Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori [1 ed.] 9781470443757, 9781470428068

Citation preview

Number 1203

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori Xiao Xiong Quanhua Xu Zhi Yin

March 2018 • Volume 252 • Number 1203 (fourth of 6 numbers)

Number 1203

Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori Xiao Xiong Quanhua Xu Zhi Yin

March 2018 • Volume 252 • Number 1203 (fourth of 6 numbers)

Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: http://dx.doi.org/10.1090/memo/1203

Memoirs of the American Mathematical Society This journal is devoted entirely to research in pure and applied mathematics. Subscription information. Beginning with the January 2010 issue, Memoirs is accessible from www.ams.org/journals. The 2018 subscription begins with volume 251 and consists of six mailings, each containing one or more numbers. Subscription prices for 2018 are as follows: for paper delivery, US$999 list, US$799.20 institutional member; for electronic delivery, US$879 list, US$703.20 institutional member. Upon request, subscribers to paper delivery of this journal are also entitled to receive electronic delivery. If ordering the paper version, add US$20 for delivery within the United States; US$80 for outside the United States. Subscription renewals are subject to late fees. See www.ams.org/help-faq for more journal subscription information. Each number may be ordered separately; please specify number when ordering an individual number. Back number information. For back issues see www.ams.org/backvols. Subscriptions and orders should be addressed to the American Mathematical Society, P. O. Box 845904, Boston, MA 02284-5904 USA. All orders must be accompanied by payment. Other correspondence should be addressed to 201 Charles Street, Providence, RI 02904-2294 USA. Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes.

Memoirs of the American Mathematical Society (ISSN 0065-9266 (print); 1947-6221 (online)) is published bimonthly (each volume consisting usually of more than one number) by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA. Periodicals postage paid at Providence, RI. Postmaster: Send address changes to Memoirs, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. c 2018 by the American Mathematical Society. All rights reserved.  R , Zentralblatt MATH, Science Citation This publication is indexed in Mathematical Reviews  R , Science Citation IndexT M -Expanded, ISI Alerting ServicesSM , SciSearch  R , Research Index  R , CompuMath Citation Index  R , Current Contents  R /Physical, Chemical & Earth Alert  Sciences. This publication is archived in Portico and CLOCKSS. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines 

established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10 9 8 7 6 5 4 3 2 1

23 22 21 20 19 18

Contents Chapter 0. Introduction Basic properties Embedding Characterizations Interpolation Multipliers

1 3 4 4 6 6

Chapter 1. Preliminaries 1.1. Noncommutative Lp -spaces 1.2. Quantum tori 1.3. Fourier multipliers 1.4. Hardy spaces

9 9 10 12 14

Chapter 2. Sobolev spaces 2.1. Distributions on quantum tori 2.2. Definitions and basic properties 2.3. A Poincar´e-type inequality 2.4. Lipschitz classes 2.5. The link with the classical Sobolev spaces

19 19 21 25 28 31

Chapter 3. Besov spaces 3.1. Definitions and basic properties 3.2. A general characterization 3.3. The characterizations by Poisson and heat semigroups 3.4. The characterization by differences 3.5. Limits of Besov norms 3.6. The link with the classical Besov spaces

35 35 42 48 51 54 55

Chapter 4. Triebel-Lizorkin spaces 4.1. A multiplier theorem 4.2. Definitions and basic properties 4.3. A general characterization 4.4. Concrete characterizations 4.5. Operator-valued Triebel-Lizorkin spaces

59 59 68 72 76 80

Chapter 5. Interpolation 5.1. Interpolation of Besov and Sobolev spaces 5.2. The K-functional of (Lp , Wpk ) 5.3. Interpolation of Triebel-Lizorkin spaces

83 83 88 91

iii

iv

CONTENTS

Chapter 6. Embedding 6.1. Embedding of Besov spaces 6.2. Embedding of Sobolev spaces 6.3. Compact embedding

93 93 95 100

Chapter 7. Fourier multiplier 7.1. Fourier multipliers on Sobolev spaces 7.2. Fourier multipliers on Besov spaces 7.3. Fourier multipliers on Triebel-Lizorkin spaces Acknowledgements

103 103 107 110 113

Bibliography

115

Abstract This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative d-torus Tdθ (with θ a skew symmetric real d × dmatrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincar´e type inequality for Sobolev spaces. We also show that the Sobolev space k (Tdθ ) coincides with the Lipschitz space of order k, already studied by Weaver W∞ in the case k = 1. We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain LittlewoodPaley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Br´ezis -Mironescu and Maz’ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov α (Tdθ ) for α > 0 is the quantum analogue of the usual Zygmund class space B∞,∞ of order α. We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple (Lp (Tdθ ), Wpk (Tdθ )), which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix θ, so coincide with those on the corresponding spaces on the usual Received by the editor July 7, 2015 and, in revised form, October 20, 2015. Article electronically published on January 29, 2018. DOI: http://dx.doi.org/10.1090/memo/1203 2010 Mathematics Subject Classification. Primary 46L52, 46L51, 46L87; Secondary 47L25, 47L65, 43A99. Key words and phrases. Quantum tori, noncommutative Lp -spaces, Bessel and Riesz potentials, (potential) Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces, Hardy spaces, characterizations, Poisson and heat semigroups, embedding inequalities, interpolation, (completely) bounded Fourier multipliers. The first author is affiliated with the Laboratoire de Math´ematiques, Universit´e de FrancheComt´ e, 25030 Besan¸con Cedex, France AND the Department of Mathematics and Statistics, University of Saskatchewan, S7N 5E6, Canada. Email: [email protected]. The second author is affiliated with the Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China AND the Laboratoire de Math´ ematiques, Universit´ e de Bourgogne Franche-Comt´ e, 25030 Besan¸con Cedex, France AND the Institut Universitaire de France. Email: [email protected]. The third author is affiliated with the Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin, 150001, China. Email: [email protected]. c 2018 American Mathematical Society

v

vi

ABSTRACT

d-torus. We also give a quite simple description of (completely) bounded Fourier multipliers on the Besov spaces in terms of their behavior on the Lp -components in the Littlewood-Paley decomposition.

CHAPTER 0

Introduction This paper is the second part of our project about analysis on quantum tori. The previous one [17] studies several subjects of harmonic analysis on these objects, including maximal inequalities, mean and pointwise convergences of Fourier series, completely bounded Fourier multipliers on Lp -spaces and the theory of Hardy spaces. It was directly inspired by the current line of investigation on noncommutative harmonic analysis. As pointed out there, very little had been done about the analytic aspect of quantum tori before [17]; this situation is in strong contrast with their geometry on which there exists a considerably long list of publications. Presumably, this deficiency is due to numerous difficulties one may encounter when dealing with noncommutative Lp -spaces, since these spaces come up unavoidably if one wishes to do analysis. [17] was made possible by the recent developments on noncommutative martingale/ergodic inequalities and the Littlewood-Paley-Stein theory for quantum Markovian semigroups, which had been achieved thanks to the efforts of many researchers; see, for instance, [31, 36, 37, 52, 56, 61, 62], and [32–34, 44, 45]. This second part intends to study Sobolev, Besov and Triebel-Lizorkin spaces on quantum tori. In the classical setting, these spaces are fundamental for many branches of mathematics such as harmonic analysis, PDE, functional analysis and approximation theory. Our references for the classical theory are [1, 42, 49, 53, 73, 74]. However, they have never been investigated so far in the quantum setting, except two special cases to our best knowledge. Firstly, Sobolev spaces with the L2 norm were studied by Spera [65] in view of applications to the Yang-Mills theory for quantum tori [66] (see also [26, 39, 58, 64] for related works). On the other hand, inspired by Connes’ noncommutative geometry [18], or more precisely, the part on noncommutative metric spaces, Weaver [78, 79] developed the Lipschitz classes of order α for 0 < α ≤ 1 on quantum tori. The fact that only these two cases have been studied so far illustrates once more the above mentioned difficulties related to noncommutativity. Among these difficulties, a specific one is to be emphasized: it is notably relevant to this paper, and is the lack of a noncommutative analogue of the usual pointwise maximal function. However, maximal function techniques play a paramount role in the classical theory of Besov and Triebel-Lizorkin spaces (as well as in the theory of Hardy spaces). They are no longer available in the quantum setting, which forces us to invent new tools, like in the previously quoted works on noncommutative martingale inequalities and the quantum Littlewood-Paley-Stein theory where the same difficulty already appeared. One powerful tool used in [17] is the transference method. It consists in transferring problems on quantum tori to the corresponding ones in the case of operatorvalued functions on the usual tori, in order to use existing results in the latter case or 1

2

0. INTRODUCTION

adapt classical arguments. This method is efficient for several problems studied in [17], including the maximal inequalities and Hardy spaces. It is still useful for some parts of the present work; for instance, Besov spaces can be investigated through the classical vector-valued Besov spaces by means of transference, the relevant Banach spaces being the noncommutative Lp -spaces on a quantum torus. However, it becomes inefficient for others. For example, the Sobolev or Besov embedding inequalities cannot be proved by transference. On the other hand, if one wishes to study Triebel-Lizorkin spaces on quantum tori via transference, one should first develop the theory of operator-valued Triebel-Lizorkin spaces on the classical tori. The latter is as hard as the former. Contrary to [17] , the transference method will play a very limited role in the present paper. Instead, we will use Fourier multipliers in a crucial way, this approach is of interest in its own right. We thus develop an intrinsic differential analysis on quantum tori, without frequently referring to the usual tori via transference as in [17]. This is a major advantage of the present methods over those of [17]. We hope that the study carried out here would open new perspectives of applications and motivate more future research works on quantum tori or in similar circumstances. In fact, one of our main objectives of developing analysis on quantum tori is to gain more insights on the geometrical structures of these objects, so ultimately to return back to their differential geometry. To describe the content of the paper, we need some definitions and notation (see the respective sections below for more details). Let d ≥ 2 and θ = (θkj ) be a real skew-symmetric d × d-matrix. The d-dimensional noncommutative torus Aθ is the universal C*-algebra generated by d unitary operators U1 , . . . , Ud satisfying the following commutation relation Uk Uj = e2πiθkj Uj Uk ,

1 ≤ j, k ≤ d.

Let U = (U1 , · · · , Ud ). For m = (m1 , · · · , md ) ∈ Zd , set U m = U1m1 · · · Udmd . A polynomial in U is a finite sum:  x= αm U m ,

αm ∈ C.

m∈Zd

For such a polynomial x, we define τ (x) = α0 . Then τ extends to a faithful tracial state on Aθ . Let Tdθ be the w*-closure of Aθ in the GNS representation of τ . This is our d-dimensional quantum torus. It is to be viewed as a deformation of the usual d-torus Td , or more precisely, of the commutative algebra L∞ (Td ). The noncommutative Lp -spaces associated to (Tdθ , τ ) are denoted by Lp (Tdθ ). The Fourier transform of an element x ∈ L1 (Tdθ ) is defined by   x (m) = τ (U m )∗ x , m ∈ Zd . The formal Fourier series of x is x∼



x (m)U m .

m∈Zd

The differential structure of Tdθ is modeled on that of Td . Let    am U m : {am }m∈Zd rapidly decreasing . S(Tdθ ) = m∈Zd

BASIC PROPERTIES

3

This is the deformation of the space of infinitely differentiable functions on Td ; it is the Schwartz class of Tdθ . Like in the commutative case, S(Tdθ ) carries a natural locally convex topology. Its topological dual S  (Tdθ ) is the space of distributions on Tdθ . The partial derivations on S(Tdθ ) are determined by ∂j (Uj ) = 2πiUj and ∂j (Uk ) = 0,

k = j, 1 ≤ j, k ≤ d.

Nd0

(N0 denoting the set of nonnegative integers), the Given m = (m1 , . . . , md ) ∈ associated partial derivation Dm is defined to be ∂1m1 · · · ∂dmd . The order of Dm is |m|1 = m1 + · · · + md . Let Δ = ∂12 + · · · + ∂d2 be the Laplacian. By duality, the derivations and Fourier transform transfer to S  (Tdθ ) too. Fix a Schwartz function ϕ on Rd satisfying the usual Littlewood-Paley decomposition property expressed in (3.1). For each k ≥ 0 let ϕk be the function whose Fourier transform is equal to ϕ(2−k ·). For a distribution x on Tdθ , define  ϕ k ∗ x = ϕ(2−k m) x(m)U m . m∈Zd

So x → ϕ k ∗ x is the Fourier multiplier with symbol ϕ(2−k ·). We can now define the four families of function spaces on Tdθ to be studied . Let 1 ≤ p, q ≤ ∞ and k ∈ N, α ∈ R, and let J α be the Bessel potential of order α: α J α = (1 − (2π)−2 Δ) 2 . • Sobolev spaces:   k Wp (Tdθ ) = x ∈ S  (Tdθ ) : Dm x ∈ Lp (Tdθ ) for each m ∈ Nd0 with |m|1 ≤ k . • Potential or fractoinal Sobolev spaces:   Hpα (Tdθ ) = x ∈ S  (Tdθ ) : J α x ∈ Lp (Tdθ ) . • Besov spaces:  1    α Bp,q (Tdθ ) = x ∈ S  (Tdθ ) : | x(0)|q + 2qkα ϕ k ∗ x qp q < ∞ . k≥0

• Triebel-Lizorkin spaces for p < ∞ :   1   Fpα,c (Tdθ ) = x ∈ S  (Tdθ ) : | x(0)|2 + 22kα |ϕ k ∗ x|2 2 p < ∞ . k≥0

Equipped with their natural norms, all these spaces become Banach spaces. Now we can describe the main results proved in this paper by classifying them into five families. Basic properties A common basic property of potential Sobolev, Besov and Triebel-Lizorkin spaces is a reduction theorem by the Bessel potential. For example, J β is an α α−β (Tdθ ) onto Bp,q (Tdθ ) for all 1 ≤ p, q ≤ ∞ and α, β ∈ R; this isomorphism from Bp,q is the so-called lifting or reduction theorem. Specifically to Triebel-Lizorkin spaces, J α establishes an isomorphism between Fpα,c (Tdθ ) and the Hardy space Hpc (Tdθ ) for any 1 ≤ p < ∞. As a consequence, we deduce that the potential Sobolev space Hpα (Tdθ ) admits a Littlewood-Paley type characterization for 1 < p < ∞. Another type of reduction for Besov and Triebel-Lizorkin spaces is that for any α (Tdθ )) iff all its partial derivatives of positive integer k, x ∈ Fpα,c (Tdθ ) (resp. Bp,q α−k,c d α−k d (Tθ ) (resp. Bp,q (Tθ )). order k belong to Fp

4

0. INTRODUCTION

Concerning Sobolev spaces, we obtain a Poincar´e type inequality: For any x ∈ Wp1 (Tdθ ) with 1 ≤ p ≤ ∞, we have x − x (0) p  ∇x p . Our proof of this inequality greatly differs with standard arguments for such results in the commutative case. k (Tdθ ) is the analogue for Tdθ of the classical Lipschitz class We also show that W∞ d of order k. For u ∈ R , define Δu x = πz (x) − x, where z = (e2πiu1 , · · · , e2πiud ) and πz is the automorphism of Tdθ determined by Uj → zj Uj for 1 ≤ j ≤ d. Then for a positive integer k, Δku is the kth difference operator on Tdθ associated to u. Note that Δku is also the Fourier multiplier with symbol eku , where eu (ξ) = e2πiu·ξ − 1. The kth order modulus of Lp -smoothness of an x ∈ Lp (Tdθ ) is defined to be ωpk (x, ε) = sup Δku x p . 00

ωpk (x, ε) ≈ εk



Dm x p .

m∈Nd 0 , |m|1 =k

In particular, we recover Weaver’s results [78, 79] on the Lipschitz class on Tdθ when p = ∞ and k = 1. Embedding The second family of results concern the embedding of the preceding spaces. A typical one is the analogue of the classical Sobolev embedding inequality for Wpk (Tdθ ): If 1 < p < q < ∞ such that 1q = p1 − kd , then Wpk (Tdθ ) ⊂ Lq (Tdθ ) continuously. Similar embedding inequalities hold for the other spaces too. Combined with real α (Tdθ ) yields the above Sobolev eminterpolation, the embedding inequality of Bp,q bedding. Our proofs of these embedding inequalities are based on Varopolous’ celebrated semigroup approach [76] to the Littlewood-Sobolev theory, which has also been developed by Junge and Mei [34] in the noncommutative setting for the study of BMO spaces on quantum Markovian semigroups. Thus the characterization of Besov spaces by Poisson or heat semigroup described below is essential for the proof of our embedding inequalities. We also establish compact embedding theorems. For instance, the previously mentioned Sobolev embedding becomes a compact one Wpk (Tdθ ) → Lq∗ (Tdθ ) for any q ∗ with 1 ≤ q ∗ < q. Characterizations The third family of results are various characterizations of Besov and TriebelLizorkin spaces. This is the most difficult and technical part of the paper. In the classical case, all existing proofs of these characterizations that we know use maximal function techniques in a crucial way. As pointed out earlier, these techniques are no longer available. Instead, we use frequently Fourier multipliers. We would like to emphasize that our results are better than those in literature even in the

CHARACTERIZATIONS

5

commutative case. Let us illustrate this by stating the characterization of Besov spaces in terms of the circular Poisson semigroup. Given a distribution x on Tdθ and k ∈ Z, let  x (m)r |m| U m Pr (x) = m∈Zd

and



Jrk Pr (x) =

Cm,k x (m)r |m|−k U m ,

0 ≤ r < 1,

m∈Zd

where | · | denotes the Euclidean norm of Rd and Cm,k = |m| · · · (|m| − k + 1) if k ≥ 0 and Cm,k =

1 if k < 0. (|m| + 1) · · · (|m| − k)

Note that Jrk is the kth derivation operator relative to r if k ≥ 0, and the (−k)th integration operator if k < 0. Then our characterization asserts that for 1 ≤ p, q ≤ ∞ and α ∈ R, k ∈ Z with k > α,  1

q dr q1 α ≈ max | x(m)|q + (1 − r)(k−α)q Jrk Pr (xk ) p , x Bp,q 1−r |m| 0. The relevant constants in all such inequalities may depend on the dimension d, the test function ϕ or ψ, etc. but never on the functions f or distributions x in consideration. The main results of this paper have been announced in [82].

CHAPTER 1

Preliminaries This chapter collects the necessary preliminaries for the whole paper. The first two sections present the definitions and some basic facts about noncommutative Lp spaces and quantum tori which are the central objects of the paper. The third one contains some results on Fourier multipliers that will play a paramount role in the whole paper. The last section gives the definitions and some fundamental results on operator-valued Hardy spaces on the usual and quantum tori. This section will be needed only starting from chapter 4 on Triebel-Lizorkin spaces. 1.1. Noncommutative Lp -spaces Let M be a von Neumann algebra equipped with a normal semifinite faithful + trace τ and SM be the set of all positive elements x in M with τ (s(x)) < ∞, where s(x) denotes the support of x, i.e., the smallest projection e such that exe = x. Let + SM be the linear span of SM . Then every x ∈ SM has finite trace, and SM is a w*-dense ∗-subalgebra of M. + (recalling Let 0 < p < ∞. For any x ∈ SM , the operator |x|p belongs to SM 1 ∗ 2 |x| = (x x) ). We define  1 x p = τ (|x|p ) p . One can check that · p is a norm or p-norm on SM according to p ≥ 1 or p < 1. The completion of (SM , · p ) is denoted by Lp (M), which is the usual noncommutative Lp -space associated to (M, τ ). For convenience, we set L∞ (M) = M equipped with the operator norm · M . The norm of Lp (M) will be often denoted simply by · p . But if different Lp -spaces appear in a same context, we will sometimes precise their norms in order to avoid possible ambiguity. The reader is referred to [57] and [83] for more information on noncommutative Lp -spaces. The elements of Lp (M) can be described as closed densely defined operators on H (H being the Hilbert space on which M acts). A closed densely defined operator x on H is said to be affiliated with M if ux = xu for any unitary u in the commutant M of M. An operator x affiliated with M is said to be measurable with respect to (M, τ ) (or simply measurable) if for any δ > 0 there exists a projection e ∈ B(H) such that e(H) ⊂ Dom(x) and τ (e⊥ ) ≤ δ, where Dom(x) defines the domain of x. We denote by L0 (M, τ ), or simply L0 (M) the family of all measurable operators. For such an operator x, we define λs (x) = τ (e⊥ s (|x|)), s > 0 where e⊥ s (x) = 1(s,∞) (x) is the spectrum projection of x corresponding to the interval (s, ∞), and μt (x) = inf{s > 0 : λs (x) < t}, t > 0. 9

10

1. PRELIMINARIES

The function s → λs (x) is called the distribution function of x and the μt (x) the generalized singular numbers of x. Similarly to the classical case, for 0 < p < ∞, 0 < q ≤ ∞, the noncommutative Lorentz space Lp,q (M) is defined to be the collection of all measurable operators x such that   ∞ 1 dt  q1 x p,q = (t p μt (x))q < ∞. t 0 Clearly, Lp,p (M) = Lp (M). The space Lp,∞ (M) is usually called a weak Lp -space, 0 < p < ∞, and 1 x p,∞ = sup sλs (x) p . s>0

Like the classical Lp -spaces, noncommutative Lp -spaces behave well with respect to interpolation. Our reference for interpolation theory is [8]. Let 1 ≤ p0 < p1 ≤ ∞, 1 ≤ q ≤ ∞ and 0 < η < 1. Then     (1.1) Lp0 (M), Lp1 (M) η = Lp (M) and Lp0 (M), Lp1 (M) η,q = Lp,q (M), η where p1 = 1−η p0 + p1 . Now we introduce noncommutative Hilbert space-valued Lp -spaces Lp (M; H c ) and Lp (M; H r ), which are studied at length in [32]. Let H be a Hilbert space and v a norm one element of H. Let pv be the orthogonal projection onto the onedimensional subspace generated by v. Then define the following row and column noncommutative Lp -spaces:

Lp (M; H r ) = (pv ⊗ 1M )Lp (B(H)⊗M), Lp (M; H c ) = Lp (B(H)⊗M)(pv ⊗ 1M ), where the tensor product B(H)⊗M is equipped with the tensor trace while B(H) is equipped with the usual trace. For f ∈ Lp (M; H c ), f Lp (M;H c ) = (f ∗ f ) 2 Lp (M) . 1

A similar formula holds for the row space by passing to adjoints: f ∈ Lp (M; H r ) iff f ∗ ∈ Lp (M; H c ), and f Lp (M;H r ) = f ∗ Lp (M;H c ) . It is clear that Lp (M; H c ) and Lp (M; H r ) are 1-complemented subspaces of Lp (B(H)⊗M) for any p. Thus they also form interpolation scales with respect to both complex and real interpolation methods: Let 1 ≤ p0 , p1 ≤ ∞ and 0 < η < 1. Then   Lp0 (M; H c ), Lp1 (M; H c ) η = Lp (M; H c ),   (1.2) Lp0 (M; H c ), Lp1 (M; H c ) η,p = Lp (M; H c ), where

1 p

=

1−η p0

+

η p1 .

The same formulas hold for row spaces too. 1.2. Quantum tori

Let d ≥ 2 and θ = (θkj ) be a real skew symmetric d × d-matrix. The associated d-dimensional noncommutative torus Aθ is the universal C ∗ -algebra generated by d unitary operators U1 , . . . , Ud satisfying the following commutation relation (1.3)

Uk Uj = e2πiθkj Uj Uk ,

j, k = 1, . . . , d.

We will use standard notation from multiple Fourier series. Let U = (U1 , · · · , Ud ). For m = (m1 , · · · , md ) ∈ Zd we define U m = U1m1 · · · Udmd .

1.2. QUANTUM TORI

A polynomial in U is a finite sum  x= αm U m

11

with αm ∈ C,

m∈Zd

that is, αm = 0 for all but finite indices m ∈ Zd . The involution algebra Pθ of all such polynomials is dense in Aθ . For any polynomial x as above we define τ (x) = α0 , where 0 = (0, · · · , 0). Then, τ extends to a faithful tracial state on Aθ . Let Tdθ be the w∗ -closure of Aθ in the GNS representation of τ . This is our d-dimensional quantum torus. The state τ extends to a normal faithful tracial state on Tdθ that will be denoted again by τ . Recall that the von Neumann algebra Tdθ is hyperfinite. Any x ∈ L1 (Tdθ ) admits a formal Fourier series:  x∼ x (m)U m , m∈Zd

where x (m) = τ ((U m )∗ x),

m ∈ Zd

are the Fourier coefficients of x. The operator x is, of course, uniquely determined by its Fourier series. We introduced in [17] a transference method to overcome the full noncommutativity of quantum tori and use methods of operator-valued harmonic analysis. Let Td be the usual d-torus equipped with normalized

Haar measure dz. Let Nθ = L∞ (Td )⊗Tdθ , equipped with the tensor trace ν = dz ⊗ τ . It is well known that for every 0 < p < ∞, Lp (Nθ , ν) ∼ = Lp (Td ; Lp (Tdθ )). The space on the right-hand side is the space of Bochner p-integrable functions from Td to Lp (Tdθ ). In general, for any Banach space X and any measure space (Ω, μ), we use Lp (Ω; X) to denote the space of Bochner p-integrable functions from Ω to X. For each z ∈ Td , define πz to be the isomorphism of Tdθ determined by (1.4)

πz (U m ) = z m U m = z1m1 · · · zdmd U1m1 · · · Udmd .

Since τ (πz (x)) = τ (x) for any x ∈ Tdθ , πz preserves the trace τ. Thus for every 0 < p < ∞, (1.5)

πz (x) p = x p , ∀x ∈ Lp (Tdθ ).

Now we state the transference method as follows (see [17]). Lemma 1.1. For any x ∈ Lp (Tdθ ), the function x  : z → πz (x) is continuous from Td to Lp (Tdθ ) (with respect to the w*-topology for p = ∞). If x ∈ Aθ , it is continuous from Td to Aθ . Corollary 1.2. (i) Let 0 < p ≤ ∞. If x ∈ Lp (Tdθ ), then x  ∈ Lp (Nθ ) and  x p = x p , that is, x → x  is an isometric embedding from Lp (Tdθ ) into Lp (Nθ ). Moreover, this map is also an isomorphism from Aθ into C(Td ; Aθ ).

12

1. PRELIMINARIES

d = { d is a von Neumann subalgebra of N and (ii) Let T x : x ∈ Tdθ }. Then T θ θ θ the associated conditional expectation is given by

   E(f )(z) = πz πw f (w) dw , z ∈ Td , f ∈ Nθ . Td

d ) Moreover, E extends to a contractive projection from Lp (Nθ ) onto Lp (T θ for 1 ≤ p ≤ ∞. d ) for every 0 < p ≤ ∞. (iii) Lp (Tdθ ) is isometric to Lp (T θ 1.3. Fourier multipliers Fourier multipliers will be the most important tool for the whole work. Now we present some known results on them for later use. Given a function φ : Zd → C, let   Mφ denote the associated Fourier multiplier on Td , namely, M φ f (m) = φ(m)f (m) for any trigonometric polynomial f on Td . We call φ a multiplier on Lp (Td ) if Mφ extends to a bounded map on Lp (Td ). Fourier multipliers on Tdθ are defined exactly in the same way, we still use the same symbol Mφ to denote the corresponding multiplier on Tdθ . Note that the isomorphism πz defined in (1.4) is the Fourier multiplier associated to the function φ given by φ(m) = z m . It is natural to ask if the boundedness of Mφ on Lp (Td ) is equivalent to that on Lp (Tdθ ). This was open until a negative answer given by Ricard [63] after the submission of the present paper. However, it is proved in [17] that the answer is affirmative if “boundedness” is replaced by “complete boundedness”, a notion from operator space theory for which we refer to [23] and [55]. All noncommutative Lp -spaces are equipped with their natural operator space structure introduced by Pisier [54, 55]. We will use the following fundamental property of completely bounded (c.b. for short) maps due to Pisier [54]. Let E and F be operator spaces. Then a linear map T : E → F is c.b. iff IdSp ⊗T : Sp [E] → Sp [F ] is bounded for some 1 ≤ p ≤ ∞. In this case, T cb = IdSp ⊗ T : Sp [E] → Sp [F ] . Here Sp [E] denotes the E-valued Schatten p-class. In particular, if E = C, Sp [C] = Sp is the noncommutative Lp -space associated to B(2 ), equipped with the usual trace. Applying this criterion to the special case where E = F = Lp (M), we see that a map T on Lp (M) is c.b. iff IdSp ⊗ T : Lp (B(2 )⊗M) → Lp (B(2 )⊗M) is bounded. The readers unfamiliar with operator space theory can take this property as the definition of c.b. maps between Lp -spaces. Thus φ is a c.b. multiplier on Lp (Tdθ ) if Mφ is c.b. on Lp (Tdθ ), or equivalently, if IdSp ⊗ Mφ is bounded on Lp (B(2 )⊗Tdθ ). Let M(Lp (Tdθ )) (resp. Mcb (Lp (Tdθ ))) denote the space of Fourier multipliers (resp. c.b. Fourier multipliers) on Lp (Tdθ ), equipped with the natural norm. When θ = 0, we recover the (c.b.) Fourier multipliers on the usual d-torus Td . The corresponding multiplier spaces are denoted by M(Lp (Td )) and Mcb (Lp (Td )). Note that in the latter case (θ = 0), Lp (B(2 )⊗Td ) = Lp (Td ; Sp ), thus φ is a c.b. multiplier on Lp (Td ) iff Mφ extends to a bounded map on Lp (Td ; Sp ). The following result is taken from [17]. Lemma 1.3. Let 1 ≤ p ≤ ∞. Then Mcb (Lp (Tdθ )) = Mcb (Lp (Td )) with equal norms.

1.3. FOURIER MULTIPLIERS

13

Remark 1.4. Note that Mcb (L1 (Td )) = Mcb (L∞ (Td )) coincides with the space of the Fourier transforms of bounded measures on Td , and Mcb (L2 (Td )) with the space of bounded functions on Zd . The most efficient criterion for Fourier multipliers on Lp (Td ) for 1 < p < ∞ is Mikhlin’s condition. Although it can be formulated in the periodic case, it is more convenient to state this condition in the case of Rd . On the other hand, the Fourier multipliers on Td used later will be the restrictions to Zd of continuous Fourier multipliers on Rd . As usual, for m = (m1 , · · · , md ) ∈ Nd0 (recalling that N0 denotes the set of nonnegative integers), we set Dm = ∂1m1 · · · ∂dmd , where ∂k denotes the kth partial derivation on Rd . Also  put |m|1 = m1 + · · · + md . The Euclidean norm of Rd is denoted by | · |: |ξ| = ξ12 + · · · + ξd2 . Definition 1.5. A function φ : Rd → C is called a Mikhlin multiplier if it is d-times differentiable on Rd \ {0} and satisfies the following condition   φ M = sup |ξ||m|1 |Dm φ(ξ)| : ξ ∈ Rd \ {0}, m ∈ Nd0 , |m|1 ≤ d < ∞. Note that the usual Mikhlin condition requires only partial derivatives up to order [ d2 ] + 1 (see, for instance, [25, section II.6] or [67, Theorem 4.3.2]). Our requirement above up to order d is imposed by the boundedness of these multipliers on UMD spaces. We refer to section 4.1 for the usual Mikhlin condition and more multiplier results on Tdθ . It is a classical result that every Mikhlin multiplier is a Fourier multiplier on Lp (Rd ) for 1 < p < ∞ (cf. [25, section II.6] or [67, Theorem 4.3.2]), so its restriction φZd is a Fourier multiplier on Lp (Td ) too. It is, however, less classical that such a multiplier is also c.b. on Lp (Rd ) or Lp (Td ). This follows from a general result on UMD spaces. Recall that a Banach space X is called a UMD space if the X-valued martingale differences are unconditional in Lp (Ω; X) for any 1 < p < ∞ and any probability space (Ω, P ). This is equivalent to the requirement that the Hilbert transform be bounded on Lp (Rd ; X) for 1 < p < ∞. Any noncommutative Lp -space with 1 < p < ∞ is a UMD space. We refer to [10, 15, 16] for more information. Before proceeding further, we make a convention used throughout the paper: Convention. To simplify the notational system, we will use the same derivation symbols for Rd and Td . Thus for a multi-index m ∈ Nd0 , Dm = ∂1m1 · · · ∂dmd , introduced previously, will also denote the partial derivation of order m on Td . Similarly, Δ = ∂12 + · · · + ∂d2 will denote the Laplacian on both Rd and Td . In the same spirit, for a function φ on Rd , we will call φ rather than φZd a Fourier multiplier on Lp (Td ) or Lp (Tdθ ). This should not cause any ambiguity in concrete contexts. Considered as a map on Lp (Td ) or Lp (Tdθ ), Mφ will be often denoted by f → φ ∗ f or x → φ ∗ x. Note that the notation φ ∗ f coincides with the usual convolution when φ is good enough. Indeed, let φ be the 1-periodization of the inverse Fourier transform of φ whenever it exists in a reasonable sense:   = φ(s) F −1 (φ)(s + m), s ∈ Rd . m∈Zd

14

1. PRELIMINARIES

Viewed as a function on Td , φ admits the following Fourier series:   φ(m)z m . φ(z) = m∈Zd

Thus for any trigonometric polynomial f ,  −1   φ ∗ f (z) = )f (w)dw, φ(zw Td

z ∈ Td .

The following lemma is proved in [43, 85] (see also [12] for the one-dimensional case). Lemma 1.6. Let X be a UMD space and 1 < p < ∞. Let φ be a Mikhlin multiplier. Then φ is a Fourier multiplier on Lp (Td ; X). Moreover, its norm is controlled by φ M , p and the UMD constant of X. The following lemma will play a key role in the whole paper. Lemma 1.7. Let φ be a function on Rd . (i) If F −1 (φ) is integrable on Rd , then φ is a c.b. Fourier multiplier on Lp (Tdθ ) for 1 ≤ p ≤ ∞. Moreover, its c.b. norm is not greater than F −1 (φ) . 1

(ii) If φ is a Mikhlin multiplier, then φ is a c.b. Fourier multiplier on Lp (Tdθ ) for 1 < p < ∞. Moreover, its c.b. norm is controlled by φ M and p. Proof. (i) Since F −1 (φ) is integrable, φ is a c.b. Fourier multiplier on L1 (Td ), so on Lp (Td ) for 1 ≤ p ≤ ∞ (see Remark 1.4). Consequently, by Lemma 1.3, φ is a c.b. Fourier multiplier on Lp (Tdθ ) for 1 ≤ p ≤ ∞. (ii) It is well known that Sp is a UMD space for 1 < p < ∞. Thus, by Lemma 1.6, φ is a Fourier multiplier on Lp (Td ; Sp ); in other words, it is a c.b. Fourier multiplier on Lp (Tdθ ).  1.4. Hardy spaces We now present some preliminaries on operator-valued Hardy spaces on Td and Hardy spaces on Tdθ . Motivated by the developments of noncommutative martingale inequalities in [36, 56] and quantum Markovian semigroups in [32], Mei [44] developed the theory of operator-valued Hardy spaces on Rd . More recently, Mei’s work was extended to the torus case in [17] with the objective of developing the Hardy space theory in the quantum torus setting. We now recall the definitions and results that will be needed later. Throughout this section, M will denote a von Neumann algebra equipped with a normal faithful tracial state τ and N = L∞ (Td )⊗M with the tensor trace. In our future applications, M will be Tdθ . A cube of Td is a product Q = I1 × · · · × Id , where each Ij is an interval (= arc) of T. As in the Euclidean case, we use |Q| to denote the normalized volume (= measure) of Q. The whole Td is now a cube too (of volume 1). We will often identify Td with the unit cube Id = [0, 1)d via (e2πis1 , · · · , e2πisd ) ↔ (s1 , · · · , sd ). Under this identification, the addition in Id is the usual addition modulo 1 coordinatewise; an interval of I is either a subinterval of I or a union [b, 1) ∪ [0, a] with 0 < a < b < 1, the latter union being the interval [b − 1, a] of I (modulo 1). So the cubes of Id are exactly those of Td . Accordingly, functions on Td

1.4. HARDY SPACES

15

and Id are identified too; they are considered as 1-periodic functions on Rd . Thus N = L∞ (Td )⊗M = L∞ (Id )⊗M. We define BMOc (Td , M) to be the space of all f ∈ L2 (N ) such that  1 1   2  2  f (z)− 1 f BMOc = max fTd M , < ∞. sup f (w)dw dz |Q| Q M Q⊂Td cube |Q| Q The row BMOr (Td , M) consists of all f such that f ∗ ∈ BMOc (Td , M), equipped with f BMOr = f ∗ BMOc . Finally, we define mixture space BMO(Td , M) as the intersection of the column and row BMO spaces: BMO(Td , M) = BMOc (Td , M) ∩ BMOr (Td , M), equipped with f BMO = max( f BMOc , f BMOr ). As in the Euclidean case, these spaces can be characterized by the circular Poisson semigroup. Let Pr denote the circular Poisson kernel of Td :  r |m| z m , z ∈ Td , 0 ≤ r < 1. (1.6) Pr (z) = m∈Zd

The Poisson integral of f ∈ L1 (N ) is   Pr (f )(z) = f(m)r |m| z m . Pr (zw−1 )f (w)dw = Td

m∈Zd

Here f denotes, of course, the Fourier transform of f :  f (z) z −m dz. f(m) = Td

It is proved in [17] that (1.7)  1   2 f (z) − 1 sup f (w)dw dz ≈ sup Pr (|f − Pr (f )|2 ) N |Q| Q M 0≤r N, m

N0  ε 2−k pk (xn ) < . 1 + pk (xn ) 2

k=0

Consequently, d(xn , 0) < ε.



Definition 2.4. A distribution on Tdθ is a continuous linear functional on S  (Tdθ ) denotes the space of distributions.

S(Tdθ ).

As a dual space, S  (Tdθ ) is endowed with the w*-topology. We will use the bracket  ,  to denote the duality between S(Tdθ ) and S  (Tdθ ): F, x = F (x) for x ∈ S(Tdθ ) and F ∈ S  (Tdθ ). We list some elementary properties of distributions: (1) For 1 ≤ p ≤ ∞, the space Lp (Tdθ ) naturally embeds into S  (Tdθ ): an element y ∈ Lp (Tdθ ) induces a continuous functional on S(Tdθ ) by x → τ (yx). (2) S(Tdθ ) acts as a bimodule on S  (Tdθ ): for a, b ∈ S(Tdθ ) and F ∈ S  (Tdθ ), aF b is the distribution defined by x → F, bxa.

2.2. DEFINITIONS AND BASIC PROPERTIES

21

(3) The partial derivations ∂j extend to S  (Tdθ ) by duality: ∂j F, x = −F, ∂j x. For m ∈ Nd0 , we use again Dm to denote the associated partial derivation on S  (Tdθ ). (4) The Fourier transform extends to S  (Tdθ ): for F ∈ S  (Tdθ ) and m ∈ Zd , F(m) = F, (U m )∗ . The Fourier series of F converges to F according to any (reasonable) summation method:  F(m)U m . F = m∈Zd

2.2. Definitions and basic properties We begin this section with a simple observation on Fourier multipliers on S(Tdθ ) and S  (Tdθ ). Let φ : Zd → C be a function of polynomial growth. Then its associated Fourier multiplier Mφ is a continuous map on both S(Tdθ ) and S  (Tdθ ). Here and in the sequel, a generic element of S  (Tdθ ) is also denoted by x. Two specific Fourier multipliers will play a key role later: they are the Bessel and Riesz potentials. Definition 2.5. Let α ∈ R. Define Jα on Rd and Iα on Rd \ {0} by α

Jα (ξ) = (1 + |ξ|2 ) 2 and Iα (ξ) = |ξ|α . Their associated Fourier multipliers are the Bessel and Riesz potentials of order α, denoted by J a and I α , respectively. By the above observation, J α is a Fourier multiplier on S  (Tdθ ), and I α is also a Fourier multiplier on the subspace of S  (Tdθ ) of all x such that x (0) = 0. Note that α α J α = (1 − (2π)−2 Δ) 2 and I α = (−(2π)−2 Δ) 2 . Definition 2.6. Let 1 ≤ p ≤ ∞, k ∈ N and α ∈ R. (i) The Sobolev space of order k on Tdθ is defined to be   Wpk (Tdθ ) = x ∈ S  (Tdθ ) : Dm x ∈ Lp (Tdθ ) for each m ∈ Nd0 with |m|1 ≤ k , equipped with the norm x Wpk =





Dm x pp

p1

.

0≤|m|1 ≤k

(ii) The potential (or fractional) Sobolev space of order α is defined to be   Hpα (Tdθ ) = x ∈ S  (Tdθ ) : J α x ∈ Lp (Tdθ ) , equipped with the norm x Hpα = J α x p . In the above definition of x Wpk , we have followed the usual convention for p = ∞ that the right-hand side is replaced by the corresponding supremum. This convention will be always made in the sequel. We collect some basic properties of these spaces in the following: Proposition 2.7. Let 1 ≤ p ≤ ∞, k ∈ N and α ∈ R. (i) Wpk (Tdθ ) and Hpα (Tdθ ) are Banach spaces. (ii) The polynomial subalgebra Pθ of Tdθ is dense in Wpk (Tdθ ) and Hpα (Tdθ ) for 1 ≤ p < ∞. Consequently, S(Tdθ ) is dense in Wpk (Tdθ ) and Hpα (Tdθ ).

22

2. SOBOLEV SPACES

(iii) For any β ∈ R, J β is an isometry from Hpα (Tdθ ) onto Hpα−β (Tdθ ). In particular, J α is an isometry from Hpα (Tdθ ) onto Lp (Tdθ ). (iv) Hpα (Tdθ ) ⊂ Hpβ (Tdθ ) continuously whenever β < α. Proof. (iii) is obvious. It implies (i) for Hpα (Tdθ ). (i) It suffices to show that Wpk (Tdθ ) is complete. Assume that {xn }n ⊂ Wpk (Tdθ ) is a Cauchy sequence. Then for every |m|1 ≤ k, {Dm xn }n is a Cauchy sequence in Lp (Tdθ ), so Dm xn → ym in Lp (Tdθ ). Particularly, {Dm xn }n converges to ym in S  (Tdθ ). On the other hand, since xn → y0 in Lp (Tdθ ), for every x ∈ S(Tdθ ) we have τ (xn Dm x) → τ (y0 Dm x). Thus {Dm xn }n converges to Dm y0 in S  (Tdθ ). Consequently, Dm y0 = ym for |m|1 ≤ k. Hence, y0 ∈ Wpk (Tdθ ) and xn → y0 in Wpk (Tdθ ). (ii) Consider the square Fej´er mean

 |m1 |

|md | FN (x) = 1− ··· 1 − x (m)U m . N +1 N +1 d m∈Z , maxj |mj |≤N

By [17, Proposition 3.1], limN →∞ FN (x) = x in Lp (Tdθ ). On the other hand, FN commutes with Dm : FN (Dm x) = Dm FN (x). We then deduce that limN →∞ FN (x) = x in Wpk (Tdθ ) for every x ∈ Wpk (Tdθ ). Thus Pθ is dense in Wpk (Tdθ ). On the other hand, FN and J α commute; so by (iii), the density of Pθ in Lp (Tdθ ) implies its density in Hpα (Tdθ ). (iv) It is well known that if γ < 0, the inverse Fourier transform of Jγ is an integrable function on Rd (see [67, Proposition V.3]). Thus, Lemma 1.7 implies that J β−α is a bounded map on Lp (Tdθ ) with norm majorized by F −1 (Jβ−α ) L (Rd ) . 1 This is the desired assertion.  The following shows that the potential Sobolev spaces can be also characterized by the Riesz potential. Theorem 2.8. Let 1 ≤ p ≤ ∞. Then p1

x Hpα ≈ | x(0)|p + I α (x − x (0)) pp , where the equivalence constants depend only on α and d. Proof. By changing α to −α, we can assume α > 0. It suffices to show (0) = 0. By [67, Lemma V.3.2], JIαα is the Fourier transform I α x p ≈ J α x p for x of a bounded measure on Rd , which, together with Lemma 1.7, yields I α x p  J α x p . To show the converse inequality, let η be an infinite differentiable function on Rd such that η(ξ) = 0 for |ξ| ≤ 12 and η(ξ) = 1 for |ξ| ≥ 1, and let φ = Jα I−α η. Then the Fourier multiplier with symbol φIα coincides with J α on the subspace of distributions on Tdθ with vanishing Fourier coefficients at the origin. Thus we are reduced to proving F −1 (φ) ∈ L1 (Rd ). To that end, first observe that for any m ∈ Nd0 ,   m 1 D φ(ξ)  . |ξ||m|1 +2 Consider first the case d ≥ 3. Choose positive integers  and k such that d2 − 2 <  < d2 and k > d2 . Then by the Cauchy-Schwarz inequality and the Plancherel

2.2. DEFINITIONS AND BASIC PROPERTIES

theorem,

 |s|0

ωpk (x, ε)  |x|Wpk . εk

The converse inequality is proved similarly to the case k = 1. Let m ∈ Nd0 with |m|1 = k. For each j with mj > 0, using the Taylor formula of the function dεej (recalling that (e1 , · · · , ed ) denotes the canonical basis of Rd ), we get m

m

Δεejj x = εmj ∂j j x + o(εmj ) , which implies d 

m

Δεejj x = εk Dm x + o(εk ) as ε → 0.

j=0

Thus by the next lemma, we deduce d  m Δεejj x p + o(1)  ε−k ωpk (x, ε) + o(1), Dm x p ≤ ε−k j=0

whence the desired converse inequality by letting ε → 0. So the theorem is proved modulo the next lemma.  Lemma 2.22. Let u1 , · · · , uk ∈ Rd . Then  Δu1 · · · Δuk = (−1)|D| TuD ΔkuD , D⊂{1,··· ,k}

where the sum runs over all subsets of {1, · · · , k}, and where  1 uj . uD = uj , uD = j j∈D

j∈D

Consequently, for ε > 0 and x ∈ Lp (Tdθ ), Δu · · · Δu x ≈ ωpk (x, ε). sup 1 k p |u1 |≤ε,··· ,|uk |≤ε

2.5. THE LINK WITH THE CLASSICAL SOBOLEV SPACES

31

Proof. This is a well-known lemma in the classical setting; see [6, Lemma 5.4.11]). We outline its proof for the convenience of the reader. The above equality is equivalent to the corresponding one with Δu and Tu replaced by du and eu , respectively. Setting   v= u and w = − u , ∈D

∈D

for each 0 ≤ j ≤ k, we have k 

d( −j)u =

=1



(−1)k−|D|



e( −j)u

∈D

D⊂{1,··· ,k}

=



(−1)k−|D| ev (ew )j .

D⊂{1,··· ,k}



The left hand side is nonzero only for j = 0. Multiply by (−1) over 0 ≤ j ≤ k; then replacing u by

u

k−j

gives the desired equality.

k j

 and sum 

Remark 2.23. It might be interesting to note that in the commutative case, the proof of Theorem 2.20 shows ωp (x, ε) ωp (x, ε) = lim = ∇x p . ε→0 ε ε 0 0 on {ξ : 2−1 < |ξ| < 2}, (3.1)  ⎪ ⎪ ⎪ ϕ(2−k ξ) = 1, ξ = 0. ⎩ k∈Z

Note that if m ∈ Zd with m = 0, ϕ(2−k m) = 0 for all k < 0, so  ϕ(2−k m) = 1, m ∈ Zd \ {0} . k≥0

On the other hand, the support of the function ϕ(2−k ·) is equal to {ξ : 2k−1 ≤ |ξ| ≤ 2k+1 }, thus supp ϕ(2−k ·) ∩ supp ϕ(2−j ·) = ∅ whenver |j − k| ≥ 2; consequently, (3.2)

ϕ(2−k ·) = ϕ(2−k ·)

k+1 

ϕ(2−j ·),

k ≥ 0.

j=k−1

Therefore, the sequence {ϕ(2−k ·)}k≥0 is a Littlewood-Paley decomposition of Td , modulo constant functions. 35

36

3. BESOV SPACES

The Fourier multiplier on S  (Tdθ ) with symbol ϕ(2−k ·) is denoted by x → ϕ k ∗x:  ϕ(2−k m) x(m)U m . ϕ k ∗ x = m∈Zd

As noted in section 1.3, the convolution here has the usual sense. Indeed, let ϕk be the function whose Fourier transform is equal to ϕ(2−k ·), and let ϕ k be its 1-periodization, that is,  ϕk (s + m). ϕ k (s) = m∈Zd

k admits the following Fourier Viewed as a function on Td by our convention, ϕ series:  ϕ k (z) = ϕ(2−k m)z m . m∈Zd

Thus for any distribution f on Td , ϕ k ∗ f (z) =



ϕ(2−k m)f(m)z m .

m∈Zd

We will maintain the above notation throughout the remainder of the paper. We can now start our study of quantum Besov spaces. Definition 3.1. Let 1 ≤ p, q ≤ ∞ and α ∈ R. The associated Besov space on Tdθ is defined by   α α Bp,q (Tdθ ) = x ∈ S  (Tdθ ) : x Bp,q q and Bp,q (Tdθ ) ⊂ Bp,r (Tdθ ) for β < α and (ii) Bp,q (Tdθ ) ⊂ Bp,r 1 ≤ r ≤ ∞. α α (Tdθ ) and Bp,c (Tdθ ) for 1 ≤ p ≤ ∞ and 1 ≤ q < ∞. (iii) Pθ is dense in Bp,q 0 α d d (iv) The dual space of Bp,q (Tθ ) coincides isomorphically with Bp−α  ,q  (Tθ ) for  1 ≤ p ≤ ∞ and 1 ≤ q < ∞, where p denotes the conjugate index of α (Tdθ ) coincides isomorphically with p. Moreover, the dual space of Bp,c 0 d Bp−α  ,1 (Tθ ). α Proof. (i) To show the completeness of Bp,q (Tdθ ), let {xn }n be a Cauchy α d xn (0)}n converges to some y(0) in C, and for every sequence in Bp,q (Tθ ). Then { k ≥ 0, {ϕ k ∗ xn }n converges to some yk in Lp (Tdθ ). Let  yk . y = y(0) + k≥0

Since yk is supported in {m ∈ Zd : 2k−1 ≤ |m| ≤ 2k+1 }, the above series converges in S  (Tdθ ). On the other hand, by (3.2), as n → ∞, we have ϕ k ∗ xn =

k+1 

ϕ k ∗ ϕ j ∗ xn →

j=k−1

k+1 

ϕ k ∗ yj = ϕ k ∗ y.

j=k−1

α α We then deduce that y ∈ Bp,q (Tdθ ) and xn → y in Bp,q (Tdθ ). (ii) is obvious. α (Tdθ ) for finite q. For N ∈ N let (iii) We only show the density of Pθ in Bp,q

xN = x (0) +

N 

ϕ j ∗ x.

j=0

k ∗ (x − xN ) = ϕ k ∗ x for k > N + 1, Then by (3.1), ϕ k ∗ (x − xN ) = 0 for k ≤ N − 1, ϕ N ∗x− ϕ N ∗ ϕ N ∗x, ϕ N +1 ∗(x−xN ) = ϕ N +1 ∗x− ϕ N +1 ∗ ϕ N ∗x. and ϕ N ∗(x−xN ) = ϕ We then deduce  ≤2 2qkα ϕ k ∗ x qp → 0 as N → ∞. x − xN qBp,q α k≥N

38

3. BESOV SPACES

α α d (iv) In this part, we view Bp,q (Tdθ ) as Bp,c (Tdθ ) when q = ∞. Let y ∈ Bp−α  ,q  (Tθ ). 0 Define y (x) = τ (xy) for x ∈ Pθ . Then k+1      |y (x)| = x (0) y (0) + τ ϕ k ∗ x ϕ j ∗ y  k≥0

≤ | x(0) y (0)| +

j=k−1

  k+1 ϕ k ∗ x p ϕ j ∗ y p k≥0

j=k−1

α y −α .  x Bp,q B   p ,q

Thus by the density of Pθ in y defines a continuous functional on α Bp,q (Tdθ ). To prove the converse, we need a more notation. Given a Banach space X, let (X) be the weighted direct sum of (C, X, X, · · · ) in the q -sense, that is, this is α q the space of all sequences (a, x0 , x1 , · · · ) with a ∈ C and xk ∈ X, equipped with the norm q1

 |a|q + 2qkα xk q . α Bp,q (Tdθ ),

k≥0 α q (X) by its 2kα xk → 0

If q = ∞, we replace subspace cα 0 (X) consisting of sequences as k → ∞. Recall that the dual space (a, x0 , x1 , · · · ) such that −α ∗ α d α d of α q (X) is q  (X ). By definition, Bp,q (Tθ ) embeds into q (Lp (Tθ )) via x → α 1 ∗ x, · · · ). Now let  be a continuous functional on Bp,q (Tdθ ) for ( x(0), ϕ 0 ∗ x, ϕ p < ∞. Then by the Hahn-Banach theorem,  extends to a continuous functional d on α q (Lp (Tθ )) of unit norm, so there exists a unit element (b, y0 , y1 , · · · ) belonging d to −α q  (Lp (Tθ )) such that  α (x) = b x(0) + τ (yk ϕ k ∗ x), x ∈ Bp,q (Tdθ ). k≥0

Let y =b+



(ϕ k−1 ∗ yk + ϕ k ∗ yk + ϕ k+1 ∗ yk ) .

k≥0 d Then clearly y ∈ Bp−α  ,q  (Tθ ) and  = y when p < ∞. The same argument works for p = ∞ too. Indeed, for  as above, there exists a unit element (b, y0 , y1 , · · · ) d ∗ belonging to −α q  (L∞ (Tθ ) ) such that  α (x) = b x(0) + yk , ϕ k ∗ x, x ∈ Bp,q (Tdθ ). k≥0

Let y be defined as above. Then y is still a distribution and 1

    q 2q kα ϕ k ∗ y qL (Td )∗ < ∞. | y (0)|q + ∞

k≥0

θ

Since it is a polynomial, ϕ k ∗ y belongs to L1 (Tdθ ); and we have ϕ k ∗ y L∞ (Tdθ )∗ = ϕ k ∗ y L1 (Tdθ ) . Thus we are done for p = ∞ too.



To proceed further, we require some elementary lemmas. Recall that Jα (ξ) = α (1 + |ξ|2 ) 2 and Iα (ξ) = |ξ|α .

3.1. DEFINITIONS AND BASIC PROPERTIES

39

Lemma 3.4. Let α ∈ R and k ∈ N0 . Then −1 F (Jα ϕk )  2αk and F −1 (Iα ϕk )  2αk . 1 1 where the constants depend only on ϕ, α and d. Consequently, for x ∈ Lp (Tdθ ) with 1 ≤ p ≤ ∞, J α (ϕ k ∗ x) p  2αk ϕ k ∗ x p and I α (ϕ k ∗ x) p  2αk ϕ k ∗ x p . Proof. The first part is well-known and easy to check. Indeed, −1 F (Jα ϕk ) = 2αk F −1 ((4−k + | · |2 ) α2 ϕ) ; 1

−k

1

the function (4 +|·| ) ϕ is a Schwartz function supported by {ξ : 2−1 ≤ |ξ| ≤ 2}, whose all partial derivatives, up to a fixed order, are bounded uniformly in k, so α sup F −1 ((4−k + | · |2 ) 2 ϕ) < ∞. 2

α 2

1

k≥0

Similarly,

−1 F (Iα ϕk ) = 2αk F −1 (Iα ϕ) . 1 1

Since ϕk = ϕk (ϕk−1 + ϕk + ϕk+1 ), by Lemma 1.7, we obtain the second part.



Given a ∈ R+ , we define Di,a (ξ) = (2πiξi )a for ξ ∈ Rd , and Dia to be the associated Fourier multiplier on Tdθ . We set Da = D1,a1 · · · Dd,ad and Da = D1a1 · · · Ddad for any a = (a1 , · · · , ad ) ∈ Rd+ . Note that if a is a positive integer, Dia = ∂ia , so there does not exist any conflict of notation. The following lemma is well-known. We include a sketch of proof for the reader’s convenience (see the proof of Remark 1 in Section 2.4.1 of [74]). Lemma 3.5. Let ρ be a compactly supported infinitely differentiable function on Rd . Assume σ, β ∈ R+ and a ∈ Rd+ such that σ > d2 , β > σ − d2 and |a|1 > σ − d2 . Then the functions Iβ ρ and Da ρ belong to H2σ (Rd ). Proof. If σ is a positive integer, the assertion clearly holds in view of H2σ (Rd ) = On the other hand, Iβ ρ ∈ L2 (Rd ) = H20 (Rd ) for β > − d2 . The general case follows by complex interpolation. Indeed, under the assumption on σ and β, we can choose σ1 ∈ N, β1 , β0 ∈ R and η ∈ (0 , 1) such that W2σ (Rd ).

d d σ1 > σ, β1 > σ1 − , β0 > − , σ = ησ1 , β = (1 − η)β0 + ηβ1 . 2 2 For a complex number z in the strip {z ∈ C : 0 ≤ Re(z) ≤ 1} define 2

Fz (ξ) = e(z−η) |ξ|β0 (1−z)+β1 z ρ(ξ). Then

sup Fib L2  Iβ0 ρ L2 and sup F1+ib H σ1  Iβ1 ρ H σ1 . b∈R

2

b∈R

2

It thus follows that Iβ ρ = Fη ∈ (L2 (Rd ), H2σ1 (Rd ))η . The second assertion is proved in the same way.



The usefulness of the previous lemma relies upon the following well-known fact. Remark 3.6. Let σ >

d 2

and f ∈ H2σ (Rd ). Then −1 F (f )  f σ . 1 H 2

40

3. BESOV SPACES

The verification is extremely easy:    −1  −1 F (f ) = F (f )(s)ds + 1 |s|≤1



k≥0



|s|≤1

  −1 F (f )(s)ds

2k 0 on {ξ : 2−1 ≤ |ξ| ≤ 2}, ⎪ ⎪ ⎪ ⎨ F −1 (ψhI−α1 ) ∈ L1 (Rd ), (3.4) ⎪ ⎪ sup 2−α0 j F −1 (ψ(2j ·)ϕ) < ∞. ⎪ ⎩ j∈N0

1

The first nonvanishing condition above on ψ is a Tauberian condition. The integrability of the inverse Fourier transforms can be reduced to a handier criterion by means of the potential Sobolev space H2σ (Rd ) with σ > d2 ; see Remark 3.6. We will use the same notation for ψ as for ϕ. In particular, ψk is the inverse Fourier transform of ψ(2−k ·) and ψk is the Fourier multiplier on Tdθ with symbol ψ(2−k ·). It is to note that compared with [74, Theorem 2.5.1], we need not require α1 > 0 in the following theorem. This will have interesting consequences in the next section.

3.2. A GENERAL CHARACTERIZATION

43

Theorem 3.9. Let 1 ≤ p, q ≤ ∞ and α ∈ R. Assume α0 < α < α1 . Let ψ α (Tdθ ) iff satisfy the above assumption. Then a distribution x on Tdθ belongs to Bp,q

 q 1q 2kα ψk ∗ x p < ∞. k≥0

If this is the case, then

 q 1q α 2kα ψk ∗ x p x Bp,q ≈ | x(0)|q +

(3.5)

k≥0

with relevant constants depending only on ϕ, ψ, α, α0 , α1 and d. Proof. We will follow the pattern of the proof of [74, Theorem 2.4.1]. Given a function f on Rd , we will use the notation that f (k) = f (2−k ·) for k ≥ 0 and f (k) = 0 for k ≤ −1. Recall that fk is the inverse Fourier transform of f (k) and fk is the 1-periodization of fk :  fk (s) = fk (s + m). m∈Zd

In the following, we will fix a distribution x on Tdθ . Without loss of generality, we α,ψ when it assume x (0) = 0. We will denote the right-hand side of (3.5) by x Bp,q is finite. For clarity, we divide the proof into several steps. α Step 1. In the first two steps, we assume x ∈ Bp,q (Tdθ ). Let K be a positive integer to be determined later in step 3. By (3.1), we have ψ (j) =

∞ 

K 

ψ (j) ϕ(k) =

k=0

k=−∞

Then (3.6)

ψ (j) ϕ(j+k) +



ψj ∗ x =

∞ 

ψ (j) ϕ(j+k) on {ξ : |ξ| ≥ 1}.

k=K

j+k ∗ x + ψj ∗ ϕ

k≤K



j+k ∗ x . ψj ∗ ϕ

k>K

For the moment, we do not care about the convergence issue of the second series above, which is postponed to the last step. Let aj,k = 2jα ψj ∗ ϕ j+k ∗ x p . Then ∞ ∞ 1



   q q q q1 (3.7) x Bp,q α,ψ ≤ aj,k + aj,k . j=0

k≤K

j=0

k>K

We will treat the two sums on the right-hand side separately. For the first one, by the support assumption on ϕ and h, for k ≤ K, we can write (3.8)

ψ (j) (ξ)ϕ(j+k) (ξ) = 2kα1

ψ (j) (ξ) (j+K) h (ξ)|2−j−k ξ|α1 ϕ(j+k) (ξ) |2−j ξ|α1

= 2kα1 η (j) (ξ)ρ(j+k) (ξ), where η and ρ are defined by ψ(ξ) η(ξ) = α1 h(K) (ξ) and ρ(ξ) = |ξ|α1 ϕ(ξ). |ξ| Note that F −1 (η) is integrable on Rd . Indeed, write (3.9)

η(ξ) =

ψ(ξ) ψ(ξ) h(ξ) + α1 (h(K) (ξ) − h(ξ)). α 1 |ξ| |ξ|

44

3. BESOV SPACES

By (3.4), the inverse Fourier transform of the first function on the right-hand side is integrable. The second one is an infinitely differentiable function with compact support, so its inverse Fourier transform is also integrable with L1 -norm controlled by a constant depending only on ψ, h, α1 and K. Therefore, Lemma 1.7 implies that each η (j) is a Fourier multiplier on Lp (Tdθ ) for all 1 ≤ p ≤ ∞ with norm controlled by a constant c1 , depending only on ψ, h, α1 and K. Therefore,   ρj+k ∗ x p = c1 2k(α1 −α) 2(j+k)α  ρj+k ∗ x p . (3.10) aj,k ≤ c1 2jα+kα1  Thus by triangular inequality and Lemma 3.4, we deduce ∞

  j=0

aj,k

q q1

≤ c1

k≤K



2k(α1 −α)

∞

  (j+k)α q q1 2  ρj+k ∗ x p j=−∞

k≤K

∞

   jα −jα1 α1 q 1q 2 2 = c1 2k(α1 −α) I ϕ j ∗ x p

≤

k≤K  α c1 x Bp,q

j=0

,

where c1 depends only on ψ, h, K, α and α1 . Step 2. The second sum on the right-hand side of (3.7) is treated similarly. Like in step 1 and by (3.2), we write (3.11) ψ (j) (ξ) (ϕ(j+k−1) + ϕ(j+k) + ϕ(j+k+1) )(ξ)|2−j−k ξ|α0 ϕ(j+k) (ξ) |2−j−k ξ|α0   ψ(2−j−k · 2k ξ) H(2−j−k ξ) ρ(j+k) (ξ), = −j−k α 0 |2 ξ|

ψ (j) (ξ)ϕ(j+k) (ξ) =

where H = ϕ(−1) + ϕ + ϕ(1) , and where ρ is now defined by ρ(ξ) = |ξ|α0 ϕ(ξ). The L1 -norm of the inverse Fourier transform of the function ψ(2−j−k · 2k ξ) H(2−j−k ξ) |2−j−k ξ|α0 is equal to F −1 (I−α0 Hψ(2k ·)) 1 . Using Lemma 3.4, we see that the last norm is majorized by −1 F (ψ(2k ·)H)  F −1 (ψ(2k ·)ϕ) . 1 1 Then, using (3.4), for k > K we get (3.12)

  ρj+k ∗ x p , aj,k ≤ c2 2k(α0 −α) 2(j+k)α 

where c2 depends only on ϕ, α0 and the supremum in (3.4). Thus as before, we get ∞

  j=0

k>K

aj,k

q 1q

≤ c2

2K(α0 −α) α , x Bp,q 1 − 2α0 −α

which, together with the inequality obtained in step 1, yields x Bp,q α,ψ  x B α . p,q

3.2. A GENERAL CHARACTERIZATION

45

Step 3. Now we prove the inequality reverse to the previous one. We first assume that x is a polynomial. We write ϕ(j) = ϕ(j) h(j+K) =

(3.13)

ϕ(j) (j+K) (j) h ψ . ψ (j)

The function ϕψ −1 is an infinitely differentiable function with compact support, so its inverse Fourier transform belongs to L1 (Rd ). Thus by Lemma 1.7, hj+K ∗ ψj ∗ x p , ϕ j ∗ x p ≤ c3  where c3 = F −1 (ϕψ −1 ) 1 . Hence, α x Bp,q ≤ c3

∞

  jα q q1 2  hj+K ∗ ψj ∗ x p . j=0

To handle the right-hand side, we let λ = 1 − h and write h(j+K) ψ (j) = ψ (j) − λ(j+K) ψ (j) . Then ∞ ∞



  jα q q1  jα  1 j+K ∗ ψj ∗ x p q q . 2  2 λ hj+K ∗ ψj ∗ x p ≤ x Bp,q α,ψ + j=0

j=0

Thus it remains to deal with the last sum. We do this as in the previous steps with ψ replaced by λψ, by writing λ(j+K) ψ (j) =

∞ 

λ(j+K) ψ (j) ϕ(j+k) .

k=−∞

The crucial point now is the fact that λ(j+K) ϕ(j+k) = 0 for all k ≤ K and all j. So  λ(j+K) ψ (j) ϕ(j+k) , λ(j+K) ψ (j) = k>K

that is, only the second sum on the right-hand side of (3.7) survives now: ∞ ∞

 1

  jα  q 1q j+K ∗ ψj ∗ x p )q q ≤ j+K ∗ ψj ∗ ϕ 2 (2jα λ λ j+k ∗ x p . j=0

j=0

k>K

(K) ψ instead Let us reexamine the argument of step 2 and formulate (3.11) with λ  of ψ. We then arrive at majorizing the norm F −1 λ(2k−K ·)ψ(2k ·)ϕ 1 : −1  k−K    F λ(2 ·)ψ(2k ·)ϕ 1 ≤ F −1 (λ) 1 F −1 ψ(2k ·)ϕ 1 .

Keeping the notation of step 2 and as for (3.12), we get   j+K ∗ ψj ∗ ϕ j+k ∗ x p ≤ cc2 2k(α0 −α) 2(j+k)α  ρj+k ∗ x p , λ where c = F −1 (λ) 1 . Thus ∞

  αj  q q1 2K(α0 −α) j+K ∗ ψj ∗ ϕ α . 2 λ j+k ∗ x p ≤ cc2 x Bp,q 1 − 2α0 −α j=0 k>K

Combining the preceding inequalities, we obtain  α x Bp,q ≤ c3 x Bp,q α,ψ + cc2

2K(α0 −α) α . x Bp,q 1 − 2α0 −α

46

3. BESOV SPACES

Choosing K so that c c2

2K(α0 −α) 1 ≤ , 1 − 2α0 −α 2

we finally deduce α x Bp,q ≤ 2c3 x Bp,q α,ψ ,

which shows (3.5) in case x is a polynomial. The general case can be easily reduced to this special one. Indeed, assume er means FN as in the proof of Proposition 2.7, x Bp,q α,ψ < ∞. Then using the Fej´ we see that FN (x) Bp,q α,ψ ≤ x α,ψ . Bp,q Applying the above part already proved for polynomials yields α FN (x) Bp,q ≤ 2c3 FN (x) Bp,q α,ψ ≤ 2c3 x α,ψ . Bp,q

However, it is easy to check that α α . lim FN (x) Bp,q = x Bp,q

N →∞

We thus deduce (3.5) for general x, modulo the convergence problem on the second series of (3.6). Step 4. We now settle up the convergence issue left previously. Each term ψj ∗ ϕ j+k ∗ x is a polynomial, so a distribution on Tdθ . We must show that the series converges in S  (Tdθ ). By (3.12), for any L > K, by the H¨older inequality (with q  the conjugate index of q), we get 2jα

L 

ψj ∗ ϕ j+k ∗ x p ≤ c2

k=K+1

L 

  2k(α0 −α) 2(j+k)α ϕ j+k ∗ x p

k=K+1

≤ c2 RK,L

L

 

2(j+k)α ϕ j+k ∗ x p

q q1

k=K+1 α ,  c2 RK,L x Bp,q

where RK,L =

L





2q k(α0 −α)

1 q

.

k=K+1

 Since α0 < α, RK,L → 0 as K tends to ∞. Thus the series k>K ψj ∗ ϕ j+k ∗ x converges in Lp (Tdθ ), so in S  (Tdθ ) too. In the same way, we show that the series α also converges in Bp,q (Tdθ ). Hence, the proof of the theorem is complete.  Remark 3.10. The infinite differentiability of ψ can be substantially relaxed without changing the proof. Indeed, we have used this condition only once to insure that the inverse Fourier transform of the second term on the right-hand side of (3.9) is integrable. This integrability is guaranteed when ψ is continuously differentiable up to order [ d2 ] + 1. The latter condition can be replaced by the following slightly weaker one: there exists σ > d2 + 1 such that ψη ∈ H2σ (Rd ) for any compactly supported infinitely differentiable function η which vanishes in a neighborhood of the origin.

3.2. A GENERAL CHARACTERIZATION

47

The following is the continuous version of Theorem 3.9. We will use similar notation for continuous parameters: given ε > 0, ψε denotes the function with Fourier transform ψ (ε) = ψ(ε·), and ψε denotes the Fourier multiplier on Tdθ associated to ψ (ε) . Theorem 3.11. Keep the assumption of the previous theorem. Then for any distribution x on Tdθ ,  1

q dε q1 α (3.14) x Bp,q ≈ | x(0)|q + ε−qα ψε ∗ x p . ε 0 The above equivalence is understood in the sense that if one side is finite, so is the other, and the two are then equivalent with constants independent of x. Proof. This proof is very similar to the previous one. Keeping the notation there, we will point out only the necessary changes. Let us first discretize the integral on the right-hand side of (3.14):  2−j  1 ∞  −α q dε  dε ≈ . ε ψε ∗ x p 2jqα ψε ∗ x qp ε ε −j−1 2 0 j=0 Now for j ≥ 0 and 2−j−1 < ε ≤ 2−j , we transfer (3.8) to the present setting: ψ (ε) (ξ)ϕ(j+k) (ξ) = 2α1 k

 ψ(2−j · 2j εξ) (j+K)  (j+k) h (ξ) ρ (ξ). |2−j ξ|α1

We then must estimate the L1 -norm of the inverse Fourier transform of the function in the brackets. It is equal to −1     F I−α ψ(2j ε·)h(K) = δ −α1 F −1 I−α ψh(δ2−K ·) , 1

1

1

1

−j −1

where δ = 2 ε . The last norm is estimated as follows: −1       F I−α1 ψh(δ2−K ·) 1 ≤ F −1 I−α1 ψh 1 + F −1 I−α1 ψ [h − h(δ2−K ·)] 1     ≤ F −1 I−α1 ψh + sup F −1 I−α1 ψ [h − h(δ2−K ·)] . 1

1

1≤δ≤2

Note that the above supremum is finite since I−α1 ψ[h − h(δ2−K ·)] is a compactly supported infinitely differentiable function and its inverse Fourier transform depends continuously on δ. It follows that for k ≤ K and 2−j−1 ≤ ε ≤ 2−j   (3.15) 2jα ψε ∗ ϕ j+k ∗ x p  2k(α1 −α) 2(j+k)α  ρj+k ∗ x p , which is the analogue of (3.10). Thus, we get the continuous analogue of the final inequality of step 1 in the preceding proof. We can make similar modifications in step 2, and then show the second part. Hence, we have proved

 1 q dε 1q α . ε−α ψε ∗ x p  x Bp,q ε 0 To show the converse inequality, we proceed as in step 3 above. By (3.4), there exists√a constant a > 2 such that ψ > 0 on {ξ : a−1 ≤ |ξ| ≤ a}. Assume also a ≤ 2 2. For j ≥ 0 let Rj = (a−1 2−j−1 , a2−j+1 ]. The Rj ’s are disjoint subintervals of (0, 1]. Now we slightly modify (3.13) as follows: (3.16)

ϕ(j) = ϕ(j) h(j+K) =

ϕ(j) (j+K) (ε) h ψ , ψ (ε)

ε ∈ Rj .

48

3. BESOV SPACES

Then −j −1  −1  ϕ(j)  −1  ϕ(δ)   < ∞. = F −1 ϕ(2 ε ·) ≤ F F sup 1 1 ψ ψ ψ (ε) 1 2a−1 ≤δ≤a2−1 Like in step 3, we deduce  ∞

  jα q dε q1 α x Bp,q 2   hj+K ∗ ψε ∗ x p ε Rj j=0  ∞

 1  q dε q1   jα q dε 1q ε−α ψε ∗ x p 2  +  hj+K ∗ ψε ∗ x p . ε ε Rj 0 j=0 The remaining of the proof follows step 3 with necessary modifications as in the first part.  Remark 3.12. Theorems 3.9 and 3.11 admit analogous characterizations for α α (Tdθ ) too. For example, a distribution x on Tdθ belongs to Bp,c (Tdθ ) iff Bp,c 0 0 ψε ∗ x p = 0. lim ε→0 εα This easily follows from Theorem 3.11 for q = ∞. The same remark applies to the characterizations by the Poisson, heat semigroups and differences in the next two sections. 3.3. The characterizations by Poisson and heat semigroups We now concretize the general characterization in the previous section to the case of Poisson and heat kernels. We begin with the Poisson case. Recall that P denotes the Poisson kernel of Rd and   ε (x) = P ε ∗ x = P e−2πε|m| x (m)U m . m∈Zd

So for any positive integer k, the kth derivation relative to ε is given by  ∂k  P (x) = (−2π|m|)k e−2πε|m| x (m)U m . ε ∂εk d m∈Z

The inverse of the kth derivation is the kth integration I k defined for x with x (0) = 0 by  ∞ ∞  ∞  ε (x) =  ε (x)dε1 · · · dεk−1 dεk P ··· Iεk P 1 ε

=

εk



ε2

(2π|m|)−k e−2πε|m| x (m)U m .

m∈Zd \{0}

In order to simplify the presentation, for any k ∈ Z, we define ∂k for k ≥ 0 and Jεk = Iε−k for k < 0. ∂εk It is worth to point out that all concrete characterizations in this section in terms of integration operators are new even in the classical case. Also, compare the following theorem with [74, Section 2.6.4], in which k is assumed to be a positive integer in the Poisson characterization, and a nonnegative integer in the heat characterization. Jεk =

3.3. THE CHARACTERIZATIONS BY POISSON AND HEAT SEMIGROUPS

49

Theorem 3.13. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z such that k > α. Then for any distribution x on Tdθ , we have  1 1

q  ε (x) q dε q . α x Bp,q ≈ | x(0)| + εq(k−α) Jεk P p ε 0  Proof. Recall that P = P1 . Thanks to P(ξ) = e−2π|ξ| , we introduce the k −2π|ξ| function ψ(ξ) = (−sgn(k)2π|ξ|) e . Then  ε (ξ). ψ(εξ) = εk Jεk e−2πε|ξ| = εk Jεk P α (Tdθ ), It follows that for x ∈ Bp,q

 ε ∗ x = εk Jεk P  ε (x) . ψε ∗ x = εk Jεk P Thus by Theorem 3.11, it remains to check that ψ satisfies (3.4) for some α0 < α < α1 . It is clear that the last condition there is verified for any α0 . For the second  So one, choosing k = α1 > α, we have I−α1 h ψ = (−sgn(k)2π)k h P. −1   F Ik−α1 h ψ 1 ≤ (2π)k F −1 (h) 1 P 1 < ∞ . 

The theorem is thus proved.

There exists an analogous characterization in terms of the heat kernel. Let Wε be the heat semigroup of Rd : Wε (s) =

1 (4πε)

d 2

e−

|s|2 4ε

.

ε be the periodization of Wε . Given a distribution x on Td let As usual, let W θ  √ ε (x) = W  ε m) ε ∗ x = W W( x(m)U m , m∈Zd

where W = W1 . Recall that 2 2  W(ξ) = e−4π |ξ| .

Theorem 3.14. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z such that k > for any distribution x on Tdθ ,  1 1

α ε (x) q dε q . α x Bp,q ≈ | x(0)|q + εq(k− 2 ) Jεk W p ε 0

α 2.

Then

Proof. This proof is similar to and simpler than the previous one. This time, 2 2 the function ψ is defined by ψ(ξ) = (−sgn(k)4π 2 |ξ|2 )k e−4π |ξ| . Clearly, it satisfies (3.4) for 2k = α1 > α and any α0 < α. Thus Theorem 3.11 holds for this choice of ψ. Note that a simple change of variables shows that the integral in (3.14) is equal to

 1 αq q dε q1 1 ε− 2 ψ√ε ∗ x p . 2− q ε 0 Then using the identity √ ε (ξ), ψ( ε ξ) = εk Jεk W we obtain the desired assertion.



50

3. BESOV SPACES

Now we wish to formulate Theorems 3.13 and 3.14 directly in terms of the circular Poisson and heat semigroups of Td . Recall that Pr denote the circular Poisson kernel of Td introduced by (1.6) and the Poisson integral of a distribution x on Tdθ is defined by  x (m)r |m| U m , 0 ≤ r < 1. Pr (x) = m∈Zd

Accordingly, we introduce the circular heat kernel W of Td :  2 r |m| z m , z ∈ Td , 0 ≤ r < 1. (3.17) Wr (z) = m∈Zd

Then for x ∈ S



(Tdθ )

we put



Wr (x) =

x (m)r |m| U m , 2

0 ≤ r < 1.

m∈Zd k

∂ As before, Jrk denotes the kth derivation ∂r k if k ≥ 0 and the (−k)th integration −k Ir if k < 0:  Jrk Pr (x) = Cm,k x (m)r |m|−k U m , m∈Zd

where Cm,k = |m| · · · (|m| − k + 1) if k ≥ 0 and

1 if k < 0. (|m| + 1) · · · (|m| − k) Jrk Wr (x) is defined similarly. Since |m| is not necessarily an integer, the coefficient Cm,k may not vanish for |m| < k and k ≥ 2. In this case, the corresponding term in Jrk Pr (x) above cause a certain problem of integrability since r (|m|−k)q is integrable on (0, 1) only when (|m| − k)q > −1. This explains why we will remove all these terms from Jrk Pr (x) in the following theorem. However, this difficulty does not occur for the heat semigroup. The following is new even in the classical case, that is, in the case of θ = 0. Cm,k =

Theorem 3.15. Let 1 ≤ p, q ≤ ∞, α ∈ R and k ∈ Z. Let x be a distribution on Tdθ . (i) If k > α, then  1

q dr q1 α x Bp,q ≈ max | x(m)|q + (1 − r)q(k−α) Jrk Pr (xk ) p , 1−r |m| 0 on {ξ : 2−1 ≤ |ξ| ≤ 2}.

=1

Now we reexamine step 3 of the proof of Theorem 3.9. To adapt it to the present setting, by an appropriate partition of unity, we decompose ϕ into a sum of infinitely differentiable functions, ϕ = ϕ1 + · · · + ϕL such that supp ϕ ⊂ Ω . Accordingly, for every j ≥ 0, let L  (j) ϕ(j) = ϕ . =1

Then we write the corresponding (3.16) with (ϕ , dnu ) in place of (ϕ, ψ) for every  ∈ {1, · · · , L}. Arguing as in step 3 of the proof of Theorem 3.9, we get L  1

 dε q1 q −qα  n ) ∗ x q x α  | x (0)| + ε . (d u ε Bp,q p ε 0 =1

 n ) ∗ x = Δn x, we deduce Since (d u ε εu  1

n q dε 1q q −qα x α  | Δεu x x (0)| + ε sup  Bp,q p ε 1≤ ≤L 0  1

dε 1q  | x(0)|q + ε−qα ωpn (x, ε)q . ε 0 

Thus the theorem is proved.

As a byproduct, the preceding theorem implies that the right-hand side of (3.18) does not depend on n with n > α, up to equivalence. This fact admits a direct simple proof and is an immediate consequence of the following analogue of Marchaud’s classical inequality which is of interest in its own right. Proposition 3.17. For any positive integers n and N with n < N and for any ε > 0, we have  ∞ N ωp (x, δ) dδ . 2n−N ωpN (x, ε) ≤ ωpn (x, ε)  εn δn δ ε N −n Proof. The argument below is standard. Using the identity ΔN Δnu , u = Δu we get N Δu (x) ≤ 2N −n Δnu (x) , p p

whence the lower estimate. The upper one is less obvious. By elementary calculations, for any u ∈ Rd , we have   j−1 n  n     n n  dn2u = 2n dnu + eiu dn+1 . eju − 2n dnu = 2n dnu + u j j j=0

j=0

In terms of Fourier multipliers, this means  j−1 n   n  n n n Δ2u = 2 Δu + Tiu Δn+1 . u j j=0

i=0

i=0

54

3. BESOV SPACES

It then follows that ωpn (x, ε) ≤

n n+1 ω (x, ε) + 2−n ωpn (x, 2ε). 2 p

Iterating this inequality yields ωpn (x, ε) ≤

J−1 n  n+1 ω (x, 2j ε) + 2−Jn ωpn (x, 2J ε), 2 j=1 p

from which we deduce the desired inequality for N = n + 1 as J → ∞. Another iteration argument then yields the general case.  In view of Definition 2.18 and Theorem 3.16, we introduce the following quantum analogue of the classical Lp -Zygmund class of order α. The case where 0 < α < 1 and p = ∞ was already studied by Weaver [79]. Definition 3.18. Let 1 ≤ p ≤ ∞, α > 0 and n be the smallest integer greater d than α. The Lp -Zygmund class of order α, Λα p (Tθ ), is defined to be the space of all distributions x such that ωpn (x) sup < ∞, εα ε>0 equipped with the norm x = | x(0)| + sup ε>0

The little Lp -Zygmund class of order α, sisting of all elements x such that

ωpn (x) . εα

d Λα p,0 (Tθ ),

d is the subspace of Λα p (Tθ ) con-

ωpn (x) = 0. ε→0 εα lim

α d α d Remark 3.19. Theorem 3.16 shows that Bp,∞ (Tdθ ) = Λα p (Tθ ) and Bp,c0 (Tθ ) = with equivalent norms. Consequently, the integer n in the above definition can be any integer greater than α. On the other hand, by the reduction Theorem 3.7, if α is not an integer and if k is the biggest integer less than α, then   α−k d  d k (Tdθ ), j = 1, · · · , d. . Λα p (Tθ ) = x ∈ S (Tθ ) : δj x ∈ Lipp d Λα p,0 (Tθ )

A similar equality holds for the little Lp -Zygmund and Lipschitz classes of order α. 3.5. Limits of Besov norms In this section we consider the behavior of the right-hand side of (3.18) as α → n. The study of this behavior is the subject of several recent publications in the classical setting; see, for instance, [4, 5, 38, 41, 75]. It originated from [14] in which Bourgain, Br´ezis and Mironescu proved that for any 1 ≤ p < ∞ and any f ∈ C0∞ (Rd ) 

p1 |f (s) − f (t)|p lim (1 − α) ds dt = Cp,d ∇f (t) p . αp+d α→1 Rd ×Rd |s − t| It is well known that

 1 p1

 ∞ |f (s) − f (t)|p Δu f p dε p . ds dt ≈ sup αp+d p ε Rd ×Rd |s − t| u∈Rd ,|u|≤ε 0

3.6. THE LINK WITH THE CLASSICAL BESOV SPACES

55

1 The right-hand side is the norm of f in the Besov space Bp,p (Rd ). Higher order extensions have been obtained in [38, 75]. The main result of the present section is the following quantum extension of these results. Let

 1 dε q1 α,ω = ε−αq ωpk (x, ε)q . (3.19) x Bp,q ε 0

Theorem 3.20. Let 1 ≤ p ≤ ∞, 1 ≤ q < ∞ and 0 < α < k with k an integer. Then for x ∈ Wpk (Tdθ ), −q α,ω ≈ q lim (k − α) q x Bp,q |x|Wpk 1

1

α→k

with relevant constants depending only on d and k. Proof. The proof is easy by using the results of section 2.4. Let x ∈ Wpk (Tdθ ) with x (0) = 0. Let A denote the limit in Lemma 2.21. Then  1  1 dε −αq k q dε q ≤A ε ωp (x, ε) ε(k−α)q , ε ε 0 0 whence



1

Aq dε ≤ . ε q α→k 0 Conversely, for any η > 0, choose δ ∈ (0, 1) such that lim sup(k − α)

ε−αq ωp (x, ε)q

ωpk (x, ε) ≥ A − η, ∀ε ≤ δ. εk Then



1

(k − α)

ε−αq ωpk (x, ε)q

0

which implies

 lim inf (k − α) α→k

Therefore,

1

(A − η)q (k−α)q dε ≥ δ , ε q

ε−αq ωpk (x, ε)q

0



dε (A − η)q ≥ . ε q

ωpk (x, ε) 1 dε = lim . α→k ε q ε→0 εk 0 So Theorem 2.20 implies the desired assertion. lim (k − α)

1

ε−αq ωpk (x, ε)q



α,ω when α → 0; Remark 3.21. We will determine later the behavior of x Bp,q see Corollary 5.20 below.

3.6. The link with the classical Besov spaces α Like for the Sobolev spaces on Tdθ , there exists a strong link between Bp,q (Tdθ ) and the classical vector-valued Besov spaces on Td . Let us give a precise definition of the latter spaces. We maintain the assumption and notation on ϕ in section 3.1. In particular, f → ϕ k ∗ f is the Fourier multiplier on Td associated to ϕ(2−k ·):  ϕ(2−k m)f(m)z m ϕ k ∗ f = m∈Zd 

for any f ∈ S (T ; X). Here X is a Banach space. d

56

3. BESOV SPACES

Definition 3.22. Let 1 ≤ p, q ≤ ∞ and α ∈ R. Define   α α (Td ; X) = f ∈ S  (Td ; X) : f Bp,q α, then for any f ∈ Bp,q (Td ; X),  1

q dr 1q α f Bp,q ≈ sup f(m) qX + (1 − r)(k−α)q Jrk Pr (fk ) L (Td ;X) , p 1−r |m|2|t|

2

k∈Z

2

The relevant constants depend only on ϕ, σ and d. The above kernel k defines a Calder´on-Zygmund operator on Rd . But we will consider only the periodic case, so we need to periodize k:   k(s + m). k(s) = m∈Zd

By a slight abuse of notation, we use  kj to denote the Calder´ on-Zygmund operator d  on T associated to kj too:    kj (f )(s) = kj (s − t)f (t)dt, Id

where we have identified T with I = [0, 1).  kj is the Fourier multiplier on Td with symbol φj : f → φj ∗ f .   We have  k = k Zd . If φ satisfies (4.2), then Lemma 4.3 implies ⎧  ⎪ ⎪ k ∞ (Zd ; 2 ) < ∞, ⎨   (4.3) ⎪ ⎪ sup  k(s − t) −  k(s) 2 ds < ∞. ⎩ t∈Id

{s∈Id :|s|>2|t|}

Now let M be a von Neumann algebra equipped with a normal faithful tracial state τ , and let N = L∞ (Td )⊗M, equipped with the tensor trace. The following lemma should be known to experts; it is closely related to similar results of [29, 46, 51], notably to [35, Lemma 2.3]. Note that the sole difference between the following condition (4.4) and (4.2) is that the supremum in (4.2) runs over all integers while the one below is restricted to nonnegative integers. Lemma 4.4. Let φ = (φj )j≥0 be a sequence of continuous functions on Rd \ {0} such that < ∞. (4.4) φ hσ = sup φ(2k ·)ϕ σ d 2

k≥0

H2 (R ; 2 )

Then for 1 < p < ∞ and any finite sequence (fj ) ⊂ Lp (N ),     1 1 |φj ∗ fj |2 2 p  φ hσ2 |fj |2 2 p . j≥0

j≥0

The relevant constant depends only on p, ϕ, σ and d.

62

4. TRIEBEL-LIZORKIN SPACES

Proof. The argument below is standard. First, note that the Fourier multiplier on Td with symbol φj does not depend on the values of φj in the open unit ball of Rd . So letting η be an infinitely differentiable function on Rd such that η(ξ) = 0 for |ξ| ≤ 12 and η(ξ) = 1 for |ξ| ≥ 1, we see that φj and ηφj induce the same Fourier multiplier on Td (restricted to distributions with vanishing Fourier coefficients at the origin). On the other hand, it is easy to see that (4.4) implies that the sequence (ηφj )j≥0 satisfies (4.2) with (ηφj )j≥0 in place of φ. Thus replacing φj by ηφj if necessary, we will assume that φ satisfies the stronger condition (4.2). We will use the Calder´on-Zygmund theory and consider  k as a diagonal matrix k is with diagonal entries ( kj )j≥1 . The Calder´on-Zygmund operator associated to  thus the convolution operator:    k(s − t)f (t)dt k(f )(s) = Id

for any finite sequence f = (fj ) (viewed as a column matrix). Then the assertion to prove amounts to the boundedness of  k on Lp (N ; c2 ).  We first show that k is bounded from L∞ (N ; c2 ) into BMOc (Td , B(2 )⊗M). Let f be a finite sequence in L∞ (N ; c2 ), and let Q be a cube of Id whose center is  = 2Q, the denoted by c. We decompose f as f = g + h with g = f 1Q , where Q cube with center c and twice the side length of Q. Setting   a= k(c − t)f (t)dt,  Id \Q

we have  k(f )(s) − a =  k(g)(s) + Thus



1 |Q|

where

 Id

( k(s − t) −  k(c − t))h(t)dt.

| k(f )(s) − a|2 ds ≤ 2(A + B),

Q

 1 | k(g)(s)|2 ds, |Q| Q   2 1   k(s − t) −  k(c − t))h(t)dt ds. B=  ( |Q| Q Id A=

The first term A is easy to estimate.   |Q|A ≤ | k(g)(s)|2 ds = Id

=

m∈Zd



     k(m)∗ g(m)∗   k(m) 2B( 2 ) | g (m)|2 k(m) g(m) ≤

m∈Zd

 ≤  k

Indeed, by (4.3) and the Plancherel formula,  2  k(m) g(m)

d ∞ (Z ; ∞ )

 ≤  k ∞ (Zd ; 2 )

m∈Zd

 

Id

Id

|g(s)|2 ds

 f 2 |g(s)|2 ds  |Q| L∞ (N ; c2 ) ,

whence A B( 2 )⊗M  f 2L∞ (N ; c2 ) .

4.1. A MULTIPLIER THEOREM

63

To estimate B, let h = (hj ). Then by (4.3), for any s ∈ Q we have  2    2        kj (s − t) −  kj (c − t))hj (t)dt  (  (k(s − t) − k(c − t))h(t)dt = Id



j

Id

j

 Id \Q

 



 Id \Q

| kj (s − t) −  kj (c − t)| |hj (t)|2 dt

 k(s − t) −  k(c − t) ∞ 

 f 2L∞ (N ; c2 )

 Id \Q



|hj (t)|2 dt

j

 k(s − t) −  k(c − t) 2 dt

 f 2L∞ (N ; c2 ) . Thus

  2 1 k(s − t) −  k(c − t))h(t)dt ds ( |Q| Q Id B( 2 )⊗M   2 1   k(s − t) −  k(c − t))h(t)dt ds =  ( |Q| Q B( 2 )⊗M Id

B B( 2 )⊗M ≤

 f 2L∞ (N ; c2 ) . Therefore,  k is bounded from L∞ (N ; c2 ) into BMOc (Td , B(2 )⊗M). We next show that  k is bounded from L∞ (N ; c2 ) into BMOr (Td , B(2 )⊗M). Let f, Q and a be as above. Now we have to estimate 1   ∗ 2   k(f )(s) − a  ds . |Q| Q B( 2 )⊗M We will use the same decomoposition f = g + h. Then   ∗ 2 1   k(f )(s) − a  ds ≤ 2(A + B  ), |Q| Q where

  ∗ 2 1   k(g)(s)  ds, |Q| Q    ∗ 2 1  ( k(s − t) −  k(c − t))h(t) dt ds. B =  |Q| Q Id A =

The estimate of B  can be reduced to that of B before. Indeed,    ∗ 1 2 ( k(s − t) −  k(c − t))h(t) dt B  B( 2 )⊗M ≤ ds |Q| Q Id B( 2 )⊗M    ∗ 2 1 ( k(s − t) −  k(c − t))h(t)dt ds = |Q| Q B( 2 )⊗M d I   2 1 = k(s − t) −  k(c − t))h(t)dt ds ( |Q| Q Id B( 2 )⊗M  f 2L∞ (N ; c2 ) .

64

4. TRIEBEL-LIZORKIN SPACES

However, A needs a different argument. Setting g = (gj ), we have ! 1    "  A B( 2 )⊗M = sup τ ki (gi )(s)kj (gj )(s)∗ a∗j ai ds , |Q| Q i,j where the supremum runs over all a = (ai ) in the unit ball of 2 (L2 (M)). Considering ai as a constant function on Id , we can write ai ki (gi ) = ki (ai gi ). Thus

 τ



Q



 ki (gi )(s)kj (gj )(s)∗ a∗j ai ds =

2  ki (ai gi )(s)

L2 (M)

Q

i,j

ds.

i

So by the Plancherel formula,   2 2    ki (ai gi )(s) L2 (M) ds ≤ ki (ai gi )(s) L2 (M) ds Q

Id

i

i

   2 = ki (m) ai gi (m) L2 (M) . i

m∈Zd

On the other hand, by the Cauchy-Schwarz inequality, (4.3) and the Plancherel formula once more, we have       2  k(m) 2 2 τ (|ai gi (m)|2 ) ki (m) ai gi (m) L2 (M) ≤ m∈Zd

i

i

m∈Zd

     τ ai gi (m)gi (m)∗ a∗i i

m∈Zd

i



    = τ ai gi (s)gi (s)∗ ds a∗i =

  τ ai i

Id

 Q

fi (s)fi (s)∗ ds a∗i



    ≤ |Q| τ ai fi 2L∞ (N ) a∗i i

 |Q| f 2L∞ (N ; c2 )



τ (|ai |2 )

i

≤

|Q| f 2L∞ (N ; c2 )

.

Combining the above estimates, we get the desired estimate of A : A B( 2 )⊗M  f 2L∞ (N ; c2 ) . Thus,  k is bounded from L∞ (N ; c2 ) into BMOr (Td , B(2 )⊗M), so is it from L∞ (N ; c2 ) into BMO(Td , B(2 )⊗M). It is clear that  k is bounded from L2 (N ; c2 ) into L2 (B(2 )⊗N ). Hence, by interpolation via (1.2) and Lemma 1.8,  k is bounded from Lp (N ; c2 ) into Lp (B(2 )⊗N ) for any 2 < p < ∞. This is the announced assertion for 2 ≤ p < ∞. The case 1 < p < 2 is obtained by duality. 

4.1. A MULTIPLIER THEOREM

65

Remark 4.5. In the commutative case, i.e., M = C, it is well known that the conclusion of the preceding lemma holds under the following weaker assumption on φ:

 12 2 (1 + |s|2 )σ F −1 (φ(2k ·)ϕ)(s) ds < ∞. (4.5) sup k≥0

∞

Rd

Like at the beginning of the preceding proof, this assumption can be strengthened to

 12 2 sup (1 + |s|2 )σ F −1 (φ(2k ·)ϕ)(s) ds < ∞. k∈Z

∞

Rd

Then if we consider k = (kj )j≥0 as a kernel with values in ∞ , Lemma 4.3 admits the following ∞ -analogue: •  k L∞ (Rd ; ∞ ) < ∞;  • sup k(s − t) − k(s) ∞ ds < ∞. t∈Rd

|s|>2|t|

Transferring this to the periodic case, we have  •  k ∞ (Zd ; ∞ ) < ∞;  • sup  k(s − t) −  k(s) ∞ ds < ∞. t∈Id

{s∈Id :|s|>2|t|}

The last two properties of the kernel  k are exactly what is needed for the estimates of A and B in the proof of Lemma 4.4, so the conclusion holds when M = C. However, we do not know whether Lemma 4.4 remains true when (4.4) is weakened to (4.5). Lemma 4.6. Let φ = (φj )j≥0 be a sequence of continuous functions on Rd \ {0} satisfying (4.4). Then for 1 ≤ p ≤ 2 and any f ∈ Hpc (Td , M),   1 |φj ∗ f |2 2 Lp (N )  φ hσ2 f Hcp . j≥0

The relevant constant depends only on ϕ, σ and d. Proof. Like in the proof of Lemma 4.4, we can assume, without loss of generality, that φ satisfies (4.2). We use again the Calder´on-Zygmund theory. Now we view  k = ( kj )j≥0 as a column matrix and the associated Calder´on-Zygmund  operator k as defined on Lp (N ):    k(s − t)f (t)dt. k(f )(s) = Id

Thus  k maps functions to sequences of functions. We have to show that  k is bounded from Hpc (Td , M) to Lp (N ; c2 ) for 1 ≤ p ≤ 2. This is trivial for p = 2. So by Lemma 1.9 via interpolation, it suffices to consider the case p = 1. The argument below is based on the atomic decomposition of H1c (Td , M) obtained in [17] (see also [44]). Recall that an Mc -atom is a function a ∈ L1 (M; Lc2 (Td )) such that • a is supported by a cube Q ⊂ Td ≈ Id ; • Q a(s)ds = 0; 1   1 |a(s)|2 ds 2 ≤ |Q|− 2 . • τ Q

66

4. TRIEBEL-LIZORKIN SPACES

Thus we need only to show that for any atom a  k(a) L1 (N ; c2 )  1. Let Q be the supporting cube of a. By translation invariance of the operator  k, we  = 2Q as before. Then can assume that Q is centered at the origin. Set Q (4.6)

k(a)1Q L1 (N ; c2 ) +  k(a)1Id \Q L1 (N ; c2 ) .  k(a) L1 (N ; c2 ) ≤ 

The operator convexity of the square function x → |x|2 implies  12

  12 |k(a)(s)|ds ≤ |Q| | k(a)(s)|2 ds .  Q

 Q

However, by the Plancherel formula,    #   | k(a)(s)|2 ds ≤ | k(a)(s)|2 ds = | k(a)(m)|2 = | k(m) a(m)|2  Q

Id



 ≤  k

∞ (Zd ; 2 )

m∈Zd

m∈Zd

 | a(m)|2 =  k ∞ (Zd ; 2 )

m∈Zd

Therefore, by (4.3)  k(a)1Q L1 (N ; c2 ) = τ

  Q

 12 τ | k(a)(s)|ds  |Q|



 |a(s)|2 ds . Q

 |a(s)|2 ds

 12 

 1.

Q

This is the desired estimate of the first term of the right-hand side of (4.6). For  we can write the second, since a is of vanishing mean, for every s ∈ Q   k(a)(s) = [ k(s − t) −  k(s)]a(t)dt. Q

Then by the Cauchy-Schwarz inequality via the operator convexity of the square function x → |x|2 , we have   | k(a)(s)|2 ≤  k(s − t) −  k(s) 2 dt ·  k(s − t) −  k(s) 2 |a(t)|2 dt. Q

Q

Thus by (4.3),

  c k(a)1Id \Q L1 (N ; 2 ) = τ | k(a)(s)|ds  Id \Q    1 τ  k(s − t) −  k(s) 2 ds dt 2  Q Id \Q    1 ·  k(s − t) −  k(s) 2 |a(t)|2 ds dt 2 d  Q I \Q   1  |a(s)|2 ds 2  1.  |Q|1/2 τ Q

Hence the desired assertion is proved.



By transference, the previous lemmas imply the following. According to our convention used in the previous chapters, the map x → φ ∗ x denotes the Fourier multiplier associated to φ on Tdθ . Lemma 4.7. Let φ = (φj )j satisfy (4.4).

4.1. A MULTIPLIER THEOREM

67

(i) For 1 < p < ∞ we have     1 1 |φj ∗ xj |2 2 p  φ hσ2 |xj |2 2 p , j≥0

xj ∈ Lp (Tdθ )

j≥0

with relevant constant depending only on p, ϕ, σ and d. (ii) For 1 ≤ p ≤ 2 we have   1 |φj ∗ x|2 2 p  φ hσ2 x Hcp , x ∈ Hpc (Tdθ ) j≥0

with relevant constant depending only on ϕ, σ and d. We are now ready to prove Theorem 4.1. Proof of Theorem 4.1. Let ζj = φj (ϕ(j−1) + ϕ(j) + ϕ(j+1) ). By (3.2) and the support assumption on φj ρj , we have φj ρj = ζj ρj , so φj ∗ ρj ∗ x = ζj ∗ ρj ∗ x for any distribution x on Tdθ . We claim that ζ = (ζj )j≥0 satisfies (4.4) in place of φ. Indeed, given k ∈ N0 , by the support assumption on ϕ in (3.1), the sequence ζ(2k ·)ϕ = (ζj (2k ·)ϕ)j≥0 has at most five nonzero terms of indices j such that k − 2 ≤ j ≤ k + 2. Thus k ζ(2 ·)ϕ

H2σ (Rd ; 2 )

k+2 

≤

ζj (2k ·)ϕ σ d . H (R ) 2

j=k−2

However, by Lemma 4.2, ζj (2k ·)ϕ σ d  sup φj (2j+k ·)ϕ σ d , H (R ) H (R ) 2

−2≤k≤2

2

k − 2 ≤ j ≤ k + 2,

where the relevant constant depends only on d, σ and ϕ. Therefore, the second condition of (4.1) yields the claim. Now applying Lemma 4.7 (i) with ζj instead of φj and xj = 2jα ϕ j ∗ x, we prove part (i) of the theorem. To show part (ii), we need the characterization of H1c (Tdθ ) by discrete square functions stated in Lemma 1.10 with ψ = I−α ρ. Let x be a distribution on Tdθ with x (0) = 0 such that   jα 1 2 | ρj ∗ x|2 2 < ∞. 1

j≥0

Let y = I α (x). Then the discrete square function of y associated to ψ is given by   |ψj ∗ y|2 = 2jα | ρj ∗ x|2 . scψ (y)2 = j≥0

So y ∈

H1c (Tdθ )

and

j≥0

  jα 1 y Hc1 ≈ 2 | ρj ∗ x|2 2 1 . j≥0

We want to apply Lemma 4.7 (ii) to y but with a different multiplier in place of φ. To that end, let ηj = 2jα I−α φj and η = (ηj )j≥0 . We claim that η satisfies (4.1) too. The support condition of (4.1) is obvious for η. To prove the second one, by (3.2), we write ηj (2j ξ)ϕ(ξ) = |ξ|−α ϕ(ξ)φj (2j ξ) = |ξ|−α [ϕ(2−1 ξ) + ϕ(ξ) + ϕ(2ξ)]ϕ(ξ)φj (2j ξ).

68

4. TRIEBEL-LIZORKIN SPACES

Since I−α (ϕ(−1) + ϕ + ϕ(1) ) is an infinitely differentiable function with compact support,    (1 + |s|2 )σ F −1 (I−α (ϕ(−1) + ϕ + ϕ(1) ))(s)ds < ∞. Rd

Thus by Lemma 4.2, ηj (2j+k ·)ϕ s d  φj (2j+k ·)ϕ s d , H (R ) H (R ) 2

2

whence the claim. As in the first part of the proof, we define a new sequence ζ by setting ζj = ηj ρj . Then the new sequence ζ satisfies (4.4) too and sup ζj (2k ·)ϕ σ d  sup ηj (2j+k ·)ϕ σ d  sup φj (2j+k ·)ϕ σ d . H2 (R )

k≥0

H2 (R )

j≥0 −2≤k≤2

j≥0 −2≤k≤2

H2 (R )

On the other hand, we have 2jα φj ∗ ρj ∗ x = ζj ∗ y . Thus we can apply Lemma 4.7 (ii) to y with this new ζ instead of φ, and as before, we get     2jα 1 1 2 |φj ∗ ρj ∗ x|2 2 1 = |ζj ∗ y|2 2 1 j≥0

j≥0

 sup ζ(2k ·)ϕ H σ (Rd ; ) y Hcp 2

k≥0



2

  jα 1 sup φj (2j+k ·)ϕ H σ (Rd ) 2 | ρj ∗ x|2 2 1 . 2

j≥0 −2≤k≤2

j≥0

Hence the proof of the theorem is complete.



4.2. Definitions and basic properties As said at the beginning of this chapter, we consider the Triebel-Lizorkin spaces on Tdθ only for q = 2. In this case, there exist three different families of spaces according to the three choices of the internal 2 -norms. Definition 4.8. Let 1 ≤ p < ∞ and α ∈ R. (i) The column Triebel-Lizorkin space Fpα,c (Tdθ ) is defined by   Fpα,c (Tdθ ) = x ∈ S  (Tdθ ) : x Fpα,c < ∞ , where

  2kα 1 x Fpα,c = | x(0)| + 2 |ϕ k ∗ x|2 2 p . k≥0

(ii) The row space Fpα,r (Tdθ ) consists of all x such that x∗ ∈ Fpα,c (Tdθ ), equipped with the norm x Fpα,r = x∗ Fpα,c . (iii) The mixture space Fpα (Tdθ ) is defined to be  α,c d if 1 ≤ p < 2, Fp (Tθ ) + Fpα,r (Tdθ ) α d Fp (Tθ ) = α,c d α,r d Fp (Tθ ) ∩ Fp (Tθ ) if 2 ≤ p < ∞,

4.2. DEFINITIONS AND BASIC PROPERTIES

equipped with    inf y Fpα,c + z Fpα,r : x = y + z x Fpα = max( x Fpα,c , x Fpα,r )

69

if 1 ≤ p < 2, if 2 ≤ p < ∞.

In the sequel, we will concentrate our study only on the column Triebel-Lizorkin spaces. All results will admit the row and mixture analogues. The following shows that Fpα,c (Tdθ ) is independent of the choice of the function ϕ. Proposition 4.9. Let ψ be a Schwartz function satisfying the same condition (3.1) as ϕ. Let ψk = ψ (k) = ψ(2−k ·). Then   2kα 1 x Fpα,c ≈ | x(0)| + 2 |ψk ∗ x|2 2 p . k≥0

Tdθ

Proof. Fix a distribution x on with x (0) = 0. By the support assumption −1 = 0) on ψ (k) and (3.2), we have (with ϕ ψk ∗ x =

1 

k+j ∗ x. ψk ∗ ϕ

j=−1

Thus by Theorem 4.1, 1    2kα   2kα 1 1 2 |ψk ∗ x|2 2 p ≤ 2 |ψk ∗ ϕ k+j ∗ x|2 2 p k≥0

j=−1

k≥0

  2kα 1  2 |ϕ k ∗ x|2 2 p . k≥0

Changing the role of ϕ and ψ, we get the reverse inequality.



Proposition 4.10. Let 1 ≤ p < ∞ and α ∈ R. (i) Fpα,c (Tdθ ) is a Banach space. (ii) Fpα,c (Tdθ ) ⊂ Fpβ,c (Tdθ ) for β < α. (iii) Pθ is dense in Fpα,c (Tdθ ) . (iv) Fp0,c (Tdθ ) = Hpc (Tdθ ). α d α,c d α d (v) Bp, min(p,2) (Tθ ) ⊂ Fp (Tθ ) ⊂ Bp, max(p,2) (Tθ ). Proof. (i) is proved as in the case of Besov spaces; see the corresponding proof of Proposition 3.3. (ii) is obvious. To show (iii), we use the Fej´er means as in the proof of Proposition 2.7. We need one more property of those means, that is, they are completely contractive. So they are also contractive on Lp (B(2 )⊗Tdθ ), in particular, on the column subspace Lp (Tdθ ; c2 ) too. We then deduce that FN is contractive on Fpα,c (Tdθ ) and limN →∞ FN (x) = x for every x ∈ Fpα,c (Tdθ ). (iv) has been already observed during the proof of Theorem 4.1. Indeed, for any distribution x on Tdθ , the square function associated to ϕ defined in Lemma 1.10 is given by 1  |ϕ k ∗ x|2 2 . scϕ (x) = k≥0

Thus x Hcp ≈ x Fp0,c . (v) follows from the following well-known property: 2 (Lp (Tdθ )) ⊂ Lp (Tdθ ; c2 ) ⊂ p (Lp (Tdθ ))

70

4. TRIEBEL-LIZORKIN SPACES

are contractive inclusions for 2 ≤ p ≤ ∞; both inclusions are reversed for 1 ≤ p ≤ 2. Note that the first inclusion is an immediate consequence of the triangular inequality  of L p2 (Tdθ ), the second is proved by complex interpolation. The following is the Triebel-Lizorkin analogue of Theorem 3.7. We keep the notation introduced before that theorem. Theorem 4.11. Let 1 ≤ p < ∞ and α ∈ R. (i) For any β ∈ R, both J β and I β are isomorphisms between Fpα,c (Tdθ ) and Fpα−β,c (Tdθ ). In particular, J α and I α are isomorphisms between Fpα,c (Tdθ ) and Hpc (Tdθ ). α−|a|1 ,c

(ii) Let a ∈ Rd+ . If x ∈ Fpα,c (Tdθ ), then Da x ∈ Fp

(Tdθ ) and

Da x F α−|a|1 ,c  x Fpα,c . p

(iii) Let β > 0. Then x ∈ Fpα,c (Tdθ ) iff Diβ x ∈ Fpα−β,c (Tdθ ) for all i = 1, · · · , d. Moreover, in this case, x Fpα,c ≈ | x(0)| +

d 

Diβ x Fpα−β,c .

i=1

Proof. (i) Let x ∈ with x (0) = 0. By Theorem 4.1,   1 J β x Fpα−β,c = 22k(α−β) |J β ∗ ϕ k ∗ x|2 2 p Fpα,c (Tdθ )

k≥0

  2kα 1  sup 2−kβ Jβ (2k ·)ϕ H2σ (Rd ) 2 |ϕ k ∗ x|2 2 p . k≥0

k≥0

However, it is easy to see that all partial derivatives of the function 2−kβ Jβ (2k ·)ϕ, of order less than a fixed integer, are bounded uniformly in k. It then follows that sup 2−kβ Jβ (2k ·)ϕ H2σ (Rd ) < ∞. k≥0

Thus J x Fpα−β,c  x Fpα,c . So J β is bounded from Fpα,c (Tdθ ) to Fpα−β,c (Tdθ ), its inverse, which is J −β , is bounded too. I β is handled similarly. If β = α, then Fpα−β,c (Tdθ ) = Fp0,c (Tdθ ) = Hpc (Tdθ ) by Proposition 4.10 (iv). (ii) This proof is similar to the previous one by replacing J β by Da and using Lemma 3.5. (iii) One implication is contained in (ii). To show the other, we follow the proof of Theorem 3.7 (iii) and keep the notation there. Since β

ϕk =

d 

χi Di,β ϕk ,

i=1

by Theorem 4.1, x

Fpα,c

d    2kα 1 ≤ 2 | χi ∗ ϕ k ∗ Dβ x|2 2 i

i=1



d 

p

k≥0

  2k(α−β) 1 sup 2kβ χi (2k ·)ϕ H2σ (Rd ) 2 |ϕ k ∗ Diβ x|2 2 p .

i=1 k≥0

k≥0

4.2. DEFINITIONS AND BASIC PROPERTIES

71

However, 2kβ χi (2k ·)ϕ H2σ (Rd ) = ψϕ H2σ (Rd ) , where

χ(2k ξi )|ξi |β 1 . χ(2k ξ1 )|ξ1 |β + · · · + χ(2k ξd )|ξd |β (2πiξi )β As all partial derivatives of ψϕ, of order less than a fixed integer, are bounded uniformly in k, the norm of ψϕ in H2σ (Rd ) are controlled by a constant independent of k. We then deduce ψ(ξ) =

x Fpα,c 

d d     2k(α−β) 1 2 |ϕ k ∗ Diβ x|2 2 p = Diβ x Fpα−β,c . i=1

i=1

k≥0



The theorem is thus completely proved.

Corollary 4.12. Let 1 < p < ∞ and α ∈ R. Then Fpα (Tdθ ) = Hpα (Tdθ ) with equivalent norms. Proof. Since J α is an isomorphism from Fpα (Tdθ ) onto Fp0 (Tdθ ), and from α Hp (Tdθ ) onto Hp0 (Tdθ ), it suffices to consider the case α = 0. But then Hp0 (Tdθ ) = Lp (Tdθ ) by definition, and Fp0 (Tdθ ) = Hp (Tdθ ) by Proposition 4.10. It remains to  apply Lemma 1.9 to conclude Fp0 (Tdθ ) = Hp0 (Tdθ ). α,c We now discuss the duality of Fpα,c (Tdθ ). For this we need to define F∞ (Tdθ ) that is excluded from the definition at the beginning of the present section. Let α 2 denote the Hilbert space of all complex sequences a = (ak )k≥0 such that   2kα 1 a = 2 |ak |2 2 < ∞. k≥0

Thus

Lp (Tdθ ; α,c 2 )

d is the column subspace of Lp (B(α 2 )⊗Tθ ).

α,c Definition 4.13. For α ∈ R we define F∞ (Tdθ ) as the space of all distributions x on Tdθ that admit a representation of the form  x= ϕ k ∗ xk with (xk )k≥0 ∈ L∞ (Tdθ ; α,c 2 ), k≥0

and endow it with the norm

1     2kα α,c = | x(0)| + inf 2 |ϕ k ∗ xk |2 2 ∞ , x F∞ k≥0

where the infimum runs over all representations of x as above. Proposition 4.14. Let 1 ≤ p < ∞ and α ∈ R. Then the dual space of Fpα,c (Tdθ ) coincides isomorphically with Fp−α,c (Tdθ ).  Proof. For simplicity, we will consider only distributions with vanishing Fourier coefficients at m = 0. We view Fpα,c (Tdθ ) as an isometric subspace of k ∗ x)k≥0 . Then the dual space of Fpα,c (Tdθ ) is identified with Lp (Tdθ ; α,c 2 ) via x → (ϕ the following quotient of the latter:    ϕ k ∗ yk : (yk )k≥0 ∈ Lp (Tdθ ; −α,c ) , Gp = y = 2 k≥0

72

4. TRIEBEL-LIZORKIN SPACES

equipped with the quotient norm  y = inf (yk )

−α,c Lp (Td ) θ ; 2

: y=



 ϕ k ∗ yk .

k≥0

−α,c The duality bracket is given by x, y = τ (xy ∗ ). If p = 1, then Gp = F∞ (Tdθ ) −α,c d by definition. It remains to show that Gp = Fp (Tθ ) for 1 < p < ∞. It is clear that Fp−α,c (Tdθ ) ⊂ Gp , a contractive inclusion. Conversely, let y ∈ Gp and   ). Then y= ϕ k ∗ yk for some (yk )k≥0 ∈ Lp (Tdθ ; −α,c 2

k ∗ ϕ k−1 ∗ yk−1 + ϕ k ∗ ϕ k ∗ yk + ϕ k ∗ ϕ k+1 ∗ yk+1 . ϕ k ∗ y = ϕ Therefore, by Lemma 4.7,   2kα 1 2 |ϕ k ∗ y|2 2

p

≤

1    −2kα 1 2 |ϕ k ∗ ϕ k+j ∗ yk+j |2 2 p j=−1

k≥0

k≥0

  −2kα 1  2 |yk |2 2 p . k≥0

Thus y ∈ Fp−α,c (Tdθ ) and y F −α,c  y Gp . 



p

Remark 4.15. (i) The above proof shows that Fpα,c (Tdθ ) is a complemented subspace of Lp (Tdθ ; α,c 2 ) for 1 < p < ∞. (ii) By duality, Propositions 4.9, 4.10 and Theorem 4.11 remain valid for p = ∞, 0,c (Tdθ ) = BMOc (Tdθ ). except the density of Pθ . In particular, F∞ We conclude this section with the following Fourier multiplier theorem, which is an immediate consequence of Theorem 4.1 for p < ∞. The case p = ∞ is obtained by duality. In the case of α = 0, this result is to be compared with Lemma 1.7 where more smoothness of φ is assumed. Theorem 4.16. Let φ be a continuous function on Rd \ {0} such that sup φ(2k ·) ϕ H σ (Rd ) < ∞ k≥0

2

for some σ > d2 . Then φ is a bounded Fourier multiplier on Fpα,c (Tdθ ) for all 1 ≤ p ≤ ∞ and α ∈ R. In particular, φ is a bounded Fourier multiplier on Hpc (Tdθ ) for 1 ≤ p < ∞ and on BMOc (Tdθ ). 4.3. A general characterization In this section we give a general characterization of Triebel-Lizorkin spaces on Tdθ in the same spirit as that given in section 3.2 for Besov spaces. Let α0 , α1 , σ ∈ R with σ > d2 . Let h be a Schwartz function satisfying (3.3). Assume that ψ is an infinitely differentiable function on Rd \ {0} such that ⎧ |ψ| > 0 on {ξ : 2−1 ≤ |ξ| ≤ 2}, ⎪ ⎪ ⎪ ⎪   ⎨ (1 + |s|2 )σ F −1 (ψhI−α1 )(s)ds < ∞, (4.7) Rd ⎪ ⎪ ⎪ ⎪ ⎩ sup 2−kα0 F −1 (ψ(2k ·)ϕ) H σ (Rd ) < ∞. k∈N0

2

4.3. A GENERAL CHARACTERIZATION

73

Writing ϕ = ϕ(ϕ(−1) + ϕ + ϕ(1) ) and using Lemma 4.2, we have    −1 2 σ  −1 k ds. F (ψ(2k ·)ϕ) σ d  F (1 + |s| ) (ψ(2 ·)ϕ)(s) H (R ) Rd

2

So the third condition of (4.7) is weaker than the corresponding one assumed in [74, Theorem 2.4.1]. On the other hand, consistent with Theorem 3.9 but contrary to [74, Theorem 2.4.1], our following theorem does not require that α1 > 0. Theorem 4.17. Let 1 ≤ p < ∞ and α ∈ R. Assume that α0 < α < α1 and ψ satisfies (4.7). Then for any distribution x on Tdθ , we have   2kα 1 (4.8) x Fpα,c ≈ | x(0)| + 2 |ψk ∗ x|2 2 p . k≥0

The equivalence is understood in the sense that whenever one side is finite, so is the other, and the two are then equivalent with constants independent of x. Proof. Although it resembles, in form, the proof of Theorem 3.9, the proof given below is harder and subtler than the Besov space case. The key new ingredient is Theorem 4.1. The main differences will already appear in the first part of the proof, which is an adaptation of step 1 of the proof of Theorem 3.9. In the following, we will fix x with x (0) = 0. By approximation, we can assume that x is α,c . a polynomial. We will denote the right-hand side of (4.8) by x Fp,ψ Given a positive integer K, we write, as before ∞ 

ψ (j) =

ψ (j) ϕ(k) =

k=0

K 

ψ (j) ϕ(j+k) +

k=−∞

∞ 

ψ (j) ϕ(j+k) .

k=K

Then α,c ≤ I + II, x Fp,ψ

(4.9) where I=

   1 22jα |ψj ∗ ϕ j+k ∗ x|2 2 p , k≤K

j

k>K

j

   1 II = 22jα |ψj ∗ ϕ j+k ∗ x|2 2 p . The estimate of the term I corresponds to step 1 of the proof of Theorem 3.9. We use again (3.8) with η and ρ defined there. Then applying Theorem 4.1 twice, we have    2(j+k)α 1 I= 2k(α1 −α) 2 | ηj ∗ ρj+k ∗ x|2 2 p j

k≤K

=



2

k≤K



 

k(α1 −α)



22jα | ηj−k ∗ ρj ∗ x|2

 12

p

j

  2jα 1 2k(α1 −α) η (−k) ϕ H σ 2 | ρj ∗ x|2 2 p 2

k≤K+2

j

   2jα 1  Iα1 ϕ H σ 2k(α1 −α) η (−k) ϕ H σ 2 |ϕ j ∗ x|2 2 p 2

= Iα1 ϕ H σ 2

k≤K+2



k≤K+2

2

2k(α1 −α) η (−k) ϕ

j

x F α,c . p Hσ 2

74

4. TRIEBEL-LIZORKIN SPACES

Being an infinitely function with compact support, Iα 1 ϕ belongs to differentiable H2σ (Rd ), that is, Iα1 ϕ H σ < ∞. Next, we must estimate η (−k) ϕ H σ uniformly 2

2

in k. To that end, for s ∈ Rd , using  2  −1 (−k) 2   F (η  ϕ)(s) =  F −1 (η)(t) ∗ F −1 (ϕ)(s − 2k t)dt Rd  −1    −1 F (η)(t) F −1 (ϕ)(s − 2k t)2 dt , ≤ F (η) 1 Rd

for k ≤ K + 2, we have  (−k) 2 η ϕH σ =



 2 (1 + |s|2 )σ F −1 (η (−k) ϕ)(s) ds     −1    F (η)(t) F −1 (ϕ)(s − 2k t)2 dtds ≤ F −1 (η)1 (1 + |s|2 )σ Rd Rd        2  F −1 (η)1 (1 + |2k t|2 )σ F −1 (η)(t) (1 + |s − 2k t|2 )σ F −1 (ϕ)(s − 2k t) dsdt d d R R        2 Kσ  −1 2 σ  −1   ≤2 (1 + |t| ) F (η)(t) dt (1 + |s|2 )σ F −1 (ϕ)(s) ds F (η) 1 d d R R    2 2 σ  −1  ≤ cϕ,σ,K (1 + |t| ) F (η)(t) dt . 2

Rd

Rd

In order to return back from η to ψ, write η = I−α1 ψh + I−α1 ψ(h(K) − h). Note that



(4.10) Rd

  (1 + |t|2 )σ F −1 (I−α1 ψ(h(K) − h))(t)dt = cψ,h,α1 ,σ,K < ∞

since I−α1 ψ(h − h) is an infinitely differentiable function with compact support. We then deduce       (1 + |t|2 )σ F −1 (η)(t)dt  (1 + |t|2 )σ F −1 (I−α1 ψh)(t)dt. (K)

Rd

Rd

The term on the right-hand side is the second condition of (4.7). Combining the preceding inequalities, we obtain    I (1 + |t|2 )σ F −1 (I−α1 ψh)(t)dt x Fpα,c . Rd

The second term II on the right-hand side of (4.9) is easier to estimate. Using (3.11), Theorem 4.1 and arguing as in the preceding part for the term I, we obtain  II  Iα0 ϕ H σ 2−2kα I−α0 ψ(2k ·)Hϕ H σ x Fpα,c 2



2

k>K−2



2−2kα I−α0 ψ(2k ·)Hϕ H σ x Fpα,c , 2

k>K−2

where H = ϕ(2−1 ·) + ϕ + ϕ(2 ·). To treat the last Sobolev norm, noting that I−α0 H is an infinitely differentiable function with compact support, by Lemma 4.2, we have    I−α0 ψ(2k ·)Hϕ σ ≤ ψ(2k ·)ϕ σ (1+|t|2 )σ F −1 (I−α0 H)(t)dt  ψ(2k ·)ϕ σ . H H H 2

2

Rd

2

4.3. A GENERAL CHARACTERIZATION

Therefore,

 22k(α0 −α) x Fpα,c II  sup 2−kα0 ψ(2k ·)ϕ H σ 2

k>K−2

(4.11)

75

k>K−2

2(α0 −α)K ≤ c sup 2−kα0 ψ(2k ·)ϕ H σ x Fpα,c 2 1 − 2α0 −α k>K−2

with some constant c independent of K. Putting this estimate together with that of I, we finally get α,c  x α,c . x Fp,ψ Fp Now we show the reverse inequality by following step 3 of the proof of Theorem 3.9 (recalling that λ = 1 − h). By (3.13) and Theorem 4.1, ∞   1 x Fpα,c  ψ −1 ϕ2 H σ 22jα | hj+K ∗ ψj ∗ x|2 2 p 2

j=0

∞   1  22jα | hj+K ∗ ψj ∗ x|2 2 p j=0 ∞   1 j+K ∗ ψj ∗ x|2 2 . α,c + ≤ x Fp,ψ 22jα |λ p j=0

Then combining the arguments in step 3 of the proof of Theorem 3.9 and (4.11) with λ(K) ψ in place of ψ, we deduce ∞  

1  j+K ∗ψ j ∗x|2 2  ≤ c  22jα |λ p j=0

  2(α0 −α)K sup 2−kα0 λ(2k−K ·)ψ(2k ·)ϕH σ xFpα,c . 2 1 − 2α0 −α k>K−2

To remove λ(2k−K ·) from the above Sobolev norm, by triangular inequality, we have k−K λ(2 ·)ψ(2k ·)ϕ H σ ≤ ψ(2k ·)ϕ H σ + h(2k−K ·)ψ(2k ·)ϕ H σ . 2

2

2

By the support assumption on h and ϕ, h(2 ·)ϕ = 0 only for k ≤ K + 2, so the second term on the right hand side above matters only for k = K +1 and k = K +2. But for these two values of k, by Lemma 4.2, we have k−K h(2 ·)ψ(2k ·)ϕ H σ ≤ c ψ(2k ·)ϕ H σ , k−K

2

2



where c depends only on h. Thus k−K λ(2 ·)ψ(2k ·)ϕ H σ ≤ (1 + c ) ψ(2k ·)ϕ H σ . 2

2

Putting together all estimates so far obtained, we deduce 2(α0 −α)K  α,c + c (1 + c ) x Fpα,c ≤ x Fp,ψ sup 2−kα0 ψ(2k ·)ϕ H σ x Fpα,c . 2 1 − 2α0 −α k≥K−2 So if K is chosen sufficiently large, we finally obtain α,c , x Fpα,c  x Fp,ψ

which finishes the proof of the theorem.



76

4. TRIEBEL-LIZORKIN SPACES

Remark 4.18. Note that we have used the infinite differentiability of ψ only to insure (4.10), which holds whenever ψ is continuously differentiable up to order 3d [ 3d 2 ] + 1. More generally, we need only to assume that there exists σ > 2 + 1 such that ψη ∈ H2σ (Rd ) for any compactly supported infinite differentiable function η which vanishes in a neighborhood of the origin. Like in the case of Besov spaces, Theorem 4.17 admits the following continuous version. Theorem 4.19. Under the assumption of the previous theorem, for any distribution x on Tdθ ,  1 dε 12 α,c x(0)| + ε−2α |ψε ∗ x|2 x Fp ≈ | . ε p 0 Proof. This proof is very similar to that of Theorem 4.17. The main idea is, of course, to discretize the continuous square function:  2−j  1 ∞  dε −2α  2 dε 2jα ≈ . ε |ψε ∗ x| 2 |ψε ∗ x|2 ε ε −j−1 2 0 j=0 We can further discretize the internal integrals on the right-hand side. Indeed, by approximation and assuming that x is a polynomial, each internal integral can be approximated uniformly by discrete sums. Then we follow the proof of Theorem 3.11 with necessary modifications as in the preceding proof. The only difference is that when Theorem 4.1 is applied, the L1 -norm of the inverse Fourier transforms of the various functions in consideration there must be replaced by the two norms of these functions appearing in (4.7). We omit the details.  4.4. Concrete characterizations This section concretizes the general characterization in the previous one in terms of the Poisson and heat kernels. We keep the notation introduced in section 3.3. The following result improves [74, Section 2.6.4] at two aspects even in the classical case: Firstly, in addition to derivation operators, it can also use integration operators (corresponding to negative k); secondly, [74, Section 2.6.4] requires k > d + max(α, 0) for the Poisson characterization while we only need k > α. Theorem 4.20. Let 1 ≤ p < ∞ and α ∈ R. (i) Let k ∈ Z such that k > α. Then for any distribution x on Tdθ ,  1 1    ε (x)2 dε 2 x(0)| + ε2(k−α) Jεk P x Fpα,c ≈ | . ε p 0 (ii) Let k ∈ Z such that k > α2 . Then for any distribution x on Tdθ ,  1 1  α  ε (x)2 dε 2 x Fpα,c ≈ | x(0)| + ε2(k− 2 ) Jεk W . ε p 0 The preceding theorem can be formulated directly in terms of the circular Poisson and heat semigroups of Tdθ . The proof of the following result is similar to that of Theorem 3.15, and is left to the reader. Theorem 4.21. Let 1 ≤ p < ∞, α ∈ R and k ∈ Z.

4.4. CONCRETE CHARACTERIZATIONS

77

(i) If k > α, then for any distribution x on Tdθ ,  1 2 dr 12  x Fpα,c ≈ max | x(m)| + (1 − r)2(k−α) Jrk Pr (xk ) , |m| d + α, the remaining case being postponed. The proof in this case is similar to and a little bit harder than the proof of Theorem 3.13. Let again ψ(ξ) = (−sgn(k)2π|ξ|)k e−2π|ξ| . As in that proof, it remains to show that ψ satisfies the second condition of (4.7) for some α1 > α and σ > d2 . Since k > d+α,  ∈ H σ1 (Rd ) for we can choose α1 such that α < α1 < k − d. We claim that Ik−α1 h P 2 d d every σ1 ∈ ( 2 , k − α1 + 2 ). Indeed, this is a variant of Lemma 3.5 with a = k − α1  The difference is that this function ρ is not infinitely differentiable and ρ = h P. at the origin. However, the claim is true if σ1 is an integer. Then by complex interpolation as in the proof of that lemma, we deduce the claim in the general case. Now choose σ such that d2 < σ < 12 (σ1 − d2 ) and set η = σ1 − 2σ. Then η > d2 , and by the Cauchy-Schwarz inequality, we have 

         1 2 σ  −1    (s)2 ds 2 Ik−α1 h P (s) ds ≤ (1 + |s| ) F (1 + |s|2 )2σ+η F −1 Ik−α1 h P Rd Rd −1    σ1 d . Ik−α1 h P  F H (R ) 2

Therefore, the second condition of (4.7) is verified. This shows part (i) in the case k > d + α.  To deal with the remaining case k > α, we need the following: Lemma 4.22. Let 1 ≤ p < ∞ and k,  ∈ Z such that  > k > α. Then for any distribution x on Tdθ with x (0) = 0,  1  1 1 1      ε (x)2 dε 2  ε (x)2 dε 2 ε2(k−α) Jεk P ε2( −α) Jε P ≈ . ε ε p p 0 0 Proof. By induction, it suffices to consider the case  = k + 1. We first show the lower estimate:  1  1 1 1      ε (x)2 dε 2  ε (x)2 dε 2 ε2(k−α) Jεk P ε2(k+1−α) Jε P  . ε ε p p 0 0 To that end, we use  ε (x) = −sgn(k) Jεk P

 ε

∞

 δ (x)dδ. Jδk+1 P

78

4. TRIEBEL-LIZORKIN SPACES

Choose β ∈ (0, k−α). By the Cauchy-Schwarz inequality via the operator convexity of the function t → t2 , we obtain −2β  ∞     k  ε (x)2 ≤ ε  δ (x)2 dδ . Jε P δ 2(1+β) Jδk+1 P 2β δ ε It then follows that  1  ∞  2 dε 2 dδ δ 2(k−α−β) dε   1 2(k−α)  k  2(1+β)  k+1    ≤ ε δ ε Jε Pε (x) Jδ Pδ (x) ε 2β 0 δ 0 ε 0  ∞   1  δ (x)2 dδ . = δ 2(k+1−α) Jδk+1 P 4β(k − α − β) 0 δ Therefore,  1  ∞ 1 2 dε 12    2(k−α)  k   δ (x)2 dδ 2  Jε Pε (x) ε δ 2(k+1−α) Jδk+1 P  ε δ p p 0 0  1 12   dδ  δ (x)2  δ 2(k+1−α) Jδk+1 P , δ p 0 as desired.  ε +ε = P ε ∗ P  ε , we have The upper estimate is harder. This time, writing P 1 2 1 2

 ∂  δ  ε  ε ∗ Jεk P = sgn(k)2k+1 εk+1 P δ k+1 Jδk+1 P ∂ε δ=2ε ε , = sgn(k)2k+1 εk φε ∗ Jεk P where φ(ξ) = −2π|ξ| e−2π|ξ| . Thus  1  12 

 2 dε 12  2 dε 12  k+1−α k+1   2(k+1−α)  k+1    J P P ε (x) = J δ  ε δ  ε δ ε ε p δ=2ε p 0 0 1 1  2    ε (x)2 dε 2 = 2k+1−α ε2(k−α) φε ∗ Jεk P ε p 0  1 12   dε  ε (x)2 ≤ 2k+1−α ε2(k−α) φε ∗ Jεk P . ε p 0 Now our task is to remove φε from the integrand on the right-hand side in the spirit of Theorem 4.1. To that end, we will use a multiplier theorem analogous to dε d Lemma 4.7. Let  H = L2 ((0, 1), ε ) and define the H-valued kernel k on R by k(s) = φε (s) 0 0. Then for a distribution x on Tdθ with x  1  1  (I β P)ε ∗ x2 dε 2 (4.13) x Hc1 ≈ . ε 1 0 Armed with this characterization, we can easily complete the proof of the lemma. Indeed, k−α P) ∗ (I α x) = (−sgn(k)2π)−k εk−α J k P  ε (x) . (I ε ε

Thus by (4.13) with β = k − α,  1 1    ε (x)2 dε 2 ε2(k−α) Jεk P ≈ I α x Hc1 . ε 1 0 It then remains to apply Lemma 4.7 (ii) to I α x to conclude that (4.12) holds for p = 1 too, so the proof of the lemma is complete.  End of the proof of Theorem 4.20. The preceding lemma shows that the norm in the right-hand side of the equivalence in part (i) is independent of k with k > α. As (i) has been already proved to be true for k > d + α, we deduce the assertion in full generality.  We end this section with a Littlewood-Paley type characterizations of Sobolev spaces. The following is an immediate consequence of Corollary 4.12 and the characterizations proved previously in this chapter. Proposition 4.23. Let ψ satisfy (4.7), k > α and 1 < p < ∞. Then for any distribution on Tdθ , xHpα ≈ | x(0)|+    ⎧    2kα 1

1   2kα 2 2  k ∗ z)∗ |2 2    ⎪ inf 2 2 | ψ ∗ y| + |( ψ k ⎪ p p ⎪ ⎨ k≥0 k≥0       2kα 1

1   ⎪ 2kα 2 2  k ∗ x)∗ |2 2    ⎪ max | ψ ∗ x| , |( ψ 2 2 ⎪ k ⎩ p p k≥0

if 1 < p < 2, if 2 ≤ p < ∞;

k≥0

and x(0)|+ xHpα ≈ | ⎧   1   1 1  1     

   ⎪ ε (y)2 dε 2  ε (z) ∗ 2 dε 2  ⎪ inf  ε2(k−α) Jεk P ε2(k−α)  Jεk P  +  ⎪ ⎪ ε ε p p ⎪ 0 0 ⎪ ⎪ ⎨ if 1 < p < 2,   1 1     1   2 dε  12  ⎪

    ⎪ 2(k−α)  k ε (x) ∗ 2 dε 2   ⎪ ε (x) , ε2(k−α)  Jεk P P max J     ⎪ ε ε ⎪ ⎪ ε ε p p 0 0 ⎪ ⎩ if 2 ≤ p < ∞.

The above infima are taken above all decompositions x = y + z.

80

4. TRIEBEL-LIZORKIN SPACES

4.5. Operator-valued Triebel-Lizorkin spaces Unlike Sobolev and Besov spaces, the study of vector-valued Triebel-Lizorkin spaces in the classical setting does not allow one to handle their counterparts in quantum tori by means of transference. Given a Banach space X, a straightforward way of defining the X-valued Triebel-Lizorkin spaces on Td is as follows: for 1 ≤ p < ∞, 1 ≤ q ≤ ∞ and α ∈ R, an X-valued distribution f on Td belongs to α (Td ; X) if Fp,q   qkα 1 α f Fp,q = f(0) X + 2 ϕ k ∗ f qX q Lp (Td ) < ∞. k≥0

A majority of the classical results on Triebel-Lizorkin spaces can be proved to be true in this vector-valued setting with essentially the same methods. Contrary to α (Td ; Lp (Tdθ )) is very different from the the Sobolev or Besov case, the space Fp,2 α,c previously studied space Fp,2 (Tdθ ). This explains why the transference method is not efficient here. α (Td ; X). Let (rk ) be a However, there exists another way of defining Fp,2 Rademacher sequence, that is, an independent sequence of random variables on a probability space (Ω, P ), taking only two values ±1 with equal probability. We α (Td ; X) to be the space of all X-valued distributions f on Td such that define Fp,rad  α f Fp,rad = f(0) X + rk 2kα ϕ k ∗ f Lp (Ω×Td ;X) < ∞. k≥0 α Fp,rad (Td ; X)

It seems that these spaces have never been studied so far in literature. They might be worth to be investigated. If X is a Banach lattice of finite concavity, then by the Khintchine inequality,   2kα 1 α f Fp,rad ≈ f(0) X + 2 |ϕ k ∗ f |2 2 Lp (Td ;X) . k≥0 α . Moreover, in this This norm resembles, in form, more the previous one f Fp,2 case, one can also define a similar space by replacing the internal 2 -norm by any q -norm. But what we are interested in here is the noncommutative case, where X is a noncommutative Lp -space, say, X = Lp (Tdθ ). Then by the noncommutative Khintchine inequality [40], we can show that for 2 ≤ p < ∞ (assuming f(0) = 0), !    1   2kα 1 " α f Fp,rad ≈ max 22kα |ϕ k ∗ f |2 2 p , 2 |(ϕ k ∗ f )∗ |2 2 p .

k≥0

k≥0

Here p is the norm of Lp (T Thus the right hand-side is closely related to the norm of Fpα (Tdθ ) defined in section 4.2. In fact, if x → x  denotes the transference map introduced in Corollary 1.2, then for 1 < p < ∞, we have d

; Lp (Tdθ )).

x Fpα (Tdθ ) ≈  x Fp,rad . α (Td ;Lp (Td θ )) This shows that if one wishes to treat Triebel-Lizorkin spaces on Tdθ via transference, α (Td ; Lp (Tdθ )). The latter ones are as one should first investigate the spaces Fp,rad α d hard to deal with as Fp (Tθ ). We would like to point out, at this stage, that the method we have developed α (Td ; Lp (Tdθ )). In view of operator-valued in this chapter applies as well to Fp,rad

4.5. OPERATOR-VALUED TRIEBEL-LIZORKIN SPACES

81

α Hardy spaces, we will call Fp,rad (Td ; Lp (Tdθ )) an operator-valued Triebel-Lizorkin space on Td . We can define similarly its column and row counterparts. We will give below an outline of these operator-valued Triebel-Lizorkin spaces in the light of the development made in the previous sections. A systematic study will be given elsewhere. In the remainder of this section, M will denote a finite von Neumann algebra M with a faithful normal tracial state τ and N = L∞ (Td )⊗M.

Definition 4.24. Let 1 ≤ p < ∞ and α ∈ R. The column operator-valued Triebel-Lizorkin space Fpα,c (Td , M) is defined to be   Fpα,c (Td , M) = f ∈ S  (Td ; L1 (M)) : f Fpα,c < ∞ , where

  2kα 1 f Fpα,c = f(0) Lp (M) + 2 |ϕ k ∗ f |2 2 L

p (N )

.

k≥0

The main ingredient for the study of these spaces is still a multiplier result like Theorem 4.1 that is restated as follows: Theorem 4.25. Assume that (φj )≥0 and (ρj )≥0 satisfy (4.1) with some σ > d2 .

(i) Let 1 < p < ∞. Then for any f ∈ S  (Td ; L1 (M)),   2jα   2jα 1 1 2 |φj ∗ ρj ∗f 2 | 2 Lp (N )  sup φj (2j+k ·)ϕ H σ 2 | ρj ∗f |2 2 Lp (N ) . j≥0 −2≤k≤2

j≥0

2

j≥0

 −j ·) for some Schwartz function ρ with supp(ρ) = {ξ : 2−1 ≤ (ii) If ρj = ρ(2 |ξ| ≤ 2}. Then the above inequality holds for p = 1 too. The proof of Theorem 4.1 already gives the above result. Armed with this multiplier theorem, we can check that all results proved in the previous sections admit operator-valued analogues with the same proofs. For instance, the dual −α,c (Td , M) analogous to the space of F1α,c (Td , M) can be described as a space F∞ one defined in Definition 4.13. However, following the H1 -BMO duality developed in the theory of operator-valued Hardy spaces in [81], we can show the following nicer characterization of the latter space in the style of Carleson measures: Theorem 4.26. A distribution f ∈ S  (Td ; L1 (M)) with f(0) = 0 belongs to α,c F∞ (Td , M) iff 1   sup 22kα |ϕ k ∗ f (s)|2 ds < ∞, |Q| Q M Q k≥log2 (l(Q))

where the supremum runs over all cubes of Td , and where l(Q) denotes the side length of Q. The characterizations of Triebel-Lizorkin spaces given in the previous two sections can be transferred to the present setting too. Let us formulate only the analogue of Theorem 4.21. Theorem 4.27. Let 1 ≤ p < ∞, α ∈ R and k ∈ Z. (i) If k > α, then for any f ∈ S  (Td ; L1 (M)),  1 2 dr 12  f Fpα,c ≈ max f(m) Lp (M) + (1 − r)2(k−α) Jrk Pr (fk ) , 1−r |m| 0, −q α,ω ≈ q x p . lim α q x Bp,q 1

1

α→0

5.3. Interpolation of Triebel-Lizorkin spaces This short section contains some simple results on the interpolation of TriebelLizorkin spaces. They are similar to those for potential Sobolev spaces presented in section 5.1. It is surprising, however, that the real interpolation spaces of Fpα,c (Tdθ ) for a fixed p do not depend on the column structure. Proposition 5.21. Let 1 ≤ p, q ≤ ∞ and α0 , α1 ∈ R with α0 = α1 . Then  α0 ,c d  α Fp (Tθ ), Fpα1 ,c (Tdθ ) η,q = Bp,q (Tdθ ), α = (1 − η)α0 + ηα1 . Similar statements hold for the row and mixture Triebel-Lizorkin spaces. Proof. The assertion is an immediate consequence of Proposition 4.10 (v) and Proposition 5.1 (i). Note, however, that Proposition 4.10 (v) is stated for p < ∞; but by duality via Proposition 4.14, it continues to hold for p = ∞.  On the other hand, the interpolation of Fpα,c (Tdθ ) for a fixed α is reduced to that of Hardy spaces by virtue of Proposition 4.10 (iv) and Lemma 1.9. Remark 5.22. Let α ∈ R and 1 < p < ∞. Then  α,c d   α,c d  F∞ (Tθ ), F1α,c (Tdθ ) 1 = Fpα,c (Tdθ ) = F∞ (Tθ ), F1α,c (Tdθ ) 1 ,p . p

p

Proposition 5.23. Let α0 , α1 ∈ R and 1 < p < ∞. Then  α0 ,c d  1 α1 F∞ (Tθ ), F1α1 ,c (Tdθ ) 1 = Fpα,c (Tdθ ), α = (1 − )α0 + . p p p Proof. This proof is similar to that of Theorem 5.6. Let x be in the unit ball of Fpα,c (Tdθ ). Then by Proposition 4.10, J α (x) ∈ Hpc (Tdθ ). Thus by Lemma 1.9, there exists a continuous function f from the strip S = {z ∈ C : 0 ≤ Re(z) ≤ 1} to H1c (Tdθ ), analytic in the interior, such that f ( p1 ) = J α (x) and such that sup f (it) BMOc ≤ c, sup f (1 + it) Hc ≤ c. t∈R

t∈R

Define

1

1 2

F (z) = e(z− p ) J −(1−z)α0 −zα1 f (z), z ∈ S. By Remark 4.15 and Lemma 5.8, for any t ∈ R, 2 1 F (it) α,c ≈ e−t + p2 J it(α0 −α1 ) f (it) ≤ c . F BMOc ∞

Similarly,

F (1 + it) α,c ≈ e−t2 +(1− p1 )2 J it(α0 −α1 ) f (1 + it) c ≤ c . F H 1

Therefore,

whence

1

 α0 ,c d  1 x = F ( ) ∈ F∞ (Tθ ), F1α1 ,c (Tdθ ) 1 , p p  α0 ,c d  Fpα,c (Tdθ ) ⊂ F∞ (Tθ ), F1α1 ,c (Tdθ ) 1 . p

The converse inclusion is obtained by duality.



CHAPTER 6

Embedding We consider the embedding problem in this chapter. We begin with Besov spaces, then pass to Sobolev spaces. Our embedding theorem for Besov spaces is complete; however, the embedding problem of W11 (Tdθ ) is, unfortunately, left unsolved at the time of this writing. The last section deals with the compact embedding. 6.1. Embedding of Besov spaces This section deals with the embedding of Besov spaces. We will follow the semigroup approach developed by Varopolous [76] (see also [20,77]). This approach can be adapted to the noncommutative setting, which has been done by Junge and Mei [33]. Here we can use either the circular Poisson or heat semigroup of Tdθ , already considered in section 3.3. We choose to work with the latter. Recall that for x ∈ S  (Tdθ ),  2 Wr (x) = x (m)r |m| U m , 0 ≤ r < 1. m∈Zd

The following elementary lemma will be crucial. Lemma 6.1. Let 1 ≤ p ≤ p1 ≤ ∞. Then (6.1)

Wr (x) p1  (1 − r) 2 ( p1 − p ) x p , x ∈ Lp (Tdθ ), 0 ≤ r < 1. d

1

1

Proof. Consider first the case p = 1 and p1 = ∞. Then   2 2 r |m| | x(m)| ≤ x 1 r |m| Wr (x) ∞ ≤ m∈Zd

= x 1



rk

k≥0

≈ (1 − r)

−d 2

 |m|2 =k

m∈Zd

1  x 1



d

(1 + k) 2 r k

k≥0

x 1 .

The general case easily follows from this special one by interpolation. Indeed, the inequality just proved means that Wr is bounded from L1 (Tdθ ) to L∞ (Tdθ ) with d norm controlled by (1 − r)− 2 . On the other hand, Wr is a contraction on Lp (Tdθ ) for 1 ≤ p ≤ ∞. Interpolating these two cases, we get (6.1) for 1 < p < p1 = ∞.  The remaining case p1 < ∞ is treated similarly. The following is the main theorem of this section. Theorem 6.2. Assume that 1 ≤ p < p1 ≤ ∞, 1 ≤ q ≤ q1 ≤ ∞ and α, α1 ∈ R such that α − dp = α1 − pd1 . Then we have the following continuous inclusion: α (Tdθ ) ⊂ Bpα11,q1 (Tdθ ) . Bp,q 93

94

6. EMBEDDING

Proof. Since Bpα11,q (Tdθ ) ⊂ Bpα11,q1 (Tdθ ), it suffices to consider the case q = q1 . On the other hand, by the lifting Theorem 3.7, we can assume max{α, α1 } < 0, so that we can take k = 0 in Theorem 3.15. Thus, we are reduced to showing

 1

 1 q dr 1q q dr q1 qα1 qα (1 − r)− 2 Wr (x) p  (1 − r)− 2 Wr (x) p . 1 1 − r 1−r 0 0   To this end, we write Wr (x) = W√r W√r (x) and apply (6.1) to get √ d 1 1 Wr (x)  (1 − r) 2 ( p1 − p ) W√r (x) . p1 p Thus

 0

1

dr 1q 1 1 − r

 1 q dr 1q qα1 √ qd 1 1  (1 − r)− 2 (1 − r) 2 ( p1 − p ) W√r (x) p 1−r 0

 1 q 2rdr q1 qd qα1 1 1 = (1 − r 2 )− 2 (1 − r) 2 ( p1 − p ) Wr (x) p 1 − r2 0 1

 1 q dr q qα  (1 − r)− 2 Wr (x) p , 1−r 0

(1 − r)−

qα1 2

Wr (x) q p



as desired.

Corollary 6.3. Assume that 1 ≤ p < p1 ≤ ∞, 1 ≤ q ≤ ∞ and α = d( p1 − p11 ). Then α α (Tdθ ) ⊂ Lp1 ,q (Tdθ ) if p1 < ∞ and Bp,1 (Tdθ ) ⊂ L∞ (Tdθ ) if p1 = ∞ . Bp,q

Proof. Applying the previous theorem to α1 = 0 and q = q1 = 1, and by Theorem 3.8, we get α (Tdθ ) ⊂ Bp01 ,1 (Tdθ ) ⊂ Lp1 (Tdθ ) . Bp,1

This gives the assertion in the case p1 = ∞. For p1 < ∞, we fix p and choose two appropriate values of α (which give the two corresponding values of p1 ); then we interpolate the resulting embeddings as above by real interpolation; finally, using  (1.1) and Proposition 5.1, we obtain the announced embedding for p1 < ∞. The preceding corollary admits a self-improvement in terms of modulus of smoothness. Corollary 6.4. Assume that 1 ≤ p < p1 ≤ ∞, α = d( p1 − p11 ) and k ∈ N such that k > α. Then  ε dδ , 0 < ε ≤ 1. δ −α ωpk (x, δ) ωpk1 (x, ε)  δ 0 Proof. Without loss of generality, assume x (0) = 0. Then by the preceding corollary and Theorem 3.16, we have  1 dδ x p1  . δ −α ωpk (x, δ) δ 0 Now let u ∈ Rd with |u| ≤ ε. Noting that   ωpk (Δu (x), δ) ≤ 2k min ωpk (x, ε), ωpk (x, δ) ≤ 2k ωpk (x, min(ε, δ)),

6.2. EMBEDDING OF SOBOLEV SPACES

we obtain



ε

95

dδ + ε−α ωpk (x, ε) δ 0  ε  ε dδ dδ + δ −α ωpk (x, δ) δ −α ωpk (x, δ)  ε δ δ 0 2  ε dδ  . δ −α ωpk (x, δ) δ 0 Taking the supremum over all u with |u| ≤ ε yields the desired inequality. Δu (x) p1 

δ −α ωpk (x, δ)



Remark 6.5. We will discuss the optimal order of the best constant of the embedding in Corollary 6.3 at the end of the next section. 6.2. Embedding of Sobolev spaces This section is devoted to the embedding of Sobolev spaces. The following is α1 our main theorem. Recall that B∞,∞ (Tdθ ) in the second part below is the quantum analogue of the classical Zygmund class of order α1 (see Remark 3.19). Theorem 6.6. Let α, α1 ∈ R with α > α1 . (i) If 1 < p < p1 < ∞ are such that α − dp = α1 −

d p1 ,

then

Hpα (Tdθ ) ⊂ Hpα11 (Tdθ ) continuously. In particular, if additionally α = k and α1 = k1 are nonnegative integers, then Wpk (Tdθ ) ⊂ Wpk11 (Tdθ ) continuously. (ii) If 1 ≤ p < ∞ is such that p(α − α1 ) > d and α1 = α − dp , then α1 Hpα (Tdθ ) ⊂ B∞,∞ (Tdθ ) continuously.

In particular, if additionally α = k ∈ N, and if either p > 1 or p = 1 and k is even, then α1 (Tdθ ) continuously. Wpk (Tdθ ) ⊂ B∞,∞

Proof. (i) By Theorem 2.9, the embedding of Wpk (Tdθ ) is a special case of that of Hpα (Tdθ ). Thus we just deal with the potential spaces Hpα (Tdθ ). On the other hand, by the lifting property of potential Sobolev spaces, we can assume α1 = 0. By Theorem 3.8 and Corollary 6.3, we have Hpα (Tdθ ) ⊂ Lq,∞ (Tdθ ). Now choose 0 < η < 1 and two indices s0 , s1 with 1 < s0 , s1
1. In the commutative case, representing a function as an indefinite integral of its derivatives, one easily checks that this embedding remains true for p = 1. However, we do not know how to prove it in the noncommutative case. A α1 (Tdθ ) in the case of odd related question concerns the embedding Wpk (Tdθ ) ⊂ B∞,∞ k which is not covered by the same part (ii). The quantum analogue of the Gagliardo-Nirenberg inequality can be also proved easily by interpolation. Proposition 6.9. Let k ∈ N, 1 < r, p < ∞, 1 ≤ q < ∞ and β ∈ Nd0 with 0 < |β|1 < k. If 1 1−η η |β|1 and = + , η= k r q p then for every x ∈ Wpk (Tdθ ) ∩ Lq (Tdθ ),   η Dm x p . Dβ x r  x 1−η q |m|=k

Proof. This inequality immediately follows from Theorem 5.13 and the wellknown relation between real and complex interpolations:     Lq (Tdθ ), Wpk (Tdθ ) η,1 ⊂ Lq (Tdθ ), Wpk (Tdθ ) η = Wr|β|1 (Tdθ ). It then follows that x W |β|1  x 1−η x ηW k . q r

p

Applying this inequality to x − x (0) instead of x and using Theorem 2.12, we get the desired Gagliardo-Nirenberg inequality.  An alternate approach to Sobolev embedding. Note that the preceding proof of Theorem 6.6 is based on Theorem 6.2, which is, in its turn, proved by Varopolous’ semigroup approach. Varopolous initially developed his method for the Sobolev embedding, which was transferred to the noncommutative setting by Junge and Mei [33]. Our argument for the embedding of Besov spaces has followed this route. Let us now give an alternate proof of Theorem 6.6 (i) by the same way. We state its main part as the following lemma that is of interest in its own right. Lemma 6.10. Let 1 ≤ p < q < ∞ such that

1 q

=

Wp1 (Tdθ ) ⊂ Lq,∞ (Tdθ ).

1 p

− d1 . Then

6.2. EMBEDDING OF SOBOLEV SPACES

97

ε Proof. We will use again the heat semigroup Wr of Tdθ . Recall that Wr = W −4π 2 ε  with r = e , where Wε is the periodization of the usual heat kernel Wε of Rd ε . In the following, we (see section 3.3). It is more convenient to work with W d −1 (0) = 0. Let Δj = Δ ∂j , 1 ≤ j ≤ d. Then assume x ∈ S(Tθ ) and x  ∞  ∞ ε (x)dε and Δj x = 4π 2 ε (∂j x) dε. W W Δ−1 x = 4π 2 0

0

We claim that for any 1 ≤ p ≤ ∞ (6.2)

ε (∂j x) ∞  ε− 2 ( p +1) x p , ε (∂j x) p  ε− 2 x p and W W 1

1

d

ε > 0.

Indeed, in order to prove the first inequality, by the transference method, it suffices to show a similar one for the Banach space valued heat semigroup of the usual d-torus. The latter immediately follows from the following standard estimate on the heat kernel Wε of Rd :    1 ∇Wε (s) ds < ∞. sup ε 2 Rd

ε>0

The second inequality of (6.2) is proved in the same way as (6.1). First, for the case p = 1, we have (recalling that x (0) = 0)  2 ε (∂j x) ∞ ≤ 2π W |mj |e−ε|m| | x(m)| m∈Zd \{0}



≤ 2π x 1

|mj |e−ε|m|

2

m∈Zd \{0}

 e−ε (1 − e−ε )−

d+1 2

x 1  ε−

d+1 2

x 1 .

Interpolating this with the first inequality for p = ∞, we get the second one in the general case. Now let ε > 0 and decompose Δj x into the following two parts:  ∞  ε δ (∂j x) dδ and z = 4π 2 δ (∂j x) dδ. y = 4π 2 W W ε

Then by (6.2),

0

 y ∞  x p

∞

δ − 2 ( p +1) dδ ≈ ε− 2 ( p −1) x p 1

d

1

d

ε

and

 z p  x p

ε

δ − 2 dδ ≈ ε 2 x p . 1

1

0

Thus for any t > 0, choosing ε such that ε− 2p = t, we deduce d

p

y ∞ + t z p  t1− d x p = tη x p , where η = 1 − dp . It then follows that Δj x q,∞ ≈ Δj x (L∞ (Tdθ ), Lp (Tdθ ))η,∞  x p . Since x=−

d  j=1

Δj ∂j x,

98

6. EMBEDDING

we finally get x q,∞ 

d 

Δj ∂j x q,∞ 

j=1

d 

∂j x p = ∇x p .

j=1



Thus the lemma is proved.

Alternate proof of Theorem 6.6 (i). For 1 < p < d, choose p0 , p1 such that 1 < p0 < p < p1 < d. Let q1i = p1i − d1 for i = 0, 1. Then by the previous lemma, Wp1i (Tdθ ) ⊂ Lqi ,∞ (Tdθ ), i = 0, 1. Interpolating these two inclusions by real method, we obtain Wp1 (Tdθ ) ⊂ Lq,p (Tdθ ). This is the embedding of Sobolev spaces in Theorem 6.6 (i) for k = 1. The case k > 1 immediately follows by iteration. Then using real interpolation, we deduce the embedding of potential Sobolev spaces.  Sobolev embedding for p = 1. Now we discuss the case p = 1 which is not covered by Theorem 6.6. The main problem concerns the following: (6.3)

W11 (Tdθ ) ⊂ L

d d−1

(Tdθ ).

At the time of this writing, we are unable, unfortunately, to prove it. However, Lemma 6.10 provides a weak substitute, namely, (6.4)

W11 (Tdθ ) ⊂ L

d d−1 ,∞

(Tdθ ).

In the classical case, one can rather easily deduce (6.3) from (6.4). Let us explain the idea coming from [77, page 58]. It was kindly pointed out to us by Marius Junge. Let f be a nice real function on Td with f(0) = 0. For any t ∈ R let ft be the indicator function of the subset {f > t}. Then f can be decomposed as an integral of the ft ’s:  +∞ (6.5) f= ft dt. −∞

By triangular inequality (with q =

d d−1 ),



f q ≤

+∞

ft q dt.

−∞

However, ft q = ft q,∞

∀ t ∈ R.

Thus by (6.4) for θ = 0, ft q  ft 1 + ∇ft 1 . It comes now the crucial point which is the following  +∞ (6.6) ∇ft 1 dt  ∇f 1 . −∞

In fact, the two sides are equal in view of Sard’s theorem. We then get the strong embedding (6.3) in the case θ = 0. Note that this proof yields a stronger embedding: (6.7)

W11 (Td ) ⊂ L

d d−1 ,1

(Td ).

6.2. EMBEDDING OF SOBOLEV SPACES

99

The above decomposition of f is not smooth in the sense that ft is not derivable even though f is nice. In his proof of Hardy’s inequality in Sobolev spaces, Bourgain [11] discovered independently the same decomposition but using nicer functions ft (see also [60]). Using (6.7) and the Hausdorff-Young inequality, Bourgain derived the following Hardy type inequality (assuming d ≥ 3):  m∈Zd

|f(m)|  f W11 (Td ) . (1 + |m|)d−1

We have encountered difficulties in the attempt of extending this approach to the noncommutative case. Let us formulate the corresponding open problems explicitly as follows: Problem 6.11. Let d ≥ 2. (i) Does one have the following embedding W11 (Tdθ ) ⊂ L

d d−1

(Tdθ ) or W11 (Tdθ ) ⊂ L

d d−1 ,1

(Tdθ ) ?

(ii) Does one have the following inequality  | x(m)|  x W11 (Tdθ ) ? (1 + |m|)d−1 d m∈Z

By the previous discussion, part (i) is reduced to a decomposition for operators in W11 (Tdθ ) of the form (6.5) and satisfying (6.6). One could attempt to do this by transference by first considering operator-valued functions on Rd . With this in mind, the following observation, due to Marius Junge, might be helpful. Given an interval I = [s, t] ⊂ R and an element a ∈ L1 (Tdθ ), we have ∂(1I ⊗ a) = δs ⊗ a − δt ⊗ a, where ∂ denotes the distribution derivative relative to R and δs is the Dirac measure at s. Let · L denote the norm of the dual space C0 (R; Aθ )∗ , which contains L1 (R; L1 (Tdθ )) isometrically. If f is a (nice) linear combination n of 1I ⊗ a’s, then we have the desired decomposition of f . Indeed, assume f = i=1 αi 1Ii ⊗ ei , where αi ∈ R+ and the 1Ii ⊗ ei ’s are pairwise disjoint projections of L∞ (R)⊗Tdθ . Let ft = 1(t,∞) (f ). Then  ∞ ft dt. f= 0

So for any q ≥ 1,



∞

f q ≤

ft q dt.

0

On the other hand, by writing explicitly ft for every t, one easily checks  ∞ ∂ft L dt. ∂f L = 0

By iteration, the above decomposition can be extended to higher dimensional case n for all functions f of the form i=1 αi 1Ri ⊗ ei , where αi ∈ R+ , Ri ’s are rectangles (with sides parallel to the axes) and 1Ri ⊗ ei ’s are pairwise disjoint projections of L∞ (Rd )⊗Tdθ . The next idea would be to apply Lemma 6.10 to these special functions. Then two difficulties come up to us, even in the commutative case. The first is that these functions do not belong to W11 ; this difficulty can be resolved quite easily by

100

6. EMBEDDING

regularization. The second one, substantial, is the density of these functions, more precisely, of suitable regularizations of them, in W11 . Uniform Besov embedding. We end this section with a discussion on the link between a certain uniform embedding of Besov spaces and the embedding of Sobolev spaces. Let 0 < α < 1, 1 ≤ p < ∞ with αp < d and 1r = p1 − αd . Then (6.8)

x pr ≤ cd,p

α(1 − α) x pBp,p α,ω , (d − αp)

α x ∈ Bp,p (Tdθ ) ,

α,ω is the Besov norm defined by (3.19). In the commutative case, this where x Bp,p inequality is proved in [13] for α close to 1 and in [41] for general α. One can show that (6.8) is essentially equivalent to the embedding of Wp1 (Tdθ ) into Lq (Tdθ ) (or Lq,p (Tdθ )) for d > p and 1q = p1 − d1 . Indeed, assume(6.8). Then taking limit in both sides of (6.8) as α → 1, by Theorem 3.20, we get

x q  |x|Wp1 for all x ∈ Wp1 (Tdθ ) with x (0) = 0. Conversely, if Wp1 (Tdθ ) ⊂ Lq (Tdθ ), then     Lp (Tdθ ), Wp1 (Tdθ ) α,p ⊂ Lp (Tdθ ), Lq (Tdθ ) α,p . Theorem 5.16 implies that   α Lp (Tdθ ), Wp1 (Tdθ ) α,p ⊂ Bp,p (Tdθ ) α with relevant constant depending only on d, here Bp,p (Tdθ ) being equipped with the α,ω . On the other hand, By a classical result of Holmstedt [28] on real norm Bp,p interpolation of Lp -spaces (which readily extends to the noncommutative case, as observed in [37, Lemma 3.7]),   Lp (Tdθ ), Lq (Tdθ ) α,p ⊂ Lr,p (Tdθ ) 1

1

with the inclusion constant uniformly controlled by α p (1 − α) q . We then deduce 1

1

α,ω . x r,p  α p (1 − α) q x Bp,p

This implies a variant of (6.8) since Lr,p (Tdθ ) ⊂ Lr (Tdθ ). Since we have proved the embedding Wp1 (Tdθ ) ⊂ Lq (Tdθ ) for p > 1, (6.8) holds for p > 1. Let us record this explicitly as follows: Proposition 6.12. Let 0 < α < 1, 1 < p < ∞ with αp < d and Then  1 α x r  α(1 − α) p x pBp,p x ∈ Bp,p (Tdθ ) α,ω ,

1 r

=

1 p

−

α d.

with relevant constant independent of α. In the case p = 1, Problem 6.11 (i) is equivalent to (6.8) for p = 1 and α close to 1. 6.3. Compact embedding This section deals with the compact embedding. The case p = 2 for potential Sobolev spaces was solved by Spera [65]: Lemma 6.13. The embedding H2α1 (Tdθ ) → H2α2 (Tdθ ) is compact for α1 > α2 ≥ 0.

6.3. COMPACT EMBEDDING

101

We will require the following real interpolation result on compact operators, due to Cwikel [19]. Lemma 6.14. Let (X0 , X1 ) and (Y0 , Y1 ) be two interpolation couples of Banach spaces, and let T : Xj → Yj be a bounded linear operator, j = 0, 1. If T : X0 → Y0 is compact, then T : (X0 , X1 )η,p → (Y0 , Y1 )η,p is compact too for any 0 < η < 1 and 1 ≤ p ≤ ∞. Theorem 6.15. Assume that 1 ≤ p < p1 ≤ ∞, 1 ≤ p∗ < p1 , 1 ≤ q ≤ q1 ≤ ∞ α (Tdθ ) → Bpα∗1,q1 (Tdθ ) is compact. and α − dp = α1 − pd1 . Then the embedding Bp,q Proof. Without loss of generality, we can assume q = q1 . First consider the case p = 2. Choose t sufficiently close to q and 0 < η < 1 such that 1−η η 1 = + . q 2 t Then by Proposition 5.1,

 α  α α B2,q (Tdθ ) = B2,2 (Tdθ ), B2,t (Tdθ ) η,q .

α1 α By Lemma 6.13, B2,2 (Tdθ ) → B2,2 (Tdθ ) is compact. On the other hand, by Theoα1 α d d rem 6.2, B2,t (Tθ ) → Bp1 ,t (Tθ ) is continuous. So by Lemma 6.14,  α1 d  α (Tdθ ) → B2,2 (Tθ ), Bpα11,t (Tdθ ) η,q is compact. B2,q

However, by the proof of Proposition 5.1 and (1.1), we have    α1 d  d d α1 d 1 B2,2 (Tθ ), Bpα11,t (Tdθ ) η,q ⊂ α q (L2 (Tθ ), Lp1 (Tθ )η,q = q (Ls,q (Tθ )), where s is determined by 1−η η 1 (1 − η)(α − α1 ) 1 = + . = + s 2 p1 p1 d Note that η tends to 1 as t tends to q. Thus we can choose η so that s > p∗ . Then Ls,q (Tdθ ) ⊂ Lp∗ (Tdθ ). Thus the desired assertion for p = 2 follows. The case p = 2 but p > 1 is dealt with similarly. Let t and η be as above. Choose r < p (r close to p). Then    α  α d d α d B2,2 (Tdθ ), Br,t (Tdθ ) η,q ⊂ α q (L2 (Tθ ), Lr (Tθ )η,q = q (Lp0 ,q (Tθ )), where p0 is determined by

1 1−η η + . = p0 2 r If η is sufficiently close to 1, then p0 < p that we will assume. Thus Lp (Tdθ ) ⊂ Lp0 ,q (Tdθ ). It then follows that  α  α α (Tdθ ) ⊂ B2,2 (Tdθ ), Br,t (Tdθ ) η,q . Bp,q

The rest of the proof is almost the same as the case p = 2, so is omitted. The remaining case p = 1 can be easily reduced to the previous one. Indeed, α (Tdθ ) into Bpα22,q (Tdθ ) for some α2 ∈ (α, α1 ) (α2 close to α) and first embed Bp,q p2 determined by α − dp = α2 − pd2 . Then by the previous case, the embedding Bpα22,q (Tdθ ) → Bpα∗1,q1 (Tdθ ) is compact, so we are done.  Theorem 6.16. Let 1 < p < p1 < ∞ and α, α1 ∈ R.

102

6. EMBEDDING

(i) If α − dp = α1 − pd1 , then Hpα (Tdθ ) → Hpα∗1 (Tdθ ) is compact for p∗ < p1 . In particular, if additionally α = k and α1 = k1 are nonnegative integers, then Wpk (Tdθ ) → Wpk∗1 (Tdθ ) is compact. α∗ (Tdθ ) (ii) If p(α − α1 ) > d and α∗ < α1 = α − dp , then Hpα (Tdθ ) → B∞,∞ is compact. In particular, if additionally α = k ∈ N, then Wpk (Tdθ ) → α∗ (Tdθ ) is compact. B∞,∞ Proof. Based on the preceding theorem, this proof is similar to that of Theorem 6.6 and left to the reader. 

CHAPTER 7

Fourier multiplier This chapter deals with Fourier multipliers on Sobolev, Besov and TriebelLizorkin spaces on Tdθ . The first section concerns the Sobolev spaces. Its main result is the analogue for Wpk (Tdθ ) of [17, Theorem 7.3] (see also Lemma 1.3) on c.b. Fourier multipliers on Lp (Tdθ ); so the space of c.b. Fourier multipliers on Wpk (Tdθ ) is independent of θ. The second section turns to Besov spaces on which Fourier multipliers behave better. We extend some classical results to the present α (Tdθ ) does not setting. We show that the space of c.b. Fourier multipliers on Bp,q depend on θ (nor on q or α). We also prove that a function on Zd is a Fourier α 0 (Tdθ ) iff it is the Fourier transform of an element of B1,∞ (Td ). multiplier on B1,q The last section deals with Fourier multipliers on Triebel-Lizorkin spaces.

7.1. Fourier multipliers on Sobolev spaces We now investigate Fourier multipliers on Sobolev spaces. We refer to [9,59] for the study of Fourier multipliers on the classical Sobolev spaces. If X is a Banach space of distributions on Tdθ , we denote by M(X) the space of bounded Fourier multipliers on X; if X is further equipped with an operator space structure, Mcb (X) is the space of c.b. Fourier multipliers on X. These spaces are endowed with their natural norms. Recall that the Sobolev spaces Wpk (Tdθ ), Hpα (Tdθ ) and the Besov α (Tdθ ) are equipped with their natural operator space structures as defined in Bp,q Remarks 2.29 and 3.25. The aim of this section is to extend [17, Theorem 7.3] (see also Lemma 1.3) on c.b. Fourier multipliers on Lp (Tdθ ) to Sobolev spaces. Inspired by Neuwirth and Ricard’s transference theorem [48], we will relate Fourier multipliers with Schur multipliers. Given a distribution x on Tdθ , we write its matrix in the basis (U m )m∈Zd :

[x] = xU n , U m 

m,n∈Zd



t ˜ = x (m − n)einθ(m−n)

m,n∈Zd

.

Here kt denotes the transpose of k = (k1 , . . . , kd ) and θ˜ is the following d × d-matrix deduced from the skew symmetric matrix θ: ⎛ 0 θ12 ⎜0 0 ⎜ ⎜ .. θ˜ = −2π ⎜ ... . ⎜ ⎝0 0 0 0

θ13 θ23 .. .

... ... .. .

0 0

... ...

103

θ1d θ2d .. .

⎞

⎟ ⎟ ⎟ ⎟. ⎟ θd−1,d ⎠ 0

104

7. FOURIER MULTIPLIER

Now let φ : Zd → C and Mφ be the associated Fourier multiplier on Tdθ . Set   ˚ = φm−n φ . Then m,n∈Zd    t ˜ (7.1) Mφ x = φm−n x (m − n)einθ(m−n) m,n∈Zd = Sφ˚([x]), ˚ where Sφ˚ is the Schur multiplier with symbol φ. k d According to the definition of Wp (Tθ ), for any matrix a = (am,n )m,n∈Zd and  ∈ Nd0 define   D a = (2πi(m − n)) am,n m,n∈Zd . If x is a distribution on Tdθ , then clearly     Mφ D x = Sφ˚ D [x] . We introduce the space ! " Spk = a = (am,n )m,n∈Zd : D a ∈ Sp (2 (Zd )), ∀  ∈ Nd0 , 0 ≤ ||1 ≤ k and endow it with the norm a Spk =





D a pSp

p1

.

0≤| |1 ≤k

ThenSpk is a closed subspace of the p -direct sum of L copies of Sp (2 (Zd )) with L = 0≤| |1 ≤k 1. The latter direct sum is equipped with its natural operator space structure, which induces an operator space structure on Spk too. If ψ = (ψm,n )m,n∈Zd is a complex matrix, its associated Schur multiplier Sψ on Spk is defined by Sψ a = (ψm,n am,n )m,n∈Zd . Let Mcb (Spk ) denote the space of all c.b. Schur multipliers on Spk , equipped with the natural norm. Theorem 7.1. Let 1 ≤ p ≤ ∞ and k ∈ N. Then Mcb (Wpk (Tdθ )) = Mcb (Spk ) with equal norms. Consequently, Mcb (Wpk (Tdθ )) = Mcb (Wpk (Td )) with equal norms. Proof. This proof is an adaptation of that of [17, Theorem 7.3]. We start with an elementary observation. Let V = diag(· · · , U n , · · · )n∈Zd . For any a = (am,n )m,n∈Zd ∈ B(2 (Zd )), let x = V (a ⊗ 1Tdθ )V ∗ ∈ B(2 (Zd ))⊗Tdθ , where 1Tdθ denotes the unit of Tdθ . Then   ˜ t x = (U m amn U −n )m,n∈Zd = am,n em,n ⊗U m U −n = am,n em,n ⊗e−inθm U m−n , m,n

m,n

where (em,n ) are the canonical matrix units of B(2 (Zd )). So,   [x] = am,n em,n m,n∈Zd , a matrix with entries in B(2 (Zd )). Since V is unitary, we have x = a ⊗ 1Tdθ Lp (B( 2 (Zd ))⊗Td ) = a Sp ( 2 (Zd )) . Lp (B( 2 (Zd ))⊗Td ) θ

Similarly, for  ∈ Nd0 , D x (7.2)

Lp (B( 2 (Zd ))⊗Td θ)

θ

= D a Sp ( 2 (Zd )) .

7.1. FOURIER MULTIPLIERS ON SOBOLEV SPACES

105

Now suppose that φ ∈ Mcb (Wpk (Tdθ )). For a = (am,n )m,n∈Zd ∈ B(2 (Zd )), define x = V (a ⊗ 1Tdθ )V ∗ as above. Then by (7.1), for  ∈ Nd0 , (IdB( 2 (Zd )) ⊗Mφ )(D x) = V (Sφ˚(D a) ⊗ 1Tdθ )V ∗ . It then follows from (7.2) that Sφ˚(a) Spk =

 

 p1

(IdB( 2 (Zd )) ⊗ Mφ )(D x) pL

d d p (B( 2 (Z ))⊗Tθ )

| |1 ≤k

≤ φ Mcb (Wpk (Tdθ ))

 

D x pL

 p1

d d p (B( 2 (Z ))⊗Tθ )

| |1 ≤k

= φ Mcb (Wpk (Tdθ )) a Spk . ˚ is a bounded Schur multiplier on S k . Considering matrices a = Therefore, φ p (am,n )m,n∈Zd with entries in Sp , we show in the same way that Mφ˚ is c.b. on ˚ is a c.b. Schur multiplier on S k and S k , so φ p

p

˚ M (S k ) ≤ φ φ . Mcb (Wpk (Td cb p θ )) To show the converse direction, introducing the following Følner sequence of Zd : ZN = {−N, . . . , −1, 0, 1, . . . , N }d ⊂ Zd , we define two maps AN and BN as follows: |ZN |

AN : Tdθ → B(2 |ZN |

where PN : B(2 (Zd )) → B(2 |ZN |

BN : B(2

) → Tdθ

)

with x → PN ([x]),

) with (am,n ) → (am,n )m,n∈ZN ; and with em,n →

t 1 −inθ(m−n) ˜ e U m−n . |ZN |

|Z |

Here B(2 N ) is endowed with the normalized trace. Both AN , BN are unital, completely positive and trace preserving, so extend to complete contractions between the corresponding Lp -spaces. Moreover, lim BN ◦ AN (x) = x in Lp (Tdθ ), ∀x ∈ Lp (Tdθ ).

N →∞ |ZN |

If we define Spk (2

|ZN |

) as before for Spk just replacing Sp (2 (Zd )) by Sp (2

), we

|Z | see that AN extends to a complete contraction from Wpk (Tdθ ) into Spk (2 N ), while |Z | BN a complete contraction from Spk (2 N ) into Wpk (Tdθ ). ˚ is a c.b. Schur multiplier on S k , then it is also a c.b. Schur Now assume that φ p k |ZN | multiplier on Sp (2 ). We want to prove that Mφ is c.b. on Wpk (Tdθ ). For any

106

7. FOURIER MULTIPLIER

x ∈ Lp (B(2 (Zd ))⊗Tdθ ), Id ⊗ Mφ (x) Lp (B( 2 (Zd ))⊗Td )    θ   = lim Id ⊗ BN ◦ Id ⊗ AN Id ⊗ Mφ (x) Lp (B( 2 (Zd ))⊗Td ) θ N      = lim Id ⊗ BN ◦ Id ⊗ Sφ˚ Id ⊗ AN (x) Lp (B( 2 (Zd ))⊗Td ) N θ ≤ lim sup Id ⊗ S˚(Id ⊗ AN (x)) |Z | N d k φ

N

Sp ( 2 (Z );Sp ( 2

≤ lim sup Sφ˚ cb Id ⊗ AN (x) S ( (Zd );S k ( |ZN | )) p 2 p 2 N ≤ S˚ x d , d φ cb

))

Lp (B( 2 (Z ))⊗Tθ )

where in the second equality we have used the fact that Id ⊗ AN (Id ⊗ Mφ (x)) = Id ⊗ Sφ˚(Id ⊗ AN (x)), which follows from (7.1). Therefore, Mφ is c.b. on Wpk (Tdθ ) and ˚ M (S k ) . φ Mcb (Wpk (Tdθ )) ≤ φ cb p 

The theorem is thus proved.

Remark 7.2. Let 1 ≤ p ≤ ∞ and α ∈ R. Since J α is a complete isometry from Hpα (Tdθ ) onto Lp (Tdθ ), we have Mcb (Hpα (Tdθ )) = Mcb (Lp (Tdθ )) with equal norms. Thus, by Lemma 1.3 Mcb (Hpα (Tdθ )) = Mcb (Hpα (Td )) with equal norms. Note that the proof of Theorem 2.9 shows that Wpk (Tdθ ) = Hpk (Tdθ ) holds completely isomorphically for 1 < p < ∞. Thus the above remark implies Corollary 7.3. Let 1 < p < ∞ and k ∈ N. Mcb (Wpk (Tdθ )) = Mcb (Lp (Td )) with equivalent norms. Clearly, the above equality still holds for p = 1 or p = ∞ if d = 1 (the commutative case) since then Wpk (T) = Lp (T) for all 1 ≤ p ≤ ∞ by the (complete) isomorphism  1 Lp (T)  x → x (0) + x (m)z m ∈ Wpk (T) . (2πim)k m∈Z\{0}

However, this is no longer the case as soon as d ≥ 2, as proved by Poornima [59] in the commutative case for Rd . Poornima’s example comes from Ornstein [50] which is still valid for our setting. Indeed, by [50], there exists a distribution T on T2 which is not a measure and such that T = ∂1 μ0 , ∂1 T = ∂2 μ1 and ∂2 T = ∂1 μ2 for three measures μi on T2 . T induces a Fourier multiplier on T2θ , which is defined by the Fourier transform of T and is denoted by x → T ∗ x. Then for any x ∈ W11 (T2θ ), T ∗ x = ∂1 μ0 ∗ x = μ0 ∗ ∂1 x ∈ L1 (T2θ ), ∂1 T ∗ x = ∂1 μ1 ∗ x = μ1 ∗ ∂2 x ∈ L1 (T2θ ), ∂2 T ∗ x = ∂2 μ2 ∗ x = μ2 ∗ ∂1 x ∈ L1 (T2θ ).

7.2. FOURIER MULTIPLIERS ON BESOV SPACES

107

Thus T ∗x ∈ W11 (T2θ ), so the Fourier multiplier induced by T is bounded on W11 (T2θ ). We show in the same way that it is c.b. too. Since T is not a measure, it does not belong to M(L1 (T2 )). 7.2. Fourier multipliers on Besov spaces It is well known that in the classical setting, Fourier multipliers behave better on Besov spaces than on Lp -spaces. We will see that this fact remains true in the quantum case. We maintain the notation introduced in section 3.1. In particular, ϕ is a function satisfying (3.1) and ϕ(k) (ξ) = ϕ(2−k ξ) for k ∈ N0 . As usual, ϕ(k) is viewed as a function on Zd too. The following is the main result of this section. Compared with the corresponding result in the classical case (see, for instance, Section 2.6 of [73]), our result is α more precise since it gives a characterization of Fourier multipliers on Bp,q (Tdθ ) in d terms of those on Lp (Tθ ). Theorem 7.4. Let α ∈ R and 1 ≤ p, q ≤ ∞. Let φ : Zd → C. Then φ is α (Tdθ ) iff the φϕ(k) ’s are Fourier multipliers on Lp (Tdθ ) a Fourier multiplier on Bp,q uniformly in k. In this case, we have φ ≈ |φ(0)| + sup φϕ(k) d d α M(Bp,q (Tθ ))

M(Lp (Tθ ))

k≥0

with relevant constants depending only on α. A similar c.b. version holds too. Proof. Without loss of generality, we assume that φ(0) = 0 and all elements x considered below have vanishing Fourier coefficients at the origin. Let α α (Tdθ )) and x ∈ Lp (Tdθ ). Then y = (ϕ k−1 + ϕ k + ϕ k+1 ) ∗ x ∈ Bp,q (Tdθ ) and φ ∈ M(Bp,q α y Bp,q ≤ cα 2kα x p with cα = 9 · 4|α| .

So

α Mφ (y) Bp,q ≤ φ M(B α

d p,q (Tθ ))

α y Bp,q ≤ cα 2kα φ M(B α

d p,q (Tθ ))

x p .

On the other hand, by (3.2), ϕ k ∗ Mφ (y) = Mφϕ(k) (x) and α Mφ (y) Bp,q ≥ 2kα ϕ k ∗ Mφ (y) p = 2kα Mφϕ(k) (x) p .

It then follows that

Mφϕ(k) (x) p ≤ cα φ M(B α

d p,q (Tθ ))

whence

x p ,

sup φϕ(k) M(Lp (Td )) ≤ cα φ M(B α

d p,q (Tθ ))

θ

k≥0

.

α Conversely, for x ∈ Bp,q (Tdθ ),

  k−1 + ϕ ϕ k ∗ Mφ (x) p = Mφϕ(k) (ϕ k + ϕ k+1 ) ∗ x p ≤ φϕ(k) k−1 + ϕ k + ϕ k+1 ) ∗ x p . d (ϕ M(Lp (Tθ ))

We then deduce Mφ (x)

α Bp,q

which implies

φ M(B α

≤ 3 · 2|α| sup φϕ(k) M(L

d p,q (Tθ ))

d p (Tθ ))

k≥0

x

α Bp,q

≤ 3 · 2|α| sup φϕ(k) M(Lp (Td )) . k≥0

θ

,

108

7. FOURIER MULTIPLIER

Thus the assertion concerning bounded multipliers is proved. The preceding argument can be modified to work in the c.b. case too. First note that for k ≥ 0, ϕ(k) is a c.b. Fourier multiplier on Lp (Tdθ ) for all 1 ≤ p ≤ ∞ with c.b. norm 1, that is, the map x → ϕ k ∗ x is c.b. on Lp (Tdθ ). So for any x ∈ Sq [Lp (Tdθ )] (the Lp (Tdθ )-valued Schatten q-class), (IdSq ⊗ Mϕ(k) )(x) ≤ x S [L (Td )] . d Sq [Lp (Tθ )]

q

Now let φ ∈ and x ∈ ϕ k + ϕ k+1 ) ∗ x. Then for k − 2 ≤ j ≤ k + 2, α Mcb (Bp,q (Tdθ ))

p

θ

Define y as above: y = (ϕ k−1 +

Sq [Lp (Tdθ )].

ϕ j ∗ y Sq [Lp (Tdθ )] ≤ 3 x Sq [Lp (Tdθ )] . It thus follows that (IdS ⊗ Mφ )(y) q Sq [B α

≤ φ Mcb (Bp,q α (Td )) y S [B α (Td )] q p,q θ θ

d p,q (Tθ )]

k+2 

≤ φ Mcb (Bp,q α (Td )) θ

2jα ϕ j ∗ y Sq [Lp (Tdθ )]

j=k−2

≤ cα 2

kα

φ Mcb (Bp,q α (Td )) x S [L (Td )] . q p θ θ

Then as before, we deduce sup φϕ(k)

Mcb (Lp (Td θ ))

k≥0

≤ cα φ M

To show the converse inequality, assume sup φϕ(k) k≥0 α Then for x ∈ Sq [Bp,q (Tdθ )], ϕ k ∗ Mφ (x) Sq [Lp (Tdθ )] ≤ φϕ(k) M

Mcb (Lp (Td θ ))

d cb (Lp (Tθ ))

d α cb (Bp,q (Tθ ))

.

≤ 1.

(ϕ k−1 + ϕ k + ϕ k+1 ) ∗ x Sq [Lp (Tdθ )]

≤ (ϕ k−1 + ϕ k + ϕ k+1 ) ∗ x Sq [Lp (Tdθ )] . Therefore, Mφ (x) Sq [Bp,q α (Td )] = θ

 q q1 2kα ϕ k ∗ Mφ (x) Sq [Lp (Tdθ )] k≥0

 q 1q 2kα (ϕ ≤ k−1 + ϕ k + ϕ k+1 ) ∗ x Sq [Lp (Tdθ )] k≥0

≤ 3 · 2|α| x Sq [Bp,q α (Td )] . θ We thus get the missing converse inequality, so the theorem is proved.



The following is an immediate consequence of the preceding theorem. α Corollary 7.5. (i) M(Bp,q (Tdθ )) is independent of α and q, up to equivalent norms. 0 (Tdθ )) = M(Bp0 ,∞ (Tdθ )), where p is the conjugate index of p. (ii) M(Bp,∞ 0 (iii) M(Bp0 ,∞ (Tdθ )) ⊂ M(Bp01 ,∞ (Tdθ )) for 1 ≤ p0 < p1 ≤ 2. α (iv) M(Lp (Tdθ )) ⊂ M(Bp,q (Tdθ )). α (Tdθ )). Similar statements hold for the spaces Mcb (Bp,q

Theorem 7.4 and Lemma 1.3 imply the following:

7.2. FOURIER MULTIPLIERS ON BESOV SPACES

109

α α Corollary 7.6. Mcb (Bp,q (Tdθ )) = Mcb (Bp,q (Td )) with equivalent norms. 0 0 Let F(B1,∞ (Td )) be the space of all Fourier transforms of functions in B1,∞ (Td ) 0 (a commutative Besov space), equipped with the norm f = f B1,∞ . α 0 (Tdθ )) = F(B1,∞ (Td )) with equivalent norms. Corollary 7.7. Mcb (B1,q 0 Proof. Let φ ∈ Mcb (B1,∞ (Tdθ )) and f be the distribution on Td such that f = φ. By Theorem 7.4 and Lemma 1.3, we have sup φϕ(k) M(L (Td )) < ∞. k≥0

1

Recall that the Fourier transform of ϕk is ϕ(k) and ϕ k is the periodization of ϕk . So ϕ k L1 (Td ) = ϕk L1 (Rd ) = ϕ L1 (Rd ) . Noting that by (3.2), ϕ k ∗ f = Mφϕ(k) (ϕ k−1 + ϕ k + ϕ k+1 ), we get (k) ϕ k ∗ f 1 ≤ φϕ M(L (Td )) ϕ k−1 + ϕ k + ϕ k+1 1 ≤ 3 ϕ L1 (Rd ) φϕ(k) M(L

d 1 (T ))

1

whence

,

0 f B1,∞ ≤ 3 ϕ L1 (Rd ) sup φϕ(k) M(L1 (Td )) . k≥0

0 0 (Td ). Let g ∈ B1,∞ (Td ). Then Conversely, assume φ = f with f ∈ B1,∞

ϕ k ∗ Mφ (g) 1 = ϕ k ∗ f ∗ g 1 = ϕ k ∗ f ∗ (ϕ k−1 + ϕ k + ϕ k+1 ) ∗ g 1 ≤ ϕ k ∗ f 1 (ϕ k−1 + ϕ k + ϕ k+1 ) ∗ g 1 0 0 ≤ 3 f B1,∞ g B1,∞ .

0 Thus Mφ (g) ∈ B1,∞ (Td ) and 0 0 0 Mφ (g) B1,∞ ≤ 3 f B1,∞ g B1,∞ ,

0 which implies that φ is a Fourier multiplier on B1,∞ (Tdθ ) and 0 ≤ 3 f B1,∞ . φ M(B1,∞ 0 (Td θ ))

Considering g with values in S∞ , we show that φ is c.b. too. Alternately, since 0 0 (Td )) = Mcb (B1,∞ (Td )), M(L1 (Td )) = Mcb (L1 (Td )), Theorem 7.4 yields M(B1,∞ which allows us to conclude the proof too.  We have seen previously that every bounded (c.b.) Fourier multiplier on Lp (Tdθ ) α is a bounded (c.b.) Fourier multiplier on Bp,q (Tdθ ). Corollary 7.7 shows that the converse is false for p = 1. We now show that it also is false for any p = 2. α Proposition 7.8. There exists a Fourier multiplier φ which is c.b. on Bp,q (Tdθ ) for any p, q and α but never belongs to M(Lp (Tdθ )) for any p = 2 and any θ.

Proof. The example constructed by Stein and Zygmund [70] for a similar circumstance can be shown to work for our setting too. Their example is a distribution on T defined as follows: ∞  n 1 μ(z) = (wn z)2 Dn (wn z) log n n=2

110

7. FOURIER MULTIPLIER

for some appropriate wn ∈ T, where Dn (z) =

n 

zj ,

z ∈ T.

j=0 0 Since Dn L1 (T) ≈ log n, we see that μ ∈ B1,∞ (T). Considered as a distribution 0 (Td ) too. Thus by Corollaries 7.5 and 7.7, φ = μ  belongs to on Td , μ ∈ B1,∞ α d Mcb (Bp,p (Tθ )) for any p, q and α. However, Stein and Zygmund proved that φ is not a Fourier multiplier on Lp (T) for any p = 2 if the wn ’s are appropriately chosen. Consequently, φ cannot be a Fourier multiplier on Lp (Tdθ ) for any p = 2 and any θ since Lp (T) isometrically embeds into Lp (Tdθ ) by an embedding that is also a c.b. Fourier multiplier. 

We conclude this section with some comments on the vector-valued case. The proof of Theorem 7.4 works equally for vector-valued Besov spaces. Recall that α (Tdθ ; E) denotes the E-valued Besov space on Tdθ (see for an operator space E, Bp,q Remark 3.25). Proposition 7.9. For any operator space E, φ ≈ |φ(0)| + sup φϕ(k) M(Lp (Td ;E)) M(B α (Td ;E)) p,q

θ

θ

k≥0

with equivalence constants depending only on α. If θ = 0, we go back to the classical vector-valued case. The above proposition explains the well-known fact mentioned at the beginning of this section that Fourier multipliers behave better on Besov spaces than on Lp -spaces. To see this, it is more convenient to write the above right-hand side in a different form: (k) φϕ = φ(2k ·)ϕ . d d M(Lp (Tθ ;E))

M(Lp (Tθ ;E))

Thus if φ is homogeneous, the above multiplier norm is independent of k, so φ is α (Td ; E) for any p, q, α and any Banach space E. In a Fourier multiplier on Bp,q α (Td ; E). particular, the Riesz transform is bounded on Bp,q The preceding characterization of Fourier multipliers is, of course, valid for Rd in place of Td . Let us record this here: Proposition 7.10. Let E be a Banach space. Then for any φ : Rd → C, φ ≈ φψ M(Lp (Rd ;E)) + sup φ(2k ·)ϕ M(L (Rd ;E)) , M(B α (Rd ;E)) p,q

k≥0

where ψ is defined by ψ(ξ) =



ϕ(2k ξ),

p

ξ ∈ Rd .

k≥1

7.3. Fourier multipliers on Triebel-Lizorkin spaces As we have seen in the chapter on Triebel-Lizorkin spaces, Fourier multipliers on such spaces are subtler than those on Sobolev and Besov spaces. Similarly to the previous two sections, our target here is to show that the c.b. Fourier multipliers on Fpα,c (Tdθ ) are independent of θ. By definition, Fpα,c (Tdθ ) can be viewed as a subspace of the column space Lp (Tdθ ; α,c 2 ), the latter is the column subspace of d α,c d Lp (B(α )⊗T ). Thus F (T ) inherits the natural operator space structure of 2 p θ θ d )⊗T ). Similarly, the row Triebel-Lizorkin space Fpα,r (Tdθ ) carries a natural Lp (B(α 2 θ

7.3. FOURIER MULTIPLIERS ON TRIEBEL-LIZORKIN SPACES

111

operator space structure too. Finally, the mixture space Fpα (Tdθ ) is equipped with the sum or intersection operator space structure according to p < 2 or p ≥ 2. Note that according to this definition, even though it is a commutative function space, the space Fpα (Td ) (corresponding to θ = 0) is endowed with three different operator space structures, the first two being defined by its embedding into Lp (Td ; α,c 2 ) and Lp (Td ; α,r 2 ), the third one being the mixture of these two. The resulting operator spaces are denoted by Fpα,c (Td ) , Fpα,r (Td ) and Fpα (Td ), respectively. Similarly, we introduce operator space structures on the Hardy space Hpc (Tdθ ), its row and mixture versions too. The main result of this section is the following: Theorem 7.11. Let 1 ≤ p ≤ ∞ and α ∈ R. Then Mcb (Fpα,c (Tdθ )) = Mcb (Fpα,c (Td )) with equal norms, Mcb (Fpα,c (Tdθ )) = Mcb (Fp0,c (Td )) with equivalent norms. Similar statements hold for the row and mixture spaces. We will show the theorem only in the case p < ∞. The proof presented below can be easily modified to work for p = ∞ too. Alternately, the case p = ∞ can be also obtained by duality from the case p = 1. Note, however, that this duality argument yields only the first equality of the theorem with equivalent norms for p = ∞. We adapt the proof of Theorem 7.1 to the present situation, by introducing the space ! " 2  1   2kα  k ∗ a 2 ∈ Sp (2 (Zd )) , 2 ϕ Spα,c = a = (am,n )m,n∈Zd : k≥0

equipped with the norm   2kα  2  1 k ∗ a 2 S , a Spα,c = 2 ϕ p

k≥0

where

  ϕ k ∗ a = ϕ(2−k (m − n)) am,n m,n∈Zd .

d Then Spα,c is a closed subspace of the column subspace of Sp (α 2 ⊗ 2 (Z )), which α,c α,c introduces a natural operator space structure on Sp . Let Mcb (Sp ) denote the space of all c.b. Schur multipliers on Spα,c , equipped with the natural norm.

Lemma 7.12. Let 1 ≤ p < ∞ and α ∈ R. Then Mcb (Fpα,c (Tdθ )) = Mcb (Spα,c ) with equal norms. Proof. This proof is similar to the one of Theorem 7.1; we point out the necessary changes. Keeping the notation there, we have for a = (am,n )m,n∈Zd ∈ Spα,c and x = V (a ⊗ 1Tdθ )V ∗ ∈ B(2 (Zd ))⊗Tdθ 2  1   2kα 2 |ϕ k ∗ x 2 k

Lp (B( 2 (Zd ))⊗Td θ)

= a S α,c . p

112

7. FOURIER MULTIPLIER

Suppose that φ ∈ Mcb (Fpα,c (Tdθ )). It then follows that   2kα  2  1 S˚(a) α,c = 2 ϕ k ∗ (V (Sφ˚(a) ⊗ 1Tdθ )V ∗ ) 2 L φ S

d d p (B( 2 (Z ))⊗Tθ )

p

k

  2kα   2  1 k ∗ (V (a ⊗ 1Tdθ )V ∗ )  2 L = 2 Mφ ϕ

d d p (B( 2 (Z ))⊗Tθ )

k

≤ φ M

α,c (Td cb (Fp θ ))

x Sp [Fpα,c (Tdθ )]

= φ Mcb (Fpα,c (Tdθ )) a Spα,c . ˚ is a bounded Schur multiplier on S α,c . Considering matrices a = Therefore, φ p (am,n )m,n∈Zd with entries in B(2 ), we show in the same way that Sφ˚ is c.b. on ˚ is a c.b. Schur multiplier on S α,c and S α,c , so φ p

p

˚ M (S α,c ) ≤ φ φ . Mcb (Fpα,c (Td cb p θ )) To show the opposite inequality, we just note that the contractive and convergence properties of the maps AN and BN introduced in the proof of Theorem 7.1 also hold on the corresponding Fpα,c (Tdθ ) or Spα,c spaces. To see this, we take AN for example. Since it is c.b. between the corresponding Lp -spaces, it is also c.b. |Z | from Lp (B(2 )⊗Tdθ ) to Lp (B(2 )⊗B(2 N )). Applying this to the elements of the form ⎛ ⎞ 0 0 ... ϕ 0 ∗ x ⎜ 2α ϕ ⎟ ⎜ 2α1 ∗ x 0 0 . . .⎟ ⎝2 ϕ 2 ∗ x 0 0 . . .⎠ · · · ... |Z |

we see that AN is completely contractive from Fpα,c (Tdθ ) to Spα,c (B(2 N )), the latter space being the finite dimensional analogue of Spα,c . We then argue as in the proof of Theorem 7.1 to deduce the desired opposite inequality.  Proof of Theorem 7.11. The first part is an immediate consequence of the previous lemma. For the second, we need the c.b. version of Theorem 4.11 (i), whose proof is already contained in section 4.1. To see this, we just note that, letting M = B(2 (Zd ))⊗Tdθ and N = B(2 (Zd ))⊗L∞ (Td )⊗Tdθ in Lemma 4.6 we obtain the c.b. version of Lemma 4.7, and that, in the same way, the c.b. version of Lemma 1.10 holds, i.e., for x ∈ Sp [Hpc (Tdθ )], x Sp [Hcp (Tdθ )] ≈  x(0) Sp + scψ (x) Sp [Lp (Tdθ )] . Finally, the previous lemma and the c.b. version of Theorem 4.11 (i) yield the desired conclusion.  Remark 7.13. The preceding theorem and the c.b. version of Theorem 4.11 (i) show that Mcb (Hpc (Tdθ )) = Mcb (Hpc (Td )) with equivalent norms. In fact, using arguments similar to the proof of the preceding theorem, we can show that the above equality holds with equal norms.

ACKNOWLEDGEMENTS

113

Acknowledgements The authors wish to thank Marius Junge for discussions on the embedding and interpolation of Sobolev spaces, Tao Mei for discussions on characterizations of Hardy spaces and Eric Ricard for discussions on Fourier multipliers and comments. The authors are also grateful to Fedor Sukochev for suggestions and comments on a preliminary version of the paper. We acknowledge the financial supports of ANR-2011-BS01-008-01, NSFC grant (No. 11271292, 11301401 and 11431011).

Bibliography [1] R. A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. MR0450957 [2] H. Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56, DOI 10.1002/mana.3211860102. MR1461211 [3] W. Arendt and S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proc. Edinb. Math. Soc. (2) 47 (2004), no. 1, 15–33, DOI 10.1017/S0013091502000378. MR2064734 [4] W. Beckner, Embedding estimates and fractional smoothness, Int. Math. Res. Not. IMRN 2 (2014), 390–417. MR3159076 [5] W. Beckner, Functionals for multilinear fractional embedding, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 1, 1–28, DOI 10.1007/s10114-015-4321-6. MR3285943 [6] C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR928802 [7] J. Bergh, On the relation between the two complex methods of interpolation, Indiana Univ. Math. J. 28 (1979), no. 5, 775–778, DOI 10.1512/iumj.1979.28.28054. MR542336 [8] J. Bergh and J. L¨ ofstr¨ om, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR0482275 [9] A. Bonami and S. Poornima, Nonmultipliers of the Sobolev spaces W k,1 (Rn ), J. Funct. Anal. 71 (1987), no. 1, 175–181, DOI 10.1016/0022-1236(87)90021-8. MR879706 [10] J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Ark. Mat. 21 (1983), no. 2, 163–168, DOI 10.1007/BF02384306. MR727340 [11] J. Bourgain. A Hardy inequality in Sobolev spaces. Vrije University, Brussels, 1981. [12] J. Bourgain, Vector-valued singular integrals and the H 1 -BMO duality, Probability theory and harmonic analysis (Cleveland, Ohio, 1983), Monogr. Textbooks Pure Appl. Math., vol. 98, Dekker, New York, 1986, pp. 1–19. MR830227 [13] J. Bourgain, H. Brezis, and P. Mironescu, Limiting embedding theorems for W s,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002), 77–101, DOI 10.1007/BF02868470. Dedicated to the memory of Thomas H. Wolff. MR1945278 [14] J. Bourgain, H. Br´ ezis and P. Mironescu. Another Look at Sobolev Spaces. In Optimal Control and Partial Differential Equations, IOS Press, Amsterdam, 2001, 439-466. [15] D. L. Burkholder, A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional, Ann. Probab. 9 (1981), no. 6, 997–1011. MR632972 [16] D. L. Burkholder, A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 270–286. MR730072 [17] Z. Chen, Q. Xu, and Z. Yin, Harmonic analysis on quantum tori, Comm. Math. Phys. 322 (2013), no. 3, 755–805, DOI 10.1007/s00220-013-1745-7. MR3079331 [18] A. Connes, Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994. MR1303779 [19] M. Cwikel, Real and complex interpolation and extrapolation of compact operators, Duke Math. J. 65 (1992), no. 2, 333–343, DOI 10.1215/S0012-7094-92-06514-8. MR1150590 [20] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR990239 [21] R. DeVore and K. Scherer, Interpolation of linear operators on Sobolev spaces, Ann. of Math. (2) 109 (1979), no. 3, 583–599, DOI 10.2307/1971227. MR534764

115

116

BIBLIOGRAPHY

[22] R. E. Edwards, Fourier series. Vol. 2, 2nd ed., Graduate Texts in Mathematics, vol. 85, Springer-Verlag, New York-Berlin, 1982. A modern introduction. MR667519 [23] E. G. Effros and Z.-J. Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR1793753 [24] T. M. Flett, Temperatures, Bessel potentials and Lipschitz spaces, Proc. London Math. Soc. (3) 22 (1971), 385–451. MR0458159 [25] J. L. Rubio de Francia, Some maximal inequalities, Recent progress in Fourier analysis (El Escorial, 1983), North-Holland Math. Stud., vol. 111, North-Holland, Amsterdam, 1985, pp. 203–214. MR848147 [26] K. Gr¨ ochenig and M. Leinert, Wiener’s lemma for twisted convolution and Gabor frames, J. Amer. Math. Soc. 17 (2004), no. 1, 1–18 (electronic), DOI 10.1090/S0894-0347-03-00444-2. MR2015328 [27] Linear and complex analysis. Problem book 3. Part II, Lecture Notes in Mathematics, vol. 1574, Springer-Verlag, Berlin, 1994. Edited by V. P. Havin [V. P. Khavin] and N. K. Nikolski [N. K. Nikolski˘ı]. MR1334346 [28] T. Holmstedt, Interpolation of quasi-normed spaces, Math. Scand. 26 (1970), 177–199. MR0415352 [29] G. Hong, L. D. L´ opez-S´ anchez, J. M. Martell, and J. Parcet, Calder´ on-Zygmund operators associated to matrix-valued kernels, Int. Math. Res. Not. IMRN 5 (2014), 1221–1252. MR3178596 [30] H. Johnen and K. Scherer, On the equivalence of the K-functional and moduli of continuity and some applications, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976), Springer, Berlin, 1977, pp. 119–140. Lecture Notes in Math., Vol. 571. MR0487423 [31] M. Junge, Doob’s inequality for non-commutative martingales, J. Reine Angew. Math. 549 (2002), 149–190, DOI 10.1515/crll.2002.061. MR1916654 [32] M. Junge, C. Le Merdy, and Q. Xu, H ∞ functional calculus and square functions on noncommutative Lp -spaces (English, with English and French summaries), Ast´erisque 305 (2006), vi+138. MR2265255 [33] M. Junge and T. Mei, Noncommutative Riesz transforms—a probabilistic approach, Amer. J. Math. 132 (2010), no. 3, 611–680, DOI 10.1353/ajm.0.0122. MR2666903 [34] M. Junge and T. Mei, BMO spaces associated with semigroups of operators, Math. Ann. 352 (2012), no. 3, 691–743, DOI 10.1007/s00208-011-0657-0. MR2885593 [35] M. Junge, T. Mei, and J. Parcet, Smooth Fourier multipliers on group von Neumann algebras, Geom. Funct. Anal. 24 (2014), no. 6, 1913–1980, DOI 10.1007/s00039-014-0307-2. MR3283931 [36] M. Junge and Q. Xu, Noncommutative Burkholder/Rosenthal inequalities, Ann. Probab. 31 (2003), no. 2, 948–995, DOI 10.1214/aop/1048516542. MR1964955 [37] M. Junge and Q. Xu, Noncommutative maximal ergodic theorems, J. Amer. Math. Soc. 20 (2007), no. 2, 385–439, DOI 10.1090/S0894-0347-06-00533-9. MR2276775 [38] G. E. Karadzhov, M. Milman, and J. Xiao, Limits of higher-order Besov spaces and sharp reiteration theorems, J. Funct. Anal. 221 (2005), no. 2, 323–339, DOI 10.1016/j.jfa.2004.06.017. MR2124867 [39] F. Luef, On spectral invariance of non-commutative tori, Operator theory, operator algebras, and applications, Contemp. Math., vol. 414, Amer. Math. Soc., Providence, RI, 2006, pp. 131– 146, DOI 10.1090/conm/414/07805. MR2277208 [40] F. Lust-Piquard and G. Pisier, Noncommutative Khintchine and Paley inequalities, Ark. Mat. 29 (1991), no. 2, 241–260, DOI 10.1007/BF02384340. MR1150376 [41] V. Mazya and T. Shaposhnikova, On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces, J. Funct. Anal. 195 (2002), no. 2, 230–238, DOI 10.1006/jfan.2002.3955. MR1940355 [42] V. G. Maz’ja, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985. Translated from the Russian by T. O. Shaposhnikova. MR817985 [43] T. R. McConnell, On Fourier multiplier transformations of Banach-valued functions, Trans. Amer. Math. Soc. 285 (1984), no. 2, 739–757, DOI 10.2307/1999461. MR752501 [44] T. Mei, Operator valued Hardy spaces, Mem. Amer. Math. Soc. 188 (2007), no. 881, vi+64, DOI 10.1090/memo/0881. MR2327840

BIBLIOGRAPHY

117

[45] T. Mei, Tent spaces associated with semigroups of operators, J. Funct. Anal. 255 (2008), no. 12, 3356–3406, DOI 10.1016/j.jfa.2008.09.021. MR2469026 [46] T. Mei and J. Parcet, Pseudo-localization of singular integrals and noncommutative Littlewood-Paley inequalities, Int. Math. Res. Not. IMRN 8 (2009), 1433–1487, DOI 10.1093/imrn/rnn165. MR2496770 [47] M. Milman, Notes on limits of Sobolev spaces and the continuity of interpolation scales, Trans. Amer. Math. Soc. 357 (2005), no. 9, 3425–3442 (electronic), DOI 10.1090/S00029947-05-03937-1. MR2146631 ´ Ricard, Transfer of Fourier multipliers into Schur multipliers and sumsets [48] S. Neuwirth and E. in a discrete group, Canad. J. Math. 63 (2011), no. 5, 1161–1187, DOI 10.4153/CJM-2011053-9. MR2866074 [49] S. M. Nikolski˘ı, Approximation of functions of several variables and imbedding theorems, Springer-Verlag, New York-Heidelberg., 1975. Translated from the Russian by John M. Danskin, Jr.; Die Grundlehren der Mathematischen Wissenschaften, Band 205. MR0374877 [50] D. Ornstein, A non-equality for differential operators in the L1 norm., Arch. Rational Mech. Anal. 11 (1962), 40–49. MR0149331 [51] J. Parcet, Pseudo-localization of singular integrals and noncommutative Calder´ on-Zygmund theory, J. Funct. Anal. 256 (2009), no. 2, 509–593, DOI 10.1016/j.jfa.2008.04.007. MR2476951 [52] J. Parcet and N. Randrianantoanina, Gundy’s decomposition for non-commutative martingales and applications, Proc. London Math. Soc. (3) 93 (2006), no. 1, 227–252, DOI 10.1017/S0024611506015863. MR2235948 [53] J. Peetre, New thoughts on Besov spaces, Mathematics Department, Duke University, Durham, N.C., 1976. Duke University Mathematics Series, No. 1. MR0461123 [54] G. Pisier, Non-commutative vector valued Lp -spaces and completely p-summing maps (English, with English and French summaries), Ast´erisque 247 (1998), vi+131. MR1648908 [55] G. Pisier, Introduction to operator space theory, London Mathematical Society Lecture Note Series, vol. 294, Cambridge University Press, Cambridge, 2003. MR2006539 [56] G. Pisier and Q. Xu, Non-commutative martingale inequalities, Comm. Math. Phys. 189 (1997), no. 3, 667–698, DOI 10.1007/s002200050224. MR1482934 [57] G. Pisier and Q. Xu, Non-commutative Lp -spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517, DOI 10.1016/S18745849(03)80041-4. MR1999201 [58] A. Polishchuk, Analogues of the exponential map associated with complex structures on noncommutative two-tori, Pacific J. Math. 226 (2006), no. 1, 153–178, DOI 10.2140/pjm.2006.226.153. MR2247860 [59] S. Poornima, Multipliers of Sobolev spaces, J. Funct. Anal. 45 (1982), no. 1, 1–28, DOI 10.1016/0022-1236(82)90002-7. MR645643 [60] S. Poornima, An embedding theorem for the Sobolev space W 1,1 (English, with French summary), Bull. Sci. Math. (2) 107 (1983), no. 3, 253–259. MR719267 [61] N. Randrianantoanina, Non-commutative martingale transforms, J. Funct. Anal. 194 (2002), no. 1, 181–212. MR1929141 [62] N. Randrianantoanina, Conditioned square functions for noncommutative martingales, Ann. Probab. 35 (2007), no. 3, 1039–1070, DOI 10.1214/009117906000000656. MR2319715 ´ Ricard, Lp -multipliers on quantum tori, J. Funct. Anal. 270 (2016), no. 12, 4604–4613. [63] E. MR3490779 [64] J. Rosenberg, Noncommutative variations on Laplace’s equation, Anal. PDE 1 (2008), no. 1, 95–114, DOI 10.2140/apde.2008.1.95. MR2444094 [65] M. Spera, Sobolev theory for noncommutative tori (English, with Italian summary), Rend. Sem. Mat. Univ. Padova 86 (1991), 143–156 (1992). MR1154104 [66] M. Spera, A symplectic approach to Yang-Mills theory for noncommutative tori, Canad. J. Math. 44 (1992), no. 2, 368–387, DOI 10.4153/CJM-1992-025-9. MR1162350 [67] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR0290095 [68] E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR1232192

118

BIBLIOGRAPHY

[69] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32. MR0304972 [70] E. M. Stein and A. Zygmund, Boundedness of translation invariant operators on H¨ older spaces and Lp -spaces, Ann. of Math. (2) 85 (1967), 337–349. MR0215121 [71] M. H. Taibleson. On the theory of Lipschitz spaces of distributions on Euclidean n-spaces, I, II. J. Math. Mech. 13 (1964), 407-479; 14 (1965), 821-839. [72] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR503903 [73] H. Triebel, Theory of function spaces, Monographs in Mathematics, vol. 78, Birkh¨ auser Verlag, Basel, 1983. MR781540 [74] H. Triebel, Theory of function spaces. II, Monographs in Mathematics, vol. 84, Birkh¨ auser Verlag, Basel, 1992. MR1163193 [75] H. Triebel, Limits of Besov norms, Arch. Math. (Basel) 96 (2011), no. 2, 169–175, DOI 10.1007/s00013-010-0214-1. MR2773218 [76] N. Th. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), no. 2, 240–260, DOI 10.1016/0022-1236(85)90087-4. MR803094 [77] N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR1218884 [78] N. Weaver, Lipschitz algebras and derivations of von Neumann algebras, J. Funct. Anal. 139 (1996), no. 2, 261–300, DOI 10.1006/jfan.1996.0086. MR1402766 [79] N. Weaver, α-Lipschitz algebras on the noncommutative torus, J. Operator Theory 39 (1998), no. 1, 123–138. MR1610310 [80] T. H. Wolff, A note on interpolation spaces, Harmonic analysis (Minneapolis, Minn., 1981), Lecture Notes in Math., vol. 908, Springer, Berlin-New York, 1982, pp. 199–204. MR654187 [81] R. Xia, X. Xiong, and Q. Xu, Characterizations of operator-valued Hardy spaces and applications to harmonic analysis on quantum tori, Adv. Math. 291 (2016), 183–227. MR3459017 [82] X. Xiong, Q. Xu, and Z. Yin, Function spaces on quantum tori (English, with English and French summaries), C. R. Math. Acad. Sci. Paris 353 (2015), no. 8, 729–734, DOI 10.1016/j.crma.2015.06.002. MR3367642 [83] Q. Xu. Noncommutative Lp -spaces and martingale inequalities. Book manuscript, 2007. [84] Q. Xu, Interactions between quantum probability and operator space theory, Quantum potential theory, Lecture Notes in Math., vol. 1954, Springer, Berlin, 2008, pp. 117–159, DOI 10.1007/978-3-540-69365-9 4. MR2463707 [85] F. Zimmermann, On vector-valued Fourier multiplier theorems, Studia Math. 93 (1989), no. 3, 201–222. MR1030488

Editorial Information To be published in the Memoirs, a paper must be correct, new, nontrivial, and significant. Further, it must be well written and of interest to a substantial number of mathematicians. Piecemeal results, such as an inconclusive step toward an unproved major theorem or a minor variation on a known result, are in general not acceptable for publication. Papers appearing in Memoirs are generally at least 80 and not more than 200 published pages in length. Papers less than 80 or more than 200 published pages require the approval of the Managing Editor of the Transactions/Memoirs Editorial Board. Published pages are the same size as those generated in the style files provided for AMS-LATEX or AMS-TEX. Information on the backlog for this journal can be found on the AMS website starting from http://www.ams.org/memo. A Consent to Publish is required before we can begin processing your paper. After a paper is accepted for publication, the Providence office will send a Consent to Publish and Copyright Agreement to all authors of the paper. By submitting a paper to the Memoirs, authors certify that the results have not been submitted to nor are they under consideration for publication by another journal, conference proceedings, or similar publication. Information for Authors Memoirs is an author-prepared publication. Once formatted for print and on-line publication, articles will be published as is with the addition of AMS-prepared frontmatter and backmatter. Articles are not copyedited; however, confirmation copy will be sent to the authors. Initial submission. The AMS uses Centralized Manuscript Processing for initial submissions. Authors should submit a PDF file using the Initial Manuscript Submission form found at www.ams.org/submission/memo, or send one copy of the manuscript to the following address: Centralized Manuscript Processing, MEMOIRS OF THE AMS, 201 Charles Street, Providence, RI 02904-2294 USA. If a paper copy is being forwarded to the AMS, indicate that it is for Memoirs and include the name of the corresponding author, contact information such as email address or mailing address, and the name of an appropriate Editor to review the paper (see the list of Editors below). The paper must contain a descriptive title and an abstract that summarizes the article in language suitable for workers in the general field (algebra, analysis, etc.). The descriptive title should be short, but informative; useless or vague phrases such as “some remarks about” or “concerning” should be avoided. The abstract should be at least one complete sentence, and at most 300 words. Included with the footnotes to the paper should be the 2010 Mathematics Subject Classification representing the primary and secondary subjects of the article. The classifications are accessible from www.ams.org/msc/. The Mathematics Subject Classification footnote may be followed by a list of key words and phrases describing the subject matter of the article and taken from it. Journal abbreviations used in bibliographies are listed in the latest Mathematical Reviews annual index. The series abbreviations are also accessible from www.ams.org/msnhtml/serials.pdf. To help in preparing and verifying references, the AMS offers MR Lookup, a Reference Tool for Linking, at www.ams.org/mrlookup/. Electronically prepared manuscripts. The AMS encourages electronically prepared manuscripts, with a strong preference for AMS-LATEX. To this end, the Society has prepared AMS-LATEX author packages for each AMS publication. Author packages include instructions for preparing electronic manuscripts, samples, and a style file that generates the particular design specifications of that publication series. Though AMS-LATEX is the highly preferred format of TEX, author packages are also available in AMS-TEX. Authors may retrieve an author package for Memoirs of the AMS from www.ams.org/ journals/memo/memoauthorpac.html. The AMS Author Handbook is available in PDF format from the author package link. The author package can also be obtained free

of charge by sending email to [email protected] or from the Publication Division, American Mathematical Society, 201 Charles St., Providence, RI 02904-2294, USA. When requesting an author package, please specify AMS-LATEX or AMS-TEX and the publication in which your paper will appear. Please be sure to include your complete mailing address. After acceptance. The source files for the final version of the electronic manuscript should be sent to the Providence office immediately after the paper has been accepted for publication. The author should also submit a PDF of the final version of the paper to the editor, who will forward a copy to the Providence office. Accepted electronically prepared files can be submitted via the web at www.ams.org/ submit-book-journal/, sent via FTP, or sent on CD to the Electronic Prepress Department, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2294 USA. TEX source files and graphic files can be transferred over the Internet by FTP to the Internet node ftp.ams.org (130.44.1.100). When sending a manuscript electronically via CD, please be sure to include a message indicating that the paper is for the Memoirs. Electronic graphics. Comprehensive instructions on preparing graphics are available at www.ams.org/authors/journals.html. A few of the major requirements are given here. Submit files for graphics as EPS (Encapsulated PostScript) files. This includes graphics originated via a graphics application as well as scanned photographs or other computergenerated images. If this is not possible, TIFF files are acceptable as long as they can be opened in Adobe Photoshop or Illustrator. Authors using graphics packages for the creation of electronic art should also avoid the use of any lines thinner than 0.5 points in width. Many graphics packages allow the user to specify a “hairline” for a very thin line. Hairlines often look acceptable when proofed on a typical laser printer. However, when produced on a high-resolution laser imagesetter, hairlines become nearly invisible and will be lost entirely in the final printing process. Screens should be set to values between 15% and 85%. Screens which fall outside of this range are too light or too dark to print correctly. Variations of screens within a graphic should be no less than 10%. Any graphics created in color will be rendered in grayscale for the printed version unless color printing is authorized by the Managing Editor and the Publisher. In general, color graphics will appear in color in the online version. Inquiries. Any inquiries concerning a paper that has been accepted for publication should be sent to [email protected] or directly to the Electronic Prepress Department, American Mathematical Society, 201 Charles St., Providence, RI 02904-2294 USA.

Editors This journal is designed particularly for long research papers, normally at least 80 pages in length, and groups of cognate papers in pure and applied mathematics. Papers intended for publication in the Memoirs should be addressed to one of the following editors. The AMS uses Centralized Manuscript Processing for initial submissions to AMS journals. Authors should follow instructions listed on the Initial Submission page found at www.ams.org/memo/memosubmit.html. 1. GEOMETRY, TOPOLOGY & LOGIC Coordinating Editor: Richard Canary, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043 USA; e-mail: [email protected] Algebraic topology, Michael Hill, Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095 USA; e-mail: [email protected] Differential geometry, Chiu-Chu Melissa Liu, Department of Mathematics, Columbia University, New York, NY 10027 USA; e-mail: [email protected] Logic, Noam Greenberg, School of Mathematics and Statistics, Victoria University of Wellington, Wellington 6140, New Zealand; e-mail: [email protected] Low-dimensional topology and geometric structures, Richard Canary 2. ALGEBRA AND NUMBER THEORY Coordinating Editor: Henri Darmon, Department of Mathematics, McGill University, Montreal, Quebec H3A 0G4, Canada; e-mail: [email protected] Algebra, Michael Larsen, Department of Mathematics, Rawles Hall, Indiana University, 831 E 3rd St., Bloomington, IN 47405 USA; e-mail: [email protected] Algebraic geometry, Lucia Caporaso, Department of Mathematics and Physics, Roma Tre University, Largo San Leonardo Murialdo, I-00146 Rome, Italy; e-mail: [email protected] Arithmetic geometry, Ted C. Chinburg, Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104-6395 USA; e-mail: [email protected] Commutative algebra, Irena Peeva, Department of Mathematics, Cornell University, Ithaca, NY 14853 USA; e-mail: [email protected] Number theory, Henri Darmon 3. GEOMETRIC ANALYSIS & PDE Coordinating Editor: Tatiana Toro, Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350 USA; e-mail: [email protected] Geometric analysis, Tatiana Toro Partial differential equations, Markus R. Keel, School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA; e-mail: [email protected] Partial differential equations and functional analysis, Alexander A. Kiselev, Department of Mathematics, MS-136, Rice University, 6100 Main Street, Houston, TX 77005 USA; e-mail: kiselev@ rice.edu Real analysis and partial differential equations, Wilhelm Schlag, Department of Mathematics, The University of Chicago, 5734 South University Avenue, Chicago, IL 60637 USA; e-mail: schlag@ math.uchicago.edu 4. ERGODIC THEORY, DYNAMICAL SYSTEMS & COMBINATORICS Coordinating Editor: Vitaly Bergelson, Department of Mathematics, Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210 USA; e-mail: [email protected] Algebraic and enumerative combinatorics, Jim Haglund, Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104 USA; e-mail: [email protected] Combinatorics and discrete geometry, Igor Pak, Department of Mathematics, University of California Los Angeles, Los Angeles, CA 90095 USA; e-mail: [email protected] Dynamical systems and ergodic theory, Ian Melbourne, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom; e-mail: [email protected] Ergodic theory and combinatorics, Vitaly Bergelson 5. ANALYSIS, LIE THEORY & PROBABILITY Coordinating Editor: Stefaan Vaes, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, B-3001 Leuven, Belgium; e-mail: [email protected] Functional analysis and operator algebras, Stefaan Vaes Harmonic analysis and complex analysis, Malabika Pramanik, Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada; e-mail: [email protected] Harmonic analysis, representation theory, and Lie theory, Erik van den Ban, Department of Mathematics, Utrecht University, P.O. Box 80 010, 3508 TA Utrecht, The Netherlands; e-mail: E.P. [email protected] Probability and statistics, Patrick J. Fitzsimmons, Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0112 USA; e-mail: [email protected] All other communications to the editors, should be addressed to the Managing Editor, ALEJANDRO ADEM, Department of Mathematics, The University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, B.C., Canada V6T 1Z2; e-mail: [email protected]

SELECTED PUBLISHED TITLES IN THIS SERIES

1193 James Damon and Ellen Gasparovic, Medial/Skeletal Linking Structures for Multi-Region Configurations, 2017 1192 R. Lawther, Maximal Abelian Sets of Roots, 2017 1191 Ben Webster, Knot Invariants and Higher Representation Theory, 2017 1190 Agelos Georgakopoulos, The Planar Cubic Cayley Graphs, 2017 1189 John McCuan, The Stability of Cylindrical Pendant Drops, 2017 1188 Aaron Hoffman, Hermen Hupkes, and E. S. Van Vleck, Entire Solutions for Bistable Lattice Differential Equations with Obstacles, 2017 1187 Denis R. Hirschfeldt, Karen Lange, and Richard A. Shore, Induction, Bounding, Weak Combinatorial Principles, and the Homogeneous Model Theorem, 2017 1186 Mikhail Ershov, Andrei Jaikin-Zapirain, and Martin Kassabov, Property (T ) for Groups Graded by Root Systems, 2017 1185 J¨ org-Uwe L¨ obus, Absolute Continuity Under Time Shift of Trajectories and Related Stochastic Calculus, 2017 1184 Nicola Gambino and Andr´ e Joyal, On Operads, Bimodules and Analytic Functors, 2017 1183 Marius Junge, Carlos Palazuelos, Javier Parcet, and Mathilde Perrin, Hypercontractivity in Group von Neumann Algebras, 2017 1182 Marco Bramanti, Luca Brandolini, Maria Manfredini, and Marco Pedroni, Fundamental Solutions and Local Solvability for Nonsmooth H¨ ormander’s Operators, 2017 1181 Donatella Danielli, Nicola Garofalo, Arshak Petrosyan, and Tung To, Optimal Regularity and the Free Boundary in the Parabolic Signorini Problem, 2017 1180 Bo’az Klartag, Needle Decompositions in Riemannian Geometry, 2017 1179 H. Hofer, K. Wysocki, and E. Zehnder, Applications of Polyfold Theory I: The Polyfolds of Gromov-Witten Theory, 2017 1178 Igor Burban and Yuriy Drozd, Maximal Cohen-Macaulay Modules Over Non-Isolated Surface Singularities and Matrix Problems, 2017 1177 Akihito Ebisu, Special Values of the Hypergeometric Series, 2017 1176 Akinari Hoshi and Aiichi Yamasaki, Rationality Problem for Algebraic Tori, 2017 1175 Matthew J. Emerton, Locally Analytic Vectors in Representations of Locally p-adic Analytic Groups, 2017 1174 Yakir Aharonov, Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa, and Jeff Tollaksen, The Mathematics of Superoscillations, 2017 1173 C. L. Fefferman, J. P. Lee-Thorp, and M. I. Weinstein, Topologically Protected States in One-Dimensional Systems, 2017 1172 Th. De Pauw, R. M. Hardt, and W. F. Pfeffer, Homology of Normal Chains and Cohomology of Charges, 2017 1171 Yves Le Jan, Michael B. Marcus, and Jay Rosen, Intersection Local Times, Loop Soups and Permanental Wick Powers, 2017 1170 Thierry Daud´ e and Fran¸ cois Nicoleau, Direct and Inverse Scattering at Fixed Energy for Massless Charged Dirac Fields by Kerr-Newman-de Sitter Black Holes, 2016 1169 K. R. Goodearl and M. T. Yakimov, Quantum Cluster Algebra Structures on Quantum Nilpotent Algebras, 2016 1168 Kenneth R. Davidson, Adam Fuller, and Evgenios T. A. Kakariadis, Semicrossed Products of Operator Algebras by Semigroups, 2016 1167 Ivan Cheltsov and Jihun Park, Birationally Rigid Fano Threefold Hypersurfaces, 2016

For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/memoseries/.

Memoirs of the American Mathematical Society

9 781470 428068

MEMO/252/1203

Number 1203 • March 2018

ISBN 978-1-4704-2806-8