parabolic systems. In the following, we first check that the assumptions of Proposition 2.1 and Theorem 2.5 for these elliptic operators are fulfilled and we prove the interpolation inequality (12). Sections 3 and 4 then directly yield the spectral inequality and the controllability results. We are concerned with the system ⎧ ∂ u + P1 u1 + au1 + bu2 = 0 ⎪ ⎨ t 1 ∂t u2 + P2 u2 + cu1 + du2 = 1ω g 0 0 u ⎪ ⎩ 1|t=0 = u1 , u2|t=0 = u2 u1 = u2 = 0

in (0, T ) × Ω, in (0, T ) × Ω, in Ω, on (0, T ) × ∂Ω,

(31)

where Ω is an open connected subset of Rn with n 3 (a compact connected Riemannian manifold with or without boundary of dimension n 3). We suppose that ∂Ω is at least of class C 2 . The function 1ω (x) stands for the characteristic function of the open subset ω ⊂ Ω and a, b, c, d ∈ L∞ (Ω). Here, Pi , i = 1, 2, denotes a positive elliptic selfadjoint operator on Ω: Pi u = − divx (ci (x)∇x u) where ci (x) is a symmetric uniformly elliptic matrix, i.e. ci ∈ W 1,∞ (Ω) and there exists C > 0 such that for every x ∈ Ω, ξ ∈ Rn , ξ · ci (x)ξ C|ξ |2 . We set H = (L2 (Ω))2 , D(A0 ) = (H 2 (Ω) ∩ H01 (Ω))2 , and

A0 =

P1 0

0 P2

and A1 =

a c

b d

.

2764


Referring to Remark 1.1, we shift A = A0 + A1 by a λ0 > 0 sufficiently large so that Eq. (7) is satisfied. We carry on the analysis with the operator A + λ0 , which we write A by abuse of notation. Then, the operator A satisfies the assumptions (a)–(c) of Proposition 2.1 with q = 0 since A1 is bounded in H . Moreover, for μ ∈ R+ the number of eigenvalues of the operator Pi lower than μ is given by the Weyl asymptotics Ni (μ) = mi μn/2 + o(μn/2 ) as μ → +∞. Thus, for A, N is given by N (μ) = N1 (μ) + N2 (μ) = (m1 + m2 )μn/2 + o(μn/2 ) and A satisfies the assumption of Proposition 2.11 and that of Theorem 2.5 with p = 2/n. The assumption θ < 1 of Theorem 4.13 is satisfied if and only if n/2 − 1 < 1, i.e., n 3. We set Y = L2 (Ω) as the control space and the operator B is given by B : g −→ (0, 1ω g)T , and is bounded from L2 (Ω) to (L2 (Ω))2 and its adjoint is B ∗ : (u1 , u2 )T −→ 1ω u2 ∈ L((L2 (Ω))2 ; L2 (Ω)). Now, it remains to prove the interpolation inequality (12) to apply Theorem 3.2. Proposition 5.1. Let T0 > 0, ζ ∈ (0, T0 /2). Suppose that there exists an open subset O ⊂ Ω, O ∩ ω = ∅ such that the coupling coefficient b ∈ L∞ (Ω) satisfies |b(x)| b0 > 0 for almost every x ∈ O. Then, there exist C > 0, ϕ ∈ C0∞ (0, T0 ) and ν ∈ (0, 1) such that for every v ∈ (H 2 ((0, T0 ) × Ω))2 , v|(0,T0 )×∂Ω = 0, we have v(H 1 ((ζ,T0 −ζ )×Ω))2 1−ν Cv(H 1 ((0,T

0 )×Ω))

2

∗ ϕB v

L2 ((0,T0 )×Ω)

+ −∂t2 + A∗ v (L2 ((0,T

0 )×Ω))

ν 2

.

(32)

Proof. We first prove (32) with ϕB ∗ vL2 ((0,T0 )×Ω) replaced by vL2 (V ) with V ⊂ (0, T0 ) × O ∩ω. In a second step, thanks to local elliptic energy estimates, we eliminate the first component and obtain (32). Here, the time variable does not play a particular role. Thus, for the sake of clarity, we simplify the notation, denoting by ∇ the time-space gradient (∂t , ∇x )T , by div the time-space divergence ∂t + divx , by Ci the time-space diffusion matrices Ci = 10 c0i , and by −i the time-space elliptic operators −i = −∂t2 + Pi = −∂t2 − divx (ci ∇x ·) = − div(Ci ∇·). We also set &∗ = −∂t2 + A∗ = A

−1 + a b

c −2 + d

.

We also denote by (·,·) the L2 scalar product on L2 ((0, T0 ) × Ω) or (L2 ((0, T0 ) × Ω))n+1 , · the associated norms, and by (·,·)i , i = 1, 2, the L2 scalar product defined by ξ, ξ i = ci ξ, ξ ,

n+1 ξ, ξ ∈ L2 ((0, T0 ) × Ω) , i = 1, 2,

and · i the associated norm. With the assumptions made on ci , the norms · i and · are equivalent on (L2 ((0, T0 ) × Ω))n+1 . &∗ . These are direct consequences of the classical We first state local Carleman estimates for A local Carleman estimates for the elliptic operators i . We first choose a local weight function φ satisfying a subellipticity condition with respect to both 1 and 2 (which can be done


2765

taking φ = eλψ for λ sufficiently large and ψ satisfying |∇ψ| C > 0, see [17, Chapter 8], [25] or [24]). Then there exists h1 > 0 and C > 0 such that for every w ∈ (C0∞ ((0, T0 ) × Ω))2 , w = (w1 , w2 )T , and 0 < h < h1 , 2 2 2 heφ/ h wi + h3 eφ/ h ∇wi Ch4 eφ/ h i wi ,

i = 1, 2

(see [25], or [11] in the case ci ∈ W 1,∞ ). Adding these two estimates and absorbing the zero&∗ : order terms for h sufficiently small, we directly obtain the same estimate for A

2 2 ∇w1 &∗ w 2 , . heφ/ h w + h3 eφ/ h ∇w Ch4 eφ/ h A ∇w = ∇w2 By optimizing in h (see [25]), these local Carleman estimates yield local interpolation estimates of the form ν 1−ν &∗ v(H 1 (B(3r)))2 Cv(H , 1 ((0,T )×Ω))2 v(H 1 (B(r)))2 + A v (L2 ((0,T )×Ω))2 0

0

where B(r) denote concentric balls of radium r. Similar estimates at the boundary (0, T0 ) × ∂Ω are also direct consequences of the Carleman estimates at the boundary for a scalar elliptic operator. Then, following [25], these local interpolation inequalities can be “propagated”, so that we obtain the following global interpolation inequality, with two observations in H 1 norm, localized in any nonempty open subset W of (ζ, T0 − ζ ) × Ω: ν 1−ν &∗ v(H 1 ((ζ,T0 −ζ )×Ω))2 Cv(H . (33) 1 ((0,T )×Ω))2 v(H 1 (W ))2 + A v (L2 ((0,T )×Ω))2 0

0

Let us take the open subsets W , V , and U such that W ⊂ V , V ⊂ U , and U ⊂ (0, T0 ) × O ∩ ω. Elliptic regularity for the operators i , i = 1, 2, shows that there exists C > 0 such that vH 1 (W ) C & A∗ v + vL2 (V ) . It finally remains to eliminate one of the two observations with energy estimates. In fact, we prove that v1 L2 (V ) C & A∗ v + v2 L2 (U ) . &∗ v, i.e. We write f = (f1 , f2 )T = A

f1 = −1 v1 + av1 + cv2 , f2 = −2 v2 + bv1 + dv2 .

(34)

Let χ be a cut-off function such that χ ∈ C0∞ (U ), 0 χ 1, and χ = 1 on V ⊂ U . We set η = χτ ,

η1 = χ τ +1 ,

η2 = χ τ −1 ,

for a real number τ > 2, so that η, η1 , η2 and χ τ −2 are also cut-off functions of the same type. We notice that ∇η1 = η(τ + 1)∇χ and ∇η = η2 τ ∇χ , where ∇χ is a bounded function.

2766


We form the scalar product of the second equation of (34) by η2 v1 2 η v1 , bv1 = η2 v1 , f2 − η2 v1 , dv2 + η2 v1 , 2 v2 . The third term can be estimated as follows, using the equivalence of the norms · 2 and · : 2 η v1 , 2 v2 = −(η∇v1 , η∇v2 )2 − v1 (2η∇η), ∇v2 2

1 1 2 2 2 2 C ε1 η1 ∇v1 + η2 ∇v2 + ε2 ηv1 + η2 ∇v2 , ε1 ε2 for every positive ε1 and ε2 , thanks to Young’s inequality. Hence,

2 1 1 2 2 2 2 η v1 , bv1 C ε1 η1 ∇v1 + η2 ∇v2 + ε2 ηv1 + η2 ∇v2 ε1 ε2 2 2 + η v1 , f2 + η v1 , dv2 .

(35)

Moreover, forming the scalar product of the first equation of (34) by δ1 η12 v1 and the second one by δ2 η22 v2 for δ1 , δ2 > 0, we obtain 0 = δ1 η12 v1 , 1 v1 + δ1 η12 v1 , f1 − δ1 η12 v1 , av1 − δ1 η12 v1 , cv2 , (36) 0 = δ2 η22 v2 , 2 v2 + δ2 η22 v2 , f2 − δ2 η22 v2 , dv2 − δ2 η22 v2 , bv1 , with 2 δ1 η12 v1 , 1 v1 = −δ1 η12 ∇v1 1 − δ1 v1 (2η1 ∇η1 ), ∇v1 1

δ1 −δ1 η1 ∇v1 21 + C δ1 ε3 η1 ∇v1 21 + ηv1 21 , ε3

(37)

and similarly

2 δ2 2 2 2 δ2 η2 v2 , 2 v2 −δ2 η2 ∇v2 2 + C δ2 ε3 η2 ∇v2 2 + v2 ∇η2 2 ε3

(38)

for all positive ε3 . Replacing (37) and (38) in (36), and adding (35) and (36), we obtain, for positive constants C0 , K0 2 1 η v1 , bv1 C0 η2 v1 , f2 + η2 v1 , dv2 + ε1 η1 ∇v1 2 + η2 ∇v2 2 + ε2 ηv1 2 ε1 1 + η2 ∇v2 2 + δ1 η12 v1 , f1 + δ1 η12 v1 , av1 + δ1 η12 v1 , cv2 ε2 + δ2 η22 v2 , f2 + δ2 η22 v2 , dv2 + δ2 η22 v2 , bv1 + δ1 ε3 η1 ∇v1 2

δ1 δ2 2 2 2 + ηv1 + δ2 ε3 η2 ∇v2 + v2 ∇η2 ε3 ε3 − K0 δ1 η1 ∇v1 2 + δ2 η2 ∇v2 2 ,


2767

where we used the equivalence between the norms · 1 , · 2 and · to write everything in terms of · . Note that all the positive parameters δi , εj have not been fixed yet. Now we suppose that b b0 > 0 on U . The case b −b0 < 0 follows the same. We thus have, b0 ηv1 2 η2 v1 , bv1 C0 {J1 + J2 } + K0 J3

(39)

where J1 contains only the terms without gradient, and ⎧ J3 = −δ1 η1 ∇v1 2 − δ2 η2 ∇v2 2 , ⎪ ⎪ ⎪ ⎪ 1 1 ⎨ J2 = ε1 η1 ∇v1 2 + η2 ∇v2 2 + η2 ∇v2 2 + δ1 ε3 η1 ∇v1 2 ε1 ε2 ⎪ ⎪ ⎪ δ2 ⎪ ⎩ + δ2 ε3 η2 ∇v2 2 + v2 ∇η2 2 . ε3 The term C0 J2 + K0 J3 in (39) is thus non-positive as soon as the conditions ⎧ + δ1 ε3 ) − K0 δ1 0, ⎨ C0 (ε

1 1 1 + + δ2 ε3 − K0 δ2 0 ⎩ C0 ε1 ε2 are satisfied. In this case, we obtain from (39) b0 ηv1 2 C0 J1 . Let us now estimate C0 J1 , using that a, b, c, d ∈ L∞ and Young’s inequality with the parameters 1, ε2 or ε4 > 0: 1 1 C0 J1 C1 ε2 ηv1 2 + f2 2 + ηv2 2 + δ1 ηv1 2 + δ1 f1 2 + δ1 ηv2 2 ε2 ε2

δ2 δ1 2 2 2 τ −2 2 2 + δ2 η2 v2 + δ2 f2 + ε4 δ2 ηv1 + v2 + ηv1 χ ε4 ε3

2 δ1 ηv1 2 + C(δi , εj ) χ τ −2 v2 + f1 2 + f2 2 . C1 ε2 + δ1 + ε4 δ2 + ε3 If we choose the parameters such that C1 (ε2 + δ1 + ε4 δ2 + b0 ηv1 2 C0 J1 , we then obtain

δ1 ε3 )

b0 2 ,

since we now have

ηv1 2L2 (V ) C(δi , εj ) v2 2L2 (U ) + f1 2 + f2 2 , A∗ v . v1 L2 (V ) C v2 L2 (U ) + & Recalling that the open subset U is chosen such that U ⊂ (0, T0 ) × O ∩ ω, we take ϕ ∈ C0∞ (0, T0 ; C), with ϕ = 1 on the time support of U and we have v2 L2 (U ) ϕB ∗ v L2 ((0,T

0 )×Ω)

The proof of the proposition is complete.

.

2768


It only remains to note that it is possible to choose the parameters δi , εj satisfying ⎧ δ1 (C0 ε3 − K0 ) + C0 ε1 0, ⎪

⎪ ⎪ ⎪ ⎨ δ (C ε − K ) + C 1 + 1 0, 2 0 3 0 0 ε1 ε2 ⎪ ⎪ δ b0 ⎪ 1 ⎪ ⎩ ε2 + δ1 + ε4 δ2 + . ε3 2C1

This can be done, fixing first ε2 + δ1 0 δ1 ( K C0

b0 6C1

0 and ε3 < min{1, K C0 }. Second, choosing ε1

− ε3 ), the first condition is satisfied. Third, we can fix δ2 sufficiently large so that the

second condition is satisfied and finally ε4 such that ε4 δ2 filled. 2

b0 6C1

and the last condition is ful-

As a consequence of Proposition 5.1, the spectral inequality (13) of Theorem 3.2, the partial controllability result of Theorem 4.10 and the null-controllability result of Theorem 4.13 hold in this case. Under the assumptions made above, in particular those of Proposition 5.1, the coupled parabolic system (31) is null-controllable in any positive time by a control function g ∈ C0∞ (0, T ; L2 (Ω)). Note that in this context, the spectral inequality (13) corresponds to the estimation of a finite sum of root vectors of A by a localized measurement of only one component of this finite sum of root vectors. Remark 5.2. In the local energy estimates made in the proof, we see that the assumption O ∩ ω = ∅ is crucial. In the case O ∩ ω = ∅, the spectral inequality and the null-controllability remain open problems to the author’s knowledge. Remark 5.3. In the case where the operator A is selfadjoint (i.e. b = c in (31)), the spectral inequality (13) is much easier to prove, once the interpolation inequality (32) holds. This spectral inequality can take the following form. We denote by (μj )j ∈N the eigenvalues of A = A∗ and {(φj , ψj )T }j ∈N the associated eigenfunctions, that form a Hilbert basis of (L2 (Ω))2 . Then, for every open subset U ⊂ (0, T ) × O, there exist C > 0 such that for every sequence (aj , bj )j ∈N ⊂ C and α > 0, we have √ √ √ 2 2 2 C α μj t − μj t + bj e |aj | + |bj | Ce aj e ψj

μj α

μj α

.

L2 (U )

Following the proof of [25] or Section 4, it yields the controllability of the coupled problem (31), without restriction on the dimension of Ω.


2769

Remark 5.4. The same proof also yields a spectral inequality, a partial controllability and a null-controllability result for the following cascade system of d equations with one control force ⎧ ∂ u + P u + 1 u = 0, t 1 1 1 ω1 2 ⎪ ⎪ ⎪ ⎪ ∂ u + P u + 1 t 2 2 2 ω2 u3 = 0, ⎪ ⎪ ⎪ ⎪ ⎪··· ⎨ ∂t ud−1 + Pd−1 ud−1 + 1ωd−1 ud = 0, ⎪ ⎪ ⎪ ∂t ud + Pd ud = 1ωd g, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ uj |t=0 = uj , j ∈ {1 · · · d}, ⎩ uj = 0 on (0, T ) × ∂Ω, j ∈ {1 · · · d}, where Pj = − div(cj (x)∇·) for some symmetric uniformly elliptic matrices cj . Note that the null-controllability result is a particular case of the article [13]. We have to suppose that %d ω =

∅. The spaces here are the same as above, the operator A0 is diag(P1 · · · Pd ) and j j =1 A1 is ⎛

⎞

0 1ω1 .. ⎜ . ⎜ ⎜ ⎝

..

.

..

.

1ωd−1 0

⎟ ⎟ ⎟. ⎠

The above analysis directly yields the spectral inequality (13) of Theorem 3.2 and the partial control result of Theorem 4.10. The null-controllability result of Theorem 4.13 in any positive time, by only one control function g ∈ C0∞ (0, T ; L2 (Ω)) holds, supposing that Ω ⊂ Rn , n 3. 6. Application to the controllability of a fractional order parabolic equation Following [29] and [28], we give here an application of the spectral inequality (15) of Theorem 3.3 to the null-controllability of the following parabolic-type problem in which the dynamics is given by a fractional power of the non-selfadjoint operator A. We only treat the “good” case, i.e., when the power ν > 0 is sufficiently large. In this case, the selfadjoint problem is nullcontrollable. We consider

∂t u + Aν u = Bg, u|t=0 = u0 ∈ H.

(40)

Here, we define (D(Aν ), −Aν ) as the infinitesimal generator of the strongly continuous semigroup 1 2iπ

e−tz RA (z) dz = SAν (t), ν

q

∂ PK

0

or equivalently one of the following expressions (see [16]),

2770


1 lim A = 2iπ t→0+

ν −tzν

ν

z e q

∂ PK

=

1 2iπ

1 RA (z) dz = (A + I ) 2iπ

q

∂ PK

0

z−ν RA (z) dz

zν (z + 1)−m RA (z) dz

m

.−1

0

,

∂Σ q−1

where m ∈ N, m ν + 1, and Σ = {z ∈ C, arg(z) arctan(K0 α0 ), Re(z) α0 } denotes a sector containing the spectrum of A. Here, we have to suppose that the operator A is positive, since Remark 1.1 does not hold in the case ν = 1. In the case ν ∈ / N, we choose the principal value of the fractional root. Hence, on each finite dimensional subspace Pk H , we can write Aν ν 1 ν in terms of the functional calculus A Pk = 2iπ γk z RA (z) dz. Moreover, from Proposition 2.8 we have (Aν Pk )∗ = (A∗ )ν Pk∗ . The same holds with Πα instead of Pk . We now assume that the spectral inequality (15) of Theorem 3.3 holds for A and we obtain the following partial controllability result for ∂t + Aν . It is the analogous of Theorem 4.9, supposing the spectral inequality (15) instead of (13). Note that in this case we have no additional restriction on the subordination number q (as opposed to the statement of Theorem 4.13). The proof follows that of [28] or [24]. However, when A is not selfadjoint, the operator Aν is not necessarily positive. As a consequence, we also have to treat the possibly non-positive low frequencies of Aν . This problem does not arise when A is positive selfadjoint since Aν is always positive. For ν > 0, we define Nν = min{k ∈ N, Re(zν ) > 0, ∀z ∈ Ik }, such that Aν (I − Παk ) (and also A∗ν (I − Πα∗k )) is a positive operator if k Nν . Proposition 6.1. Let α αNν . The partial control problem ∂t u + Aν u = Πα Bg, u|t=0 = u0 ∈ Πα H,

(41)

is null-controllable in any positive time T by a control function satisfying gL2 (0,T ;H ) CT −1/2 eCT αNν +Cα u0 H . ν

θ

ν

Note that the additional cost eCT αNν of the control function is needed to handle the exponentially increasing low frequencies. Proof. The adjoint system of (41) is

−∂t w + A∗ν w = 0, w|t=T = wT ∈ Πα∗ H.

∗ν

Thus w(0) ∈ Πα∗ H and w(t) = etA w(0). We first estimate −tA∗ν ∗ 1 −tzν e ∗ Παk L(H ) = e R (z) dz A 2iπ L(H ) Γk

1 2π

−tzν e RA∗ (z)

ΓαN

ν

L(H ) dz +

αNν θ . For every T > 0, for every u0 ∈ H , there exists a control function g ∈ L2 (0, T ; Y ) such that the solution u of the problem (40) satisfies u(T ) = 0. In the case where A is a second order selfadjoint elliptic operator, the spectral inequality (15) always holds for θ = 1/2, and ν > 1/2 is necessary and sufficient for the null-controllability (see [29] and [28]). Here, with the estimations we have proved, the case 1/2 < ν θ is open. Remark 6.3. Using arguments of measure theory given in [34], Proposition 6.2 still holds if we replace the control operator B in (40) by 1E B, for any subset of positive measure E ⊂ (0, T ). This means that the control function g given by Proposition 6.2 can be chosen so that g(t) = 0 for t ∈ / E. Example 6.4. For Ω ⊂ Rn and ω a non-empty subset of Ω, we take H = Y = L2 (Ω), D(A0 ) = H 2 ∩ H01 (Ω), A0 = − and A1 = b · ∇ + c with b ∈ W 1,∞ (Ω; Cn ), c ∈ L∞ (Ω; C) chosen so that A is positive. Here, B ∗ a localized observation, i.e. B ∗ = B = 1ω ∈ L(L2 (Ω)). Under the conditions above, Proposition 2.1 is valid with q = 1/2 and the assumption of Theorem 2.5 is satisfied for p = 2/n. Moreover, the interpolation inequality (14) is well known in this case. In fact, it originates from Carleman inequalities [25], which form is invariant under changes in the operator by lower order terms. Hence, the spectral inequality (15) of Theorem 3.3 holds for

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θ = max{1/2, n−1 2 }. For any time T > 0 and E ⊂ (0, T ) satisfying meas(E) > 0, Proposition 6.2 and Remark 6.3 give the null-controllability of the problem ∂t u + (− + b · ∇ + c)ν u = 1E×ω g in (0, T ) × Ω, in Ω, u|t=0 = u0 u=0 on (0, T ) × ∂Ω, for any n ∈ N, ν > max{1/2, n−1 2 }. 7. Application to level sets of sums of root functions Following Jerison and Lebeau [20], we give here an application of the spectral inequality (15) of Theorem 3.3 to the measurement of the level sets of finite sums of root functions in term of the largest eigenvalue. The operator involved here is a non-selfadjoint perturbation of the Laplace operator. Let Ω be a bounded open set in Rn (or a n-dimensional Riemannian compact manifold with or without boundary). We set H = Y = L2 (Ω). Let − be the Laplace operator on Ω and P (x, D) a differential operator of order d ∈ {0, 1}, such that A1 = P (x, D) is a relatively compact perturbation of A0 = −, with dirichlet boundary conditions. We set A = A0 + A1 = − + P (x, D) and take for B ∗ a localized observation, i.e. B ∗ = B = 1ω ∈ L(L2 (Ω)) for some nonempty open subset ω ⊂ Ω. First note that under the conditions above, Proposition 2.1 is valid with q = d/2 < 1 and assumption (a) of Theorem 2.5 is satisfied for p = 2/n. Moreover, the interpolation inequality (14) holds in this case (see Example 6.4 above). Note that a function ϕ is a sum of root functions of the operator A associated with eigenvalues of real part lower than max{αk ; αk α} if ϕ ∈ Πα L2 (Ω). From Theorem 3.3, we have the following spectral inequality: there exists positive constants C, D such that for every positive α, for all ϕ ∈ Πα L2 (Ω) (the dual space Πα∗ L2 (Ω) does not play any role here),

1 n+d Dα θ −1 . (42) ϕL2 (ω) , θ = max ; ϕL2 (Ω) Ce 2 2 Assume now that Ω is real-analytic, and, moreover the differential operator P (x, D) has realanalytic coefficients. Under these conditions, the operator −∂t2 + A is real-analytic hypoelliptic on R × Ω [33, Theorem 5.4], and ϕ ∈ Πα L2 (Ω) implies that ϕ is real-analytic. We denote by Hn−1 the (n − 1)-dimensional Hausdorff measure on Ω. We can now state the analogous of the result of Jerison and Lebeau [20] for the class of non-selfadjoint elliptic operators we consider. Theorem 7.1. For every level set K ∈ R, there exist positive constants C1 , C2 such that for all α > 0 and ϕ ∈ Πα L2 (Ω),

1 n+d −1 . (43) θ = max ; Hn−1 {ϕ = K} C1 α θ + C2 , 2 2 The proof follows exactly the same of [20] and uses arguments from [7] and [8]. This estimations (42) and (43) are known to be optimal for the Laplace operator, with θ = 1/2, see [20]. As a consequence, one cannot hope to have better estimates in the cases where n+d 1 2 − 1 2 , i.e. n 3 if A = − + c(x) and n 2 if A = − + b(x) · ∇ + c(x). 1 However in the case n+d 2 − 1 > 2 , the results (42), (43) do not seem to be optimal.


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Acknowledgments The author wishes to thank O. Glass and J. Le Rousseau for very fruitful discussions, helpful comments and encouragements, and also L. Robbiano for discussions about the article [25]. The author was partially supported by l’Agence Nationale de la Recherche under grant ANR-07JCJC-0139-01. Appendix A A.1. Properties of the Gevrey function e ∈ Gσ , 1 < σ < 2 Here, we prove the existence of the Gevrey function e that is needed in the proof of Proposition 4.5. Lemma A.1. For every σ > 1, B0 > 0 and κ 1 − such that

1 2σ

, there exists a Gevrey function e ∈ Gσ

(i) supp(e) = [0, 1] and 0 < e(t) 1 for all t ∈ (0, 1); 1

(ii) |e(z)| ˆ c1 e−c2 |z| σ if Im(z) 0; 1

(iii) |e(z)| ˆ c3 e−c4 |z| σ in −iPBκ 0 where the constants ci are positive and −iPBκ 0 = −i{z ∈ C, Re(z) 0, | Im(z)| < B0 Re(z)κ }. Proof. The function −1 −1 e0 (t) = exp −t σ −1 − (1 − t) σ −1 is in Gσ and satisfies the properties (i) and (ii). We aim to prove a lower bound for |eˆ0 (z)| as |z| → ∞ in the parabola −iPBκ 0 . To have a precise estimation, we develop in detail the Laplace method, following [10]. Let κ and B0 two positive integers. For β < B0 , we estimate eˆ0 −i s + iβs κ =

1

−1 −1 exp −t σ −1 − (1 − t) σ −1 exp − s + iβs κ t dt

0 σ −1

s σ

=

1 −1 exp s σ −u σ −1 − u

0

σ −1 σ −1 − 1 1 × exp − 1 − s − σ u σ −1 − iβus κ+ σ −1 s − σ du after the rescaling change of variable t = s − −1 h(u) = −u σ −1 − u and

σ −1 σ

1

u. We then set ω = s σ the increasing parameter,

− 1 gβ (ω, u) = exp − 1 − ω−(σ −1) u σ −1 − iβuω1+σ (κ−1) ,

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such that we write eˆ0 −i s + iβs κ = I (ω, β) =

ωσ −1

ω−(σ −1) eωh(u) gβ (ω, u) du.

0

The function h(u) is negative on R+∗ , concave and h(u) < h(a) < 0 for u = a with a = σ −1 (σ − 1)− σ > 0. Following the Laplace method, we then split the integral I (ω, β) in three parts. The most important contribution comes from the region where h reaches its maximum. We write (44) I (ω, β) = ω−(σ −1) I1 (ω, β) + I2 (ω, β) + I3 (ω, β) , with a−η I1 (ω, β) = eωh(u) gβ (ω, u) du,

a+η I2 (ω, β) = eωh(u) gβ (ω, u) du, a−η

0 ωσ −1

I3 (ω, β) =

eωh(u) gβ (ω, u) du

(45)

a+η

for η > 0, sufficiently small, that will be fixed below. We first treat the main contribution I2 : Morse Lemma [15] implies that for η sufficiently small, there exists two positive constants ν1 and ν2 and a diffeomorphism H : (a − η, a + η) −→ 2 (−ν1 , ν2 ) such that h ◦ H−1 (x) = h(a) − x2 for x ∈ (−ν1 , ν2 ). Moreover, the jacobian J (x) = | det(dH−1 )|(x) satisfies J (0)2 = |h (a)|−1 . With this change of variable, we obtain ν2 I2 (ω, β) =

ωx 2 gβ ω, H−1 (x) eωh(a)− 2 J (x) dx.

−ν1

Setting y =

/

ω 2 x,

we obtain

0 I2 (ω, β) =

2 ωh(a) e ω

0

1(−ν1 ,ν2 ) R

0 0

2 2 2 2 −1 y J y gβ ω, H y e−y dy. ω ω ω

The modulus of the integrand is clearly bounded on R by Ce−y , independent of ω and integrable. Let us study the asymptotics of the integrand as ω → +∞. 2

gβ ω, H

−1

0

2 y ω

0 − 1

σ −1 2 −(σ −1) −1 = exp − 1 − ω y H ω

0 2 y ω1+σ (κ−1) . × exp −iβH−1 ω


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The first exponential converges when ω → +∞. In fact, setting

0 ψ(ω, y) = 1(−ν1 ,ν2 )

0 − 1 0

σ −1 2 2 2 −(σ −1) −1 , y J y exp − 1 − ω y H ω ω ω

we have − 1 ψ(ω, y) −→ e−1 J (0) = e−1 h (a) 2 ,

as ω → +∞.

For the second exponential in gβ (the oscillating part), as H−1 (0) = a, we write H−1 (x) = a + xK(x) where K ∈ C ∞ (R) and we have

0

2 −1 1+σ (κ−1) exp −iβH y ω ω 0

0

2 2 1+σ (κ−1) a+ yK y = exp −iβω ω ω

0 √ 1 2 1+σ (κ−1) +σ (κ−1) 2 = exp −iβω a exp −i 2βω yK y . ω We may thus write 0 I2 (ω, β) =

2 ωh(a) e exp −iβω1+σ (κ−1) a I$2 (ω, β), ω

where I$2 (ω, β) =

R

0 √ 1 2 2 +σ (κ−1) y e−y dy. ψ(ω, y) exp −i 2βω 2 yK ω

The integrand in I$2 (ω, β) converges as ω → +∞ under the condition summated convergence, we have I$2 (ω, β) −→ L(β) =

√ π − 12 h (a) , e

if

1 2

+ σ (κ − 1) 0. By

1 + σ (κ − 1) < 0, 2

and I$2 (ω, β) −→ L(β) =

√

π − 12 β 2 K(0)2 h (a) , exp − e 2

if

1 + σ (κ − 1) = 0. 2

Moreover, similar arguments show that I$2 (ω, β) − L(β) is C 1 ([−B0 , B0 ]) with respect to the variable β. Then there exists k0 such that for every ω sufficiently large ∂(I$2 − L) (ω, β) k0 , ∂β

β ∈ [−B0 , B0 ].

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Thus, I$2 − L is uniformly Lipschitz with respect to the variable β and tends to zero for every fixed β as ω −→ +∞. Lemma A.2 below implies that for all ε > 0, there exists ω0 > 0 such that |I$2 (ω, β) − L(β)| < ε for ω > ω0 and β ∈ [−B0 , B0 ]. We now address the terms I1 and I3 in (45). As h(u) < h(a) < 0 for u = a, we can write h(a − η) = h(a) − C− and h(a + η) = h(a) − C+ with C+ , C− > 0, depending only on η. Because h increases on (0, a], decreases on [a, +∞), and |gβ | 1, we have I1 (ω, β) aeωh(a) e−ωC− , Finally, for every fixed κ 1 − I (ω, β) =

√

1 2σ

I3 (ω, β) ωσ −1 eωh(a) e−ωC+ .

, we can write

1 2ω 2 −σ eωh(a) exp −iβω1+σ (κ−1) a I$2 (ω, β) + D(ω, β)

with |D(ω, β)| Cωσ − 2 e−C± ω and I$2 converging to a non-zero limit uniformly in β. As a consequence, there exist C1 , C2 > 0 and ω0 > 0 such that 1

I (ω, β) C1 e−C2 ω ,

ω > ω0 , β ∈ [−B0 , B0 ].

Switching back to the variable s = ωσ , we then have for some s0 > 0 1 eˆ0 −i s + iβs κ C1 e−C2 s σ ,

s > s0 , β ∈ [−B0 , B0 ].

To conclude the lemma, we now set e(t) = e−s0 t e0 (t) that is also in Gσ and satisfies property (i). For z ∈ C, we have e(z) ˆ = eˆ0 (z − is0 ) and (ii) holds. Property (iii) follows from what precedes. 2 Lemma A.2. Let K be a compact set and I (ω, x) a function defined on R+ × K, that is uniformly Lipschitz on R+ × K with respect to the variable x ∈ K, i.e., ∃k0 > 0,

I (ω, x2 ) − I (ω, x1 ) k0 |x2 − x1 |,

ω ∈ R+ , x1 , x2 ∈ K.

If for every x ∈ K, limω→+∞ I (ω, x) = 0, then limω→+∞ maxx∈K I (ω, x) = 0. A.2. A Paley–Wiener-type theorem Here we prove a Paley–Wiener-type theorem adapted to the situation of Proposition 4.5 Proposition A.3. Let Y be a separable Hilbert space and f ∈ H(C; Y ) satisfying for positive constants CN , Cε : f (z) Cε eε|z| eT Im(z) , ∀ε > 0, z ∈ C; Y f (τ ) CN 1 + |τ | −N , ∀N ∈ N, τ ∈ R. Y ˆ = f (z), z ∈ C. Then, there exists u ∈ C0∞ (0, T ; Y ) such that u(z)

(46) (47)


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Proof. Let (ej )j ∈N be a Hilbert basis of Y . For every j , z → (f (z), ej )Y ∈ H(C; C). Eq. (46) gives |(f (z), ej )Y | Cε eε|z| eT Im(z) , ∀ε > 0, z ∈ C, and the Paley–Wiener theorem [18, Theorem 15.1.5] then implies that there exists an analytic functional uj carried by (0, T ) (see [19, Chapter 9] for a precise definition) such that uˆ j (z) = (f (z), ej )Y , z ∈ C. Moreover, (47) yields |uˆ j (τ )| CN (1 + |τ |)−N , ∀N ∈ N, τ ∈ R and thus, uj ∈ C0∞ (0, T ; C). We now set u = j ∈N uj ej and observe that u ∈ L2 (R; Y ) u2L2 (R;Y ) =

j ∈N

uj 2L2 (R) =

1 1 f (·), ej 2 2 uˆ j 2L2 (R) = Y L (R) 2π 2π j ∈N

j ∈N

1 f 2L2 (R;Y ) . = 2π We note that supp(u) ⊂ (0, T ) since supp(uj ) ⊂ (0, T ) for all j ∈ N. Hence the Fourier–Laplace transform of u is an entire function, satisfying, for z ∈ C u(z), ˆ ek Y =

T

uj (t)e

−itz

=

ej dt, ek

0 j ∈N

Y

T

uj (t)e−itz (ej , ek )Y dt = uˆ k (z).

0 j ∈N

Thus, u(z) ˆ =

j ∈N

uˆ j (z)ej =

f (z), ej Y ej = f (z)

j ∈N

and f is the Fourier–Laplace of u. Finally, (47) yields u(τ ˆ )Y CN (1+|τ |)−N , ∀N ∈ N, τ ∈ R ∞ and thus u ∈ C . 2 References [1] F. Ammar Khodja, A. Benabdallah, C. Dupaix, Null controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl. 320 (2006) 928–943. [2] F. Ammar Khodja, A. Benabdallah, C. Dupaix, M. González-Burgos, Controllability for a class of reaction-diffusion systems: The generalized kalman’s condition, C. R. Acad. Sci. Paris, Ser I. 345 (2007) 543–548. [3] G. Alessandrini, L. Escauriaza, Null-controllability of one-dimensional parabolic equations, ESAIM Control Optim. Calc. Var. 14 (2008) 284–293. [4] M.S. Agranovich, Summability of series in root vectors of non-selfadjoint elliptic operators, Funct. Anal. Appl. 10 (1976) 165–174. [5] M.S. Agranovich, On series with respect to root vectors of operators associated with forms having symmetric principal part, Funct. Anal. Appl. 28 (1994) 151–167. [6] F. Boyer, F. Hubert, J. Le Rousseau, Discrete carleman estimates for elliptic operators and uniform controllability of semi-discretized parabolic equations, preprint, http://hal.archives-ouvertes.fr/hal-00366496/fr/, 2008. [7] H. Donnelly, C. Fefferman, Nodal sets of eigenfunctions on riemannian manifolds, Invent. Math. 93 (1988) 161– 183. [8] H. Donnelly, C. Fefferman, Nodal sets of eigenfunctions: Riemannian manifolds with boundary, in: Analysis, Et Cetera, Academic Press, Boston, MA, 1990, pp. 251–262. [9] L.S. Dzhanlatyan, Basis properties of the system of root vectors for weak perturbations of a normal operator, Funct. Anal. Appl. 28 (1994) 204–207. [10] A. Erdélyi, Asymptotic Expansions, Dover Publications, Inc., New York, 1956. [11] A. Fursikov, O.Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser., vol. 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

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[12] M. González-Burgos, R. Pérez-García, Controllability results for some nonlinear coupled parabolic systems by one control force, Asymptot. Anal. 46 (2006) 123–162. [13] M. González-Burgos, L. de Teresa, Controllability results for cascade systems of m coupled parabolic pdes by one control force, Port. Math. (2009), in press. [14] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Non-selfadjoint Operators, Transl. Math. Monogr., vol. 18, Amer. Math. Soc., Providence, RI, 1969. [15] A. Grigis, J. Sjöstrand, Microlocal Analysis for Differential Operators, Cambridge University Press, Cambridge, 1994. [16] M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Verlag, Basel, 2006. [17] L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, Berlin, 1963. [18] L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. II, Springer-Verlag, Berlin, 1983. [19] L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I, second ed., Springer-Verlag, Berlin, 1990. [20] D. Jerison, G. Lebeau, Nodal sets of sums of eigenfunctions, in: Harmonic Analysis and Partial Differential Equations, Chicago, IL, 1996, in: Chicago Lectures in Math., Univ. Chicago Press, Chicago, IL, 1999, pp. 223–239. [21] V.E. Katsnel’son, Conditions under which systems of eigenvectors of some classes of operators form a basis, Funct. Anal. Appl. 1 (1967) 122–132. [22] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, 1980. [23] O. Kavian, L. de Teresa, Unique continuation principle for systems of parabolic equations, ESAIM Control Optim. Calc. Var. (2009), doi:10.1051/cocv/2008077. [24] J. Le Rousseau, G. Lebeau, On carleman estimates for elliptic and parabolic operators, applications to unique continuation and control of parabolic equations, preprint, http://hal.archives-ouvertes.fr/hal-00351736/fr/, 2009. [25] G. Lebeau, L. Robbiano, Contrôle exact de l’équation de la chaleur, Comm. Partial Differential Equations 20 (1995) 335–356. [26] G. Lebeau, E. Zuazua, Null-controllability of a system of linear thermoelasticity, Arch. Ration. Mech. Anal. 141 (1998) 297–329. [27] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Monogr., vol. 71, Amer. Math. Soc., Providence, RI, 1988. [28] L. Miller, On the controllability of anomalous diffusions generated by the fractional laplacian, Math. Control Signals Systems 3 (2006) 260–271. [29] S. Micu, E. Zuazua, On the controllability of a fractional order parabolic equation, SIAM J. Control Optim. 44 (2006) 1950–1979. [30] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. [31] D.L. Russell, A unified boundary controllability theory for hyperbolic and parabolic partial differential equations, Stud. Appl. Math. 52 (1973) 189–221. [32] L. de Teresa, Insensitizing controls for a semilinear heat equation, Comm. Partial Differential Equations 25 (2000) 39–72. [33] F. Treves, Introduction to Pseudodifferential Operators and Fourier Integral Operators, vol. I, Plenum Press, New York, 1980. [34] G. Wang, L∞ -null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim. 47 (2008) 1701–1720.


Positive convolution structure for a class of Heckman–Opdam hypergeometric functions of type BC Margit Rösler Institut für Mathematik, TU Clausthal, Erzstr. 1, D-38678 Clausthal-Zellerfeld, Germany Received 27 July 2009; accepted 14 December 2009 Available online 29 December 2009 Communicated by P. Delorme

Abstract In this paper, we derive explicit product formulas and positive convolution structures for three continuous classes of Heckman–Opdam hypergeometric functions of type BC. For specific discrete series of multiplicities these hypergeometric functions occur as the spherical functions of non-compact Grassmann manifolds G/K over one of the skew fields F = R, C, H. We write the product formula of these spherical functions in an explicit form which allows analytic continuation with respect to the parameters. In each of the three cases, we obtain a series of hypergroup algebras which include the commutative convolution algebras of K-biinvariant functions on G as special cases. The characters are given by the associated hypergeometric functions. © 2009 Elsevier Inc. All rights reserved. Keywords: Hypergeometric functions associated with root systems; Heckman–Opdam theory; Hypergroups; Grassmann manifolds

1. Introduction There is a well-established theory of hypergeometric functions associated with root systems due to Heckman, Opdam and Cherednik which generalizes and completes the theory of spherical functions on Riemannian symmetric spaces in many respects; see [13,6,14,17] as well as the literature cited there. In rank one, i.e. for root systems of type BC1 , these hypergeometric functions are known as Jacobi functions and were studied by Flensted-Jensen and Koornwinder in a series of papers in the 1970s. A comprehensive exposition is given in [9]. In generalization E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.12.007

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M. Rösler / Journal of Functional Analysis 258 (2010) 2779–2800

of the one-variable case, hypergeometric functions associated with root systems are indexed by continuous parameters (the multiplicities) on a given root system. They build up the solutions of the joint eigenvalue problem for an associated system of commuting differential operators which generalize the radial parts of all invariant differential operators on a Riemannian symmetric space G/K of the non-compact type. In such geometric cases, the root system and multiplicity function are given in terms of the root space data of (G, K). In fact, the harmonic analysis associated with such hypergeometric functions is only the Weyl-group invariant part of a more general harmonic analysis associated with a commuting family of differential-reflection operators of Dunkl type, the so-called Cherednik operators. The associated integral transform, which generalizes the spherical transform on symmetric spaces, is studied in detail in [13]. There are, in particular, a Paley–Wiener theorem and a Plancherel theorem established for this transform. In the geometric cases (G, K) is a Gelfand pair, and the corresponding spherical functions satisfy a product formula which is intimately connected to the harmonic analysis on the commutative algebra of K-biinvariant measures on G. In the rank one case, a positive product formula and harmonic analysis for Jacobi functions associated with general non-negative multiplicities were established by Flensted-Jensen and Koornwinder, see [9]. However, apart from theses cases, the existence of a positive product formula for multivariable hypergeometric functions and a positivity-preserving convolution which would allow for a general Lp -theory are still open in general. A natural idea to extend the convolution from particular geometric cases to general multiplicities is analytic continuation of the product formula with respect to the multiplicities. There are only three classes of geometric cases with an infinite discrete series of multiplicities when the rank is fixed, namely the non-compact Grassmann manifolds SO0 (p, q)/SO(p) × SO(q), SU(p, q)/S(U (p) × U (q)) and Sp(p, q)/Sp(p) × Sp(q). Their real rank is q and the spherical functions are hypergeometric functions of type BC with multiplicities depending on p. In the present paper, we carry out the interpolation program in these cases. We give an explicit product formula for the spherical functions which allows analytic extension with respect to the multiplicity parameter p. This yields a product formula for three continuous classes of hypergeometric functions of type BC interpolating the group cases. Based on the product formula, we obtain a complete picture of harmonic analysis within the framework of commutative hypergroups on the associated Weyl chamber. In particular, the hypergeometric transform becomes an interpretation as a hypergroup Fourier transform. The paper is organized as follows: In Section 2, we calculate the product formula for the spherical functions on the Grassmann manifolds. Section 3 gives a short account on Heckman–Opdam theory as well as the identification of the spherical functions on Grassmann manifolds as hypergeometric functions of type BCq . The extension of the product formula to a continuous range of multiplicities interpolating the dimension parameter p is carried out in Section 4, and Section 5 is devoted to the study of the associated hypergroup algebras on the Weyl chamber. A central part of this section is the characterization of the bounded multiplicative functions which generalizes well-known results for spherical functions. The reasoning here is, however, not based on an integral representation but on exponential bounds for the Heckman–Opdam hypergeometric functions and their generalized Harish–Chandra expansion. 2. Spherical functions on Grassmann manifolds and their product formula We consider the Grassmann manifolds G/K where G is one of the indefinite orthogonal, unitary or symplectic groups SO0 (p, q), SU(p, q) or Sp(p, q) with maximal compact subgroup K = SO(p) × SO(q), S(U (p) × U (q)) or Sp(p) × Sp(q), respectively. For a unified point of


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view we also consider K as subgroup of U (p; F) × U (q, F), where U (p; F) is the unitary group over F = R, C or H. In the same way G is a subgroup of the indefinite unitary group U (p, q; F), which is the isometry group for the quadratic form |x1 |2 + · · · + |xp |2 − |xp+1 |2 − · · · − |xp+q |2 on Fp+q . To avoid exceptions which will be irrelevant lateron, we shall exclude the case p = q and assume that p > q 1. It is well known that (G, K) is a Gelfand pair (this follows from Corollary 1.5.4 of [4]). The spherical functions of this pair are characterized as the non-zero K-biinvariant continuous functions ϕ : G → C which satisfy the product formula ϕ(g)ϕ(h) =

ϕ(gkh) dk

for all g, h ∈ G

(2.1)

K

where dk denotes the normalized Haar measure of K. This means that the space of continuous, K-biinvariant compactly supported functions on G is a commutative subalgebra of the convolution algebra Cc (G). The space Cc (G//K) on the double coset space G//K therefore inherits the structure of a commutative topological algebra. The spherical functions of (G, K) provide exactly the non-zero continuous characters of this algebra, via f → G f (x)ϕ(x) dx. To make the product formula explicit, we recall the KAK-decomposition of G. Let g and k denote the Lie algebras of G and K. g has the Cartan decomposition g = k ⊕ p with p consisting of the (p + q)-block matrices

X , 0

0 X∗

X ∈ Mp,q (F).

Let a be a maximal abelian subalgebra of p. Then G = KAK with A = exp a. The spherical functions of (G, K) are therefore determined by their values on A. Actually, they are already determined by their values on the topological closure A+ = exp(a+ ) if a+ is the positive Weyl chamber associated with an (arbitrary) choice of positive roots within the restricted root system = (a, g) of g with respect to a. We may choose for a the set of all matrices Ht ∈ Mp+q (F) of the form

0p×p

Ht = t

0q×(p−q)

t

0(p−q)×q 0q×q

where t := diag(t1 , . . . , tq ) is the q × q diagonal matrix corresponding to t = (t1 , . . . , tq ) ∈ Rq (here R is considered as a subfield of C and H in the usual way). The real rank of G is q, and the restricted root system = (a, g) is of type BCq with the understanding that zero is allowed as a multiplicity on the long roots. In this way the limiting case Bq , which occurs for F = R, is included. We identify a with Rq via Ht → t, where the coordinates are with respect to the standard basis e1 , . . . , eq of Rq . Then the Killing form on a becomes the standard Euclidean inner product on Rq . Here is a comprehensive table of the roots α and their (geometric) multiplicities m(α), that is the dimensions of the corresponding root spaces; cf. Table 9 of [12]. The constant d denotes the dimension of F as an R-vector-space, i.e. d = 1, 2, 4 for F = R, C, H.

2782


root α

multiplicity m(α) = mp,d (α)

α(t) = ±ti ; 1 i q α(t) = ±2ti ; 1 i q α(t) = ±ti ± tj ; 1 i < j q

d(p − q) d −1 d

(2.2)

Thanks to our restriction p > q, the Weyl group of (a, g) is the hyperoctahedral group in all cases, and as a Weyl chamber we may choose a+ := Ht : t = (t1 , . . . , tq ) ∈ R with t1 > t2 > · · · > tq > 0 . In our identification of a with Rq , the closed chamber a+ corresponds to the set C := t ∈ Rq : t1 t2 · · · tq 0 . A short calculation gives

A+ = at =

cosh t

0q×(p−q) Ip−q 0q×(p−q)

0(p−q)×q sinh t

sinh t 0(p−q)×q cosh t

∈ Mp+q (F): t ∈ C .

Consider now g=

u˜ 0 a v t 0

u 0

0 v˜

∈ Kat K.

To obtain t back from g, we write g in (p × q)-block notation as g=

A(g) C(g)

B(g) . D(g)

A short calculation gives D(g) = v cosh t v. ˜

(2.3)

Let specs (x) denote the singular spectrum of x ∈ Mq (F), that is, specs (x) =

spec x ∗ x = (λ1 , . . . , λq ) ∈ Rq

with the singular values λi of x ordered by size: λ1 · · · λq 0. Eq. (2.3) shows that the singular spectrum of D(g) is given by specs (D(g)) = (cosh t1 , . . . , cosh tq ) =: cosh t. Therefore

for each g ∈ Kat K, t ∈ C, t = arcosh specs D(g)

(2.4)

where arcosh is also taken componentwise. (Observe that D(g) Iq and therefore all its singular values are 1.)


2783

Let us now evaluate the product formula (2.1) for the spherical functions of (G, K) explicitly. As spherical functions are K-biinvariant, it suffices to calculate the product formula for arguments g = at , h = as ∈ A+ . Write at ∈ A+ in (p × q)-block notation: At Bt . at = Ct D t Then for at , as ∈ A+ and k =

u 0 0v

∈ K we obtain

at kas =

∗ ∗

∗ Ct uBs + Dt vDs

and therefore D(at kas ) = Ct uBs + Dt vDs = (sinh t | 0)u

sinh s 0

+ cosh t v cosh s.

With the block matrix σ0 :=

Iq 0

∈ Mp,q (F)

this can be written as D(at kas ) = sinh t σ0∗ uσ0 sinh s + cosh t v cosh s. Notice that σ0∗ uσ0 ∈ Mq (F) is a truncation of u given by the upper left (q × q)-block of σ . Let ϕ be a spherical function of (G, K) and put ϕ(t) ˜ := ϕ(at ) for t ∈ C. Then according to formula (2.4) it satisfies

(2.5) ϕ(t) ˜ ϕ(s) ˜ = ϕ˜ arcosh specs D(at kas ) dk. K

In order to achieve a simplification of this formula we first extend the integral over K to an integral over U (p; F) × U0 (q; F) =: K0 , where U0 (q; F) denotes the connected component of the identity in U (q; F). If F = H then K = K0 , but in the other cases K is a proper normal subgroup of K0 . More precisely, let T := {z ∈ F: |z| = 1} and H the group of diagonal matrices H = {dz : z ∈ T} ⊂ Mp+q (F) where the diagonal entries of dz are equal 1 apart from the entry in position (p, p), which is z. Then K0 = H K ∼ = T K. Suppose f is a continuous function on K0 of the form

∗ u 0 ˜ f (k0 ) = f σ0 uσ0 , v for k0 = . 0 v Then f (dz k) = f (k) for all z ∈ T and k ∈ K and thus by Weyl’s formula, f (k0 ) dk0 = f (dz k) dk dz = f (k) dk K0

T

K

K

2784


where on each of the involved groups, integration is with respect to the normalized Haar measure. Thus

ϕ˜ arcosh specs sinh t σ0∗ uσ0 sinh s + cosh t v cosh s du dv

ϕ(t) ˜ ϕ(s) ˜ = U (p,F) U0 (q,F)

with du and dv the normalized Haar measures on U (p, F) and U0 (q, F) respectively. Here the integrand depends only on v and the truncation σ0∗ uσ0 , which is contained in the closure of the ball Bq := w ∈ Mq (F): w ∗ w γ − 1, put

μ−γ I − w∗ w dw.

κμ =

(2.6)

Bq

Here (x) denotes the determinant of x ∈ Mq (F), which is defined as the usual determinant for F = R or C, while for F = H we choose the Dieudonné determinant, i.e. (x) = (detC (x))1/2 when x is considered as a complex matrix in the usual way. Lemma 2.1. Suppose that p 2q. Then for continuous f : B q → C,

f σ0∗ uσ0 du =

1

κpd/2

pd/2−γ f (w) I − w ∗ w dw.

Bq

U (p,F)

Proof. Consider the action of the unitary group U (p, F) on Mp,q (F) by left multiplication, (u, x) → ux. The orbit of the matrix σ0 under this action is the Stiefel manifold Σp,q = x ∈ Mp,q (F): x ∗ x = Iq . Consider further the map U (p, F) → Σp,q , u → uσ0 . The image measure of du under this map coincides with the normalized U (p, F)-invariant measure dσ on Σp,q . Therefore U (p,F)

f σ0∗ uσ0 du =

Σp,q

f σ0∗ σ dσ.


2785

But σ0∗ σ is the q × q matrix given by the first q rows of σ only. According to Corollary 3.3 of [15],

pd/2−γ 1 f σ0∗ σ dσ = f (w) I − w ∗ w dw, (2.7) κpd/2 Σp,q

which finishes the proof.

Bq

2

We thus obtain Proposition 2.2. Suppose that p 2q. Then the spherical functions ϕ(t) ˜ = ϕ(at ) satisfy the product formula

1 ϕ˜ arcosh specs (sinh t w sinh s + cosh t v cosh s ) ϕ(t) ˜ ϕ(s) ˜ = κpd/2 Bq U0 (q,F)

pd/2−γ · I − w∗ w dv dw. Notice that the dependence on p now occurs only in the density, not in the domain of integration. 3. The spherical functions as BCq -hypergeometric functions In this section, we first provide the necessary background on hypergeometric functions associated with root systems. For an introduction to the subject, we refer to [13,14] and part I of [6]. In a second part, we identify the spherical functions on Grassmann manifolds within this framework. Let a be a finite-dimensional Euclidean space with inner product . , . which is extended to a complex bilinear form on the complexification aC of a. We identify a with its dual space a∗ = Hom(a, R) via the given inner product. Let R ⊂ a be a (not necessarily reduced) root system and let W be the Weyl group of R. For α ∈ R we write α ∨ = 2α/ α, α and denote by σα (x) = x − x, α ∨ α the orthogonal reflection in the hyperplane perpendicular to α. A multiplicity function on R is a function k : R → C which is W -invariant, i.e. k(wα) = k(α) for all α ∈ R. We denote by K the vector space of multiplicity functions on R and fix a positive subsystem R+ of R. For k ∈ K we put ρ(k) :=

1 k(α)α. 2 α∈R+

The Cherednik operator in direction ξ ∈ a is the differential-reflection operator on aC defined by Tξ (k) = ∂ξ +

α∈R+

k(α) α, ξ

1 (1 − σα ) − ρ(k), ξ −α 1−e

where ∂ξ is the usual directional derivative and eλ (ξ ) := e λ,ξ for λ, ξ ∈ aC . For fixed multiplicity k, the operators {Tξ (k), ξ ∈ aC } commute. Therefore the assignment ξ → Tξ (k) uniquely

2786


extends to a homomorphism on the symmetric algebra S(aC ) over aC , which may be identified with the algebra of complex polynomials on aC . The differential-reflection operator which in this way corresponds to p ∈ S(aC ) will be denoted by T (p, k). Let S(aC )W denote the subalgebra of W -invariant elements in S(aC ). Then for each p ∈ S(aC )W , the Cherednik operator T (p, k) coincides with a W -invariant differential operator on C ∞ (a)W , the W -invariant functions from C ∞ (a). The following theorem establishes hypergeometric functions associated with root systems. It was proved by Heckman and Opdam in a series of papers, see [6] as well as [13]. Theorem 3.1. There exists an open regular set K reg ⊆ K with {k ∈ K: Re k 0} ⊆ K reg such that for each k ∈ K reg and each spectral parameter λ ∈ aC , the hypergeometric system T (p, k)f = p(λ)f

∀p ∈ S(aC )W

(3.1)

has a unique W -invariant solution f (t) = F (λ, k; t) which is analytic on a and satisfies f (0) = 1. Moreover, there is a W -invariant tubular neighborhood U of a in aC such that F extends to a (single-valued) holomorphic function on aC × K reg × U , which is called the hypergeometric function associated with R. F (λ, k; t) is W -invariant both in λ and t. Suppose that k is real. Then for W -invariant polynomials p with real coefficients, we have T (p, k)F (λ, k; . ) = p(λ)F (λ, k; . ) which shows that F (λ, k; t) = F (λ, k; t) ∀t ∈ a.

(3.2)

The uniqueness of the solution to the hypergeometric system also implies the equivalence

F (λ, k; . ) = F λ , k; .

⇐⇒

λ ∈ W.λ.

Let Cc∞ (a)W denote the W -invariant functions from Cc∞ (a). The hypergeometric transform of f ∈ Cc∞ (a)W is defined by F f (λ) =

f (t)F (−λ, k; t) dω(t) a

where the measure ω = ωk on a is given by dω(t) =

e α,t /2 − e− α,t /2 k(α) dt

(3.3)

α∈R

(dt denotes the Lebesgue measure on a). There are Paley–Wiener and Plancherel theorems for this transform which are obtained by Weyl-group symmetrization of the (non-symmetric) Cherednik transform studied in [13]; see also [14]. Define the measure ν = νk on ia by dν(λ) =

1 dλ |c(λ, k)|2


2787

where dλ denotes the Lebesgue measure on ia and c( . , k) is the c-function on aC , c(λ, k) =

Γ ( λ, α ∨ + 12 k( α2 ))

α∈R+

Γ ( λ, α ∨ + 12 k( α2 ) + k(α))

with the convention that k( α2 ) = 0 if

α 2

Γ ( ρ(k), α ∨ + 1 k( α ) + k(α)) 2 2

·

α∈R+

Γ ( ρ(k), α ∨ + 12 k( α2 ))

(3.4)

∈ / R.

Theorem 3.2. (See [13, Theorems 8.6 and 9.13].) (1) The hypergeometric transform F is an isomorphism from Cc∞ (a)W onto the W -invariant Paley–Wiener space PW(aC )W , where PW(aC ) consists of all holomorphic functions f on aC satisfying the growth condition ∃R > 0, ∀N ∈ N:

N sup 1 + |λ| e−R|Re λ| f (λ) < ∞. λ∈aC

The inverse of F : Cc∞ (a)W → PW(aC )W is given by F

−1

h(t) =

h(λ)F (λ, k; t) dν(λ). ia

(2) Let f, g ∈ Cc∞ (a)W and let a+ be the Weyl chamber of W corresponding to R+ . Then

f (t)g(t) dω(t) = c a+

F f (λ)F g(λ) dν(λ)

ia+

where c > 0 is a normalization constant. According to Proposition 6.1 of [13], F (λ, k; t) |W |1/2 · e|Re λ||t|

for t ∈ a, λ ∈ aC .

Thus for f ∈ Cc∞ (a)W and fixed s ∈ a, the function λ → F f (λ)F (λ, k; s) belongs to PW(aC )W , and we obtain the following Corollary 3.3. For s ∈ a and f ∈ Cc∞ (a)W , the generalized translate τs f (t) :=

F f (λ)F (λ, k; s)F (λ, k; t) dν(λ) ia

again belongs to Cc∞ (a)W . Moreover, F (τs f )(λ) = F (λ, k; s)F f (λ).

2788


Let us now turn to the spherical functions on the Grassmann manifolds G/K. They are identified with hypergeometric functions of type BCq , as follows: Consider a = Rq with the standard inner product . , . and regard the restricted root system of G/K as a subset of Rq as described in Section 2. With our convention including the case F = R, it is given by BCq = {±ei , ±2ei , 1 i q} ∪ {±ei ± ej , 1 i < j q} where (e1 , . . . , eq ) denotes the standard basis of Rq . The corresponding Weyl group W is the hyperoctahedral group, which is generated by permutations and sign changes of the ei . Put R := {2α: α ∈ BCq } and R+ := {2ei , 4ei , 1 i q} ∪ {2(ei ± ej ), 1 i < j q} and denote the associated hypergeometric function by FBCq . Let m = mp,d be one of the multiplicity functions on BCq in the geometric cases according to table (2.2) and define k = kp,d on R by 1 kp,d (2α) = mp,d (α), 2

α ∈ BCq .

Writing k in the form k = (k1 , k2 , k3 ) where k1 and k2 are the values on the roots ±2ei and ±4ei , respectively and k3 is the value on the roots 2(±ei ± ej ), we have

kp,d = d(p − q)/2, (d − 1)/2, d/2 . The spherical functions of G/K are then indexed by spectral parameters λ ∈ Cq and given by ϕλ (at ) = ϕ˜ λ (t) = FBCq (iλ, kp,d ; t),

t ∈ C.

This follows from the fact that for k = kp,d , the commutative algebra {D(p, k); p ∈ S(Cq )W } just represents the radial parts of the algebra of all invariant differential operators on G/K, see Remark 2.3 of [5]. Example 3.4 (The rank one case). Here R+ = {2, 4} ⊂ R. We have multiplicities k1 , k2 and ρ = ρ(k) = k1 + 2k2 . According to the example in [13, p. 89f], the associated hypergeometric function is given by FBC1 (λ, k; t) = 2 F1 With α := k1 + k2 − 12 , β := k2 − as

1 2

λ + ρ −λ + ρ 1 2 , , k1 + k2 + ; − sinh t . 2 2 2 (α,β)

and the Jacobi functions ϕλ

(α,β)

FBC1 (iλ, k; t) = ϕλ

as in [9], this can be written

(t).

d The geometric cases correspond to α = dp 2 − 1, β = 2 − 1. In Proposition 2.2, the U0 (1)-integral −1 cancels (use the coordinate transform w˜ := v w), and the product formula reduces to


ϕ(t) ˜ ϕ(s) ˜ =

1 κpd/2

=

2789

pd −γ ϕ˜ arcosh |cosh t cosh s + w sinh t sinh s| · 1 − |w|2 2 dw

B1

ϕ˜ arcosh |cosh t cosh s + x1 sinh t sinh s| dσ (x)

Σp,1 (α,β)

d with α = pd where ϕ˜ = ϕλ 2 − 1, β = 2 − 1. The second identity is obtained by formula p (2.7) for the sphere Σp,1 = {x ∈ F : |x| = 1}. In view of relation (5.24) in [9], this formula just coincides with the product formula in rank 1 given in Section 7 of [9],

(α,β) (α,β) ϕλ (t)ϕλ (s) = cα,β

1 π

(α,β)

arcosh |cosh t cosh s + reiψ sinh t sinh s|

ϕλ 0 0

α−β−1 2β+1 · 1 − r2 r (sin ψ)2β r dr dψ

(3.5)

which degenerates for β = −1/2 (i.e. F = R) to an integral over [−1, 1] with respect to (1 − r 2 )α−1/2 dr. In fact, formula (3.5) was established in [3] for arbitrary α β − 12 with (α, β) = (− 12 , − 12 ), i.e. arbitrary non-negative root multiplicities different from zero. 4. Continuation of the product formula In the following, q and d = dimR (F) are fixed. For μ ∈ C with Re μ > γ − 1 and spectral parameter λ ∈ Cq define μ

ϕλ (t) = FBCq (iλ, kμ ; t),

t ∈ Rq ,

with multiplicity

kμ = μ − dq/2, (d − 1)/2, d/2 . If μ = pd/2, then kμ = kp,d as in the previous section. μ

Theorem 4.1. For μ ∈ C with Re μ > γ − 1, the hypergeometric functions ϕλ satisfy the product formula

μ μ μ ϕλ (t)ϕλ (s) = (δt ∗μ δs ) ϕλ with the probability measures 1 (δt ∗μ δs )(f ) = κμ

Bq U0 (q,F)

μ−γ f d(t, s; v, w) I − w ∗ w dv dw

2790


where κμ is given by (2.6) and the argument is

d(t, s; v, w) = arcosh specs (sinh t w sinh s + cosh t v cosh s ) . This is a partial generalization of formula (3.5) by Flensted-Jensen and Koornwinder for BC1 to higher rank. Proof of Theorem 4.1. The basic idea is analytic continuation with respect to the parameter μ in the right half-plane by use of Carlson’s theorem. (See e.g. [18, p. 186].) Let f be a function which is holomorphic in a neighborhood of {z ∈ C: Re z 0} satisfying f (z) = O(ec|z| ) for some constant c < π . Suppose that f (n) = 0 for all n ∈ N0 . Then f is identically zero. A direct application of Carlson’s theorem would require moderate exponential growth of the hypergeometric function with respect to the relevant multiplicity parameter k1 in a right halfplane. So far however, sufficient exponential estimates are available only for real, non-negative multiplicities (Proposition 6.1 of [13], and the results of [17]). We therefore proceed in two steps. First, we restrict to a discrete set of spectral parameters, for which the hypergeometric function is a Jacobi polynomial and the required growth properties are guaranteed. In a second step, we fix a non-negative multiplicity and carry out analytic continuation with respect to the spectral parameter, using known bounds on the hypergeometric function for non-negative multiplicities. To go into detail, let R ∨ = {α ∨ : α ∈ R} be the root system dual to R, Q∨ = Z.R ∨ the coroot lattice and P = {λ ∈ Rq : λ, α ∨ ∈ Z ∀α ∈ R} the weight lattice of R. Further, denote by P+ = {λ ∈ P : λ, α ∨ 0 ∀α ∈ R+ } the set of dominant weights associated with R+ . Then for k ∈ K reg and λ ∈ P+ ,

FBCq λ + ρ(k), k; t = c λ + ρ(k), k Pλ (k; t) where c(λ, k) is the c-function (3.4) which is meromorphic on Cq × K, and the Pλ are the Heckman–Opdam Jacobi polynomials of type BCq ; see [6, Eq. (4.4.10)]. In our case, ρ(k) is given by ρ(k) = (k1 + 2k2 )

q

q ei + 2k3 (q − i)ei

i=1

= μ−

dq +d −1 2

i=1

q i=1

ei + d

q

(q − i)ei .

i=1

Using the asymptotics of the gamma function, one checks that for fixed λ ∈ P+ , the function c(λ + ρ(kμ ), kμ ) is bounded away from zero as μ → ∞ in the right half-plane H = {μ ∈ C: Re μ > γ − 1}.


2791

Indeed, for ρ = ρ(k) with k = (k1 , k2 , k3 ) one has q q λ +ρ ρ Γ (λi + ρi )Γ (ρi + k1 ) Γ ( i 2 i + 12 k1 )Γ ( 2i + 12 k1 + k2 ) · c(λ + ρ, k) = λi +ρi ρi 1 1 Γ (λi + ρi + k1 )Γ (ρi ) i=1 i=1 Γ ( 2 + 2 k1 + k2 )Γ ( 2 + 2 k1 )

·

Γ ( λi +ρi −λj −ρj )Γ ( ρi −ρj + k3 ) Γ ( λi +ρi +λj +ρj )Γ ( ρi +ρj + k3 ) 2 2 2 2 · . λi +ρi −λj −ρj ρi −ρj λi +ρi +λj +ρj ρi +ρj + k )Γ ( ) Γ ( + k )Γ ( ) 3 3 i<j Γ ( i<j 2 2 2 2

q As k1 → ∞ in the half-plane Re k1 > 0, the first product is asymptotically equal to i=1 ( 12 )λi , the second one is asymptotically equal to 1, the third product is independent of k1 , and the last product is asymptotically equal to 1. Thus for fixed λ, c(λ + ρ, k) is bounded away from zero. According to Proposition 2.2, the Pλ (kμ ; . ) with μ = pd/2 (p 2q) satisfy the product formula Pλ (kμ ; t)Pλ (kμ ; s)

μ−γ 1 1 Pλ kμ ; d(t, s; v, w) I − w ∗ w = dv dw κμ c(λ + ρ(kμ ), kμ )

(4.1)

Bq U0 (q,F)

for all t, s ∈ Rq . The Jacobi polynomials Pλ (k; . ) have rational coefficients in k with respect to the monomial basis eν , ν ∈ P . This is shown in Par. 11 of [10], but it also follows from the explicit determinantal construction in [1, Theorem 5.4]. Moreover, as derived in the proof of Theorem 3.6 of [15], the normalized integral 1 |κμ |

I − w ∗ w μ−γ dw

Bq

converges exactly if Re μ > γ − 1 and is of polynomial growth as μ → ∞ in H . Thus for fixed t, s, both sides of (4.1) are holomorphic in μ ∈ H and of polynomial growth as μ → ∞ in H . Moreover, they coincide for all half-integer values μ = pd/2, p 2q. Application of Carlson’s theorem yields that formula (4.1) holds for all μ ∈ H . This proves the stated result for spectral parameters λ + ρ(k) with λ ∈ P+ and k = kμ , μ ∈ H . Denote again by C ⊂ Rq the closed Weyl chamber associated with R+ . In order to extend the product formula with respect to the spectral parameter, we fix s, t ∈ C as well as k = kμ and restrict to real μ > γ − 1 first. Then k is non-negative, and we have the following exponential estimate for FBCq from [13, Proposition 6.1]: FBC (λ, k; t) |W |1/2 emaxw∈W Re wλ,t . q Let H := {λ ∈ Cq : Re λ ∈ C 0 }. Then for λ ∈ H and all w ∈ W , Re wλ, t Re λ, t .

2792


Choose a constant vector a ∈ C 0 so large that d(t, s; v, w) − a is contained in the negative chamber −C for all v ∈ U (q) and all w ∈ Bq . Then consider F˜ (λ, k; t) := e− λ,a+t FBCq (λ, k; t). The function F˜ is bounded as a function of λ ∈ H . If the spectral parameter is of the form λ = λ˜ + ρ(kμ ) with λ˜ ∈ P+ , then by our previous results we have the product formula F˜ (λ, kμ ; t)F˜ (λ, kμ ; s) μ−γ

1 e λ,d(t,s;v,w)−a−s−t F˜ λ, kμ ; d(t, s; v, w) I − w ∗ w dv dw. = κμ

(4.2)

Bq U0 (q,F)

Both sides are holomorphic and bounded in λ ∈ H . We are now going to carry out analytic extension with respect to λ. For this, we choose a set of fundamental weights {λ1 , . . . , λq } ⊂ P+ q and write λ ∈ H as λ = i=1 zi λi with coefficients zi ∈ {z ∈ C: Re z > 0}. Successive holomorphic extension with respect to z1 , . . . , zq by use of Carlson’s theorem then yields the validity of (4.2) for all λ ∈ H , and thus, by W -invariance and continuity, for all λ ∈ Cq . This proves the stated product formula for real μ > γ − 1. Analytic continuation finally gives it for all μ ∈ H , which finishes the proof of Theorem 4.1. 2 5. Hypergroup algebras associated with FBC The positive product formula of Theorem 4.1 leads to three continuous series (d = 1, 2, 4) of positivity-preserving convolution algebras on the Weyl chamber C which are parametrized by μ. We shall describe them as commutative hypergroups, having Heckman–Opdam hypergeometric functions as characters. In the group cases, which correspond to the discrete values μ = pd/2, these hypergroup algebras are just given by the double coset convolutions associated with the Gelfand pairs (G, K) as in Section 2. In the rank one case, they coincide with the well-known one-variable Jacobi hypergroups. Let us first briefly recall some key notions and facts from hypergroup theory. For a detailed treatment, the reader is referred to [8]. Hypergroups generalize the convolution algebras of locally compact groups, with the convolution product of two point measures δx and δy being in general not a point measure again but a probability measure depending on x and y. Definition 5.1. A hypergroup is a locally compact Hausdorff space X with a weakly continuous, associative convolution ∗ on the space Mb (X) of regular bounded Borel measures on X, satisfying the following properties: 1. The convolution product δx ∗ δy of two point measures is a compactly supported probability measure on X, and supp(δx ∗ δy ) depends continuously on x and y with respect to the socalled Michael topology on the space of compact subsets of X (see [8]). 2. There is a neutral element δe satisfying δe ∗ δx = δx = δx ∗ δe for all x ∈ X. 3. There is a continuous involution x → x¯ on X such that for all x, y ∈ X, e ∈ supp(δx ∗ δy ) is equivalent to x = y, ¯ and δx¯ ∗ δy¯ = (δy ∗ δx )− . Here for μ ∈ Mb (X), the measure μ− is given − − by μ (A) = μ(A ) for Borel sets A ⊂ X.


2793

Due to the weak continuity, the convolution of measures on a hypergroup is uniquely determined by the convolution of point measures. If the convolution is commutative, then (Mb (X), ∗) becomes a commutative Banach-∗algebra with identity δe . Moreover, there exists an (up to a multiplicative factor) unique Haar measure ω, that is a positive Radon measure on X satisfying f (x ∗ y) dω(y) = f (y) dω(y) for all x ∈ X, f ∈ Cc (X), X

X

where f (x ∗ y) = (δx ∗ δy )(f ). The multiplicative functions of a commutative hypergroup X are given by χ(X) = ϕ ∈ C(X): ϕ = 0, ϕ(x ∗ y) = ϕ(x)ϕ(y) ∀x, y ∈ X . The decisive object for harmonic analysis is the dual space of X, defined by Xˆ := ϕ ∈ χ(X): ϕ is bounded and ϕ(x) = ϕ(x) ∀x ∈ X . The elements of Xˆ are called characters. As in the case of LCA groups, the dual of a commutative hypergroup is a locally compact Hausdorff space with the topology of locally uniform convergence. It is naturally identified with the symmetric part of the spectrum of the convoluˆ tion algebra L1 (X, ω). In contrast to the group case, X is often a proper subset of χ(X). The 1 ˆ Fourier transform on L (X, ω) is defined by f (ϕ) := X f ϕ dω. It is injective, and there exists ˆ called the Plancherel measure of (X, ∗), such that a unique positive Radon measure π on X, ˆ π). As for groups, there f → fˆ extends to an isometric isomorphism from L2 (X, ω) onto L2 (X, are convolutions between functions from various classes of Lp -spaces (or measures) on a hypergroup with Haar measure ω. For example, if 1 p ∞ and f ∈ L1 (X, ω), g ∈ Lp (X, ω), then the convolution product f ∗ g(x) = f (x ∗ y)g(y) dω(y) X

belongs to Lp (X, ω) and satisfies f ∗ gp,ω f 1,ω gp,ω . Let us come back to the situation of Section 3. With the notions from there, we can now state our main theorem: Theorem 5.2. (1) Let μ > γ − 1. Then the probability measures given by

μ−γ 1 f d(s, t; v, w) I − w ∗ w dv dw (δs ∗μ δt )(f ) = κμ

(5.1)

Bq U0 (q,F)

for s, t ∈ C define a commutative hypergroup structure Cμ = (C, ∗μ ) on the chamber C∼ = a+ . The neutral element is 0 and the involution is the identity mapping. The support of δs ∗μ δt satisfies

2794


supp(δs ∗μ δt ) ⊆ r ∈ C: r∞ s∞ + t∞ where . ∞ is the maximum norm in Rq . (2) A Haar measure of the hypergroup Cμ is given by the weight function (3.3) of the corresponding hypergeometric transform, dωμ (t) = const ·

q

|sinh ti |2μ−d(q−1)−1 |cosh ti |d−1 ·

i=1

cosh(2ti ) − cosh(2tj )d dt.

1i<j q

Proof. (1) It is clear that δs ∗μ δt is a probability measure on C with

supp(δs ∗μ δt ) = d(s, t; v, w) = arcosh specs (sinh s w sinh t + cosh s v cosh t ) , v ∈ U0 (q, F), w ∈ Bq . For the support statement, we denote by A the spectral norm of A ∈ Fq×q , that is A = specs (A)∞ (the biggest singular value of A). By the submultiplicativity of . we obtain for v and w within the relevant range the estimate sinh s w sinh t + cosh s v cosh t sinh ssinh t + cosh scosh t = sinh s∞ · sinh t∞ + cosh s∞ · cosh t∞

= cosh s∞ + t∞ . This implies the stated support inclusion. For the weak continuity of the convolution ∗μ on Mb (C), it suffices to verify that for each f ∈ Cb (C), the mapping (s, t) → f (s ∗μ t) is continuous. But this is immediate because d(s, t; v, w) depends continuously on its arguments. To see that ∗μ is commutative, we note that specs (A) = specs (A∗ ) for A ∈ Fq×q , and hence d(t, s; v, w) = d(s, t; v ∗ , w ∗ ). As the integral in (5.1) is invariant under the substitution v → v ∗ = v −1 , w → w ∗ , it follows that δt ∗μ δs = δs ∗μ δt . For the associativity of ∗μ it suffices to verify that δr ∗μ (δs ∗μ δt )(f ) = (δr ∗μ δs ) ∗μ δt (f ) for all f ∈ Cc∞ (Rq )W and all r, s, t ∈ C. In view of the Paley–Wiener theorem for the hypergeometric transform, both sides are equal to F f (λ)F (λ, k; r)F (λ, k; s)F (λ, k; t) dν(λ). iRq

This proves the assertion. From the explicit form of the convolution it is obvious that 0 is neutral. In the discrete cases μ = pd/2 coming from Gelfand pairs, ∗μ is the convolution of a double coset hypergroup. Moreover, supp(δs ∗μ δt ) is independent of μ. In order to see that the identity mapping is a hypergroup involution for all μ, it therefore suffices (by uniqueness of an involution) to show


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that the zero matrix 0 is contained in supp(δt ∗μ δt ). But

d(t, t; Iq , −Iq ) = arcosh specs −(sinh t )2 + (cosh t )2 = arcosh(Iq ) = 0, which proves the claim. (2) Let f, g ∈ Cc∞ (Rq )W . Notice first that f (t) =

F f (λ)F (λ, k; t) dν(λ)

iRq

by the inversion theorem for the hypergeometric transform (Theorem 3.2). As F (λ, k; s ∗μ t) = F (λ, k; s)F (λ, k; t) for all s, t ∈ C we obtain, with the notation of Corollary 3.3, f (s ∗μ t) = τs f (t). The Plancherel formula (Theorem 3.2) further gives

(τs f )g dωμ = c

C

F (τs f )F g dν = c

iC

=c

F f (λ)F (λ, k; s)F g(λ) dν(λ)

iC

F f (λ)F (τs g)(λ) dν(λ) =

iC

f (τs g) dωμ C

with a constant c > 0. It was used here that F (λ, k; s) = F (−λ, k; s) = F (λ, k; s) for λ ∈ iC. Choose now a sequence gn ∈ Cc∞ (Rq )W , n ∈ N, such that gn ↑ 1 pointwise. Then also τs (gn ) ↑ 1, and the monotonic convergence theorem shows that

(τs f ) dωμ =

C

f dωμ . C

This proves that ωμ is a Haar measure of Cμ .

2

Lemma 5.3. Suppose that ϕ : Cμ → C is continuous and multiplicative, i.e. ϕ(s)ϕ(t) = ϕ(s ∗μ t) for all s, t ∈ C. μ

Then ϕ = ϕλ with some λ ∈ Cq . Proof. The proof follows standard arguments. For abbreviation, we write k = kμ , ω = ωμ and ∗ = ∗μ . In a first step, consider g ∈ Cc∞ (Rq )W . Let p ∈ S(Cq )W be a W -invariant polynomial and T (p) = T (p, kμ ) the associated Cherednik operator. As g(s ∗ t) = iRq

F g(λ)F (λ, k; s)F (λ, k; t) dν(λ)

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for all s, t ∈ C, we obtain T (p)s g(s ∗ t) =

F g(λ)p(λ)F (λ, k; s)F (λ, k; t) dν(λ) = T (p)t g(s ∗ t)

(5.2)

iRq

and T (p)s g(s ∗ t)|s=0 = T (p)g(t). Suppose now ϕ is continuous, non-zero and multiplicative on C. Notice first that ϕ(0) = 1, q ∞ q because 0 is neutral. We extend ϕ to a W -invariant function on R and choose g ∈ Cc (R ) with C ϕg dω = 1. Recall that the involution of the hypergroup Cμ is the identity. Thus ϕ(s ∗ t)g(t) dω(t) = ϕ(s)

ϕ ∗ g(s) = C

and therefore ϕ(s) = ϕ ∗ g(s) =

ϕ(t)τt g(s) dω(t), C

which belongs to C ∞ (Rq ) because τt g ∈ Cc∞ (Rq ) for all t according to Lemma 3.3. Further, ϕ(r)τr g(s ∗ t) dω(r)

ϕ(s ∗ t) = C

and therefore T (p)s ϕ(s ∗ t) =

ϕ(r)T (p)s τr g(s ∗ t) dω(r) =

C

ϕ(r)T (p)t τr g(s ∗ t) dω(r)

C

= T (p)t ϕ(s ∗ t). In particular, T (p)ϕ(t) = T (p)t ϕ(s ∗ t)|s=0 = T (p)s ϕ(s ∗ t)|s=0 = σϕ (p) · ϕ(t) with σϕ (p) = (T (p)ϕ)(0). The mapping p → σϕ (p) is obviously multiplicative and linear on S(Cq )W . According to a well-known result form invariant theory (see e.g. [7, Chap. III.4, Lemma 3.11]), it coincides with a point evaluation, that is, ∃λ ∈ Cq :

σϕ (p) = p(λ)

W ∀p ∈ S Cq .

It is thus shown that ϕ satisfies the hypergeometric system (3.1) with spectral parameter λ, corresponding to R = BCq and k = kμ . By uniqueness of the solution, it follows that ϕ = μ FBCq (λ, kμ ; . ) = ϕ−iλ . 2


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Theorem 5.4. The set of multiplicative functions and the dual space of the hypergroup Cμ are given by μ χ(Cμ ) = ϕλ = ϕλ : λ ∈ C + iC , μ = ϕλ ∈ χ(Cμ ): λ ∈ W.λ and Im λ ∈ co(W.ρ) C where ρ = ρ(kμ ) and co(W.ρ) denotes the convex hull of the Weyl group orbit W.ρ. The second part of this theorem is in accordance with the characterization of the bounded spherical functions of a Riemannian symmetric space of non-compact type, see [7, Chap. IV, Theorem 8.1]. In our more general context, we shall not work with an integral representation but proceed by using estimates on the hypergeometric function given in [17] as well as the generalized Harish–Chandra expansion of [13]. We mention at this point that for the Grassmann manifolds over F = R, there is an explicit integral formula for the spherical functions given in [16] which could probably also be used after analytic extension. Proof of Theorem 5.4. The identification of χ(Cμ ) is furnished by the previous lemma. For the identification of the dual space, note first that ϕλ = ϕλ as a consequence of (3.2). Thus ϕλ is real if and only if λ ∈ W.λ. It remains to identify those functions from χ(Cμ ) which are bounded. For this, we observe first that the set A = {λ ∈ Cq : μ } is closed in Cq . Indeed, suppose that (λi )i∈N is a sequence in A which converges to ϕλ ∈ C λ0 ∈ Cq . Being members of a hypergroup dual, the ϕλi are uniformly bounded by 1 (see [8]). As (λ, t) → ϕλ (t) is continuous, it follows by a standard compactness argument that the sequence ϕλi converges to ϕλ0 locally uniformly on C (see e.g. [2, Chap. XII, Section 8]). This implies μ as well. that ϕλ0 belongs to C Next recall that ϕλ (t) = FBCq (iλ, kμ ; t) =: Fiλ (t) and notice that F−λ = Fλ . We thus have to prove that Fλ is bounded if and only if Re λ ∈ co(W.ρ). We may assume that λ = ξ + iη with ξ, η ∈ C. By Corollary 3.1 of [17], Fλ (t) Fξ (t)

∀t ∈ C.

(5.3)

Further, according to Remark 3.1 of [17], Fξ behaves asymptotically (for large arguments in C) as Fξ (t) e ξ −ρ,t ·

1 + α, t .

(5.4)

α∈R0+ | α,ξ =0

Here R0+ are the indivisible positive roots, in our case R0+ = {2ei , 2(ei ± ej ), 1 i < j q}. Consider now λ = ξ + iη with ξ = Re λ ∈ co(W.ρ). We claim that Fλ is bounded. By closedness

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of A, it suffices to assume that ξ is actually contained in the open interior of co(W.ρ). Then there exists a constant 0 < s < 1, s = 1 − , such that ξ ∈ co(W.sρ). We use the characterization w x − C∗ (5.5) co(W.x) = w∈W

C∗

for x ∈ C, where = {x ∈ t, x 0 ∀t ∈ C} is the closed dual cone of C; see e.g. [7, Lemma IV.8.3]. This shows that sρ − ξ ∈ C ∗ and therefore Rq :

ξ − ρ, t = ξ − sρ, t − ρ, t − ρ, t ∀t ∈ C. Note that ρ, t > 0 for all t ∈ C \ {0}, because our multiplicity is non-negative and different from zero. Hence ρ, t c|t| for some constant c > 0. Together with estimates (5.3) and (5.4), this proves boundedness of Fλ as claimed. For the converse inclusion, we have to show that Fλ is unbounded if ξ = Re λ ∈ / co(W.ρ). For real λ = ξ ∈ C we use again (5.4). According to (5.5), there exists some t ∈ C such that ξ − ρ, t > 0 (recall that ξ, t wξ, t for all w ∈ W ). This implies that Fξ is unbounded in C. In case η = Im λ = 0 we employ the Harish–Chandra expansion of Fλ (see [13]) in the interior C ◦ of C. It is of the form c(wλ)e wλ−ρ,t

Γq (wλ)e− q,t

Fλ (t) = q∈Q+

w∈W

with (unique) coefficients Γq (wλ) ∈ C, where Γ0 (wλ) = 1. Here Q+ is the positive lattice generated by R+ and c(λ) = c(λ, kμ ) denotes the c-function. As ξ ∈ C \co(W.ρ), there is some t ∈ C and hence also some t0 ∈ C ◦ such that ξ −ρ, t0 > 0. Fix t0 and consider Fλ (st0 ) for s ∈ R, s → +∞. As the imaginary part of λ is non-zero, Lemma 4.2.2 in Part I of [6] implies that there exist constants Mwλ > 0 (depending on t0 ) such that Γq (wλ) Mwλ e q,t0

for all q ∈ Q+ .

For s ∈ R, s > 0 we may therefore estimate − q,st0 Γ (wλ)e q Mwλ q∈Q+ \{0}

e(1−s) q,t0 ,

q∈Q+ \{0}

which tends to zero as s → +∞. Thus c(wλ)e wλ−ρ,st0 = c(wλ)es wξ −ρ,t0 eis wη,t0

Fλ (st0 ) w∈W

as s → +∞.

w∈W

Notice that c(λ) = 0. Moreover, ξ − ρ, t0 wξ − ρ, t0 for all w ∈ W where equality can only occur if wξ = ξ . Therefore, the leading term of the last sum is es ξ −ρ,t0 ·

w∈Wξ

c(wλ)eis wη,t0


2799

with Wξ = {w ∈ W : wξ = ξ }. Application of Lemma 5.5 below now implies that s → Fλ (st0 ) is unbounded as s → +∞. This finishes the proof. 2 iλk s on R with constants a > 0, c ∈ C which are not all Lemma 5.5. Let f (s) = eas · N k k=1 ck e zero, and distinct λk ∈ R. Then f is unbounded on [0, ∞). Proof. Let T > 0. Then according to Corollary 2 of [11], 2 T N N

iλk s −1 c e ds = T + 2πθδ |ck |2 k 0

k=1

k=1

with a constant δ > 0 depending on the λk and |θ | 1. If f were bounded on [0, ∞), say |f | M, this would imply that 2 T T n M2 iλk s 2 , ck e ds M e−2as ds 2a 0

a contradiction.

k=1

0

2

Notice that only the first part of our proof of Theorem 5.4 uses uniform boundedness of hypergroup characters in order to settle boundedness of Fλ in the case where Re λ is contained in the boundary of co(W.ρ). The rest of the proof works equally for arbitrary root systems R and arbitrary non-negative multiplicities k 0, k = 0, and the case k = 0 is classical. Actually, we have Corollary 5.6. Let R ⊂ a be an arbitrary root system, k 0 a non-negative multiplicity function and ρ = ρ(k). Then the associated hypergeometric function t → F (λ, k; t) is unbounded on a if Re λ ∈ / co(W.ρ). Moreover, t → F (λ, k; t) is bounded on a if Re λ is contained in the interior of co(W.ρ). μ of the hypergroup Cμ We return to our specific BC-cases and identify the dual space C q with a subset of C via ϕλ → λ. Due to the condition λ ∈ {w.λ, w ∈ W } it is contained in the union of finitely many hyperplanes in Cq ∼ = R2q of real dimension q. Note that the chamber C is a proper subset of Cμ . The following is an immediate consequence of Opdam’s Plancherel theorem (Theorem 3.2): Proposition 5.7. The Plancherel measure of the hypergroup Cμ is given by the measure dπμ (λ) =

1 dλ |c(iλ, kμ )|2

μ ⊂ Cq . Its support coincides with the chamber C. on C

2800


Acknowledgment It is a pleasure to thank Angela Pasquale for fruitful discussions at an early stage of this paper. References [1] J.F. van Diejen, L. Lapointe, J. Morse, Determinantal construction of orthogonal polynomials associated with root systems, Compos. Math. 140 (2004) 255–273. [2] J. Dugundji, Topology, Allyn and Bacon Inc., 1966. [3] M. Flensted-Jensen, T. Koornwinder, The convolution structure for Jacobi function expansions, Ark. Mat. 11 (1973) 245–262. [4] R. Gangolli, V.S. Varadarajan, Harmonic Analysis of Spherical Functions on Real Reductive Groups, SpringerVerlag, Berlin, Heidelberg, 1988. [5] G. Heckman, Dunkl operators, in: Séminaire Bourbaki 828, 1996–97, Astérisque 245 (1997) 223–246. [6] G. Heckman, H. Schlichtkrull, Harmonic Analysis and Special Functions on Symmetric Spaces, Perspect. Math., vol. 16, Academic Press, California, 1994. [7] S. Helgason, Groups and Geometric Analysis, Math. Surveys Monogr., vol. 83, American Mathematical Society, 2000. [8] R.I. Jewett, Spaces with an abstract convolution of measures, Adv. Math. 18 (1975) 1–101. [9] T. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, in: R. Askey, T. Koornwinder, W. Schempp (Eds.), Special Functions: Group Theoretical Aspects and Applications, Reidel, Dordrecht, 1984, pp. 1–85. [10] I.G. Macdonald, Orthogonal polynomials associated with root systems, Sém. Lothar. Combin. 45 (2000), Article B45a. [11] H. Montgomery, R.C. Vaughan, On Hilbert’s inequality, J. London Math. Soc. (2) 8 (1974) 73–81. [12] A.L. Onishchik, E.B. Vinberg, Lie Groups and Algebraic Groups, Springer-Verlag, Berlin, Heidelberg, 1990. [13] E.M. Opdam, Harmonic analysis for certain representations of graded Hecke algebras, Acta Math. 175 (1) (1995) 75–121. [14] E.M. Opdam, Lecture Notes on Dunkl Operators for Real and Complex Reflection Groups, MSJ Mem., vol. 8, Mathematical Society of Japan, Tokyo, 2000. [15] M. Rösler, Bessel convolutions on matrix cones, Compos. Math. 143 (2007) 749–779. [16] P. Sawyer, Spherical functions on SO0 (p, q)/SO(p) × SO(q), Canad. Math. Bull. 42 (1999) 486–498. [17] B. Schapira, Contributions to the hypergeometric function theory of Heckman and Opdam: sharp estimates, Schwartz space, heat kernel, Geom. Funct. Anal. 18 (2008) 222–250. [18] E.C. Titchmarsh, The Theory of Functions, Oxford Univ. Press, London, 1939.


Carleson–Sobolev measures for weighted Bloch spaces ✩ Evgueni Doubtsov St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia Received 29 July 2009; accepted 30 October 2009 Available online 12 November 2009 Communicated by Paul Malliavin

Abstract Carleson–Sobolev measures for weighted Bloch spaces on the unit ball of Cn are described. The classical Carleson measures for growth spaces on special circular domains are characterized. © 2009 Elsevier Inc. All rights reserved. Keywords: Carleson measure; Bloch space; Growth space; Bounded circular domain

1. Introduction Let H (Bn ) denote the space of holomorphic functions on the unit ball Bn = {z ∈ Cn : |z| < 1}. 1.1. Weighted Bloch spaces For α 0, the weighted Bloch space B α (Bn ) consists of those f ∈ H (Bn ) for which 1+α sup Rf (z) 1 − |z| < ∞, z∈Bn

where ✩

This research was supported by RFBR (grant No. 08-01-00358-a). E-mail address: [email protected].


(1)

2802

E. Doubtsov / Journal of Functional Analysis 258 (2010) 2801–2816

Rf (z) =

n j =1

zj

∂f (z), ∂zj

z ∈ Bn ,

is the radial derivative of f . In particular, B 0 (Bn ) is the classical Bloch space B(Bn ). If α > 0, then B α (Bn ) coincides with the growth space A−α (Bn ) (see, e.g., [10, Corollary 8.4]). Recall that A−α (Bn ) consists of those f ∈ H (Bn ) for which α sup f (z) 1 − |z| < ∞. z∈Bn

For α < 0, the weighted Bloch space B α (Bn ) consists of those f ∈ H (Bn ) for which j +α sup Rj f (z) 1 − |z| < ∞,

(2)

z∈Bn

where j is the least positive integer such that j + α > 0. Note that B α (Bn ), α < 0, coincides with the holomorphic Lipschitz–Zygmund space Λ−α (Bn ) (see, e.g., [20, Chapter 7]). Let I denote the identity operator. It is well known that we obtain the same spaces B α (Bn ), α ∈ R, if we replace R by R + I in (1) and (2). The above parametrization of the scale {B α (Bn )}α∈R is not standard. In fact, the present paper is motivated by [10, Theorem 9.3], so, we adopt the notation used in [10]. 1.2. Carleson measures Definition 1.1. Let X ⊂ H (Bn ) and let 0 < q < ∞. A positive Borel measure μ on Bn is called q-Carleson for X provided that I : X → Lq (μ) is a bounded operator. Carleson [2] solved the problem when n = 1 and X is the Hardy space H q (B1 ). By now, characterizations of q-Carleson measures are known for various spaces X of holomorphic functions. The first result of the present paper is the following theorem. Theorem 1.2. Assume that μ is a positive Borel measure on Bn , 0 < q < ∞ and α > 0. Then the following properties are equivalent: μ is a q-Carleson measure for B α (Bn ) = A−α (Bn ); −αq 1 − |z| dμ(z) < ∞;

(3) (4)

Bn

I : B (Bn ) = A−α (Bn ) → Lq (μ) is a compact operator. α

(5)

If α < 0, then 1 ∈ B α (Bn ) and B α (Bn ) ⊂ H ∞ (Bn ). So, the following assertion holds. Proposition 1.3. Assume that μ is a positive Borel measure on Bn , 0 < q < ∞ and α < 0. Then μ is a q-Carleson measure for B α (Bn ) if and only if μ is finite.


2803

For α = 0, we have the following partial result. Proposition 1.4. Assume that μ is a positive Borel measure on Bn and 0 < q < ∞. 1. If Bn

e log 1 − |z|

q dμ(z) < ∞,

then μ is a q-Carleson measure for B(Bn ). 2. Let μ be a q-Carleson measure for B(Bn ). Then Bn

e log 1 − |z|

q

2

dμ(z) < ∞.

As far as the author is aware, for n = 1, Proposition 1.4 was obtained in [12]; see also [9]. The author is grateful to S. Stević for reference [12]. Also, we obtain a characterization of the radial q-Carleson measures for B(Bn ) (see Proposition 2.3 below). The logarithmic growth space A− log (Bn ) consists of those f ∈ H (Bn ) for which f − log = sup

z∈Bn

|f (z)| < ∞. log(e/(1 − |z|))

It is well known that B(Bn ) ⊂ A− log (Bn ); moreover, this embedding is, in a sense, optimal. For A− log (Bn ), we have the following analogue of Theorem 1.2. Theorem 1.5. Assume that μ is a positive Borel measure on Bn and 0 < q < ∞. Then the following properties are equivalent: (i) μ is a q-Carleson measure for A− log (Bn ); e (ii) Bn (log 1−|z| )q dμ(z) < ∞; (iii) I : A− log (Bn ) → Lq (μ) is a compact operator. For n = 1, Theorem 1.5 was obtained in [9]. 1.3. Radial differential operators and Carleson–Sobolev measures u For u, s ∈ R, Kaptano˘

∞ glu [10] introduced radial differential operators Ds : H (Bn ) → H (Bn ) as follows. Let f = k=0 Fk be the homogeneous expansion of a function f ∈ H (Bn ). Then

Dsu f (z) =

∞ k=0

u s dk Fk (z),

z ∈ Bn .

2804


The formulas for us dk are given in [10, Section 2]. Note that us dk = 0 and us dk ∼ k u as k → ∞. In fact, we will not use the explicit values of us dk . The main property that we need is Theorem 1.7 below. −u . Note that each operator Dsu is invertible on H (Bn ) with two-sided inverse (Dsu )−1 = Ds+u 1 0 Also, we have D−n = R + I and Ds = I for any s ∈ R. Given s, u ∈ R and f ∈ H (Bn ), put u Isu f (z) = 1 − |z| Dsu f (z),

z ∈ Bn .

Definition 1.6. Assume that X ⊂ H (Bn ), 0 < q < ∞ and u, s ∈ R. A positive Borel measure μ on Bn is called (q; u, s)-Carleson–Sobolev for X provided that Isu : X → Lq (μ) is a bounded operator. In particular, the (q; 0, s)-Carleson–Sobolev measures coincide with the q-Carleson measures. The following result allows to translate the results about the q-Carleson measures for B α (Bn ) into results about the (q; u, s)-Carleson–Sobolev measures for weighted Bloch spaces. Theorem 1.7. (See [10, Corollary 8.5].) For any α, u, s ∈ R, Dsu (B α (Bn )) = B α+u (Bn ) is an isomorphic isomorphism under the equivalence of norms. For example, we have the following corollary of Theorem 1.2. Corollary 1.8. Assume that ρ is a positive Borel measure on Bn , 0 < q < ∞, α, u, s ∈ R and α + u > 0. Then the following properties are equivalent: (i) ρ is a (q; u, s)-Carleson–Sobolev measure for B α (Bn ); (ii) Bn (1 − |z|)−αq dρ(z) < ∞; (iii) Isu : B α (Bn ) → Lq (ρ) is a compact operator. Proof. Let (i) hold. Then Dsu (B α (Bn )) ⊂ Lq ((1 − |z|)uq dρ(z)). By Theorem 1.7, we have α α+u Dsu (B (Bn )) = B−αq (Bn ). Recall that α + u > 0, hence, the implication (3) ⇒ (4) guarantees dρ(z) < ∞. that Bn (1 − |z|) Now, assume that (ii) holds. Represent the operator Isu : B α (Bn ) → Lq (ρ) as the following composition: uq M(1−|z|)u Dsu I B α (Bn ) −−→ B α+u (Bn ) − → Lq 1 − |z| dρ(z) −−−−−→ Lq (ρ), where M(1−|z|)u is the operator of multiplication by (1 − |z|)u . Clearly, M(1−|z|)u is bounded; Dsu : B α (Bn ) → B α+u (Bn ) is bounded by Theorem 1.7. Finally, we have α + u > 0, hence, property (ii) and the implication (4) ⇒ (5) guarantee that I : B α+u (Bn ) → Lq ((1 − |z|)uq dρ(z)) is a compact operator. So, property (iii) holds. Clearly, (iii) ⇒ (i), so, the proof is complete. 2 Corollary 1.9. Assume that τ is a positive Borel measure on Bn , α, u, s ∈ R and α + u > 0. Then τ is a finite measure if and only if


2805

αq Isu : B α (Bn ) → Lq 1 − |z| dτ (z) is a bounded operator for some q > 0. Proof. Put dρ(z) = (1 − |z|)αq dτ (z) and apply Corollary 1.8.

2

Note that Corollary 1.9 coincides with [10, Theorem 9.3]. Unfortunately, the proof of Theorem 9.3 in [10] is incorrect for n 2, since it uses the following assertion: There exist holomorphic homogeneous polynomials Fk , deg Fk = mk > 0, such that mαk Fk (z) Cmαk |z|mk

for some C > 0 and all z ∈ Bn .

(6)

Let n 2. Note that Fk (0) = 0, hence, for any r ∈ (0, 1), there exists a point ζ ∈ ∂Bn such that Fk (rζ ) = 0. In other words, estimate (6) is false for n 2. Using Propositions 1.3 and 1.4 in the place of Theorem 1.2, we obtain the following corollaries. Corollary 1.10. Assume that ρ is a positive Borel measure on Bn , 0 < q < ∞, α, u, s ∈ R and α + u < 0. Then ρ is a (q; u, s)-Carleson–Sobolev measure for B α (Bn ) if and only if

uq 1 − |z| dρ(z) < ∞.

Bn

Corollary 1.11. Assume that ρ is a positive Borel measure on Bn , 0 < q < ∞, α, u, s ∈ R and α + u = 0. 1. If log Bn

e 1 − |z|

q

uq 1 − |z| dρ(z) < ∞,

then μ is a (q; u, s)-Carleson–Sobolev measure for B α (Bn ). 2. Let μ be a (q; u, s)-Carleson–Sobolev measure for B α (Bn ). Then Bn

e log 1 − |z|

q

2

uq 1 − |z| dρ(z) < ∞.

1.4. Organization of the paper Carleson measures for B(Bn ) are investigated in Section 2; in particular, we prove Proposition 1.4. In Sections 3 and 4, we prove generalizations of Theorems 1.2 and 1.5. Namely, we investigate the growth spaces A−α (Ω), α > 0, and A− log (Ω), where Ω is a circular domain of a special type. The key technical result is Lemma 3.3. Note that Lemma 3.3 is also useful in the study of weighted composition operators and extended Cesàro operators (see [5–7]). Some results of this paper have been announced in [5].

2806


2. Carleson measures for B(Bn ) In this section, we use the following norm on the Bloch space B(Bn ): f B(Bn ) = f (0) + sup 1 − |z| Rf (z) < ∞. z∈Bn

Also, recall that the space A− log (Bn ) consists of those f ∈ H (Bn ) for which f − log = sup

z∈Bn

|f (z)| < ∞. log(e/(1 − |z|))

2.1. Aleksandrov–Ryll–Wojtaszczyk polynomials Ryll and Wojtaszczyk [17] constructed holomorphic polynomials which proved to be very useful for many problems of function theory in the unit ball (see, e.g., [16]). The results of the present section are based on the following improvement of the Ryll–Wojtaszczyk theorem. Theorem 2.1. (See Aleksandrov [1, Theorem 4].) Let n ∈ N. Then there exist δ = δ(n) ∈ (0, 1) and J = J (n) ∈ N with the following property: For every d ∈ N, there exist holomorphic homogeneous polynomials Wj [d] of degree d, 1 j J , such that Wj [d]

L∞ (∂Bn )

1

(7)

and max Wj [d](ζ ) δ

1j J

for all ζ ∈ ∂Bn .

(8)

2.2. Proof of Proposition 1.4 Part 1. If log Bn

e 1 − |z|

q dμ(z) < ∞,

then A− log (Bn ) ⊂ Lq (Bn , μ) by the definition of A− log (Bn ). Recall that B(Bn ) ⊂ A− log (Bn ), so, μ is a q-Carleson measure for B(Bn ). Part 2. Assume that 0 < q < ∞ and μ is a q-Carleson measure for B(Bn ). Let the constant δ ∈ (0, 1) and the polynomials Wj [d], 1 j J , d ∈ N, be those provided by Theorem 2.1. For k ∈ Z+ , let Rk denote the Rademacher function: Rk (t) = sign sin 2k+1 πt , For each non-diadic t ∈ [0, 1], consider the functions

t ∈ [0, 1].


Fj,t (z) =

∞

Rk (t)Wj 2k (z),

2807

z ∈ Bn , 1 j J.

k=0

Estimate (7) guarantees that ∞ ∞ k 2k 1 − |z| (RFj,t )(z) 1 − |z| 2 |z| 2 1 − |z| |z|m 2 k=0

m=1

for all z ∈ Bn . We have (RFj,t )(0) = 0, hence, Fj,t B(Bn ) 2. By assumption, B(Bn ) ⊂ Lq (Bn , μ), thus, applying the closed graph theorem, we obtain

Fj,t (z)q dμ(z) C Fj,t q

B(Bn )

C,

1 j J.

Bn

Changing the order of integration, we have 1

Fj,t (z)q dt dμ(z) =

Bn 0

1

Fj,t (z)q dμ(z) dt C,

1 j J.

0 Bn

Ref. [21, Chapter V, Theorem 8.4] guarantees that q 1 ∞ k 2 2 Wj 2 (z) Fj,t (z)q dt dμ(z) C. dμ(z) C Bn

k=0

Bn 0

Given positive numbers aj , 1 j J , we have

J

q

2

aj

Cq,n

j =1

J

q/2

aj .

j =1

Hence, q ∞ q J J ∞ 2 2 k 2 2 Wj 2 (z) Wj 2k (z) dμ(z) C dμ(z) Bn

j =1 k=0

j =1B

n

k=0

CJ C. Since Wj [2k ] is a homogeneous polynomial of degree 2k , estimate (8) guarantees that

2808

E. Doubtsov / Journal of Functional Analysis 258 (2010) 2801–2816 ∞ ∞ J k 2 k+1 Wj 2 (z) δ 2 |z|2 k=0 j =1

k=0

δ2

∞ |z|2m 1 = δ 2 log , m 1 − |z|2

z ∈ Bn .

m=1

So, log Bn

1 1 − |z|2

q

2

dμ(z) < ∞.

Finally, remark that 1 ∈ B(Bn ), thus, μ is a finite measure. Hence, log Bn

e 1 − |z|2

q

2

dμ(z) < ∞,

as required. 2.3. Radial Carleson measures Let σn denote the normalized Lebesgue measure on the sphere ∂Bn . The following lemma is essentially known (see, e.g., [20, Exercise 3.19]). Lemma 2.2. Let 0 < q < ∞. Then

f (rζ )q dσn (ζ ) Cf B(B

n)

∂Bn

e log 1−r

q

2

,

0 r < 1,

(9)

for all f ∈ B(Bn ). Proof. Let f ∈ B(Bn ). Given ζ ∈ Bn , put fζ (λ) = f (λζ ) for λ ∈ B1 . So, fζ ∈ H (B1 ). Remark that (Rf )(λζ ) = λfζ (λ), hence, max fζ (λ) 4f B(Bn )

|λ|1/2

by the maximum principle. Also, we have sup 1/2 0. Then there exist functions fm ∈ A−α (Ω), 0 m M, such that M fm (z) m=0

1 , (1 − rΩ (z))α

z ∈ Ω.

(14)

2. There exist functions hm ∈ A− log (Ω), 0 m M, such that M hm (z) log

e , 1 − rΩ (z)

m=0

z ∈ Ω.

(15)

For Ω = B1 , the above lemma is known. Namely, the first part of Lemma 3.3 was proved by Ramey and Ullrich [14] for Ω = B1 and α = 1; see [8,19] for the case Ω = B1 and α > 0. The second part of Lemma 3.3 was proved in [9] for Ω = B1 . Proof of Lemma 3.3. Part 1. Let the constants δ ∈ (0, 1), J ∈ N and the polynomials Wj [d], 1 j J , d ∈ N, be those provided by Theorem 3.2. Put fj (z) =

∞

Qαk Wj Qk (z),

z ∈ Ω, 1 j J,

k=0

where Q ∈ N is sufficiently large. For z ∈ Ω, by (12), we have ∞ ∞ −α αk Qk

fj (z) Q rΩ (z) C

α−1 rΩ (z) C 1 − rΩ (z) . k=0

=1

In other words, fj ∈ A−α (Ω), 1 j J . Claim. For all Q ∈ N large enough, we have J fj (z) j =1

C (1 − rΩ (z))α

for 1 − Q−k rΩ (z) 1 − Q−(k+1/2) ,

(16) k ∈ N.

(17)

Proof. The argument below is similar to that used in the proof of [14, Proposition 5.4], where Ω = B1 . For any z ∈ Ω, we have J J k−1 ∞ k Qm Qm fj (z) Qαk Wj Q (z) − J Qαm rΩ (z) − J Qαm rΩ (z) j =1

j =1

= Σ0 − Σ− − Σ+ ,

m=0

k ∈ N.

m=k+1

2812


Remark that, by (13), Qk

Σ0 δQαk rΩ (z),

k ∈ N.

(18)

Below we assume that (17) holds. So, we have Qk Qk+1/2 Q−1/2 Qk 1 − Q−k rΩ (z) 1 − Q−(k+1/2) ,

k ∈ N.

Thus, if Q is large enough, then Qk

1/3 rΩ (z) 2−Q

−1/2

,

k ∈ N.

(19)

Therefore, Σ0 δQαk /3 by (18). Also, we have Σ− J

k−1

Qαm

m=0

J Qαk . Q−1

Now, consider the third term. Remark that Qm (Q−1) Qk+1 (Q−1) rΩ (z) rΩ (z)

for m k + 1, z ∈ Ω.

So, the ratio of two successive terms in Σ+ is not greater that the ratio of the first two terms. Hence, the series Σ+ is dominated by the geometric series having the same first two terms. Qk Thus, putting x = rΩ (z), we obtain Q Σ+ /J Q(k+1)α rΩ

k+1

(z)

∞ α (Qk+2 −Qk+1 ) m Q rΩ (z) m=0

k+1

=

Q Q(k+1)α rΩ (z) k+2 k+1 1 − Qα (rΩ (z))(Q −Q )

= Qkα Q

kα

Qα x Q 1 − Qα x (Q

2 −Q)

Qα 2−Q 1 − Qα 2(Q

1/2

1/2 −Q3/2 )

by (19). In sum, we have J 1 fj (z) δ Qαk = δ Qα(k+1/2) δ α/2 α/2 4 4Q 4Q (1 − rΩ (z))α j =1

if Q is sufficiently large and z satisfies (17). The proof of the claim is complete.

2


2813

Similarly, let fJ +j (z) =

∞

Qα(k+1/2) Wj Qk+1/2 (z),

z ∈ Ω, 1 j J,

k=0

where Q = q 2 and q ∈ N. If q is sufficiently large, then fJ +j ∈ A−α (Ω) and J fJ +j (z) j =1

C (1 − rΩ (z))α

(20)

for 1 − Q−(k+1/2) rΩ (z) 1 − Q−(k+1) ,

k ∈ N.

(21)

The proof of the above estimate is analogous to that of the claim; so, we omit it. Now, fix Q so large that (16) and (20) hold under assumptions (17) and (21), respectively. Put M = 2J and multiply the functions fm , 1 m M, by a sufficiently large constant. Then M fm (z) m=1

1 (1 − rΩ (z))α

for 1 − Q−1 rΩ (z) < 1.

It remains to define f0 ≡ Q. The proof of part 1 is complete.

2

Proof of Lemma 3.3. Part 2. Put gj (z) =

∞

k Qk Wj QQ (z),

z ∈ Ω, 1 j J,

k=0

where the notation from the proof of the first part of Lemma 3.3 is used. Then gj ∈ A− log (Ω), 1 j J , by Theorem 12 from [9]. The argument used in the proof of Theorem 2 from [9] guarantees that J gj (z) C log j =1

1 1 − rΩ (z)

for 1 − Q−Q rΩ (z) 1 − Q−Q k

(k+1/2)

if Q ∈ N is large enough (see also the proof of the first part of Lemma 3.3). Similarly, let gJ +j (z) =

∞

(k+1/2) (z), Q(k+1/2) Wj QQ

z ∈ Ω, 1 j J,

k=0

where Q = q 2 and q ∈ N. If q is large enough, then gJ +j ∈ A− log (Ω) and

, k ∈ N,

2814

E. Doubtsov / Journal of Functional Analysis 258 (2010) 2801–2816 J gJ +j (z) C log j =1

for 1 − Q−Q

(k+1/2)

1 1 − rΩ (z)

rΩ (z) 1 − Q−Q

(k+1)

, k ∈ N.

To finish the proof, put M = 2J and f0 ≡ C > 0, where the constant C is sufficiently large. 2 4. Carleson measures for the growth spaces The following compactness criterion is well known (see, e.g., [4, Proposition 3.11], [18, Lemma 3.7]). Lemma 4.1. Let X = A−α (Ω), α > 0, or X = A− log (Ω) and let Y be a linear metric space with translation invariant metric. Consider a linear operator T : X → Y . Then the following implication holds. Assume that {T hj } converges to zero in the metric of Y for any bounded in X sequence {hj } such that hj → 0 uniformly on compact subsets of Ω. Then T is a compact operator. We have the following generalization of Theorems 1.2 and 1.5. Theorem 4.2. Let 0 < q < ∞ and let μ be a positive Borel measure on Ω. 1. Assume that α > 0. Then μ is a q-Carleson measure for A−α (Ω) if and only if Ω

dμ(z) 0}, Ω − (u) = Ω\Ω + (u), and F (u) = ∂Ω + (u)∩Ω which is called the free boundary of u. Ω + (u) and Ω − (u) are the positive and negative phases. A free boundary point x0 ∈ F (u) is said to be regular if there is a ball Bρ ⊂ Ω + (u) or Bρ ⊂ Ω − (u) that touches F (u) at x0 . If this is the case, we denote by ν the radial direction at the tangent point x0 that points inward of Ω + (u). The free boundary problem of phase transition we consider is formulated in a PDE form as ⎧ ⎪ ⎨ u = 0 − u+ ν = g(uν ) ⎪ ⎩ u=σ

in Ω + (u) ∪ Ω − (u), along F (u), on ∂Ω,

¯ or variationally as minimizing the functional where u ∈ C(Ω), J (u) =

|∇u|2 + λ2 (u) dx, Ω

¯ such that limx∈Ω→a u(x) = σ (a) for every a ∈ ∂Ω, where u ∈ W 1,2 (Ω) ∩ C(Ω) λ (u) = 2

and λ22 − λ21 > 0.

λ21

if u 0,

λ22

if u > 0,

G. Lu, P. Wang / Journal of Functional Analysis 258 (2010) 2817–2833

2819

We define a viscosity solution of the free boundary problem as follows. Definition 1.1. A continuous function u is called a viscosity sub-solution of the elliptic twophase free boundary problem, if it verifies the following conditions. 1. u 0 in Ω + (u) ∪ Ω − (u) in the viscosity sense. 2. ∀x0 ∈ F (u) := ∂Ω ∩ Ω, if there exists a ball Bρ ⊂ Ω + (u) that touches F (u) at x0 , then there exists β > 0 such that u(x) α x − x0 , ν+ − β x − x0 , ν− + ◦ |x − x0 | for all x near x0 , where α = g(β) and ν is the radial direction of ∂Bρ at x0 pointing to Ω + (u). A continuous function v is a viscosity super-solution of the elliptic two-phase free boundary problem in Ω, if it verifies the following conditions. 1. v 0 in Ω + (v) ∪ Ω − (v) in the viscosity sense. 2. ∀x0 ∈ F (v) := ∂Ω + (v) ∩ Ω, if there exists a ball Bρ ⊂ Ω − (v) that touches F (v) at x0 , then there exists β > 0 such that v(x) α x − x0 , ν+ − β x − x0 , ν− + ◦ |x − x0 | for all x near x0 , where α = g(β) and ν is the radial direction of ∂Bρ at x0 pointing to Ω + (v). A continuous function u is a viscosity solution of the elliptic two-phase free boundary problem if it is both a viscosity sub-solution and viscosity super-solution. Remark 1.1. According to Caffarelli’s theory [5–7], a viscosity solution of the free boundary problem is indeed an as classical as possible solution of the free boundary problem. In particular, the set of singular free boundary points is of (n − 1)-Hausdorff measure 0. Nevertheless, in the following we still adopt the term “viscosity solutions” instead of “classical solutions” to distinguish them from minimizers. Contrary to the properties of viscosity solutions of the Dirichlet problem for the Laplace equation, the following facts about a viscosity solution of the free boundary problem deserve mentioning. That u is a viscosity solution does not imply −u is also a viscosity solution. That u is a viscosity solution does not imply u + C is also a viscosity solution for a constant C. That u and v are both viscosity solutions does not imply u + v or u − v is also a viscosity solution. The uniqueness problem about the phase transition is formulated either in a PDE way as “Is there a unique viscosity solution of the free boundary problem, given a continuous boundary date σ ?” or variationally as “Is there a unique minimizer of the functional J (u), given a continuous boundary data σ ?” This paper answers these questions with counter-examples. On the

2820


σ (b) u v v=0

a

c

b

Fig. 1. The basic picture.

other hand, a uniqueness theorem of an evolutionary problem is proved. We propose a plausible explanation of non-uniqueness in the elliptic problem. However, many detailed questions about non-uniqueness in the elliptic free boundary problem are still open. Some of them are summarized in the last section. We organize the paper in the following order. In the next three sections, we provide counterexamples to the uniqueness question in various cases in 1D and multi-dimensions, followed by a section devoted to the 1D bifurcation phenomenon. In the last section, we prove the uniqueness theorem for the ε-evolutionary problem. We conclude the paper with a list of open questions about uniqueness in the elliptic case. 2. Counter-examples in 1D In this section, we provide counter-examples to the uniqueness problem in 1D of various kinds of free boundaries and boundary data. We start with the basic picture. Take Ω = (a, b), and the boundary data σ (a) = 0 and (b) σ (b) > 0, where a is taken so that a < c := b − σg(0) . Recall that g : [0, ∞) → (0, ∞) is the function that prescribes the free boundary condition. Define u : Ω → R by u(x) = x−a b−a σ (b), and v : Ω → R by

v(x) =

0 g(0)(x − c)

if x ∈ [a, c], if x ∈ (c, b].

Then u is harmonic on Ω with no free boundary point. Thus it is a viscosity solution of the free boundary problem. v has exactly one free boundary point c at which vν+ = g(0) and vν− = 0. So the free boundary condition vν+ = g(vν− ) is verified at the free boundary point c. Therefore v is also a viscosity solution and v = u on ∂Ω. u and v are two viscosity solutions of the free boundary problem with equal boundary condition. Fig. 1 illustrates the counterexample. We now modify the basic picture to obtain a counter-example in which both u and v have free boundary points. In fact, we glue two pieces of the basic picture with the roles of u and v switched in the two cases. More precisely, let Ω = (a, b) ∪ (b, c), σ (a) > 0, σ (b) = 0, and (a) (x) σ (c) > 0, where d := a + σg(0) < b < e := c − σg(0) by taking a small enough and c large enough. Define u, v : Ω → R by ⎧ ⎪ ⎨ −g(0)(x − d) if x ∈ [a, d], if x ∈ (d, b], u(x) = 0 ⎪ ⎩ x−b σ (c) if x ∈ (b, c], c−b

G. Lu, P. Wang / Journal of Functional Analysis 258 (2010) 2817–2833 σ (a)

v u u v

a

d

u=0

b

v=0

e

2821

σ (c)

c

Fig. 2. Both with free boundary. σ (a)

σ (c)

v u u v u=v a d u=0 b v=0 e c g f

σ (f )

Fig. 3. With changing sign. σ (b)

σ (b)

v c

u a d b v u

σ (a)

u v d v c a v e b u σ (a)

Fig. 4. Impossible pictures.

and ⎧ x−b ⎪ ⎨ a−b σ (a) v(x) = 0 ⎪ ⎩ g(0)(x − e)

if x ∈ [a, b], if x ∈ (b, e], if x ∈ (e, c].

Then u and v have both exactly one free boundary point, namely d and e respectively, at which the free boundary condition is readily verified. So u and v are two different viscosity solutions satisfying the same boundary condition which have free boundary points. See Fig. 2. At last, we give counter-examples in case the boundary data change sign. We simply attach a viscosity solution to the two distinct viscosity solutions obtained in the preceding case. Take Ω = (a, b) ∪ (b, c) ∪ (c, f ), σ (a) > 0, σ (b) = 0, σ (c) > 0, σ (f ) < 0, and take d, e, and g as in the previous case so that the free boundary condition is verified. So u and v are distinct viscosity solutions of the free boundary problem with the same boundary data as illustrated in Fig. 3. On the other hand, the pictures in Fig. 4 are impossible due to the monotonicity of the free − boundary condition u+ ν = g(uν ), where Ω = (a, b).

2822

G. Lu, P. Wang / Journal of Functional Analysis 258 (2010) 2817–2833 σ (a)

u

v

v

a

v=0

c

σ (b)

d

b

Fig. 5. Minimum principle fails. + The picture of two viscosity solutions on the left is impossible as u+ ν (c) < vν (d), − + − + − > vν (d), uν (c) = g(uν (c)), vν (d) = g(vν (d)), and g is strictly increasing. For similar reasons, the picture of two viscosity solutions on the right is not possible, either. (a) + Another counter-example is worth mentioning. If σ (a) > 0, σ (b) > 0, and b − a > σg(0)

u− ν (c)

σ (b) g(0) ,

then we have the counter-example in Fig. 5. In words, u(x) =

x−a b−x σ (a) + σ (b), b−a b−a

for x ∈ [a, b],

and ⎧ ⎨ g(0)(c − x) if a x < c, if c x < d, v(x) = 0 ⎩ g(0)(x − d) if d x b, (a) (b) (a) (b) where c = a + σg(0) and d = b − σg(0) . Notice that d > c as b − a > σg(0) + σg(0) . On account of this counter-example, the minimum principle does not hold if σ > 0 on ∂Ω. The following two sections give counter-examples in multi-dimensions. Using these counterexamples and attach more annuli or shells in the same way as in 1D, we may have counterexamples of various cases as above. In fact, we can construct counter-examples in any dimension in this way. One should be convinced that the non-uniqueness is a physical phenomenon instead of a problem arising from mathematical modeling.

3. A counter-example to the uniqueness of a viscosity solution in multi-dimensions Similar to the 1D case, even in the simplest form of a two-phase free boundary problem in multi-dimensions, the uniqueness of a viscosity solution is false, as shown by the following example. Indeed, we consider the uniqueness of a viscosity solution of the following two-phase free boundary problem. ⎧ + u = 0 ⎪ ⎪ ⎪ ⎨ u− = 0 ⎪ u+ 2 − u− 2 = 1 ⎪ ⎪ ν ⎩ ν u=σ

in Ω + (u), in Ω − (u), along F (u) := ∂Ω + (u) ∩ Ω, on ∂Ω.

We take Ω = B1 (0), the unit ball of Rn . Here we assume n > 2 for simplicity. The example also works in dimension two with proper modification in the formula of the function constructed.


Pick any value in (0, 1) for a number s. We take a constant function σ (x) = ¯ by Define a function u0 ∈ C(Ω) u0 (x) ≡

2823 n−2 s−s n−1

> 0 on ∂Ω.

n−2 s − s n−1

¯ Clearly u0 is a viscosity solution of the two-phase free boundary problem, as it does not on Ω. even have a free boundary. ¯ is defined by the formula The second function u1 ∈ C(Ω) u1 (x) = a|x|2−n + b n−1

with a = − sn−2 < 0 and b =

s n−2

> 0.

s−s n−1 n−2

Then u1 = a + b = on ∂Ω and u1 = as 2−n + b = 0 on ∂Bs (0). Clearly u1 = 0 in B1 \B¯s as u1 is basically the fundamental solution of the Laplacian. Furthermore, along F (u1 ) = ∂Bs (0), 1−n ∂r u+ =1 1 = a(2 − n)s

while ∂r u− 1 = 0. So the free boundary condition + 2 − 2 ∂r u1 − ∂r u1 = 1 is verified in the classical sense and hence in the viscosity sense. So, for the same boundary data σ ∈ C(∂Ω), one obtains two distinct viscosity solutions u0 and u1 , for any s with 0 < s < 1, of the same two-phase free boundary problem. 4. A counter-example of the uniqueness of a minimizer in multi-dimensions One might think though there are more than one viscosity solution of a two-phase free boundary problem, there is probably only one minimizer of a corresponding variational problem. Well, we give a counter-example to the uniqueness of a minimizer of the following simplest variational problem. For simplicity, we assume again the dimension n > 2. A similar counter-example may be constructed in two dimensions or in 1D. Let Ω = B1 , the unit ball of Rn as in the previous section. We take λ1 > 0 and λ2 > 0 such that Λ = λ22 − λ21 > 0, but otherwise the values of λ1 and λ2 are free to pick. n n−2 − 1, for s ∈ [0, 1]. Then h(0) = −1 < 0 < n−2 = h(1). So there Define h(s) = n−2 n s +s n exists an s0 ∈ (0, 1) such that h(s0 ) = 0, i.e. n−2 n s + s0n−2 − 1 = 0. n 0 Then n − 2 1 − s0n−2 s02−n − 1 = = . n s0n s02

2824


Take Λ = s02−n −1 s02

(n−2)n s0n (s02−n −1)

> 0. It follows that

(n−2)n s02−n −1

= Λs0n . This equality combined with

n−2 n

=

implies that 2 (n − 2)2 = s0n−2 s02−n − 1 . Λ

Now let g(s) = (n − 2)2 − Λs n−2 (s 2−n − 1)2 , s ∈ (0, 1]. For s ∈ (0, 1), g (s) = −(n − 2)Λs n−3 (s 2−n − 1)2 + 2(n − 2)Λs −1 (s 2−n − 1) = (n − 2)Λs −1 (s 2−n − 1)(1 + s n−2 ) > 0. So g is an increasing function, and g(1) = (n − 2)2 > 0 and lims→0+ g(s) = −∞. In addition, 2 the choice of s0 implies that g(s0 ) = 0 as (n−2) = s0n−2 (s02−n − 1)2 . So Λ ⎧ ⎨ < 0 for 0 < s < s0 , g(s) = 0 for s = s0 , ⎩ > 0 for s0 < s < 1. 1−n

ns 2 n Define f (s) = s(n−2)n 2−n −1 + λ2 − Λs , where 0 s < 1. For 0 < s < 1, f (s) = (s 2−n −1)2 g(s). So f attains its absolute minimum at s = s0 , according to our analysis of the function g. Note that f (0) = λ22 − (n − 2)n and lims→1− f (s) = +∞. We minimize the functional J (u) = |∇u|2 + λ2 (u) dx, Ω

with u = 1 on ∂Ω, where λ2 (s) = λ22 , if s > 0, and λ2 (s) = λ21 , if s 0. If there were only one minimizer u under the condition u = 1 on ∂Ω, then u must be radial as all the rotation of u around the origin are minimizers of the same boundary data. Now suppose u(x) = 0 for |x| = s for some s ∈ [0, 1]. As a result of the maximum principle of harmonic functions, u(x) = 0 for all x ∈ Bs . Therefore, for some s ∈ [0, 1], u(x) > 0 and u = 0 in s < |x| < 1, while u(x) = 0 in |x| s. This forces u to take the form u(x) = a|x|2−n + b. The boundary data give conditions on a and b, i.e.

2−n

a + b = 1, as 2−n + b = 0.

s So a = 1−s12−n < 0 and b = s 2−n 0. And ∇u = a(2 − n)r 1−n x, ˆ where xˆ = −1 If we denote the measure of the unit ball by σ = |B1 |, then

1 |∇u| dx = 2

B\Bs

s

x |x| .

(n − 2)n σ a 2 (n − 2)2 ρ 1−n dρ nσ = a 2 (n − 2)nσ s 2−n − 1 = 2−n s −1


2825

and λ22 |B\Bs | + λ21 |Bs | = λ22 σ − Λs n σ. Therefore J (u) =

(n − 2)n σ + λ22 σ − Λs n σ = f (s)σ. s 2−n − 1

The minimizer u0 of J (u) should be the one corresponding to s = s0 ∈ (0, 1). Then J (u0 ) = σ + λ22 σ − Λs0n σ . f (s0 )σ = (n−2)n 2−n s0

−1

¯ In this case, J (u1 ) = λ2 σ . On the other hand, if we define u1 (x) ≡ 1 in Ω. 2 Then (n − 2)n J (u0 ) − J (u1 ) = 2−n − Λs0n σ = 0 s0 − 1 as a result of

(n−2)n s02−n −1

= Λs0n . So both u0 and u1 are minimizers of the functional J (u) with the

equal boundary data. Of course, as 0 < s0 < 1, they are distinct minimizers, under the assumption there is a unique minimizer. We are done. 5. A bifurcation phenomenon in 1D In 1D, an open

set is the disjoint union of at most countably many open intervals. Thus in 1D, we write Ω = j ∈Λ Ij , where Ij = (aj , bj ) is an interval. Lemma 5.1 (Maximum–minimum principle for the free boundary problem). Let Ω be a bounded domain in Rn , and u a continuous viscosity solution of the two-phase free boundary problem in Ω. sup u = max u

(a)

Ω

∂Ω

holds, while infΩ u may be smaller than min∂Ω u. (b) If, in addition, min∂Ω u 0, then inf u = min u Ω

∂Ω

holds. Proof. Both (a) and (b) follow from a simple argument by contradiction and the maximum– minimum principle for the Laplacian in either phase. 2 Now we show, in 1D, a bifurcation phenomenon. We may restrict to every component Ij = (aj , bj ) of Ω. We also omit the subscript j . First assume σ (a) > 0 and σ (b) > 0. An obvious solution is the one without any free boundσ (a)+σ (b) x−a , then there cannot ary point, namely u(x) = b−x b−a σ (a) + b−a σ (b), x ∈ I . If b − a g(0)

2826


a

c

u=0

d

u

σ (b) > 0

b

u

σ (a) < 0 Fig. 6. Picture for c < d.

be a free boundary point for a solution. So the affine function just found is the only viscosity solution of the free boundary problem. Otherwise, assume c is the least value in (a, b) at which a viscosity solution u = 0 and d is the largest value in (a, b) at which u = 0. Then u is a viscosity solution of the two-phase free boundary problem in (c, d) with zero boundary data. The maximum–minimum principle implies that u = 0 on (c, d). So there is only one more viscosity solution of the free boundary problem on (a, b) other than the affine solution, namely the solution v defined by ⎧ ⎨ g(0)(c − x) if a x < c, if c x < d, v(x) = 0 ⎩ g(0)(x − d) if d x b, (a) (b) where c = a + σg(0) and d = b − σg(0) . If σ (a)σ (b) < 0, say σ (a) < 0 and σ (b) > 0, then u has at least one free boundary point. (Just keep in mind that there might not even be a viscosity solution u if (a, b) is too short with respect to σ (b).) Suppose there exist two (or more) points x1 and x2 such that u(x1 ) = u(x2 ) = 0. Then the maximum–minimum principle implies u = 0 on [x1 , x2 ]. Define c := inf{x ∈ (a, b): u(x) = 0} and d := sup{x ∈ (a, b): u(x) = 0}. Clearly u(c) = u(d) = 0.

Step 1. We claim c = d. Suppose c < d. We then have the picture in Fig. 6. σ (a) σ (a) − At the free boundary point c, u+ ν = 0 and uν = − c−a . Then 0 = g(− c−a ) > g(0) > 0, which is impossible. So c = d. Step 2. u is unique. Suppose there are two viscosity solutions u and v, and u = v on ∂Ω, as shown in Fig. 7. Without loss of generality, we assume c < d. σ (b) σ (a) − + − At c, u+ ν = g(uν ) where uν = b−c and uν = − c−a . σ (b) (a) . At d, vν+ = g(vν− ) where vν+ = b−d and vν− = − σd−a + and u− > v − , while the monotonicity of g implies that u+ = g(u− ) > < v Note u+ ν ν ν ν ν ν g(vν− ) = vν+ , which is a contradiction. So u is unique if σ (a)σ (b) < 0.


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σ (b)

u v c

d

a b u

v

σ (a) Fig. 7. u is unique.

u

a

σ (b)

b Fig. 8. The unique solution when (a, b) is short.

The critical case, σ (a)σ (b) = 0. If σ (a) = σ (b) = 0, then u = 0 everywhere. Otherwise, we may assume σ (a) = 0 and σ (b) > 0. (b) If σb−a > g(0), then there cannot be any free boundary point. The only viscosity solution is x−a u(x) = b−a σ (b) as shown in Fig. 8. (b) = g(0), still there is no free boundary point strictly between a and b. The unique If σb−a viscosity solution is u(x) = g(0)(x − a). (b) < g(0), we have seen the counter-example in Section 2 that declines the uniqueness of If σb−a a viscosity solution. In fact, let u(x) = x−a b−a σ (b) for x ∈ [a, b] and v be the function defined by

v(x) =

0 g(0)(x − c)

if x ∈ [a, c], if x ∈ (c, b],

(b) . Both u and v are viscosity solutions of the free boundary problem on [a, b]. where c = b − σg(0) We show that v is stable under perturbations of boundary data from below, and the perturbations of boundary data from above cause two perturbed solutions which converge to u and v respectively. (b) (b) Indeed, as σb−a < g(0), ∃!c such that a < c < b and σb−c = g(0). Let σε (a) = −ε and σε (b) → σ (b) as ε → 0+. Let uε be the unique solution of the free boundary problem with uε = σε on the boundary. The uε has a unique free boundary point cε . Obviously, cε > c. So it is easy to see that uε converges to v uniformly on [a, b], as ε → 0. Let σ ε (a) = ε and σ ε (b) → σ (b) as ε → 0+. Then there are two solutions of the free boundary with boundary data σ ε . Let uε1 be the solution without a free boundary and uε2 the solution x−a ε with a free boundary. Clearly, uε1 → u as uε1 (x) = b−x b−a ε + b−a σ (b). Also,

⎧ ε ⎨ g(0)(c − x) ε u2 (x) = 0 ⎩ g(0)(x − d ε )

if a x < cε , if cε x < d ε , if d ε x b,

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G. Lu, P. Wang / Journal of Functional Analysis 258 (2010) 2817–2833 ε

(b) ε where cε = a + g(0) and d ε = b − σg(0) . Clearly, cε → a and d ε → c as ε → 0. So uε2 → u uniformly. In case σ (a) < 0 and σ (b) < 0, the maximum–minimum principle implies supΩ u < 0. So a solution has only one phase. Therefore there is only one viscosity solution u(x) = b−x b−a σ (a) + x−a b−a σ (b), x ∈ (a, b).

6. Uniqueness for the ε-evolutionary problem Heuristically, the elliptic free boundary problem describes the limiting stationary state of the corresponding evolutionary free boundary problem. Unlike the elliptic free boundary problem, the evolutionary problem seems to enjoy the uniqueness of a viscosity solution with prescribed initial–boundary data. In fact, if we smooth the free boundary condition in a very small scale, we can prove the uniqueness of a viscosity solution of the smoothed parabolic ε-evolutionary problem. In the following, we prove the uniqueness of a viscosity solution of the following εevolutionary problem, ⎧ ε ⎨ H w = wt − Lw + βε (w) = 0 in Ω × T , w(x, t) = σ (x) on ∂Ω × T , ⎩ ¯ w(x, 0) = w0 (x) on Ω, where Lw = F (∇w, D 2 w) is a degenerate linear or non-linear elliptic operator such that F (p, On×n ) = 0 such as the Laplacian or p-Laplacian, βε (w) = 1ε β( wε ), β : → [0, ∞) is a compactly supported around origin, smooth non-negative function with β(0) > 0, Ω is a bounded domain in n , T = (0, T ) with T possibly being infinity, and the compatibility condition σ = w0 on ∂Ω is verified. Here the partial differential equation is verified in the viscosity sense, namely if a smooth function ϕ satisfies ϕ w (or ϕ w) in a neighborhood of (x0 , t0 ) and ϕ(x0 , t0 ) = w(x0 , t0 ), which is usually denoted by w ≺(x0 ,t0 ) ϕ, then H ε ϕ(x0 , t0 ) 0 (or H ε ϕ(x0 , t0 ) 0). The parabolic sub- and super-jets P 2,− w(x0 , t0 ) and P 2,+ are defined by P 2,− w(x0 , t0 ) =

ϕt (x0 , t0 ), ∇ϕ(x0 , t0 ), D 2 ϕ(x0 , t0 ) ϕ ≺(x0 ,t0 ) w

(6.1)

P 2,+ w(x0 , t0 ) =

ϕt (x0 , t0 ), ∇ϕ(x0 , t0 ), D 2 ϕ(x0 , t0 ) w ≺(x0 ,t0 ) ϕ .

(6.2)

and

The “closures” of the semi-jets are defined by P¯ 2,− w(x0 , t0 ) = (b, p, X) ∈ × n × Sn×n ∃(xk , tk , bk , pk , Xk ) ∈ Ω × T × × n × Sn×n such that (bk , pk , Xk ) ∈ P 2,− w(xk , tk ) and (xk , tk , bk , pk , Xk ) → (x0 , t0 , b, p, X)

(6.3) (6.4) (6.5)

and P¯ 2,+ w(x0 , t0 ) = (b, p, X) ∈ × n × Sn×n ∃(xk , tk , bk , pk , Xk )

(6.6)


∈ Ω × T × × n × Sn×n such that (bk , pk , Xk ) ∈ P 2,+ w(xk , tk ) and (xk , tk , bk , pk , Xk ) → (x0 , t0 , b, p, X) ,

2829

(6.7) (6.8)

where Sn×n is the set of symmetric n × n matrices. We also require σ and w0 to be continuous on ∂Ω and Ω¯ respectively. Note that w → −Lw + βε (w) is not a proper elliptic operator in the sense of Crandall–Ishii– Lions. As there is no confusion, we will skip the superscript and subscript ε, and write H for H ε and β for βε . Lemma 6.1. For T > 0 small enough, if H w 0 H w2 in Ω ×T and w < w2 on ∂p (Ω ×T ), then w w2 in Ω × T . Proof. As β is compactly supported and smooth, it is globally Lipschitz continuous for some Lipschitz constant K. For any given small number δ > 0, we define a new function w1 by w1 (x, t) = w(x, t) −

δ , T −t

where x ∈ Ω¯ and 0 t < T . In order to prove w w2 in Ω × T , it suffices to prove w1 w2 in Ω × T for all small δ > 0. Clearly, w1 < w2 on ∂p (Ω × T ), and limt→T w1 (x, t) = −∞ uniformly on Ω. Moreover, δ δ 2 − F ∇w, D w + β w − H w1 = w t − T −t (T − t)2 δ δ − β(w) = Hw − +β w− T −t (T − t)2 δ δ due to the Lipschitz continuity of β +K T −t (T − t)2 δ 1 δ 1 Hw − so that 2K + for T 2 2 2K T −t (T − t) 2(T − t) δ δ = Hw − − 2(T − t)2 2(T − t)2 δ − 2 < 0. 2T Hw −

The above differential equalities and inequalities are all in the viscosity sense. Every step can be made rigorous in the viscosity sense. We leave the work to the reader. Define, for j = 1, 2, vj (x, t) = e−λt wj (x, t), where λ > 2K. So wj (x, t) = eλt vj (x, t). Obviously, w1 w2 in Ω × T is equivalent to v1 v2 in Ω × T . A simple computation shows that in the viscosity sense, H wj = eλt H˜ vj , where H˜ v = vt − e−λt F (eλt ∇v, eλt D 2 v) + λv + e−λt β(eλt v). Then, in the viscosity sense, H˜ v1 − 2Tδ 2 e−λt − 2Tδ 2 e−λT < 0 and ¯ H˜ v2 0. Furthermore, v1 < v2 on ∂p (Ω × T ), and limt→T − v1 (x, t) = −∞ uniformly on Ω.

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Suppose supΩ×T (v1 − v2 ) > 0. Then supΩ×T (v1 − v2 ) is a maximum and is assumed exclusively in Ω × (0, T ), due to the last two conditions on v1 and v2 . Let M0 = supΩ×T (v1 − v2 ) = maxΩ×T (v1 − v2 ). For any small ε > 0, we define uε (x, y, t) = v1 (x, t) − v2 (y, t) −

1 |x − y|2 , 2ε

¯ t ∈ [0, T ). x, y ∈ Ω,

(6.9)

ε ¯ We observe first that maxΩ× ¯ Ω×[0,T ¯ ) u (x, y, t) exists as limt→T v1 (x, t) = −∞ uniformly on Ω. ε ε ε ε ε ε ε ε ¯ Let Mε = u (x , y , t ) = maxΩ× ¯ Ω×[0,T ¯ ) u , where x , y ∈ Ω and t ∈ [0, T ) ⊂ [0, T ) for some T < T independent of ε. Clearly, Mε M0 > 0. According to Proposition 3.7 in [9], 1 |x ε − y ε |2 = 0 hold. a generalization of Lemma 3.1 in [9], limε↓0 Mε = M0 and limε↓0 2ε ε ε ε We claim that x , y ∈ Ω and t > 0 for all sufficiently small ε. Suppose not. There exists a sequence εj → 0 such that either (x εj , t εj ) ∈ ∂p (Ω × T ) or (y εj , t εj ) ∈ ∂p (Ω × T ), and without loss of generality {x εj }, {y εj }, {t εj } converge. As 1 εj εj 2 εj εj εj εj εj 2εj |x − y | → 0 implies |x − y | → 0, we may assume x → x0 , y → x0 , t → t0 , where (x0 , t0 ) ∈ ∂p (Ω × T ), and t0 T < T . So

0 < M0 lim sup Mεj = v1 (x0 , t0 ) − v2 (x0 , t0 ) < 0 j

as (x0 , t0 ) ∈ ∂p (Ω × T ), which is an obvious contradiction. For any small ε > 0, Theorem 8.3 in [9] implies that there exist X, Y ∈ Sn×n , and b ∈ such ε ε ε ε ¯ 2,+ v1 (x ε , t ε ), (b, x −y , Y ) ∈ P¯ 2,− v2 (y ε , t ε ), and that (b, x −y ε , X) ∈ P ε X 0 −I 3 I 3 . − I ε ε −I 0 Y I The last inequality implies that X Y , while the first two inclusion conditions imply that ε ε δ ε ε −λt ε λt ε x − y λt ε b−e , e X + λv1 x ε , t ε + e−λt β eλt v1 x ε , t ε − 2 e−λT < 0 F e ε 2T (6.10) and

ε ε ε εx −y ε ε ε b − e−λt F eλt , eλt Y + λv2 y ε , t ε + e−λt β eλt v2 y ε , t ε 0. ε ε

That F is degenerate elliptic implies that F (eλt x As a result of the three preceding inequalities,

ε −y ε

ε

ε

, eλt X) F (eλt

ε

x ε −y ε λt ε ε , e Y ).

ε ε ε 0 > λ v1 x ε , t ε − v2 x ε , t ε + e−λt β eλt v1 x ε , t ε − β eλt v2 y ε , t ε λ v 1 x ε , t ε − v 2 x ε , t ε − K v 1 x ε , t ε − v 2 y ε , t ε as β is Lipschitz continuous with Lipschitz constant K λ λ v1 x ε , t ε − v2 x ε , t ε − v1 x ε , t ε − v2 y ε , t ε as λ > 2K. 2

(6.11)


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On account of the reasons that justify the preceding claim, we know that there exists a sequence εj → 0 such that x εj → x0 , y εj → x0 , t εj → t0 , and x0 ∈ Ω, 0 < t0 T < T . In addition, Proposition 3.7 in [9] implies v1 (x0 , t0 ) − v2 (x0 , t0 ) = M0 . Taking limits in 0 λ(v1 (x εj , t εj ) − v2 (y εj , t εj )) − λ2 |v1 (x εj , t εj ) − v2 (y εj , t εj )|, we obtain, since v1 (x0 , t0 ) − v2 (x0 , t0 ) = M0 > 0, that 0

λ v1 (x0 , t0 ) − v2 (x0 , t0 ) > 0, 2

which is an obvious contradiction. We are done.

2

We now loose the strict inequality restriction to a non-strict one. Lemma 6.2. For T > 0 sufficiently small, if H w1 0 H w2 in Ω × T and w1 w2 on ∂p (Ω × T ), then w1 w2 on Ω × T . ˜ where the value of δ˜ > 0 will be taken in the following. Proof. For any δ > 0, let w = w1 −δt − δ, Then w < w1 w2 on ∂p (Ω × T ), and ˜ H w = H w1 − δ − βε (w1 ) + βε (w1 + δt + δ) −δ + Kδt + K δ˜ −δ + KδT + K δ˜ 1 1 < −δ + δ + δ 2 4 1 = − δ < 0. 4

for T small and δ˜

δ 4K

Again, the above differential equality and inequalities are in the viscosity sense and can be made rigorous. The preceding lemma implies w w2 on Ω × T for small T , for any δ > 0. Therefore w1 w2 on Ω × T . 2 The following comparison principle for the ε-evolutionary problem follows quite easily. Lemma 6.3. For any T > 0 including ∞, if H ε w1 0 H ε w2 in Ω × T and w1 w2 on ∂p (Ω × T ), then w1 w2 on Ω × T . Proof. Let T0 > 0 be any small value of T in the preceding lemma so that the conclusion of the preceding lemma holds. Then w1 w2 on Ω × (0, T0 ). In particular, w1 w2 on ∂p (Ω × (T0 , 2T0 )). The preceding lemma may be applied again to conclude that w1 w2 on Ω × (T0 , 2T0 ). And so on. In the end, we see that w1 w2 on Ω × T . 2 The uniqueness of a viscosity solution of the ε-evolutionary problem is the straightforward corollary of the preceding comparison result.

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Theorem 6.1. The ε-evolutionary problem ⎧ ε ⎨ H w = wt − Lw + βε (w) = 0 in Ω × T , w(x, t) = σ (x) on ∂Ω × T , ⎩ ¯ w(x, 0) = w0 (x) on Ω, possesses at most one viscosity solution. A feasible explanation of the non-uniqueness of a viscosity solution of the elliptic free boundary problem versus the uniqueness of a viscosity solution of the ε-evolutionary problem is that different physical evolutionary processes with the same boundary condition may end up with different steady states. For example, if the melting of ice and solidification of water observe the physical laws described by the mathematical model so far discussed, we may have the following phenomenon. We manage to keep the temperature distribution on the surface of a closed container fixed as time goes by (however, the distribution in general is non-constant, somewhere above the freezing point and somewhere below). If ice or water is put in the container, after a long time, the temperature distribution inside the container reaches a steady state. Even though the boundary temperature distribution is the same for either case, the steady states resulted may differ from each other depending on the initial state. It needs rigorous mathematical justification and is the subject of the authors’ following study. For now, we content ourselves with some questions about the elliptic free boundary problem for which the uniqueness of a solution fails. Let S(σ ) be the set of solutions of the elliptic free boundary problem with continuous initial and boundary data ⎧ ⎨ u = 0 − + ⎩ uν = g uν u=σ

in Ω + (u) ∪ Ω − (u), along F (u), on ∂Ω.

We ask the following questions about the set of solutions S(σ ). How to determine, from the initial value, to which viscosity solution in S(σ ) do viscosity solutions of the evolutionary free boundary problem converge as time goes to infinity? Is S(σ ) a finite set? Are there a largest element and a least element of S(σ ) in general? Does S(σ ) contain only two solutions in general, which model the stationary states resulting from the melting of ice and the solidification of water respectively? And under what condition do they coincide with each other? Acknowledgments The authors would like to thank Professor Luis Caffarelli and Professor Michael Crandall for their helpful suggestions. References [1] H.W. Alt, L.A. Caffarelli, A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. 282 (2) (1984) 431–461.


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[2] I. Anthanasopoulos, L. Caffarelli, S. Salsa, Regularity of the free boundary in parabolic phase transition problems, Acta Math. 176 (2) (1996) 245–282. [3] I. Anthanasopoulos, L. Caffarelli, S. Salsa, Caloric functions in Lipschitz domains and the regularity of solutions to phase transition problems, Ann. of Math. (2) 143 (3) (1996) 413–434. [4] I. Anthanasopoulos, L.A. Caffarelli, S. Salsa, Phase transition problems of parabolic type: Flat free boundaries are smooth, Comm. Pure Appl. Math. 51 (1) (1998) 77–112. [5] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part 1: Lipschitz free boundaries are C 1,α , Rev. Mat. Iberoamericana 3 (2) (1987) 139–162. [6] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part 3: Existence theory, compactness, and dependence on X, Ann. Sc. Norm. Super. Pisa Cl. Sci. 15 (4) (1988) 583–602. [7] L.A. Caffarelli, A Harnack inequality approach to the regularity of free boundaries. Part 2: Flat free boundaries are Lipschitz, Comm. Pure Appl. Math. 42 (1) (1989) 55–78. [8] L.A. Caffarelli, D. Jerison, C.E. Kenig, Some new monotonicity theorems with applications to free boundary problems, Ann. of Math. 155 (2002) 369–404. [9] M.G. Crandall, H. Ishii, P.L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992) 1–67. [10] F. Ferrari, Two-phase problems for a class of fully nonlinear elliptic operators. Lipschitz free boundaries are C 1,γ , Amer. J. Math. 128 (2006) 541–571. [11] P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Part 1: Lipschitz free boundaries are C 1,α , Comm. Pure Appl. Math. 53 (7) (2000) 799–810. [12] P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order. Part 2: Flat free boundaries are Lipschitz, Comm. Partial Differential Equations 27 (7–8) (2002) 1497–1514. [13] P. Wang, Existence of solutions of two-phase free boundaries problems for fully nonlinear elliptic equations of second order, J. Geom. Anal. 13 (4) (2003) 715–738.


Calderón–Zygmund operators on product Hardy spaces ✩ Yongsheng Han a , Ming-Yi Lee b , Chin-Cheng Lin b,∗ , Ying-Chieh Lin b a Department of Mathematics, Auburn University, Auburn, AL 36849-5310, USA b Department of Mathematics, National Central University, Chung-Li, Taiwan 320, ROC

Received 31 July 2009; accepted 26 October 2009 Available online 7 November 2009 Communicated by C. Kenig

Abstract Let T be a product Calderón–Zygmund singular integral introduced by Journé. Using an elegant rectangle atomic decomposition of H p (Rn × Rm ) and Journé’s geometric covering lemma, R. Fefferman proved the remarkable H p (Rn × Rm ) − Lp (Rn × Rm ) boundedness of T . In this paper we apply vector-valued singular integral, Calderón’s identity, Littlewood–Paley theory and the almost orthogonality together with Fefferman’s rectangle atomic decomposition and Journé’s covering lemma to show that T is bounded on n , m } < p 1 if and only if T ∗ (1) = T ∗ (1) = 0, where ε is the product H p (Rn × Rm ) for max{ n+ε m+ε 1 2 regularity exponent of the kernel of T . © 2009 Elsevier Inc. All rights reserved. Keywords: Calderón–Zygmund operators; Journé’s class; Littlewood–Paley function; Product Hardy spaces

1. Introduction The product Hardy space was first introduced by M.P. Malliavin and P. Malliavin [11] and Gundy and Stein [8]. Chang and R. Fefferman [3] provided the atomic decomposition of ✩

Research by the first author supported in part by NCU Center for Mathematics and Theoretical Physics. Research by the second and third authors supported by both National Science Council and National Center for Theoretical Sciences, Republic of China. * Corresponding author. E-mail addresses: [email protected] (Y. Han), [email protected] (M.-Y. Lee), [email protected] (C.-C. Lin), [email protected] (Y.-C. Lin). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.10.022

Y. Han et al. / Journal of Functional Analysis 258 (2010) 2834–2861

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H p (R2+ × R2+ ). However, atomic decomposition of the product Hardy space H p (Rn × Rm ) is more complicated than the classical H p (Rn ). Indeed it was conjectured that the product atomic Hardy space H p (Rn × Rm ) could be characterized by rectangle atoms (see definition below). This conjecture, however, was disproved by Carleson [2] based on a counterexample. This leads that the role of cubes in the classical atomic Hardy space H p (Rn ) was replaced by arbitrary open sets on Rn × Rm with finite measures. It was quite surprising that using the rectangle atomic decomposition of H p (Rn × Rm ) and a geometric covering lemma due to Journé [10], R. Fefferman [5] proved the remarkable H p (Rn × Rm ) − Lp (Rn × Rm ) boundedness of product singular integrals introduced by Journé. Nevertheless, the H p (Rn × Rm ) boundedness of Journé’s product singular integrals is still open. The purpose of the current article is to study this issue. Let us recall the classical H p (Rn ) boundedness of singular integrals. We first begin with recalling the definition of a Calderón–Zygmund kernel. Definition 1. A continuous complex-valued function K(x, y) defined on Rn × Rn \ {x = y} is called a Calderón–Zygmund kernel if there exist constants C > 0 and a regularity exponent ε ∈ (0, 1] such that (i) |K(x, y)| C|x − y|−n , (ii) |K(x, y) − K(x , y)| C|x − x |ε |x − y|−n−ε (iii) |K(x, y) − K(x, y )| C|y − y |ε |x − y|−n−ε

if |x − x | |x − y|/2, if |y − y | |x − y|/2.

The smallest such constant C is denoted by |K|CZ . We say that an operator T is a Calderón–Zygmund operator if the operator T is a continuous linear operator from C0∞ (Rn ) into its dual associated with a Calderón–Zygmund kernel K(x, y) given by Tf, g =

g(x)K(x, y)f (y) dy dx

for all test functions f and g with disjoint supports and T is bounded on L2 (Rn ). If T is a Calderón–Zygmund operator associated with a kernel K, its Calderón–Zygmund operator norm is defined by T CZ = T L2 →L2 + |K|CZ . Given 0 < p 1, let n ∞ C0,0 R = ψ ∈ C ∞ Rn : ψ has a compact support and

ψ(y)y α dy = 0 for 0 |α| Np,n ,

Rn ∞ (Rn ) satisfy the condition where Np,n is a large integer depending on p and n. Let ψ ∈ C0,0

∞ dt ψ(tξ ˆ )2 = 1 for all ξ = 0. t 0

(1.1)

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For t > 0 and x ∈ Rn , set ψt (x) = t −n ψ(x/t). The Littlewood–Paley square function of f ∈ S (Rn ) is defined by ∞ 1/2 2 dt ψt ∗ f (x) . g(f )(x) = t 0

The classical Hardy space H p (Rn ) can be defined by

H p Rn = f ∈ S Rn : g(f ) ∈ Lp Rn with f H p (Rn ) := g(f )Lp (Rn ) . The criterion for the H p (Rn ) boundedness of Calderón–Zygmund operators is given as follows. Theorem A. Let T be a Calderón–Zygmund operator associated to a kernel with regularity n exponent ε. Then T is bounded on H p (Rn ), n+ε < p 1, if and only if T ∗ (1) = 0. Here, T ∗ (1) = 0 means that

Rn Rn

∞ (Rn ). K(x, y)ψ(y) dy dx = 0 for all ψ ∈ C0,0

Remark 1. Theorem A still holds for any given p, 0 < p 1, if one requires more regularity conditions on the kernel of T and high order cancellation conditions on T (see [7] for more details). One proof of Theorem A was shown in terms of atomic decomposition together with the maximal function characterization of H p (Rn ) (see [12, p. 115, Theorem 4]). Another proof was given by molecule decomposition of H p (Rn ) (see [7, p. 335, Theorem 7.18]). These methods, however, cannot be carried out to the product Hardy space H p (Rn × Rm ). To see this, let us recall the definition and atomic decomposition of H p (Rn × Rm ). Let n1 = n, n2 = ∞ (Rni ) supported in the unit ball of Rni , and ψ i satisfy condition (1.1), i = 1, 2. For m, ψ i ∈ C0,0 ti > 0 and (x1 , x2 ) ∈ Rn × Rm , set ψtii (xi ) = ti−ni ψ(xi /ti ) and ψt1 t2 (x1 , x2 ) = ψt11 (x1 )ψt22 (x2 ). The product Littlewood–Paley square function of f ∈ S (Rn × Rm ) is defined by ∞∞ ψt g(f )(x1 , x2 ) = 0 0

2 dt1 dt2 1 t2 ∗ f (x1 , x2 ) t1 t2

1/2 .

For 0 < p 1, the product Hardy space H p (Rn × Rm ) can be defined by

H p Rn × Rm = f ∈ S Rn × Rm : g(f ) ∈ Lp Rn × Rm with f H p (Rn ×Rm ) := g(f )Lp (Rn ×Rm ) . A function a(x1 , x2 ) defined in Rn × Rm is called an H p (Rn × Rm ) atom if a(x1 , x2 ) is supported in an open set Ω ⊂ Rn × Rm with finite measure and satisfies the following conditions: (i) a2 |Ω|1/2−1/p ,


2837

(ii) a can further be decomposed as a(x1 , x2 ) = R∈M(Ω) aR (x1 , x2 ), where aR are supported on the double of R = I × J (I a dyadic cube in Rn , J a dyadic cube in Rm ) and M(Ω) is the collection of all maximal dyadic rectangles contained in Ω,

1/2 aR 22

|Ω|1/2−1/p ,

R∈M(Ω)

aR (x1 , x2 )x1α dx1 = 0

(iii)

for all x2 ∈ Rm , 0 |α| Np,n ,

2I

β

aR (x1 , x2 )x2 dx2 = 0 for all x1 ∈ Rn , 0 |β| Np,m , 2J ∞. where Np,n and Np,m are given in the definition of C0,0

Chang and R. Fefferman [3] provided the following atomic decomposition of H p (Rn × Rm ).

Theorem B. A distribution f ∈ H p (Rn × Rm ) if and only if f = j λj aj , where aj are

H p (Rn × Rm ) atoms, j |λj |p < ∞, and the series converges in the distribution sense. More

p over, f H p is equivalent to inf{ j |λj |p : for all f = j λj aj }. The fact that the support of H p (Rn × Rm ) atom is an open set prevents from applications of atomic decomposition of H p (Rn × Rm ). However, it was quite surprising that R. Fefferman [5] proved the following remarkable result. Theorem C. Let 0 0 and δ > 0 such that, for any H p (Rn × Rm ) rectangle atom a supported on R,

T a(x1 , x2 )p dx1 dx2 Cγ −δ

for all γ 2,

(1.2)

Rn ×Rm \γ R

where γ R denotes the concentric γ -fold dilation of R. Then T is a bounded operator from H p (Rn × Rm ) to Lp (Rn × Rm ). Here a function a(x1 , x2 ) supported on a rectangle R = I × J (I a cube in Rn , J a cube in Rm ) is called an H p (Rn × Rm ) rectangle atom provided (i) a2 |R|1/2−1/p , (ii) a(x1 , x2 )x1α dx1 = 0, I

β

a(x1 , x2 )x2 dx2 = 0,

(iii) J

0 |α| Nn,p

for all x2 ∈ J,

0 |β| Nm,p

for all x1 ∈ I.

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Definition 2. A singular integral operator T is said to be in Journé’s class if Tf (x1 , x2 ) =

K(x1 , x2 , y1 , y2 )f (y1 , y2 ) dy1 dy2 ,

Rn ×Rm

where the kernel K(x1 , x2 , y1 , y2 ) satisfies the following conditions. For each x1 , y1 ∈ Rn , set 1 (x1 , y1 ) to be the singular integral operator acting on functions on Rm with the kernel K 2 (x2 , y2 )(x1 , y1 ) = K(x1 , x2 , y1 , y2 ). K 1 (x1 , y1 )(x2 , y2 ) = K(x1 , x2 , y1 , y2 ), and similarly, K There exist constants C > 0 and ε ∈ (0, 1] such that (A1 ) T is bounded on L2 (Rn+m ), 1 (x1 , y1 )CZ C|x1 − y1 |−n , (A2 ) K 1 (x1 , y )CZ C|y1 − y |ε |x1 − y1 |−(n+ε) for |y1 − y | |x1 − y1 |/2, 1 (x1 , y1 ) − K K 1 1 1 1 (x , y1 )CZ C|x1 − x |ε |x1 − y1 |−(n+ε) for |x1 − x | |x1 − y1 |/2, 1 (x1 , y1 ) − K K 1 1 1 2 (x2 , y2 )CZ C|x2 − y2 |−m , (A3 ) K 2 (x2 , y )CZ C|y2 − y |ε |x2 − y2 |−(m+ε) for |y2 − y | |x2 − y2 |/2, 2 (x2 , y2 ) − K K 2

2

2

2 (x , y2 )CZ C|x2 − x |ε |x2 − y2 |−(m+ε) for |x2 − x | |x2 − y2 |/2. 2 (x2 , y2 ) − K K 2 2 2 R. Fefferman [4] further proved that product singular integrals in Journé class satisfy the estimate (1.2), and hence such product singular integrals are bounded from H p (Rn × Rm ) to Lp (Rn+m ). Suppose that T is a singular integral in Journé’s class. Then by a result in [4] T is ∞ (Rn ) and ϕ 2 ∈ C ∞ (Rm ) then bounded from H 1 (Rn × Rm ) to L1 (Rn+m ). Note that if ϕ 1 ∈ C0,0 0,0 1 2 1 n m 1 2 ϕ (y1 )ϕ (y2 ) ∈ H (R × R ). Therefore, T (ϕ ϕ )(x1 , x2 ) ∈ L1 (Rn × Rm ). This implies that T (ϕ 1 ϕ 2 )(x1 , x2 ), as a function of x1 is a integrable function on Rn . Similarly, T (ϕ 1 ϕ 2 )(x1 , x2 ), as a function of x2 is a integrable function on Rm . Now we say that T1∗ (1) = 0 if

K(x1 , x2 , y1 , y2 )ϕ 1 (y1 )ϕ 2 (y2 ) dy1 dy2 dx1 = 0

Rn Rn ×Rm ∞ (Rn ), ϕ 2 ∈ C ∞ (Rm ), and x ∈ Rm . Similarly, T ∗ (1) = 0 if for all ϕ 1 ∈ C0,0 2 0,0 2

K(x1 , x2 , y1 , y2 )ϕ 1 (y1 )ϕ 2 (y2 ) dy1 dy2 dx2 = 0

Rm Rn ×Rm ∞ (Rn ), ϕ 2 ∈ C ∞ (Rm ), and x ∈ Rn . for all ϕ 1 ∈ C0,0 1 0,0 The main result of this paper is the following

Theorem 1. Let T be a singular integral operator in Journé’s class with regularity exponent n m ε. Then T is bounded on H p (Rn × Rm ) for max{ n+ε , m+ε } < p 1 if and only if T1 ∗ (1) = T2 ∗ (1) = 0.


2839

Remark 2. As in the classical H p (Rn ), Theorem 1 still holds for any given p, 0 < p 1, if the kernel of T satisfies more regularity conditions and T satisfies high order cancellation conditions. We leave these details to the reader. The approach used in this paper is even new for the classical H p (Rn ). Therefore, we would like first to describe that how one can use this approach to prove the classical H p (Rn ) boundedness. This approach includes the following steps. p Step 1. Reduce the H p (Rn ) boundedness to H p (Rn ) − LH1 (Rn ) boundedness: we first introduce the Hilbert space H1 ([0, ∞), dtt ) by H1

dt [0, ∞), t

∞

= {ht }t>0 : {ht }H

dt 1 ([0,∞), t )

=

2 dt

|ht |

t

1/2

0 by Tt (f )(x) = ψt ∗ Tf (x),

t > 0. p

Therefore, the H p boundedness of T is equivalent to the H p − LH1 boundedness of L. Step 2. The almost orthogonal estimates and decomposition of Tt : this step is crucial. We first start with a function f ∈ H p (Rn ) ∩ L2 (Rn ). By the classical Calderón identity, ∞ f (x) =

ψt ∗ ψt ∗ f (x)

f ∈ L2 R n ,

dt , t

0 ∞ satisfies condition (1.1). Since T is bounded on L2 (Rn ), we rewrite where ψ ∈ C0,0

∞ Tt (f )(x) = ψt ∗ T

∞ ds ds ψs ∗ ψs ∗ f (·) (x) = ψt ∗ T ψs ∗ ψs ∗ f (·) (x) . s s

0

0

Denote Tt (x, y) to be the kernel of Tt . Then ∞ Tt (x, y) =

ψt (x − u)K(u, v)ψs (v − w)ψs (w − y) du dv dw 0

ds . s

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The almost orthogonal estimate says that there exists a constant C such that ε (s ∨ t)ε ψt (x − z)ψs (z − y) dz C t ∧ s , s t ((s ∨ t) + |x − y|)n+ε where s ∨ t = max{s, t} and s ∧ t = min{s, t}. Suppose that K(x, y) is a Calderón–Zygmund kernel with regularity exponent ε. Then the following almost orthogonal estimates still hold: for 0 < ε < ε, ψt (x − u) − ψt (x − y) K(u, v)ψs (v − y) du dv C|K|CZ

ε s tε t (t + |x − y|)n+ε

(1.3)

for s t, and for t s, (x − u)K(u, v) ψ (v − y) − ψ (x − y) du dv ψ t s s ε t sε C|K|CZ . s (s + |x − y|)n+ε

(1.4)

Suppose T ∗ (1) = 0. These considerations lead to the following decomposition t Tt (x, y) =

ψt (x − u)K(u, v)ψs (v − w)ψs (w − y) du dv dw

ds s

0

∞ +

ds ψt (x − u)K(u, v) ψs (v − w) − ψs (x − w) ψs (w − y) du dv dw s

t

+ ψt ∗ T (1)(x)φt (x − y) := Tt1 (x, y) + Tt2 (x, y) + Tt3 (x, y), ∞ where φt (x − y) = t ψs ∗ ψs (x − y) ds s . The almost orthogonal estimate (1.3) can be used to estimate the kernel of Tt1 because T ∗ (1) = 0. The estimate of the kernel of Tt2 then follows immediately from (1.4). We remark that Tt3 is an H1 -valued para-product operator and the estimate of Tt3 (x, y) then easily follows n 1 from the facts that ψt ∈ H R whose norm is bounded uniformly for t > 0 and T (1) ∈ BMO. All these estimates together with the fact that the L2 boundedness of T implies the L2 − L2H1 boundedness of L yield that the kernel of L satisfies the size condition (i) and the smoothness condition (iii) for variable y in Definition 1 with the norm replaced by H1 -valued norm. Finally, p the H p − LH1 boundedness of L then follows from the following: p Step 3. The H p − LH1 boundedness via atoms: suppose that L is an L2 − L2H1 bounded operator. A general result of the boundedness says that L extends to a bounded operator from


2841

p

H p into LH1 if and only if LaLp is uniformly bounded for any H p (Rn ) atom a. This is nonH1 trivial, in the light of the Meyer–Bownik example [1]. Let L be a L2 − L2H1 bounded operator and its kernel satisfies the condition (iii) of Definition 1 with the norm replaced by H1 -valued norm. We then have the uniform boundedness of LaLp for any H p (Rn ) atom a and hence L H1

p

is bounded from H p to LH1 by the above general result of the boundedness. In the next section, we will carry out these steps to the product Hardy space H p (Rn × Rm ). 2. The proof of Theorem 1 The necessary conditions of Theorem 1 follow from the classical results. To see this, let f (x1 , x2 ) ∈ H p (Rn × Rm ) ∩ L2 (Rn × Rm ) and f ∗ (x1 , x2 ) be the maximal function of f defined in [8]. By the maximal function characterization of H p (Rn × Rm ), f ∗ (x1 , x2 ) ∈ Lp (Rn × Rm ) (see [8, Theorem 1]). Denote f1∗ (x1 , x2 ) by the maximal function of f (x1 , x2 ), as the function of variable x1 when x2 is fixed. Then f1∗ (x1 , x2 ) Cf ∗ (x1 , x2 ) for fixed x2 . This implies that f1∗ (x1 , x2 ) ∈ Lp (Rn ) and hence f (x1 , x2 ) ∈ H p (Rn ) for fixed x2 . By a classical result ∞ (Rn ) and ϕ 2 ∈ C ∞ (Rm ), let on H p (Rn ), f (x1 , x2 ) dx1 = 0 for fixed x2 . Now, for ϕ 1 ∈ C0,0 0,0 1 2 g(x1 , x2 ) = ϕ (x1 )ϕ (x2 ). It is easy to see that gH p (Rn ×Rm ) Cϕ 1 H p (Rn ) ϕ 2 H p (Rm ) and hence g ∈ H p (Rn × Rm ). Thus, by the L2 (Rn × Rm ) boundedness and H p (Rn × Rm ) boundedness of T , the above explanation yields T g(x1 , x2 ), as a function of x1 , is in H p (Rn ). This 1 2 implies that K(x1 , x2 , y1 , y2 )ϕ (y1 )ϕ (y2 ) dy1 dy2 dx1 = T g(x1 , x2 ) dx1 = 0 for fixed x2 , which yields T1∗ (1) = 0. Similarly, T2∗ (1) = 0. We now prove the sufficiency; that is, if T1∗ (1) = T2∗ (1) = 0, then T is bounded on p H (Rn × Rm ). As in the step 1 of Section 1, we define the Hilbert space H by

H = {ht,s }t,s>0 : {ht,s }H =

∞∞

ds |ht,s | t s 2 dt

1/2

0 .

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Note that the L2 (Rn+m ) boundedness of T and the product Littlewood–Paley estimate [6] imply that L is bounded from L2 (Rn+m ) to L2H (Rn+m ). Let ε be the regularity exponent satisfying (A2 ) and (A3 ). We will prove that {Tt,s (x1 , x2 , y1 , y2 )}t,s>0 satisfies the following estimates: (B1 ) {Tt,s (x1 , x2 , y1 , y2 )}H C|x1 − y1 |−n |x2 − y2 |−m , (B2 ) for ε < ε, (i) {Tt,s (x1 , x2 , y1 , y2 ) − Tts (x1 , x2 , y1 , y2 )}H C if |y1 − y1 | |x1 − y1 |/2,

|y1 −y1 |ε |x2 |x1 −y1 |n+ε

(ii) {Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , y1 , y2 )}H C

− y2 |−m

|y2 −y2 |ε |x1 |x2 −y2 |m+ε

− y1 |−n

if |y2 − y2 | |x2 − y2 |/2, (B3 ) for ε < ε, {[Tt,s (x1 , x2 , y1 , y2 )−Tt,s (x1 , x2 , y1 , y2 )]−[Tt,s (x1 , x2 , y1 , y2 )−Tt,s (x1 , x2 , y1 , y2 )]}H C

|y1 −y1 |ε |y2 −y2 |ε n+ε |x1 −y1 | |x2 −y2 |m+ε

if |y1 − y1 | |x1 − y1 |/2, |y2 − y2 | |x2 − y2 |/2.

We would like to point out that the above estimates (B1 )–(B3 ) show that L is an H-valued singular integral operator. However, we will use the estimate (B3 ) only. See Lemma 3 below for more regularities of L from L2 to L2H . To this end, according to the almost orthogonal estimates as we mentioned in the step 2 for the classical case, we decompose Tt,s (x1 , x2 , y1 , y2 ) as follows. Here and throughout, we denote Rn ×Rm du1 du2 simply by du1 du2 , and similarly for dv1 dv2 and dz1 dz2 . Tt,s (x1 , x2 , y1 , y2 ) t s =

ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) 0 0

× ψt1 ∗ ψt1 (v1 − y1 )ψs2 ∗ ψs2 (v2 − y2 ) du1 du2 dv1 dv2

dt ds t s

t ∞ +

ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) 0

s

Rm

dt ds × ψt1 ∗ ψt1 (v1 − y1 ) ψs2 (v2 − z2 ) − ψs2 (x2 − z2 ) ψs2 (z2 − y2 ) du1 du2 dv1 dv2 dz2 t s t + ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) 0

× ψt1 ∗ ψt1 (v1 − y1 ) du1 du2 dv1 dv2 ∞s + t

0

Rn

dt 2 φ (x2 − y2 ) t s

ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) ψt1 (v1 − z1 ) − ψt1 (x1 − z1 )


× ψt1 (z1 − y1 )ψs2 ∗ ψs2 (v2 − y2 ) du1 du2 dv1 dv2 dz1

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dt ds t s

s +

ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) 0

× ψs2 ∗ ψs2 (v2 − y2 ) du1 du2 dv1 dv2 ∞∞ + t

ds 1 φ (x1 − y1 ) s t

ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) ψt1 (v1 − z1 ) − ψt1 (x1 − z1 )

s

dt ds × ψs2 (v2 − z2 ) − ψs2 (x2 − z2 ) ψt1 (z1 − y1 )ψs2 (z2 − y2 ) du1 du2 dv1 dv2 dz1 dz2 t s ∞ + ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) Rm

s

ds × ψs2 (v2 − z2 ) − ψs2 (x2 − z2 ) ψs2 (z2 − y2 ) du1 du2 dv1 dv2 dz2 φt1 (x1 − y1 ) s ∞ + ψt1 (x1 − u1 )ψs2 (x2 − u2 )K(u1 , u2 , v1 , v2 ) ψt1 (v1 − z1 ) − ψt1 (x1 − z1 ) Rn

t

dt 2 φ (x2 − y2 ) t s + ψt,s ∗ T (1)(x1 , x2 )φt,s (x1 − y1 , x2 − y2 ) × ψt1 (z1 − y1 ) du1 du2 dv1 dv2 dz1

:=

9

j

Tt,s (x1 , x2 , y1 , y2 ),

j =1

∞ ∞ 1 2 where φt1 = t ψt1 ∗ ψt1 (·) dtt , φs2 = s ψs2 ∗ ψs2 (·) ds s and φt,s = φt φs . By a result in [10], T (1) ∈ BMO(Rn ×Rm ) and hence ψt,s ∗T (1)(x1 , x2 ) makes sense because ψt,s ∈ H 1 (Rn ×Rm ). It is also easy to see that φt,s satisfy the same size and smoothness conditions as ψt,s . The estimates of (B1 )–(B3 ) for {Tt,s (x1 , x2 , y1 , y2 )}t,s>0 will follow easily by the following lemma. Lemma 2. For 1 j 9 and t, s > 0, there exists a constant C such that (D1 ) for ε < ε, |Tt,s (x1 , x2 , y1 , y2 )| C j

(D2 ) for

ε

< ε ,

(i) |Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , y1 , y2 )| C j

j

if |y1 − y1 | t/2,

(ii) |Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , y1 , y2 )| C j

if |y2 − y2 | s/2,

j

tε sε (t+|x1 −y1 |)n+ε (s+|x2 −y2 |)m+ε

|y1 −y1 | ε t

|y2 −y2 | ε s

,

tε sε (t+|x1 −y1 |)n+ε (s+|x2 −y2 |)m+ε tε sε (t+|x1 −y1 |)n+ε (s+|x2 −y2 |)m+ε

2844


(D3 ) for ε < ε , j j j j |[Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , y1 , y2 )] − [Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , y1 , y2 )]| |y −y | ε |y2 −y2 | ε tε sε C 1t 1 n+ε m+ε s (t+|x1 −y1 |)

if |y1 − y1 | t/2 and |y2 − y2 | s/2.

(s+|x2 −y2 |)

Before proving Lemma 2, we recall the orthogonal estimates on Rn (cf. [9, Lemma 4.3] for details). Let S be a Calderón–Zygmund operator with regularity exponent ε associated with a ∞ (Rn ), the following almost orthogonal kernel S(z, w) and satisfy S ∗ (1) = 0. Then, for ψ ∈ C0,0 estimates hold: for ε < ε < ε, ψt (x − z)S(z, w)ψs (w − u) dz dw ε s tε CSCZ t (t + |x − u|)n+ε and

for s t,

ψt (x − z)S(z, w) ψs (w − u) − ψs (x − u) dz dw ε t sε CSCZ for t < s. s (s + |x − u|)n+ε

(2.3)

(2.4)

The estimate (2.3) and the size condition on ψs imply ψt (x − z)S(z, w)ψs ∗ ψs (w − y) dz dw ε s tε CSCZ t (t + |x − y|)n+ε Similarly,

for s t.

ψt (x − z)S(z, w) ψs (w − u) − ψs (x − u) ψs (u − y) dz dw du ε t sε CSCZ for t < s. s (s + |x − y|)n+ε

(2.5)

(2.6)

For s t and ε < ε < ε , by the almost orthogonal estimate (2.3) and the smoothness condition on ψs , ψ u − y dz dw du (x − z)S(z, w)ψ (w − u) ψ (u − y) − ψ t s s s CSCZ

ε s tε du ψs (u − y) − ψs u − y n+ε t (t + |x − u|) Rn

ε −ε s |y − y | ε tε CSCZ t t (t + |x − y|)n+ε

for y − y t/2.

(2.7)


2845

Similarly, for t < s, by the estimate (2.4) and the smoothness condition on ψs , ψt (x − z)S(z, w) ψs (w − u) − ψs (x − u) ψs (u − y) − ψs u − y dz dw du CSCZ

ε t |y − y | ε sε s s (s + |x − y|)n+ε

for y − y t/2.

We now return to the proof of Lemma 2. Proof of Lemma 2. The main idea is that the iteration method can be applied to reduce the 1 (x , x , y , y ) satisfies product case to the classical case. To be precise, let us first prove that Tt,s 1 2 1 2 the estimates (D1 )–(D3 ). For fixed t, x1 and y1 , set t K2 (u2 , v2 ) =

2 (u2 , v2 )(u1 , v1 )ψ 1 ∗ ψ 1 (v1 − y1 ) du1 dv1 ψt1 (x1 − u1 )K t t

0 Rn Rn

dt . t

2 (u2 , v2 )(u1 , v1 ) = K(u1 , u2 , v1 , v2 ) is a Calderón– Note that when u2 and v2 are fixed, K n n 2 (u2 , v2 )CZ C|u2 − v2 |−m . By T ∗ (1) = 0 Zygmund kernel on R × R with the norm K 1 2 (u2 , v2 )(u1 , v1 ) with fixed (u2 , v2 ), and the almost orthogonal estimate (2.5) for the kernel K K2 (u2 , v2 ) C

tε −m . |u2 − v2 | (t + |x1 − y1 |)n+ε

(2.8)

Similarly, when u2 , u2 and v2 are fixed, we have K2 (u2 , v2 ) − K2 u2 , v2 =

t

2 (u2 , v2 )(u1 , v1 ) − K 2 u2 , v2 (u1 , v1 ) ψt1 (x1 − u1 ) K

0 Rn Rn

× ψt1 ∗ ψt1 (v1 − y1 ) du1 dv1

dt . t

2 (u , v2 )(u1 , v1 ) is a Calderón–Zygmund kernel on 2 (u2 , v2 )(u1 , v1 ) − K Note again that K 2 n n 2 (u2 , v2 ) − K 2 (u , v2 )CZ C|u2 − u |ε |u2 − v2 |−m−ε for R × R with the norm K 2 2 |u2 − u2 | 12 |u2 − v2 |. The same argument as (2.8) gives K2 (u2 , v2 ) − K2 u , v2 C 2

|u2 − v2 | for u2 − u2 . 2

|u2 − v2 | . for v2 − v2 2

|u2 − u2 |ε tε (t + |x1 − y1 |)n+ε |u2 − v2 |m+ε

A same process shows K2 (u2 , v2 ) − K2 u2 , v C 2

|v2 − v2 |ε tε (t + |x1 − y1 |)n+ε |u2 − v2 |m+ε

2846


These imply that K2 (u2 , v2 ) is a Calderón–Zygmund kernel on Rm × Rm and

|K2 |CZ C

tε . (t + |x1 − y1 |)n+ε

(2.9)

Note that if S is an operator associated with the kernel K2 (u2 , v2 ), then the condition T2∗ (1) = 0 implies S ∗ (1) = 0. Therefore, first writing s 1 Tt,s (x1 , x2 , y1 , y2 ) =

ψs2 (x2 − u2 )K2 (u2 , v2 )ψs2 ∗ ψs2 (v2 − y2 ) du2 dv2 0 Rm Rm

ds s

and then applying the orthogonal estimate (2.5) for K2 (u2 , v2 ) with the norm estimate in (2.9) imply 1 T (x1 , x2 , y1 , y2 ) C t,s

tε sε . (t + |x1 − y1 |)n+ε (s + |x2 − y2 |)n+ε

1 (x , x , y , y ) satisfies (D ). This shows that Tt,s 1 2 1 2 1 To check (D2 ) (i), we write

1 1 Tt,s x1 , x2 , y1 , y2 (x1 , x2 , y1 , y2 ) − Tt,s s =

ψs2 (x2 − u2 )K2,2 (u2 , v2 )ψs2 ∗ ψs2 (v2 − y2 ) du2 dv2 0 Rm Rm

ds , s

where for fixed t, x1 , y1 , y1 , t K2,2 (u2 , v2 ) =

2 (u2 , v2 )(u1 , v1 )ψ 1 (v1 − z1 ) ψt1 (x1 − u1 )K t

0 Rn Rn Rn

dt × ψt1 (z1 − y1 ) − ψt1 z1 − y1 du1 dv1 dz1 . t 2 (u2 , v2 )CZ C|u2 − v2 |−m , By the estimate of (2.7) and the fact that K ε tε 1 K2,2 (u2 , v2 ) C |y1 − y1 | n+ε t |u2 − v2 |m (t + |x1 − y1 |)

for y1 − y1 t/2.

2 (u2 , v2 ) − K 2 (u , v2 )CZ C|u2 − u |ε |u2 − v2 |−m−ε yield that, A similar argument and K 2 2 1 for |u2 − u2 | 2 |u2 − v2 | and |y1 − y1 | t/2,


2847

K2,2 (u2 , v2 ) − K2,2 u , v2 2

t 2 (u2 , v2 )(u1 , v1 ) − K 2 u2 , v2 (u1 , v1 ) = ψt1 (x1 − u1 ) K 0 Rn Rn Rn

1 dt 1 − z1 ) ψt (z1 − y1 ) − ψt z1 − y1 du1 dv1 dz1 t |y1 − y1 | ε |u2 − u2 |ε tε C . t (t + |x1 − y1 |)n+ε |u2 − v2 |m+ε × ψt1 (v1

Similarly, for |v2 − v2 | 12 |u2 − v2 | and |y1 − y1 | t/2, ε |v2 − v2 |ε tε K2,2 (u2 , v2 ) − K2,2 u2 , v C |y1 − y1 | . 2 t (t + |x1 − y1 |)n+ε |u2 − v2 |m+ε Hence, K2,2 (u2 , v2 ) is a Calderón–Zygmund kernel and |y1 − y1 | ε tε |K2,2 |CZ C t (t + |x1 − y1 |)n+ε

for y1 − y1 t/2.

(2.10)

Applying the estimate of (2.5) to K2,2 (u2 , v2 ) together with the estimate of (2.10) yields s ds 2 2 2 ψs (x2 − u2 )K2,2 (u2 , v2 )ψs ∗ ψs (v2 − y2 ) du2 dv2 s 0 Rm Rm

C

|y1 − y1 | t

ε

tε sε n+ε (t + |x1 − y1 |) (s + |x2 − y2 |)m+ε

for y1 − y1 t/2,

1 (x , x , and hence (D2 ) (i) follows. The proof of (D2 ) (ii) is the same. To prove (D3 ) for Tt,s 1 2 y1 , y2 ), we write 1 1 1 1 x1 , x2 , y1 , y2 − Tt,s x1 , x2 , y1 , y2 − Tt,s x1 , x2 , y1 , y2 Tt,s (x1 , x2 , y1 , y2 ) − Tt,s

t s =

2 (u2 , v2 )(u1 , v1 )ψ 1 (v1 − z1 ) ψt1 (x1 − u1 )ψs2 (x2 − u2 )K t

0 0

× ψt1 (z1 − y1 ) − ψt1 z1 − y1 ψs2 (v2 − z2 ) ψs2 (z2 − y2 ) − ψs2 z2 − y2 × du1 du2 dv1 dv2 dz1 dz2 s =

ds dt s t

ψs2 (x2 − u2 )K2,2 (u2 , v2 )ψs2 (v2 − z2 ) ψs2 (z2 − y2 ) − ψs2 z2 − y2

0 Rm Rm Rm

× du2 dv2 dz2

ds . s

2848


1 (x , x , y , y ) satisfies (D ). The proofs By the estimate (2.7) for the kernel K2,2 (u2 , v2 ), Tt,s 1 2 1 2 3 j

for Tt,s , j = 2, 4, 6, are similar provided replacing (2.3) and (2.5) by (2.4) and (2.6), so we leave details to the reader. j 3 (x , x , Since the proofs for Tt,s (x1 , x2 , y1 , y2 ), j = 3, 5, 7, 8, are similar, we estimate Tt,s 1 2 y1 , y2 ) only. For fixed x2 , set 1 (u1 , v1 )(u2 , v2 ) du2 dv2 . ψs2 (x2 − u2 )K K1 (u1 , v1 ) = Rm Rm

1 (u1 , v1 )(u2 , v2 ) dv2 , as a function of the variable u2 , is a Note that for fixed (u1 , v1 ), Rm K 2 BMO function and ψs (x2 − u2 ) is a function in H 1 (Rm ) with H 1 (Rm )-norm uniformly bounded for all x2 and s. Moreover, 1 K (u1 , v1 ) C|u1 − v1 |−n , 1 (u1 , v1 )(·, v2 ) dv2 C K CZ BMO(Rm )

Rm

which implies K1 (u1 , v1 ) C|u1 − v1 |−n . Similarly, for |u1 − u1 | 12 |u1 − v1 |, we have 1 1 K (u1 , v1 )(·, v2 ) − K u1 , v1 (·, v2 ) dv2

BMO(Rm )

Rm

1 (u1 , v1 ) − K 1 u1 , v1 C K CZ ε C u1 − u1 |u1 − v1 |−n−ε ,

and hence K1 (u1 , v1 ) − K1 u , v1 1 1 2 1 = ψs (x2 − u2 ) K (u1 , v1 )(u2 , v2 ) − K u1 , v1 (u2 , v2 ) du2 dv2 Rm Rm

ε C u1 − u1 |u1 − v1 |−n−ε . The estimate |K1 (u1 , v1 ) − K1 (u1 , v1 )| can be obtained by the same manner. Thus, K1 (u1 , v1 ) is a Calderón–Zygmund kernel and |K1 |CZ C. Note that t 3 Tt,s (x1 , x2 , y1 , y2 ) =

ψt1 (x1 − u1 )K1 (u1 , v1 )ψt1 ∗ ψt1 (v1 − y1 ) du1 dv1 0 Rn Rn

dt 2 φ (x2 − y2 ). t s

Applying the almost orthogonal estimate of (2.5) to K1 (u1 , v1 ) together with the size condition 3 (x , x , y , y ). The estimates of (D ) and (D ) can be proved by the on φs2 leads to (D1 ) for Tt,s 1 2 1 2 2 3 same way.


2849

Finally, note that K(u1 , u2 , v1 , v2 ) dv1 dv2 , as a function of variables u1 and u2 , belongs to BMO(Rn × Rm ), and ψt1 (x1 − u1 )ψs2 (x2 − u2 ), as function of (u1 , u2 ), is in H 1 (Rn × Rm ) with the bounded norm uniformly for all t, s and x1 , x2 . Thus, ψts ∗ T (1)(x1 , x2 ) is uniformly 9 (x , x , y , y ) are bounded for all t, s and x1 , x2 . Therefore, the estimates (D1 )–(D3 ) for Tt,s 1 2 1 2 the same as those for φt1 (x1 − y1 )φs2 (x2 − y2 ), which can be immediately obtained. The proof of Lemma 2 is completed. 2 Now we demonstrate the regularity of the operator Tt,s mapping from L2 into L2H . Lemma 3. Let Tt,s be defined in (2.2) and ε be the regularity exponent of T . For ε < ε, (i) if |y1 − xI | |x1 − xI |/2, then Tt,s (x1 , ·, y1 , y2 ) − Tt,s (x1 , ·, xI , y2 ) f (y2 ) dy2 Rm

L2H (Rm )

C

|y1 − xI |ε f 2 ; |x1 − xI |n+ε

C

|y2 − yJ |ε f 2 . |x2 − yJ |n+ε

(ii) if |y2 − yJ | |x2 − yJ |/2, then T (·, x , y , y ) − T (·, x , y , y ) f (y ) dy t,s 2 1 2 t,s 2 1 J 1 1 Rn

L2H (Rn )

Proof. The proofs of (i) and (ii) are the same, so we show the case (i) only. We will use 0 < ε < ε < ε < ε through the proof. Note that 2 T (x , ·, y , y ) − T (x , ·, x , y ) f (y ) dy t,s 1 1 2 t,s 1 I 2 2 2 2

LH (Rm )

Rm

2 Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , xI , y2 ) f (y2 ) dy2 = dx2 . Rm

H

Rm

We write Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , xI , y2 ) f (y2 ) dy2 Rm

∞∞ =

ψt,s (x1 − u1 , x2 − u2 )k(u1 , u2 , v1 , v2 )

Rm

0 0

× ψt1 ∗ ψt1 (v1 − y1 ) − ψt1 ∗ ψt1 (v1 − xI ) ψs2 ∗ ψs2 (v2 − y2 )f (y2 ) × du1 du2 dv1 dv2

dt ds dy2 t s

∞∞ =

ψt,s (x1 − u1 , x2 − u2 )K(u1 , u2 , v1 , v2 ) 0 0

2850


dt ds × ψt1 ∗ ψt1 (v1 − y1 ) − ψt1 ∗ ψt1 (v1 − xI ) ψs2 ∗ ψs2 ∗ f (v2 ) du1 du2 dv1 dv2 t s ∞ = ψt,s (x1 − u1 , x2 − u2 )K(u1 , u2 , v1 , v2 ) 0

dt × ψt1 ∗ ψt1 (v1 − y1 ) − ψt1 ∗ ψt1 (v1 − xI ) f (v2 ) du1 du2 dv1 dv2 t ∞ = ψs2 ∗ ψt1 (x1 − u1 )K(u1 , ·, v1 , v2 ) Rn

0

1 dt 1 1 1 × ψt ∗ ψt (v1 − y1 ) − ψt ∗ ψt (v1 − xI ) f (v2 ) du1 dv1 dv2 (x2 ), t where we first write ψs2 ∗ ψs2 (v2 − y2 )f (y2 ) dy2 = ψs2 ∗ ψs2 ∗ f (v2 ) and then use the Calderón ∞ 2 identity 0 ψs ∗ ψs2 ∗ f (v2 ) ds s = f (v2 ). The Littlewood–Paley estimate gives 2 T (x , x , y , y ) − T (x , x , x , y ) f (y ) dy t,s 1 2 1 2 t,s 1 2 I 2 2 2 dx2 Rm

H

Rm

∞ ∞ C 0 Rm

ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )

0 Rn ×Rm Rn

2 1 dt dt 1 1 1 × ψt ∗ ψt (v1 − y1 ) − ψt ∗ ψt (v1 − xI ) f (v2 ) du1 dv1 dv2 dx2 . (2.11) t t Dividing the integral with respect to t into three parts, we obtain ∞ ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )

Rm

0

Rn

2 1 dt 1 1 1 × ψt ∗ ψt (v1 − y1 ) − ψt ∗ ψt (v1 − xI ) f (v2 ) du1 dv1 dv2 dx2 t

t C ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )ψt1 (v1 − z1 ) Rm

0

Rn Rn

2 1 dt 1 × ψt (z1 − y1 ) − ψt (z1 − xI ) f (v2 ) dz1 du1 dv1 dv2 dx2 t


2851

∞ +C ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 ) Rm

t

Rn Rn

Rm

t

Rn Rn

× ψt1 (v1 − z1 ) − ψt1 (x1 − zI ) ψt1 (z1 − y1 ) − ψt1 (z1 − xI ) f (v2 ) 2 dt × dz1 du1 dv1 dv2 dx2 t ∞ +C ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 ) × ψt1 (x1

2 1 dt 1 − z1 ) ψt (z1 − y1 ) − ψt (z1 − xI ) f (v2 ) dz1 du1 dv1 dv2 dx2 t

:= E + F + G. We first consider the item G and write ∞ G=C ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 ) Rm

Rn

t

2 1 dt × ψt ∗ ψt1 (x1 − y1 ) − ψt1 ∗ ψt1 (x1 − xI ) f (v2 ) du1 dv1 dv2 dx2 t 2 = C φt1 (x1 − y1 ) − φt1 (x1 − xI ) 2 × ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )f (v2 ) du1 dv1 dv2 dx2 Rm

Rn

= C sup

g2 1

2 ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )f (v2 )g(x2 ) du1 dv1 dv2 dx2

Rm

Rn

2 × φ 1 (x1 − y1 ) − φ 1 (x1 − xI ) , t

t

∞ where φt1 (·) = t ψt1 ∗ ψt1 (·) dtt . For fixed u1 and v1 , set K(u1 , v1 ) = K(u1 , x2 , v1 , v2 )f (v2 )g(x2 ) dv2 dx2 . Rm Rm

Then the operator associated to the kernel K(u1 , v1 ) is a Calderón–Zygmund operator with operator norm Cf 2 g2 . Since Rn K(u1 , v1 ) dv1 is a BMO function for u1 , ψ 1 (x1 − u1 ) K(u1 , v1 ) dv1 du1 Cf 2 g2 uniformly for x1 . t Rn

Rn

2852


Hence, for |y1 − xI | t/2, 2 G C φt1 (x1 − y1 ) − φt1 (x1 − xI ) f 22 t 2ε |y1 − xI | 2ε C f 22 . t (t + |x1 − xI |)2(n+ε)

(2.12)

To estimate E, we consider two cases {|x1 − z1 | > 8t} and {|x1 − z1 | 8t}. For the case {|x1 − z1 | > 8t}, we will use the kernel estimates of K. To be precise, using the cancellation properties of ψt1 and duality, we get t ψt1 (x1 − u1 ) K(u1 , x2 , v1 , v2 ) − K(u1 , x2 , z1 , v2 ) E=C Rm

0 Rn

× ψt1 (v1

Rn

2 1 dt 1 − z1 ) ψt (z1 − y1 ) − ψt (z1 − xI ) f (v2 ) du1 dv1 dv2 dz1 dx2 t

= C sup

h2 1

t h(x2 )

Rm

0 Rn

ψt1 (x1 − u1 ) K(u1 , x2 , v1 , v2 ) − K(u1 , x2 , z1 , v2 )

Rn

dt × ψt1 (v1 − z1 ) ψt1 (z1 − y1 ) − ψt1 (z1 − xI ) f (v2 ) du1 dv1 dv2 dz1 dx2 t

2 .

Note that the facts |x1 − z1 | > 8t, |x1 − u1 | < t and |v1 − z1 | < t t easily imply |v1 − z1 | |u1 − v1 |/2 and |u1 − v1 | > |x1 − z1 |/2. We apply (A2 ) to obtain, for |y1 − xI | t/2, t E

1/2

C sup h2 f 2 h2 1

0 Rn Rn Rn

|v1 − z1 |ε 1 ψt (x1 − u1 ) n+ε |u1 − v1 |

dt × ψt1 (v1 − z1 )ψt1 (z1 − y1 ) − ψt1 (z1 − xI ) du1 dv1 dz1 t t ε −ε t |y1 − xI | ε tε Cf 2 (t + |x1 − z1 )|n+ε t t

0 Rn

(t )ε (t )ε × + (t + |z1 − y1 |)n+ε (t + |z1 − xI |)n+ε tε |y1 − xI | ε C f 2 . t (t + |x1 − xI |)n+ε

dz1

dt t

For the case {|x1 − z1 | 8t}, by the condition on the support of ψt1 , we write

(2.13)


t E=C Rm

0 Rn

2853

ψt1 (x1 − u1 )K(u1 , x2 , v1 , v2 )ψt1 (v1 − z1 ) |u1 −z1 |9t

2 1 dt × ψt (z1 − y1 ) − ψt1 (z1 − xI ) f (v2 ) du1 dv1 dv2 dz1 dx2 . t Note that E = 0 if |z1 − y1 | > t and |z1 − xI | > t. It implies |x1 − xI | 10t provided |x1 − z1 | 8t and |y1 − xI | t/2. This fact will be used later. Now let η0 ∈ C ∞ (Rn ) be 1 on the unit ball and 0 outside the ball B(0, 2). Set η1 = 1 − η0 . We use T1∗ 1 = 0 to obtain t E=C Rm

0 Rn

t C 0 Rn

|u1 −z1 |9t

η0 |u1 −z1 |9t

u1 − z1 1 ψt (x1 − u1 ) − ψt1 (x1 − z1 ) K(u1 , x2 , v1 , v2 ) 4t


× ψt1 (v1

t +C Rm

1 ψt (x1 − u1 ) − ψt1 (x1 − z1 ) K(u1 , x2 , v1 , v2 )


× ψt1 (v1

Rm

0 Rn

× ψt1 (v1

η1 |u1 −z1 |9t

u1 − z1 1 ψt (x1 − u1 ) − ψt1 (x1 − z1 ) K(u1 , x2 , v1 , v2 ) 4t


:= E1 + E2 . By duality and the L2 (Rn × Rm ) boundedness of T , for |y1 − xI | t/2, 1/2 E1

= C sup

h2 1

t

ft ,z1 (u1 )K(u1 , ·, v1 , v2 )gt ,z1 (v1 )f (v2 )

h, 0

Rn

|u1 −z1 |9t

dt × du1 dv1 dv2 dz1 t

t h2 ft ,z1 2 gt ,z1 2 f 2 dz1

C sup

h2 1

0 Rn

dt t

2854


t C

t n/2 |y1 − xI | ε −n/2 dt t t f 2 t t t n+1

0

C|y1 − xI |ε t −n−ε f 2 , where ft ,z1 (u1 ) = η0 ( u14t−z 1 )[ψt1 (x1 − u1 ) − ψt1 (x1 − z1 )] and gt ,z1 (v1 ) = ψt1 (v1 − z1 ) × [ψt1 (z1 − y1 ) − ψt1 (z1 − xI )]. To estimate E2 , we use the cancellation property of ψt1 and write t E2 = C Rm

0

η1

Rn

|u1 −z1 |9t

u1 − z1 1 ψt (x1 − u1 ) − ψt1 (x1 − z1 ) 4t

× K(u1 , x2 , v1 , v2 ) − K(u1 , x2 , z1 , v2 ) ψt1 (v1 − z1 )

2 1 dt × ψt (z1 − y1 ) − ψt1 (z1 − xI ) f (v2 ) du1 dv1 dv2 dz1 dx2 . t By duality again,

1/2

E2

= C sup

h2 1 Rm

t

h(x2 )

η1 0 Rn

|u1 −z1 |9t

u1 − z1 1 1 ψ (x − u ) − ψ (x − z ) 1 1 1 1 t t 4t

× K(u1 , x2 , v1 , v2 ) − K(u1 , x2 , z1 , v2 ) ψt1 (v1 − z1 ) dt × ψt1 (z1 − y1 ) − ψt1 (z1 − xI ) f (v2 ) du1 dv1 dv2 dz1 dx2 . t By the conditions on the supports of η1 and ψt1 , we have |u1 − z1 | 4t and |v1 − z1 | < t . This gives |v1 − z1 | |u1 − z1 |/2. Applying (A2 ) with the estimate 1 ψ (x1 − u1 ) − ψ 1 (x1 − z1 ) C |u1 − z1 | , t t t n+1 we obtain, for |y1 − xI | t/2, 1/2 E2

t

C 0 Rn 4t |u1 −z1 |9t Rn

|u1 − z1 | |v1 − z1 |ε f 2 t n+1 |u1 − z1 |n+ε

dt × ψt1 (v1 − z1 )ψt1 (z1 − y1 ) − ψt1 (z1 − xI ) dv1 du1 dz1 t ε t |u1 − z1 | |y1 − xI | ε (t )ε −n Ct f 2 t |u1 − z1 |n+ε t 0 Rn 4t |u1 −z1 |9t


×

(t )ε (t )ε + (t + |z1 − y1 |)n+ε (t + |z1 − xI |)n+ε

du1 dz1

2855

dt t

Ct −n−ε |y1 − xI |ε f 2 .

Thus, for |x1 − z1 | 8t and |y1 − xI | t/2, we have E Ct −n−ε |y1 − xI |ε f 2 , which together with the fact |x1 − xI | 10t as mentioned before implies

|y1 − xI | EC t

2ε

t 2ε 2 f 2 . (t + |x1 − xI |)2(n+ε )

(2.14)

The estimate of F is the same as the estimate of E. It follows from (2.12)–(2.14) that, for |y1 − xI | t/2, ∞

Rm

ψt1 (x1 − u1 )k(u1 , x2 , v1 , v2 )

0 Rn Rn ×Rm

2 1 dt × ψt ∗ ψt1 (v1 − y1 ) − ψt1 ∗ ψt1 (v1 − xI ) f (v2 ) du1 dv1 dv2 dx2 t

|y1 − xI | C t

2ε

t 2ε 2 f 2 . (t + |x1 − xI |)2(n+ε )

(2.15)

Inserting (2.15) into (2.11), we obtain the desired result (i) of Lemma 3. Hence, the proof of Lemma 3 is completed. 2 To finish the proof of Theorem 1, as mentioned in step 3 of Section 1, we show the following general result. Proposition 4. Let L be a bounded operator from L2 (Rn+m ) to L2H (Rn+m ). Then, for 0 < p p 1, L extends to be a bounded operator from H p (Rn × Rm ) to LH (Rn+m ) if and only if L(a)Lp (Rn+m ) C for all H p (Rn × Rm ) atoms a, where the constant C is independent of a. H

Proof. We only need to show the sufficiency. This follows from a special atomic decomposition. p n m To be precise,

for f ∈ H (R × R ), Chang and R. Fefferman [3] gave an atomic decomposition f = j λj aj , but the series converges only in the sense of distributions. In general, to

p estimate the LH (Rn+m ) norm of L(f ), one cannot get L(f ) = j λj L(aj ). However, we prove n m 2 n+m ), this to be true if f ∈ H p (Rn × Rm ) ∩ L2 (Rn+m ). Indeed, for f ∈ H p (R

× R ) ∩ L (R we will provide an atomic decomposition of f such that f (x1 , x2 ) = j λj aj (x1 , x2 ), where

p aj are H p (Rn × Rm ) atoms and j |λj |p Cf H p (Rn ×Rm ) . The crucial point is that the series converges in both H p (Rn × Rm ) and L2 (Rn+m ). Assuming this atomic decomposition for the moment, since L is bounded from L2 (Rn+m ) to L2H (Rn+m ) and the series in atomic de composition of f converges in L2 (Rn+m ), thus L(f )(x1 , x2 ) = j λj L(aj )(x1 , x2 ). Moreover, this series also converges in L2 (Rn+m ) and hence a subsequence (written in the same indices) converges almost everywhere. Therefore,

2856


L(f )p p

LH

(Rn+m )

j

C

p |λj |p L(aj ) t,s>0 Lp

H (R

n+m )

p

|λj |p Cf H p (Rn ×Rm ) .

j

Since H p (Rn × Rm ) ∩ L2 (Rn+m ) is dense in H p (Rn × Rm ), so L can be extended to a bounded p operator from H p (Rn × Rm ) to LH (Rn+m ). To prove the atomic decomposition above, we recall the proof of atomic decomposition of H p (Rn × Rm ) given by Chang and R. Fefferman [3]. Given f ∈ H p (Rn × Rm ) ∩ L2 (Rn+m ), set Ωk = {(x1 , x2 ) ∈ Rn × Rm : S(f )(x1 , x2 ) > 2k } where S(f ) is the double S-function defined in [3, p. 456], and set Bk = {dyadic rectangle R = I × J : |R ∩ Ωk | > 12 |R| and |R ∩ Ωk+1 | 1 n m 2 |R|} where I and J are cubes in R and R , respectively. By the classical Calderón identity on L2 (Rn+m ), ∞∞ f (x1 , x2 ) =

ψt,s ∗ ψt,s ∗ f (x1 , x2 )

dt ds t s

0 0

=

ψt,s (x1 − y1 , x2 − y2 )ψt,s ∗ f (y1 , y2 ) dy1 dy2

k∈Z R∈Bk R

dt ds , t s

(2.16)

= {(x1 , t, x2 , s): R = I × J, x1 ∈ I, x2 ∈ J, (I ) t < (I ), (J ) s < (J )} is the where R 2 2 tent of R. Chang and R. Fefferman [3] proved that (2.16) provided an atomic decomposition of H p (Rn × Rm ) and the series converges in the sense of distribution. We would like to point out that the series (2.16) converges in L2 (Rn+m ) as well. To see this, let g ∈ L2 (Rn+m ) with g2 = 1. By the duality argument, dt ds ψt,s (· − y1 , · − y2 )ψt,s ∗ f (y1 , y2 ) dy1 dy2 t s 2 k∈Z R∈Bk R

dt ds , g = sup ψt,s (· − y1 , · − y2 )ψt,s ∗ f (y1 , y2 ) dy1 dy2 t s g2 1 k∈Z R∈Bk R

dt ds = sup , ψt,s ∗ g(y1 , y2 )ψt,s ∗ f (y1 , y2 ) dy1 dy2 t s g2 1 k∈Z R∈Bk R

t,s (x1 , x2 ) = ψt,s (−x1 , −x2 ). By Schwarz’s inequality and the L2 boundedness of the where ψ Littlewood–Paley square function, dt ds Cf 2 , ψ (· − y , · − y )ψ ∗ f (y , y ) dy dy t,s 1 2 t,s 1 2 1 2 t s k∈Z R∈Bk R

2

which implies that the series (2.16) converges in L2 (Rn+m ). Hence the proof is completed. We now return to the proof of Theorem 1.

2


2857

n m Proof of Theorem 1. Given max{ n+ε , m+ε } 0 is bounded from L2 (Rn+m ) to L2H (Rn+m ), to show Theorem 1, by Proposition 4, we only need to prove

Tt,s (a) p C for all H p Rn × Rm atoms a, t,s>0 L (Rn+m ) H

where the constant C is independent of a. To do this, we follow R. Fefferman’s idea [5]. Suppose that a is an H p (Rn × Rm ) atom supn m ported

on an open set Ω ⊂ R × R with finite measure. Furthermore, a can be decomposed as a = R∈M(Ω) aR , where M(Ω) is the collection of all maximal dyadic subrectangles contained in Ω, each aR is supported on 2R = 2I × 2J , the double of R = I × J , 2I aR (x1 , x2 ) dx1 = 0 for all x2 ∈ 2J , and 2J aR (x1 , x2 ) dx2 = 0 for all x1 ∈ 2I . Here the higher order moments vanishn m , m+ε } 4−n−m n−n/2 m−m/2 }, where Ms is the strong maximal function defined by 1 f (y1 , y2 ) dy1 dy2 , Ms (f )(x1 , x2 ) = sup (x1 ,x2 )∈P |P | P

where the supremum is taken over all rectangles P (a product of a cube in Rn with a cube in Rm ) C|Ω|. containing (x1 , x2 ). It follows from the strong maximal theorem that |Ω| We now estimate {Tt,s (a)}t,s>0 Lp (Rn+m ) as follows. Write Ω = {(x1 , x2 ) ∈ Rn × Rm : H Then Ms (χΩ )(x1 , x2 ) > 4−n−m n−n/2 m−m/2 } and similarly for Ω.

Tt,s (a) (x1 , x2 )p dx1 dx2 H p

Tt,s (a) (x1 , x2 )p dx1 dx2 . Tt,s (a) (x1 , x2 ) H dx1 dx2 + = H Ω

c (Ω)

By Hölder’s inequality, the L2 − L2H boundedness of L, and the size condition of a,

Tt,s (a) (x1 , x2 )p dx1 dx2 H

Ω

Tt,s (a) (x1 , x2 )2 dx1 dx2 H

p 2

p 1− 2 |Ω|

Ω p

p

Ca2 |Ω|1− 2 C. Therefore it remains to deal with

Tt,s (a) (x1 , x2 )p dx1 dx2 H c (Ω)

R∈M(Ω) c (Ω)

Tt,s (aR ) (x1 , x2 )p dx1 dx2 , H

where we use the inequality (α + β)p α p + β p for p 1.

2858


= I× J such that I is the largest For each R = I × J ∈ M(Ω), we set a larger rectangle R dyadic cube containing I and I × J ⊂ Ω. Similarly, R = I × J where J is the largest dyadic Let M (Ω) denote the collection of all dyadic subrectangles cube containing J and I× J⊂ Ω. 1 R ⊂ Ω, R = I × J that are maximal in the x1 direction. It is clear that R ∈ M(Ω) implies √ √ ∈ M1 (Ω). Define M2 (Ω) similarly. Also note that 4 nI× 4 mJ ⊂ Ω. R ∈ M2 (Ω) and R Then

Tt,s (aR ) (x1 , x2 )p dx1 dx2 H c (Ω)

√ (4 nI)c ×Rm

Tt,s (aR ) (x1 , x2 )p dx1 dx2 + H

√ Rn ×(4 mJ)c

Tt,s (aR ) (x1 , x2 )p dx1 dx2 H

:= U (R) + V (R). I) We define γ1 (R) = γ1 (R, Ω) = ( (I ) and γ2 (R) = γ2 (R, Ω) = length of I . To estimate U (R), we write

U (R) = √ √ (4 nI)c ×4 mJ

(J) (J ) ,

where (I ) denotes the side


+ √ √ (4 nI)c ×(4 mJ )c


:= U1 (R) + U2 (R). By Hölder’s inequality and Minkowski’s inequality, 1− p2

U1 (R) C|J |

√ (4 nI)c

Tt,s (aR ) (x1 , x2 )2 dx2 H

p 2

(2.17)

dx1 .

Rm

The cancellation condition of aR yields Tt,s (aR )(x1 , x2 ) =

Tt,s (x1 , x2 , y1 , y2 )aR (y1 , y2 ) dy1 dy2

=

Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , xI , y2 ) aR (y1 , y2 ) dy1 dy2 ,

where xI denotes the center of I . Now we apply Schwarz’s inequality to get

Tt,s (aR ) (x1 , x2 )2 H 2 C|I | (x , x , y , y ) − T (x , x , x , y ) a (y , y ) dy T t,s 1 2 1 2 t,s 1 2 I 2 R 1 2 2 dy1 . 2I

2J

H


2859

√ This estimate and Lemma 3 imply that, for x1 ∈ (4 nI)c and y1 ∈ 2I , Rm

Tt,s (aR ) (x1 , x2 )2 dx2 H 2 dx2 dy1 T C|I | (x , x , y , y ) − T (x , x , x , y ) a (y , y ) dy t,s 1 2 1 2 t,s 1 2 I 2 R 1 2 2 2I Rm

C|I |

H

2J

(I )ε |x1 − xI |n+ε

2 aR 22 .

Inserting the estimate above into (2.17) shows 1− p2

U1 (R) C|J |

|I |

p 2

p aR 2

√ (4 nI)c

(I )ε |x1 − xI |n+ε

p dx1

n−(n+ε )p p C|J |1− 2 |I | 2 aR 2 (I )ε p (I) n−(n+ε )p 1− p p = C γ1 (R) |R| 2 aR 2 . p

p

(2.18)

To estimate U2 (R), we use the cancellation conditions of aR to write Tt,s (aR )(x1 , x2 ) = Tt,s (x1 , x2 , y1 , y2 )aR (y1 , y2 ) dy1 dy2 =

Tt,s (x1 , x2 , y1 , y2 ) − Tt,s (x1 , x2 , xI , y2 ) − Tt,s (x1 , x2 , y1 , xJ ) + Tt,s (x1 , x2 , xI , xJ )

× aR (y1 , y2 ) dy1 dy2 , √ √ where xJ is the center of J . For x1 ∈ (4 n I)c , x2 ∈ (4 mJ )c , y1 ∈ 2I , and y2 ∈ 2J , we have |y1 − xI | 12 |x1 − xI | and |y2 − xJ | 12 |x2 − xJ |. Thus, the estimate (B3 ) gives

Tt,s (aR ) (x1 , x2 ) C H

1/2 2 |y2 − xJ |ε |y1 − xI |ε a (y , y ) dy dy . |R| R 1 2 1 2 |x1 − xI |n+ε |x2 − xJ |m+ε

Hence, U2 (R) C

p

|R| 2

√ √ (4 nI)c ×(4 mJ )c

(I )ε p (J )ε p p aR 2 dx1 dx2 |x1 − xI |(n+ε )p |x2 − xJ |(m+ε )p

n−(n+ε )p p C|R| 2 (I )ε p (I) (J )ε p (J )m−(m+ε )p aR 2 n−(n+ε )p 1− p p C γ1 (R) |R| 2 aR 2 . p

(2.19)

2860


Both estimates (2.18) and (2.19) give n−(n+ε )p 1− p p U (R) C γ1 (R) |R| 2 aR 2 . The estimate for V (R), though slightly different from U (R), can be handled in much the same manner so that p m−(m+ε )p |R|1− 2 aR p . V (R) C γ2 (R) 2

Summing over R gives

R∈M(Ω) c (Ω)


C

R∈M(Ω)

C

R∈M(Ω)

−δ |R| γ1 (R) 1

aR 22

1− p2

+

γ2 (R) −δ2 |R|

1− p 2

M1 (Ω) R∈

R∈M2 (Ω)

×

p m−(m+ε )p |R|1− 2 aR p γ2 (R) 2

n−(n+ε )p 1− p p γ1 (R) |R| 2 aR 2 + C

p 2

,

R∈M(Ω)

)p] )p] where δ1 = 2[n−(n+ε > 0 and δ2 = 2[m−(m+ε > 0. p−2 p−2 To estimate the last part above, we use the following

Journé’s lemma. R∈M2 (Ω) |R|(γ1 (R))−δ Cδ |Ω| and R∈M1 (Ω) |R|(γ2 (R))−δ Cδ |Ω| for any δ > 0, where Cδ is a constant depending on δ only. Journé’s lemma and the size condition of aR imply

p p Tt,s (aR ) (x1 , x2 )p dx1 dx2 C|Ω|1− 2 |Ω| 2 −1 C. H R∈M(Ω) c (Ω)

This is the desired result, and hence the proof of Theorem 1 is completed.

2

References [1] M. Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc. 133 (2005) 3535–3542. [2] L. Carleson, A counterexample for measures bounded on H p for the bi-disc, Mittag Leffler Report No. 7 (1974). [3] S.Y. Chang, R. Fefferman, A continuous version of duality of H 1 with BMO on bi-disc, Ann. of Math. 112 (1980) 179–201. [4] R. Fefferman, Calderón–Zygmund theory for product domain-H p theory, Proc. Nat. Acad. Sci. USA 83 (1986) 840–843. [5] R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. 126 (1987) 109–130.


2861

[6] R. Fefferman, E.M. Stein, Singular integrals on product spaces, Adv. Math. 45 (1982) 117–143. [7] J. Garcia-Cuerva, J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. [8] R.F. Gundy, E.M. Stein, H p theory for the poly-disc, Proc. Nat. Acad. Sci. USA 76 (1979) 1026–1029. [9] Y. Han, E.T. Sawyer, Para-accretive functions, the weak boundedness property and the T b theorem, Rev. Mat. Iberoamericana 6 (1990) 17–41. [10] J.L. Journé, Calderón–Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985) 55–91. [11] M.P. Malliavin, P. Malliavin, Intégrales de Lusin–Calderón pour les fonctions biharmoniques, Bull. Sci. Math. 101 (1977) 357–384. [12] E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, NJ, 1993.


Corrigendum

Corrigendum to “The Dunkl intertwining operator” [J. Funct. Anal. 256 (8) (2009) 2697–2709] M. Maslouhi a , E.H. Youssfi b,∗ a Classes Préparatoires aux Grandes Écoles d’Ingénieurs, Moulay Youssef, Rabat, Morocco b LATP, UMR CNRS 6632, CMI, Université de Provence, 39 Rue F-Joliot-Curie, 13453 Marseille Cedex 13, France

Received 22 November 2009; accepted 25 November 2009

Communicated by Paul Malliavin

Abstract In this note we correct the statement related to the regularity characterization of parameter functions and give some related new results. © 2008 Elsevier Inc. All rights reserved.

1. Correction to Theorem 2.3 in [1] Unlike the one-dimensional setting, in the case of higher dimensions d 2, there is a slip in the proof of Theorem 2.3 in [1]. The main goal of this is to correct this. Using the notations in [1], for n 1 consider the Euler operator Wn = Wn (k) defined in Pn by Wn (p) = (n + γ )p −

k(α)Lσα p,

p ∈ Pn ,

α∈R+

where R+ is a positive root system and for g ∈ G, Lg is defined in the space Pn of homogeneous polynomials of degree n by (Lg p)(x) = p(gx),

x ∈ Rd .

DOI of original article: 10.1016/j.jfa.2008.09.018. * Corresponding author.

E-mail address: [email protected] (E.H. Youssfi). 0022-1236/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.11.022

M. Maslouhi, E.H. Youssfi / Journal of Functional Analysis 258 (2010) 2862–2864

2863

We denote by M ∗ the set of all parameter functions k for which −n is not an eigenvalue of Wn (k) for 2 n 2 α∈R+ |kα |. Then in the statement of Theorem 2.1 (and hence in the “only if”-part of the statement of Theorem A) in [1], the set M reg should be replaced by M reg ∩ M ∗ . This is not a significant restriction since in general M ∗ contains all the elements k of M with real part greater than or equal to 0 as shown in the next section below. Using Lemma 2.2 of [1], we see that under these assumptions all the statement of Theorem 2.3 of [1] (and hence in the “only if”-part of the statement of Theorem A) is correct. 2. Invertibility of Euler operator An easy calculation shows that Tx (p)(x) = Wn (p)(x)

(2.1)

for all p ∈ Pn and x ∈ Rd . Proposition 2.1. The following are equivalent. (i) Wn is invertible. (ii) There exists cn : G → C, satisfying

cn (g)Tgx (p)(gx) = p(x),

∀p ∈ Pn , ∀x ∈ Rd .

(2.2)

g∈G

Proof. From (2.1) we have clearly that (ii) ⇒ (i). Conversely, suppose that Wn = Wn (k) is invertible in Pn . Since Wn is contained in the algebra spanned by Lg , g ∈ G then Wn−1 lies in the same algebra and then there exists cn : G → C such that Wn−1 =

cn (g)Lg .

g∈G

This leads to

cn (g)Lg Wn (p) = p,

∀p ∈ Pn ,

g∈G

and this shows that (i) ⇔ (ii).

2

To give interesting examples showing that the class M ∗ is very large, we consider the complex vector space C|G| equipped with its canonical basis (eg )g∈G . For g ∈ G let Ag be the endomorphism of C|G| defined by Ag (eg ) = egg , g ∈ G. We consider the endomorphism Bk = α∈R+ kα (1 − Aσα ) of C|G| . Proposition 2.2. Suppose that n is a positive integer. If −n is not an eigenvalue of the matrix Bk , then Wn is invertible.

2864

M. Maslouhi, E.H. Youssfi / Journal of Functional Analysis 258 (2010) 2862–2864

Proof. An easy calculation shows that

cn (g)Tgx (p)(gx) =

g∈G

(n + γ )cn (g) −

g∈G

k(α)cn (σα g) p(gx)

(2.3)

α∈R+

d for p ∈ Pn and x ∈ R . If −n is not an eigenvalue of the matrix Bk , then there is an element Xn = g∈G cn (g)eg of C|G| such that (n + Bk )Xn = eI . From this it follows that

(n + γ )cn (g) −

k(α)cn (σα g) = δ1,g ,

∀g ∈ G,

(2.4)

α∈R+

so by (2.3) we see that

cn (g)Tgx (p)(gx) = p(x)

g∈G

for p ∈ Pn and x ∈ Rd . By Proposition 2.1 we see that Wn is invertible.

2

Suppose that λ is an eigenvalue of B(k) and let Eλ denote the corresponding eigensubspace of C|G| . It follows that the dimension dλ of Eλ satisfies 1 dα . Since B(k) commutes with the Aσα , α ∈ R, it follows that Eλ is an invariant subspace of 1 − Aσα for all α ∈ R+ . In addition, 1 2 (1 − Aσα ) is a projection and thus the trace of its restriction πλ,α to Eλ is equal to the dimension dλ,α of the image πλ,α (Eλ ). Therefore, λ=

k(α)

α∈R+

dλ,α . dλ

(2.5)

By Proposition 2.2 and the latter equality it follows: Corollary 2.3. The Euler operator Wn (k) is invertible for all n 1 in each of the following cases: (i) Re k(α) 0, for all α ∈ R+ . (ii) Im k(α) > 0, for all α ∈ R+ . (iii) Im k(α) < 0, for all α ∈ R+ . In particular, in each of the above cases the parameter function k must be regular. References [1] M. Maslouhi, E.H. Youssfi, The Dunkl intertwining operator, J. Funct. Anal. 256 (8) (2009) 2697–2709.

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