Characterization of function spaces and

boundedness of bilinear pseudodifferential

operators through Gabor frames

A ThesisPresented to

The Academic Faculty

by

Kasso Akochaye Okoudjou

In Partial Fulfillmentof the Requirements for the Degree

Doctor of Philosophy

School of MathematicsGeorgia Institute of Technology

April 2003

Characterization of function spaces and

boundedness of bilinear pseudodifferential

operators through Gabor frames

Approved by:

Professor Christopher E. Heil, Adviser

Professor Jeffrey Geronimo

Professor Gerd Mockenhaupt

Professor Yang Wang

Professor Anthony YezziECE

Date Approved

To my family.

iii

ACKNOWLEDGEMENTS

First and foremost, I would like to thank my advisor, Professor Chris Heil for intro-

ducing me to the fascinating theory of time-frequency analysis, and for guiding and

supporting me during my studies. I also thank him for setting very high standards

for me in research as well as in writing.

I wish to thank Professor Wilfrid Gangbo, not only for offering me the opportunity

to apply to Georgia Tech, but also for being so supportive during all the years I have

spent here.

I would like to thank Professors Jeffrey Geronimo, Gerd Mockenhaupt, Yang

Wang, and Anthony Yezzi for serving as members of my defense committee. I am

also thankful to Professor Michael Loss for his support while I was working on my

thesis.

Over the last five years I have had the opportunity to meet several mathematicians

who have helped me in the course of my research. I am especially grateful to Professor

Hans G. Feichtinger, Head of the Numerical Harmonic Analysis Group —NuHAG—

at the University of Vienna, for inviting me to spend a summer with his group, and

to Professor Michael Loss for providing the financial support that made this visit

possible. I really appreciated his enthusiasm for sharing his deep understanding of

mathematics. I was lucky that Professor Karlheinz Grochenig was also visiting the

NuHAG at the same time. I thank him for suggesting one of the problems I present

in this thesis, and for helping me gain some insights in time-frequency analysis. I

also wish to thank all the other members of the NuHAG group who made my stay in

Vienna wonderful and memorable. My thanks in particular go to Massimo Fornasier,

Monika Dorfler, Norbert Kaiblinger, Tobias Werther, as well as to Erik Alapaa and

iv

Bernard Keville who were visiting the NuHAG at the same time.

Several people have made my stay here at Georgia Tech a wonderful one and I

would like to thank them. Ms. Cathy Jacobson, English Language Consultant, has

been especially helpful from the first day I came to Georgia Tech till now. I thank her

for helping me improve my oral expression and my writing techniques in English as

well as for the wonderful teaching seminars she organized during the academic year

1998-1999. I wish also to thank my colleague, Brody Johnson, for all his help while

I was writing this thesis.

I have been enriched by my friends and colleagues in the School of Mathematics as

we shared so many good moments together. In particular, I am thankful to Martial

Agueh, Claudia Antonini, Gianluigi Del Magno, Jose Enrique Figueroa-Lopez, Luis

Hernandez-Urena, Armel Kelome, Hamed Maroofi, Victor Morales Duarte, Jose Miguel

Renom and Jorge Viveros for their friendship. I will miss you all.

I would like to thank my family for their steady support and love. Last, but not

least, I would like to thank my wife, Rookhiyath. My life has positively changed for

the better since we got married. Thank you for being so patient and understanding

with me, as I sometimes put my research before you.

v

TABLE OF CONTENTS

DEDICATION iii

ACKNOWLEDGEMENTS iv

SUMMARY viii

I PRELIMINARIES 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

II GABOR FRAMES AND MODULATION SPACES 9

2.1 Weight Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Submultiplicative weights . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Moderate weights . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Gabor frames in L2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Modulation spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Definition and basic properties . . . . . . . . . . . . . . . . . 15

2.3.2 Gabor frames on modulation spaces . . . . . . . . . . . . . . 19

III GABOR ANALYSIS IN WEIGHTED AMALGAM SPACES 22

3.1 Weighted amalgam spaces . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.1.2 Duality and convergence . . . . . . . . . . . . . . . . . . . . 25

3.1.3 Sequence spaces . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.2 Boundedness of the analysis and synthesis operators . . . . . . . . . 29

3.2.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2 Boundedness of the synthesis operator . . . . . . . . . . . . . 32

3.2.3 Boundedness of the analysis operator . . . . . . . . . . . . . 35

3.2.4 The Walnut representation of the Gabor frame operator onamalgam spaces . . . . . . . . . . . . . . . . . . . . . . . . . 38

vi

3.3 Gabor expansions in the amalgam spaces . . . . . . . . . . . . . . . 39

3.4 Convergence of Gabor expansions . . . . . . . . . . . . . . . . . . . 42

3.5 Necessary conditions on the window . . . . . . . . . . . . . . . . . . 45

IV EMBEDDINGS OF BESOV, TRIEBEL-LIZORKIN SPACES INTOMODULATION SPACES 51

4.1 The Besov and Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . . 52

4.2 Embedding of Besov, Triebel-Lizorkin spaces into modulation spaces 58

V BILINEAR PSEUDODIFFERENTIAL OPERATORS ON MOD-ULATION SPACES 70

5.1 Bilinear operators on modulation spaces . . . . . . . . . . . . . . . . 71

5.1.1 Definition and background . . . . . . . . . . . . . . . . . . . 71

5.1.2 Bilinear operators . . . . . . . . . . . . . . . . . . . . . . . . 73

5.1.3 A discrete model . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.1.4 Boundedness of bilinear pseudodifferential operators . . . . . 78

5.2 Linear Hilbert transform on the modulation spaces . . . . . . . . . . 81

REFERENCES 87

VITA 92

vii

SUMMARY

A frame in a separable Hilbert space H is a sequence of vectors fnn∈I which

provides a basis-like expansion for any vector in H . However, this representation is

usually not unique, since most useful frames are over-complete systems, and, hence,

are not bases. Furthermore, frames with particular structures—wavelet frames, expo-

nential frames, or Gabor frames—have proven very useful in numerous applications.

Gabor frames, also known as Weyl-Heisenberg frames, are generated by time-

frequency shifts of a single function which is called the window function or the gener-

ator. Not only do Gabor frames characterize any square integrable function, but they

also provide a precise characterization of a class of Banach spaces called modulation

spaces.

One objective of this thesis is to extend the theory of Gabor frames to other

Banach spaces which are not included in the class of the modulation spaces. In

particular, we will prove that Gabor frames do characterize a class of Banach spaces

called amalgam spaces, which include the Lebesgue spaces and play important roles in

sampling theory. Moreover, we will study the behavior of various operators connected

to the theory of Gabor frames on the amalgam spaces.

Another objective of this thesis is to formulate and prove sufficient conditions

on a function to belong to a particular modulation space. Modulation spaces have a

rather implicit definition, yet they are the natural setting for time-frequency analysis.

Consequently it is important to give sufficient conditions for membership in them. We

will prove that certain classical Banach spaces such as the Besov and Triebel-Lizorkin

spaces are embedded in the modulation spaces. These embeddings provide us with

sufficient conditions for membership in the modulation spaces.

viii

Finally, we will use the theory of Gabor frames to formulate certain boundedness

results for bilinear pseudodifferential operators with non-smooth symbols on products

of modulation spaces. More precisely, we use the Gabor frame expansions of func-

tions in the modulation spaces to convert the boundedness of these operators to the

boundedness of an infinite matrix acting on sequence spaces associated to the modu-

lation spaces. A particular modulation space known as the Feichtinger algebra turns

out to be a class of non-smooth symbols that yield the boundedness of the bilinear

pseudodifferential operators on products on modulation spaces. Additionally, we use

the same decomposition techniques to study the boundedness of the (linear) Hilbert

transform on the modulation spaces in the one dimensional case.

ix

CHAPTER I

PRELIMINARIES

1.1 Introduction

In 1946 D. Gabor [31] proposed a decomposition of signals that displays simultane-

ously the local time and frequency content of the signal, as opposed to the classical

Fourier transform which displays only the global frequency content for the entire sig-

nal. He used building blocks generated by time-frequency shifts of a Gaussian, i.e.,

building blocks of the form gm,n = e2πim· g(· − n) where g(x) = e−πx2, and sought an

orthonormal basis for L2(R) made up of these elementary functions. However, the

Balian-Low theorem [17, Chapter 2] shows that no orthonormal basis can be obtained

in this fashion with any function g as ”nice” as the Gaussian. However, by relaxing

the orthonormal basis requirement, and seeking a representation that preserves the

main features of the signal —such as its energy— and that allows stable reconstruc-

tion, Gabor’s idea turns out to yield positive results. More precisely, one can obtain

using Gabor’s scheme some very good and useful substitutes for orthonormal bases:

Gabor frames.

Frames were introduced by R. J. Duffin and A. C. Schaeffer [15] in 1952 while

working on some problems in nonharmonic Fourier series, but they were used little

until the dawn of the wavelet era. Formally, a frame in a separable Hilbert space H

is a sequence fnn∈I for which there exist constants 0 < A,B < ∞ —called frame

bounds— such that

A ‖f‖2H ≤

∑n∈I

|〈f, fn〉|2 ≤ B‖f‖2H ∀f ∈ H.

It is remarkable that the above inequalities imply the existence of a (canonical) dual

1

frame fnn∈I , such that the following reconstruction formula holds for every f ∈ H :

f =∑n∈I

〈f, fn〉 fn.

In particular, any orthonormal basis for H is a frame. However, in general, a frame

need not be a basis and, in fact, most useful frames are over-complete. The redun-

dancy that frames carry is what makes them very useful in many applications.

Gabor frames and wavelet frames are examples of “easily constructible” frames,

and have played important roles in applications as well as in pure mathematics over

the last two decades [14, 46]. From the abstract frame theory of Duffin and Schaeffer

and Gabor’s original idea, the theory of Gabor frames has grown to become a field on

its own right. Notwithstanding the fact that many questions concerning the existence

of Gabor frames remain unsolved, there exist numerous “constructible” examples of

such frames whose generators are well-localized in the time-frequency plane. In such

cases the frames are necessarily over-complete.

A more remarkable property of Gabor frames lies in the characterizations they

provide for a whole class of Banach function spaces. Indeed, a deep result in the area,

due to H. G. Feichtinger and K. Grochenig [23, 24], is the atomic decomposition of

the class of Banach spaces known as the modulation spaces via Gabor frames. These

spaces were introduced by Feichtinger [21] and can be seen as the proper tools to

quantify the time-frequency content of functions. The modulation spaces have since

then found numerous applications. In particular, they appear quite naturally in the

theory of pseudodifferential operators. A pseudodifferential operator is a formalism

that assigns to a distribution σ ∈ S ′(R2d) —the symbol of the operator— a linear

operator Tσ : S(Rd) → S ′(Rd) in such a way that properties of the symbol can be

inferred from properties of the operator. Moreover, pseudodifferential operators are

encountered in engineering, where they are known as time-varying filters, as well as

in quantum mechanics, where they appear under the name of quantization rules. We

2

refer to cf. [41, Sect. 14.1] and the references therein for more background on pseudod-

ifferential operators, as well as for their connections with partial differential equations.

A natural question one could ask is to find conditions on σ under which Tσ can be

extended to a bounded operator on L2, or on more general Banach spaces. Sym-

bols in the so-called Hormander class are known to yield bounded pseudodifferential

operators on various Banach spaces, cf. [26, Chapter 2]. In particular, Calderon-

Vaillancourt [10] proved that if σ is smooth enough and has enough decay, then Tσ

can be extended to a bounded operator on L2. Grochenig and Heil [42] recovered

and extended this result using non-smooth symbols with only a mild time-frequency

concentration as measured by a modulation space norm.

In this thesis we consider three problems centered around the theory of Gabor

frames and the modulation spaces.

The first problem we consider is an extension of the theory of Gabor frames from

its “natural setting” (the modulation spaces) to other spaces. Indeed, because the

Lp spaces are not modulation spaces if p 6= 2 [25], it was not known if these spaces

could be characterized via Gabor frames. In a joint work with K. Grochenig and

C. Heil [39], we show that Gabor frames do characterize a class of Banach spaces

called the amalgam spaces, which include the Lebesgue spaces. Amalgam spaces

are spaces that amalgamate local and global criterion for membership. They appear

naturally in sampling theory, where they are the “right” setting for different problems

[1]. Additionally, we will prove a weak necessary condition on the Gabor frame’s

generator, thereby extending a result due to R. Balan [2].

The second problem we consider is concerned with the modulation spaces. In spite

of being the “right” spaces for time-frequency analysis, their rather implicit definition

makes it very difficult to decide if a function belongs to a particular modulation

space. We formulate sufficient conditions for membership in the modulation spaces by

proving embeddings of certain Banach spaces such as the Besov and Triebel-Lizorkin

3

spaces into some modulation spaces [50]. The class of Besov and Triebel-Lirzorkin

spaces, includes some well-known Banach spaces such as the Lebesgue, the Holder-

Lipschitz, the Sobolev spaces, and is equipped with a wide variety of equivalent norms.

We refer to [55, Chapter 4], [58, Chapters 1–2] for background on these spaces. The

embedding results we prove can be seen as a comparison among certain properties of

functions, i.e., smoothness and decay versus time-frequency concentration.

The last problem we consider can be viewed as an application of the theory of

Gabor frames. More precisely, we consider the boundedness of bilinear pseudodiffer-

ential operators on modulation spaces. One of the motivation of investigating the

bilinear pseudodifferential operators on the modulation spaces comes from the recent

developments of their linear counterpart in the realm of the modulation spaces. More

precisely, because the Weyl correspondence —which is a particular way of assigning

to a symbol a pseudodifferential operator corresponding to the Weyl quantization

rule— can be expressed as superposition of time-frequency shifts, which are the main

objects used in defining the modulation spaces, it was natural to study the linear

pseudodifferential operators on these spaces. Bilinear pseudodifferential operators

are defined through their symbols as bilinear operators from S(Rd) × S(Rd) into

S ′(Rd), and are not just generalizations of their linear counterparts, but are impor-

tant tools in many problems in analysis [49]. A natural question in this context is

again to find sufficient conditions on the symbols that guarantee the boundedness of

the corresponding operators on products of certain Banach spaces. Smoothness and

decay of the symbols are often the conditions needed to prove the boundedness of

these operators [11, 49, 35, 36]. In a joint work with A. Benyi [6], we prove that if the

symbols are in a particular modulation space —the so-called Feichtinger algebra—

then the corresponding bilinear pseudodifferential operators are bounded on products

of modulation spaces. As particular cases, we obtain boundedness results on prod-

ucts of certain Lebesgue spaces using non-smooth symbols. Finally, we prove that

4

the Hilbert transform is bounded on the modulation spaces, using a discrete tool via

the atomic decomposition of these spaces by Gabor frames and by relying on the L2

theory of the Hilbert transform.

1.2 Outline of the Thesis

The thesis is organized as follows. Chapter 1 contains a brief survey of the notations

and definitions that will be used in the sequel.

Chapter 2 is mostly expository. In particular, it contains the definition of the

basic tools of time-frequency analysis. Moreover, it contains the definition of the

Gabor frames, as well as their main properties. Additionally, the definition of the

modulation spaces and their atomic decompositions by Gabor frames as well as their

main properties is given in the chapter.

Chapter 3 is devoted to the first main result of the thesis, namely the characteriza-

tion of the weighted amalgam spaces by Gabor frames. To obtain this characterization

we study the behavior of the various operators connected with (Gabor) frames the-

ory. Additionally we prove a weak necessary condition on the generator of the Gabor

frames.

Chapter 4 contains the embedding results of certain Besov and Triebel-Lizorkin

spaces into the modulation spaces. These results provide some sufficient conditions

for membership in the modulation spaces. To obtain these results, we rely on the

numerous equivalent norms defining the Besov and Triebel-Lizorkin spaces as well as

the properties of the short time Fourier transform (STFT).

Finally, Chapter 5 contains some applications of the theory of Gabor frames to

the study of bilinear pseudodifferential operators. In particular, we present in Section

5.1 a boundedness result for bilinear pseudodifferential operators using a discrete

approach via Gabor frames. Section 5.2 is devoted to prove the boundedness of the

Hilbert transform on the modulation spaces again using a discrete approach.

5

1.3 Notations

The usual dot product of x, y ∈ Rd is denoted by x · y = x1y1 + x2y2 + . . . + xnyn.

The length of x is |x|.We use the notation |a| to denote the magnitude of a complex number a.

We use the notation f ∗(t) = f(−t).The convolution of f and g is defined formally as (f ∗ g)(x) =

∫f(x− t) g(t) dt.

The Fourier transform of a function f is defined formally by

Ff(ω) = f(ω) =

∫Rd

f(t) e−2πit·ω dt for ω ∈ Rd.

Similarly, the inverse Fourier transform of f is defined formally by

F−1f(t) = f(t) =

∫Rd

f(ω) e2πiω·t dω for t ∈ Rd.

If E ⊂ Rd is a measurable set, χE is the characteristic function of E, and we

denote its Lebesgue measure by |E|.If a > 0, we deonote Qa the cube in Rd with side length a, i.e., Qa = [0, a[d.

Let X be a Banach space, then the norm of u ∈ X will be denoted ‖u‖X or simply

‖u‖ when the appropriate space is clear from the context. Moreover, if two norms

‖ · ‖1 and ‖ · ‖2, are equivalent on a Banach space X we will write ‖u‖1 ‖u‖2 to

mean the existence of two positive constant C1, C2 such that

C1‖u‖1 ≤ ‖u‖2 ≤ C2‖u‖1 ∀u ∈ X.

The dual of a Banach space X is denoted X∗. We write 〈f, g〉 for the action of

f ∈ X ′ on g ∈ X.

The adjoint of an operator T is denoted by T ∗.

For 1 ≤ p ≤ ∞, p′ will denote the conjugate of p, i.e., 1p

+ 1p′ = 1.

Lp(Rd) is the Banach space of complex-valued functions f on Rd with norm

‖f‖p = ‖f‖Lp =

(∫Rd

|f(x)|p dx)1/p

,

6

for 1 ≤ p <∞. If p = ∞, the norm is given by

‖f‖∞ = ess supx∈Rd

|f(x)|.

Similarly, `p(Zd) is the Banach space of complex-valued sequences c on Zd with

norm

‖c‖p = ‖c‖`p =

(∑n∈Zd

|cn|p)1/p

,

for 1 ≤ p <∞. If p = ∞, the norm is given by

‖c‖∞ = supn∈Zd

|cn|.

We will also consider weighted mixed-norm spaces Lp,qν (R2d), which are Banach

spaces of complex-valued functions f on R2d with norm

‖f‖Lp,qν

=

(∫Rd

(∫Rd

|f(x, y)|p ν(x, y)p dx

)q/p

dy

)1/q

,

for 1 ≤ p, q < ∞, with obvious modifications if p = ∞, or q = ∞. The weight

function ν will be described in the following chapters.

We define similarly the discrete weighted mixed-norm spaces `p,qν as the Banach

space of complex-valued sequences c on Z2d with norm

‖c‖`p,qν

=

(∑l∈Zd

(∑k∈Zd

|ck,l|p ν(k, l)p

)q/p)1/q

,

for 1 ≤ p, q < ∞, with usual modifications if p = ∞ or q = ∞. The weight ν is an

appropriate sample of the weight function ν, typically ν(k, l) = ν(αk, βl) for some

α, β > 0.

We use the notation ωs for the function ωs(x) = (1 + |x|2)s/2 for s > 0.

If E is a measurable subset of Rd, we let

‖f‖p,E = ‖f χE‖p

denote the norm of the function f restricted to the set E.

7

If T is a bounded linear operator from a Banach space X to a Banach space Y , we

denote the operator norm of T by ‖T‖X→Y , or simply by ‖T‖ if there is no confusion.

For a multi-index α = (α1, . . . , αd), we write |α| =∑d

i=1 αi. The differentiation

operator Dα and the multiplication operator Xβ are defined by

Dαf(x) =d∏

i=1

(∂xi)αi f(x), and Xβf(x) =

d∏i=1

xβi

i f(x).

S(Rd) is the Schwartz space of all infinitely differentiable functions f for which

the seminorms

‖f‖(M,N) =∑

|α|≤M

∑|β|≤N

‖DαXβ f‖∞

are finite for all non-negative integers M,N . Its topological dual, S ′(Rd), is the space

of tempered distributions.

More details on the basic properties of the Fourier transform and more generally,

on some of the theory from real and functional analysis that we will systematically

used in the sequel can be found in many standard analysis texts, e.g., [27], [48], [52].

8

CHAPTER II

GABOR FRAMES AND MODULATION

SPACES

A Gabor frame G(g, α, β) = e2πiβn·g(· − αk)k,n∈Zd for L2(Rd) provides basis-like

series representations of functions in L2, with unconditional convergence of the series.

However, unless the frame is a Riesz basis (and hence, by the Balian–Low theorem

has poor time-frequency localization), these representations will not be unique. Still,

a canonical and computable representation exists, and Gabor frames have found a

wide variety of applications in mathematics, science, and engineering [14, 17, 18, 41].

An important fact is that Gabor frames provide much more than just a means to

recognize square-integrability of functions. If the window function g is reasonably

well-localized in time and frequency, then Gabor frame expansions are valid not only

in L2 but in an entire range of associated spaces Mp,qν known as the modulation

spaces. The frame expansions converge unconditionally in the norm of those spaces,

and membership of a tempered distribution in Mp,qν is characterized by membership

of its sequence of Gabor coefficients in a weighted sequence space `p,qν . We refer to [41]

for a recent detailed development of time-frequency analysis and modulation spaces.

In this chapter, we review some of the key results regarding Gabor expansions of

L2 functions. We then define the modulation spaces and their atomic decompositions

by Gabor frames, which will be used often throughout this thesis.

2.1 Weight Functions

Before delving into Gabor analysis per se, we introduce here a class of weight functions

that appear in most of the subsequent chapters.

9

2.1.1 Submultiplicative weights

A submultiplicative weight function ω is a positive, symmetric, and continuous func-

tion which satisfies

∀ x, y ∈ Rd, ω(x+ y) ≤ ω(x)ω(y).

The prototypical example of a submultiplicative weight is the polynomially-growing

function ωs(x) = (1+ |x|2)s/2, where s > 0. We also consider weight functions defined

on R2d by making the obvious changes in the definition.

2.1.2 Moderate weights

A positive, symmetric, and continuous function ν is called ω-moderate function if

there exists a constant Cν > 0 such that

∀ x, y ∈ Rd, ν(x+ y) ≤ Cν ω(x) ν(y). (1)

If ν is ω-moderate with respect to ω = ωs, for some s > 0, we say that ν is s-moderate.

For example, ν(x) = (1 + |x|)t is moderate with respect to ωs(x) = (1 + |x|2)s/2,

where s > 0, exactly for |t| ≤ s.

If desired, the assumptions of continuity and symmetry of ω and ν could be

removed, but there would be no increase in the generality of the results. For if ω is

a positive, submultiplicative function, then there exists a continuous weight function

ω1 such that 0 < A ≤ ω(x)/ω1(x) ≤ B < ∞ for all x, and similarly for ω-moderate

functions ν, cf. [41, Sect. 11.1].

If ν is ω-moderate, then by manipulating (1) we see that

1

ν(x+ y)≤ Cν ω(x)

1

ν(y),

so 1/ν is also ω-moderate (with the same constant). Thus, the class of ω-moderate

weights is closed under reciprocals, and consequently the class of spaces Lpν using

ω-moderate weights is closed under duality (with the usual exception for p = ∞).

This would not be the case if we restricted only to submultiplicative weights.

10

Given an ω-moderate weight ν on Rd, we will often use the notation ν to denote

the weight on Zd defined by ν(k) = ν(αk), and for a weight ν on R2d we define

ν(k, n) = ν(αk, βn), or ν(k) = ν(k/β), the particular choice being clear from context,

2.2 Gabor frames in L2

Before defining Gabor frames we first introduce two operators that play important

roles in time-frequency analysis.

Definition 2.2.1. Given a, b ∈ Rd, the translation and modulation operators are

defined respectively by

Taf(t) = f(t− a), Mbf(t) = e2πit·bf(t)

for any function f defined on Rd.

Additionally, for c ∈ R, c 6= 0 we define the dilation operator acting on a function

f defined on Rd by

Dcf(t) = |c|−d/2 f(t/c).

It is easily seen that the translation, modulation, and dilation operators are uni-

tary on L2, and that they map S and S ′ isomorphically onto themselves. The following

lemma collects some basic facts about the translation and modulation operators.

The proof of the following lemma is immediate from the definition so we omit it.

Lemma 2.2.2. Let a, b ∈ Rd, c ∈ R, c 6= 0, and f be a function defined on Rd. The

following statements hold.

a. TaMb f = e−2πia·bMb Taf.

b. DcTaf = TcaDcf.

c. DcMbf = M bcDcf.

d. Ta f = M−a f , and Mb f = Tbf .

11

e. Dcf = D 1cf .

f. Mb Ta f = e2πia·bM−a Tb f .

We are now in position to define Gabor frames.

Definition 2.2.3. Given a window function g ∈ L2(Rd) and given α, β > 0, we say

that

G(g, α, β) = MβnTαkgk,n∈Zd = e2πiβn·g(· − αk)k,n∈Zd

is a Gabor frame for L2(Rd) if there exist constants A, B > 0 (called frame bounds)

such that for all f ∈ L2(Rd),

A ‖f‖2L2 ≤

∑k,n∈Zd

|〈f,MβnTαkg〉|2 ≤ B ‖f‖2L2. (2)

We should point out that due to the commutation relations between the transla-

tion and modulation operators, it is trivial to see that the order of the translation and

modulation in the definition of a Gabor frame is irrelevant. Moreover, the image of a

Gabor frame G(g, α, β) under the Fourier transform is another Gabor frame, namely

G(g, β, α).

Definition 2.2.4. Consider the collection of time-frequency shifts G(g, α, β) gener-

ated by g ∈ L2(Rd), and α, β > 0.

a. The analysis operator associated with G(g, α, β) is the operator Cg : L2(Rd) →`2(Z2d) defined by Cgf =

(〈f,MβnTαkg〉)

k,n∈Zd, for f ∈ L2.

b. The synthesis operator associated with G(g, α, β) is the operator Rg : `2(Z2d) →L2(Rd) defined formally by Rgc =

∑k,n∈Zd cknMβnTαkg, for c = (ck,n)k,n∈Zd ∈ `2.

The basic properties of Gabor frames are laid out in the following result; we refer

to [14], [41], or [46] for more extensive treatments of frames and Gabor frames.

12

Theorem 2.2.5. Let G(g, α, β) be a Gabor frame for L2(Rd) with frame bounds A,

B. Then the following statements hold.

a. The analysis operator Cgf =(〈f,MβnTαkg〉

)k,n∈Zd is a bounded mapping from

L2 into `2, and we have the norm equivalence ‖f‖2 ‖Cgf‖`2.

b. The synthesis operator Rgc =∑

k,n∈Zd cknMβnTαkg is a bounded mapping from

`2 into L2. The series defining Rgc converges unconditionally in L2 for every

c ∈ `2.

c. Rg = C∗g , and the frame operator Sg = RgCg : L2 → L2 is strictly positive.

d. The dual window γ = S−1g g generates a Gabor frame G(γ, α, β) for L2(Rd) with

frame bounds 1/B, 1/A.

e. RγCg = I on L2(Rd), i.e., we have the Gabor expansions

f = RγCgf =∑

k,n∈Zd

〈f,MβnTαkγ〉MβnTαkg (3)

for f ∈ L2(Rd), with unconditional convergence of the series.

Proof. a. The fact that Cg : L2 → `2 is bounded follows from the second part of

(2); moreover, (2) is precisely the statement that ‖f‖2 ‖Cgf‖`2.

b. Let c = (ckn)k,n∈F , where F is a finite subset of Z2d. For f ∈ L2, we have

|〈Rgc, f〉| = |∑

k,n∈F

ckn〈MβnTαkg, f〉|

≤∑

k,n∈F

|ckn||〈f,MβnTαkg〉|

≤( ∑

kn∈F

|ckn|2)1/2 ( ∑

kn∈Zd

|〈f,MβnTαkg〉|2)1/2

≤√B ‖f‖L2

( ∑kn∈F

|ckn|2)1/2

,

13

where we have used the second inequality in (2). By duality we obtain

‖Rgc‖L2 = sup‖f‖L2=1

|〈Rgc, f〉|

≤√B

( ∑kn∈F

|ckn|2)1/2

for all sequences with finite support. A standard density argument shows that

Rg is bounded from `2 to L2, and that the series defining Rg converges uncon-

ditionally.

c. For f ∈ L2 and c = (ckn)k,n∈Zd, we have

〈Rgc, f〉 =∑

k,n∈Zd

ckn〈f,MβnTαkg〉〈c, Cgf〉.

Hence, Rg = C∗g . The frame operator Sg = RgCg is clearly bounded on L2;

moreover, the first part of inequality (2) implies that Sg is strictly positive.

d. The frame inequality (2) can be rewritten as A ‖f‖2L2 ≤ 〈Sgf, f〉 ≤ B ‖f‖2

L2 for

all f ∈ L2, or equivalently in operator notation as AI ≤ Sg ≤ B I. Moreover,

the above operator inequalities are preserved when multiplied by operators that

commute with each of the terms appearing in the inequalities. Thus, we obtain

that B−1 I ≤ S−1g ≤ A−1 I. Moreover, by some easy computations, one can

show that Sg commutes with the translation and modulation operators Tαk and

Mβn, and so does S−1g . Hence, S−1

g = Sγ , which together with the last operator

inequality concludes the proof of this part.

e. Follows from the fact that f = SgSγf , and that Sg and Sγ commute with the

translation and modulation operators Tαk and Mβn.

In brief, if G(g, α, β) is a frame for L2(Rd) then the `2-norm of the sequence of

Gabor coefficients (〈f,MβnTαkg〉)k,n∈Zd is an equivalent norm for L2, and the Gabor

14

expansions given by (3) hold in L2. Moreover, for our purposes it is important to

note that once the analysis and synthesis operators are defined, the statement “Gabor

expansions converge in L2” is equivalent to the statement that the identity operator

on L2 factorizes as I = RγCg.

In all these statements, and throughout this thesis, the roles of g and γ may be

interchanged.

2.3 Modulation spaces

2.3.1 Definition and basic properties

The modulation spaces introduced by Feichtinger are spaces of tempered distributions

defined by imposing some decay condition on their short-time Fourier transforms,

which we next define.

Definition 2.3.1. The Short-Time Fourier Transform (STFT) of a function f ∈ L2

with respect to a window g ∈ L2 is

Vgf(x, y) = 〈f,MyTxg〉 =

∫Rd

e−2πiy·t g(t− x) f(t) dt.

Remark 2.3.2. From the above definition, it is clear that the STFT can be defined

whenever f and g are in dual spaces. In particular, the STFT is well-defined and

scalar-valued when f ∈ S ′ and g ∈ S. Moreover, analogously to the Fourier trans-

form, the STFT extends in a distributional sense to f , g in the space of tempered

distributions S ′, cf. [26, Prop. 1.42].

The next proposition, whose proof is immediate and will be omitted, collects some

different definitions of the STFT that will be used throughout this thesis.

15

Proposition 2.3.3. If f, g ∈ L2(Rd), then the following statements are true:

Vgf(x, y) = 〈f,MyTxg〉

=(f · Txg

)∧(y)

= e2πix·y f ∗ (Myg

∗)(x)= 〈f , TyM−xg〉

= e−2πix·y 〈f ,M−xTyg〉

= e−2πix·y Vgf(y,−x).

The next result sheds some light on the behavior of the STFT on L2.

Proposition 2.3.4. Let g ∈ L2(Rd), and assume that g 6= 0. Then for all f ∈ L2(Rd)

we have that

‖Vgf‖L2 = ‖f‖L2 ‖g‖L2.

That is, Vg is a multiple of an isometry from L2(Rd) into L2(R2d).

Proof. First assume that f ∈ S then, f · Txg ∈ L2(Rd) for almost all x ∈ Rd.

Therefore, we have

‖Vgf‖2L2 =

∫∫R2d

|Vgf(x, ω)|2 dx dω

=

∫∫R2d

| (f · Txg)(ω)|2 dω dx

=

∫∫R2d

|(f · Txg)(t)|2 dt dx

=

∫∫R2d

|f(t)|2 |g(t− x)|2 dt dx

= ‖f‖2L2 ‖g‖2

L2.

Thus ‖Vgf‖L2 = ‖f‖L2 ‖g‖L2 for all f ∈ S, and a standard density argument extends

the result to all f ∈ L2.

16

The previous proposition shows that on L2 the STFT is an isometry (up to a

constant), and hence does not provide any new information beside the conservation

of the energy. However, by imposing other norms on the STFT, we can better quantify

the time-frequency concentration of functions. More precisely, we have the following

definition.

Definition 2.3.5. Let ν be an ω-moderate weight on R2d, and let 1 ≤ p, q ≤ ∞.

Given a window function g ∈ S, the modulation space Mp,qν (Rd) is the space of all

distributions f ∈ S ′ for which the following norm is finite:

‖f‖Mp,qν

=

(∫Rd

(∫Rd

|Vgf(x, ξ)|p ν(x, ξ)pdx

)q/p

dξ

)1/q

= ‖Vgf‖Lp,qν, (4)

with the usual modifications when p or q is infinite.

For background and information on the basic properties of the modulation spaces

we refer to [21], [23], [24], [41]. The definition of the modulation space is independent

of the choice of the window g in the sense of equivalent norms. More precisely, the

following result whose prove may be found in [41, Proposition 11.3.1].

Proposition 2.3.6. Assume that ν is ω-moderate and that g1, g2 ∈ S(Rd) and that

gi 6= 0 when i = 1, 2. If 1 ≤ p, q ≤ ∞, let ‖f‖gi

Mp,qν

denote the norm of f in the

modulation space Mp,qν as measured by the window gi when i = 1, 2. Then there exist

two constants C1, C2 > 0 such that

C11

‖Vg2g1‖L1ω

‖f‖g2

Mp,qν

≤ ‖f‖g1

Mp,qν

≤ C2 ‖Vg1g2‖L1ω‖f‖g2

Mp,qν.

The next theorem collects some basic facts on the modulation spaces, its proof

may be found in [41].

Theorem 2.3.7. Let ν be an ω-moderate weight.

a. For 1 ≤ p, q ≤ ∞, Mp,qν is a Banach space.

17

b. If p, q < ∞, S is a dense subspace of Mp,qν . Moreover, the dual of Mp,q

ν is the

modulation space Mp′,q′1/ν . More precisely we have that

‖f‖Mp,qν

= sup‖g‖

Mp′,q′1/ν

=1

|〈f, g〉|.

c. If p1 ≤ p2, and q1 ≤ q2, then Mp1,q1ν ⊂Mp2,q2

ν .

d. If p, q <∞, then M1ω is a dense subspace of Mp,q

ν .

Remark 2.3.8. a. If p = q we denote the modulation space Mp,pν simply by Mp

ν .

Moreover, if ν = 1 we simply denote Mp,qν by Mp,q.

b. Among the modulation spaces are certain well-known spaces:

• if ν(x, ξ) = 1, and p = q = 2, it is easy to see that M2 = L2,

• if ν(x, ξ) = (1 + |x|2)s/2 where s > 0, and p = q = 2, then M2ν = L2

s, a

weighted-L2 space,

• if ν(x, ξ) = (1 + |ξ|2)s/2 where s > 0, and p = q = 2, then M2ν = H2

s , the

standard Sobolev space.

c. Lp for p 6= 2 does not coincide with any modulation space [25].

d. The modulation M1 has several properties that deserve to be mentioned. It is

a Banach algebra under both pointwise multiplication and convolution. It is

the smallest Banach space that is isometrically invariant under translation and

modulation. Moreover, it is a Segal algebra known as the Feichtinger algebra,

and often denoted S0, and plays an important role in time-frequency analysis.

We refer to [41] and the references therein for more detail on the Feichtinger

algebra and its weighted version.

The next result, whose proof can be found in [41, Proposition 11.3.1], provides a

characterization of S and its dual S ′ in terms of the modulation spaces.

18

Proposition 2.3.9. Let vs be the weight function defined on R2d by vs(z) = (1 +

|z|)s, z ∈ R2d. Then we have

S(Rd) =⋂s≥0

M∞vs

and S ′(Rd) =⋃s≥0

M∞1/vs

.

The next proposition, due to Feichtinger [21], on complex interpolation of mod-

ulation spaces will be used in the proof of our results in the following chapters. For

more background on complex interpolation we refer to [8, Chapter 4].

Proposition 2.3.10. Let 1 ≤ p0 < ∞, 1 ≤ q0 < ∞, 1 ≤ p1 ≤ ∞, 1 ≤ q1 ≤ ∞, and

θ ∈ (0, 1). If 1p

= 1−θp0

+ θp1

, and 1q

= 1−θq0

+ θq1

then

(Mp0,q0,Mp1,q1

)[θ]

= Mp,q. (5)

2.3.2 Gabor frames on modulation spaces

Under stronger assumptions on g, the expansions in (3) are valid not only in L2 but

in the entire class of the modulation spaces.

The following result summarizes some basic facts on Gabor frames in the modu-

lation spaces, cf. [41, Ch. 12]. The theorem is not stated in its weakest possible form;

for example, the boundedness of the analysis and synthesis operators requires only

the assumption g ∈ M1ω, and does not require that g generate a frame for L2. Recall

that the mixed-norm sequence space `p,qν consists of all sequences c = (ckn)k,n∈Zd such

that

‖c‖`p,qν

=

(∑n∈Zd

(∑k∈Zd

|ckn|p ν(k, n)p

)q/p)1/q

< ∞,

where ν(k, n) = ν(αk, βn), with the usual adjustments when p = ∞ or q = ∞.

Theorem 2.3.11. Let ν be an ω-moderate weight on R2d, and let 1 ≤ p, q ≤ ∞.

Let g ∈ M1ω be such that G(g, α, β) is a Gabor frame for L2(Rd). Then the following

statements hold.

19

a. The analysis operator defined by Cgf =(〈f,MβnTαkg〉

)k,n∈Zd is a bounded map-

ping from Mp,qν to `p,q

ν , and we have the norm equivalence

‖f‖Mp,qν

‖Cgf‖`p,qν.

b. The synthesis operator defined by Rgc =∑

k,n∈Zd cknMβnTαkg is a bounded map-

ping from `p,qν to Mp,q

ν . The series defining Rgc converges unconditionally in the

norm of Mp,qν for every c ∈ `p,q

ν (weak* unconditionally in M∞,∞1/ω if p = ∞ or

q = ∞).

c. The frame operator Sg = RgCg is a continuously invertible mapping of Mp,qν

onto itself.

d. The dual window γ = S−1g g lies in M1

ω.

e. RγCg = I on Mp,qν , i.e., we have the Gabor expansions

f = RγCgf =∑

k,n∈Zd

〈f,MβnTαkγ〉MβnTαkg (6)

for f ∈Mp,qν , with unconditional convergence of the series if p, q <∞, and with

unconditional weak* convergence otherwise.

f. A distribution f ∈ M∞,∞ν belongs to Mp,q

ν if and only if Cgf ∈ `p,qν . If g ∈ S,

then a tempered distribution f ∈ S ′(Rd) belongs to Mp,qν if and only if Cgf ∈ `p,q

ν .

In brief, the `p,qν norm of the Gabor coefficients (〈f,MβnTαkg〉)k,n∈Zd is an equiva-

lent norm forMp,qν , and the Gabor expansions (3) are valid in Mp,q

ν with unconditional

convergence of that series in the norm of Mp,qν . Moreover, there is a strong statement

made in part f of Theorem 2.3.11 that is not usually observed in the standard list of

Gabor frame properties in L2 (Theorem 2.2.5), namely that ‖Cgf‖`p,qν

is not only an

equivalent norm for Mp,qν , but membership of f in the modulation space is character-

ized by membership of its sequence of Gabor coefficients Cgf in `p,qν . In particular,

20

only the magnitude of these coefficients is important in determining whether a given

distribution lies in Mp,qν .

The proof of Theorem 2.3.11 requires deep analysis. In particular, the invertibility

of Sg on M1ω for arbitrary values of α, β was only recently proved in [43].

In summary, once the analysis and synthesis operators have been correctly defined,

the fact that Gabor expansions converge in the modulation spaces is simply the

statement that the identity operator on Mp,qν factorizes as I = RγCg.

21

CHAPTER III

GABOR ANALYSIS IN WEIGHTED

AMALGAM SPACES

Some results on Gabor analysis outside of the modulation spaces were obtained by

Walnut in [59]. In particular, he introduced what is now known as the Walnut repre-

sentation of the frame operator, and considered the boundedness of the frame operator

on Lp. Recently, it was independently observed in [34] and [38] that Gabor expansions

actually converge in Lp(Rd) when 1 < p < ∞. Since Lp is not a modulation space

when p 6= 2, it was known that Gabor expansions could not converge unconditionally

in Lp [25].

In this chapter we consider a much larger class of spaces than the Lp spaces,

namely, we consider the weighted amalgam spaces W (Lp, Lqν). These spaces amalga-

mate a local criteria for membership with a global criteria. We will show that not

only do Gabor expansions converge for the special case Lp = W (Lp, Lp), but that

they converge in the entire range of weighted amalgam spaces. Moreover, member-

ship in the amalgam space is characterized by membership of the Gabor coefficients

in an appropriate sequence space. In the course of obtaining these results, we prove

several results of independent interest on the behavior of the analysis and synthesis

operators associated with the Gabor frame, and on the Walnut representation, which

is an extremely useful tool in Gabor frame theory. Moreover, we include the cases

p = 1,∞ or q = 1,∞ in our consideration. In particular, we show that Gabor ex-

pansions exist even in L1 and in a weak sense in L∞, given the right interpretation

of “expansion.” Additionally, we obtain some necessary conditions on the window

22

g, extending weaker necessary conditions obtained by Balan in [2] for the particular

case W (L2, L∞).

3.1 Weighted amalgam spaces

3.1.1 Definition

Given an ω-moderate weight ν on Rd and given 1 ≤ p, q ≤ ∞, the weighted amalgam

space W (Lp, Lqν) is the Banach space of all measurable functions on Rd for which the

norm

‖f‖W (Lp,Lqν) =

(∑k∈Zd

‖f · TαkχQα‖qp ν(αk)

q

)1/q

(7)

is finite, with the usual adjustment if q = ∞.

The first use of amalgam spaces was by Wiener, who introduced the spaces

W (L1, L2) and W (L2, L1) in [60] and W (L∞, L1) and W (L1, L∞) in [61], [62], in

connection with his development of the theory of generalized harmonic analysis. The

space W (L∞, L1) is sometimes called the Wiener algebra (although this term is some-

times used to denote FL1), cf. [51]. It was shown in [59] that W (L∞, L1) is a conve-

nient and general class of windows for Gabor analysis within L2.

Since any cube Qα in Rd can be covered by a finite number of translates of a

cube Qβ, the space W (Lp, Lqν) is independent of the value of α used in (7) in the

sense that each different choice of α yields an equivalent norm for W (Lp, Lqν). A wide

variety of other equivalent norms is provided by Feichtinger’s theory of amalgam

spaces [20, 19, 22]. We refer to [44] for an exposition of the “continuous” norms on

the amalgam spaces.

The following lemma provides some useful inclusions among the amalgam spaces.

Lemma 3.1.1. For each ω-moderate weight ν, we have the following inclusion rela-

tions: if p1 ≥ p2, and q1 ≤ q2, then

W (Lp1, Lq1ω ) ⊂W (Lp1, Lq1

ν ) ⊂ W (Lp2, Lq2ν ) ⊂W (Lp2, Lq2

1/ω).

23

In particular, the inclusions W (L∞, L1ω) ⊂ W (Lp, Lq

ν) ⊂ W (L1, L∞1/ω) hold for all

1 ≤ p, q ≤ ∞ and all ω-moderate weights ν. In this sense W (L∞, L1ω) is the smallest

and W (L1, L∞1/ω) is the largest amalgam space in the class of amalgam spaces with

ω-moderate weight functions.

Proof. The fact that ν is ω-moderate implies in particular that ν(x) ≤ Cω(x) for

some positive constant C (this follows immediately from (1). Hence the inclusion

W (Lp1, Lq1ω ) ⊂W (Lp1 , Lq1

ν ) follows from the (7).

Now let f ∈W (Lp1, Lq1ν ). Then

‖f‖W (Lp2 ,Lq2ν ) =

(∑k∈Zd

‖f · TαkχQα‖q2p2ν(αk)q2

)1/q2

≤ C

(∑k∈Zd

‖f · TαkχQα‖q2p1ν(αk)q2

)1/q2

≤ C

(∑k∈Zd

‖f · TαkχQα‖q1p1ν(αk)q1

)1/q1

,

where we have used the inclusions `q1(Zd) ⊂ `q2(Zd) (because q1 ≤ q2), as well as

Lp1(K) ⊂ Lp2(K) for p1 ≥ p2 and K a compact subset of Rd. Thus we obtain

W (Lp1, Lq1ν ) ⊂W (Lp2 , Lq2

ν ).

The last inclusion, W (Lp2, Lq2ν ) ⊂W (Lp2, Lq2

1/ω) follows again from the fact that ν

is ω-moderate, and so is 1/ν, and so 1ν(x)

≤ Cω(x) for all x ∈ Rd.

The last part of the lemma is just an application of the above with p1 = ∞, p2 =

p, q1 = 1, and q2 = q.

Remark 3.1.2. For p, q <∞, the Schwartz class S and the space of functions with

compact support are dense in W (Lp, Lqν).

24

3.1.2 Duality and convergence

We will need to be precise about the meaning of convergence of series. For general

references we refer to the text of Singer [54], and for references on Banach function

spaces we refer to the text of Bennett and Sharpley [5].

The following lemma characterizing unconditional convergence will be useful.

Lemma 3.1.3. Let X be a Banach space with dual space X∗, and let fk ∈ X for

k ∈ J . Then the following statements are equivalent.

a.∑

k∈J fk converges unconditionally in X, i.e., it converges with respect to every

ordering of the index set J .

b. There exists f ∈ X such that for each ε > 0, there exists a finite F0 ⊂ J such

that

∀ finite F ⊃ F0,∥∥∥f −

∑k∈F

fk

∥∥∥X< ε.

c. For every ε > 0, there exists a finite F0 ⊂ J such that

∀ finite F ⊃ F0, sup

∑k/∈F

|〈fk, h〉| : h ∈ X∗, ‖h‖X∗ = 1

< ε.

Now let X be a Banach function space in the sense of [5]. In particular, this

includes the amalgam spaces W (Lp, Lqν). The Kothe dual of X (or the associated

space, as it is called in [5]), is the space X consisting of all measurable functions

h such that fh ∈ L1 for each f ∈ X. By [5, Thm. 1.2.9], X is a closed, norm-

fundamental subspace of X∗, so in particular,

∀ f ∈ X, ‖f‖X = sup|〈f, h〉| : h ∈ X, ‖h‖X = 1

.

By [5, Cor. 1.5.3], X is complete in the σ(X, X) topology, i.e., the weak topology onX

generated by X. In particular, a series∑

k∈J fk converges in the σ(X, X) topology if∑k∈J〈fk, h〉 converges for each h ∈ X. It converges unconditionally in that topology

25

if the convergence is independent of the ordering of J , and since the terms 〈fk, h〉 are

scalars, this occurs if and only if

∀h ∈ X,∑k∈J

|〈fk, h〉| < ∞.

Remark 3.1.4. a. If X1 and X2 are two Banach function spaces such that X1 ⊂X2 then X2 ⊂ X1. Indeed, let f ∈ X2 and g ∈ X1 with ‖g‖X1 = 1, then by the

definition of the Kothe dual, we have that f g ∈ L1, which implies that f ∈ X1,

and, moreover,

|〈f, g〉| =∣∣∣∫

Rd

f(x) g(x) dx∣∣∣

≤ ‖f‖X2‖g‖X1

≤ ‖f‖X2.

Thus,

‖f‖X1≤ ‖f‖X2

. (8)

b. For every Banach function space X we have ˜X = X, cf. [5, Theorem 2.7].

The dual and Kothe dual of the amalgam spaces are given in the next lemma.

Lemma 3.1.5. Let ν be an ω-moderate weight.

a. For 1 ≤ p, q <∞, the dual space of W (Lp, Lqν) is W (Lp′, Lq′

1/ν).

b. For 1 ≤ p, q ≤ ∞, the Kothe dual of W (Lp, Lqν) is W (Lp′, Lq′

1/ν).

Proof. a. We refer to [28, 19] for the proof of this part.

b. If 1 ≤ p, q < ∞, the result follows from part a. Now assume that p = ∞ or

q = ∞. We divide the proof in three parts.

26

Case I: 1 ≤ p <∞ and q = ∞. Let f ∈W (Lp′, L11/ν), and g ∈ W (Lp, L∞

ν ) with

‖g‖W (Lp,L∞ν ) = 1. Then we have

|〈f, g〉| =∣∣∣∫

Rd

f(x) g(x) dx∣∣∣

≤∫

Rd

|f(x)| |g(x)| dx

=∑k∈Zd

∫αk+Qα

|f(x)| |g(x)| dx

≤∑k∈Zd

‖f · TαkχQα‖p′ ‖g · TαkχQα‖p

=∑k∈Zd

‖f · TαkχQα‖p′1

ν(αk)ν(αk) ‖g · TαkχQα‖p

≤ supk∈Zd

‖g · TαkχQα‖p ν(αk)∑k∈Zd

‖f · TαkχQα‖p′1

ν(αk)

= ‖g‖W (Lp,L∞ν ) ‖f‖W (Lp′ ,L1

1/ν)

= ‖f‖W (Lp′ ,L11/ν

).

Thus,

‖f‖W (Lp,L∞ν ) ≤ ‖f‖W (Lp′ ,L1

1/ν),

and consequently, W (Lp′, L11/ν) ⊂ W (Lp, L∞

ν ). From part a and (8), as well as

Remark 3.1.4, we obtain the reverse inclusion, which completes the proof in

this case.

Case II: if p = ∞ and 1 ≤ q <∞ then the proof is very similar to the above so

we omit it.

Case III: if p = q = ∞, we easily see that L11/ν ⊂ (L∞

ν )˜, and the proof follows

from the same arguments as above.

27

3.1.3 Sequence spaces

Before stating our results, we must define the sequence spaces that will be associated

with Gabor expansions in the amalgam spaces. We begin by recalling that the Fourier

transform of f ∈ L1(Q1/β) is the sequence f defined by

f(n) = Ff(n) = βd

∫Q1/β

f(t) e−2πiβn·t dt, n ∈ Zd.

For 1 ≤ p ≤ ∞, let FLp(Q1/β) denote the image of Lp(Q1/β) under the Fourier

transform. Since Fourier coefficients are unique in Lp, if c = (cn)n∈Zd ∈ FLp(Q1/β)

then there exists a unique function m ∈ Lp(Q1/β) such that m(n) = cn for every n,

and the norm on FLp(Q1/β) is defined by

‖c‖FLp(Q1/β) = ‖m‖p,Q1/β. (9)

For 1 < p < ∞, Littlewood–Paley theory can be used to give an equivalent norm

for (9), cf. [16, Ch. 7]. The ongoing development motivates the following definition.

Definition 3.1.6. Let α, β > 0 be given. Then Sp,qν = `qν(FLp(Q1/β)) will denote

the space of all FLp(Q1/β)-valued sequences which are `qν-summable. That is, a

doubly-indexed sequence c = (ckn)k,n∈Zd lies in Sp,qν if for each k ∈ Zd there exists

mk ∈ Lp(Q1/β) such that

mk(n) = ckn, k, n ∈ Zd,

and such that

‖c‖Sp,qν

=

(∑k∈Zd

‖mk‖qp,Q1/β

ν(k)q

)1/q

< ∞,

with the usual change if q = ∞.

When 1 < p <∞, we can write mk as a Fourier series

mk(x) =∑n∈Zd

ckn e2πiβn·x, (10)

28

in the sense that the square partial sums of (10) converge to mk in the norm of

Lp(Q1/β), cf. [48], [63]. Hence, for 1 < p <∞ and 1 ≤ q <∞ we can write the norm

on Sp,qν as

‖c‖Sp,qν

=

(∑k∈Zd

(∫Q1/β

∣∣∣∣∑n∈Zd

ckn e2πiβn·x

∣∣∣∣p dx)q/p

ν(k)q

)1/q

.

Remark 3.1.7. A Banach function space X is called a solid space if f ∈ X and

|g| ≤ |f | implies that g ∈ X and, moreover, ‖g‖ ≤ ‖f‖.Note that for p = 2, we have via the Plancherel theorem that S2,q

ν = `2,qν , thus is

a solid space. However, for general p 6= 2, Sp,qν is not a solid space. In particular,

changing the phases of the ckn can change the norm of c.

3.2 Boundedness of the analysis and synthesis

operators

In this section, we prove the boundedness of the analysis and synthesis operators on

the amalgam spaces. Moreover, we show that the Walnut representation, which is

an extremely useful tool in Gabor analysis, holds on the amalgam spaces. However,

before presenting these results, we give here some Lemmas that will be needed in the

sequel.

3.2.1 Lemmas

The following lemmas will be important in the sequel. The first lemma is simply a

counting argument.

Lemma 3.2.1. Let α, β > 0 be given. Let Kαβ be the maximum number of 1βZd-

translates of Q1/β required to cover any αZd-translate of Qα, i.e.,

Kαβ = maxk∈Zd

#` ∈ Zd :

∣∣( `β

+Q1/β) ∩ (αk +Qα)∣∣ > 0

.

29

Then given 1 ≤ p ≤ ∞, we have for any 1/β-periodic function m ∈ Lp(Q1/β) and

any k ∈ Zd that

‖m‖p,αk+Qα ≤ K1/pαβ ‖m‖p,Q1/β

,

where K1/∞αβ = 1.

Proof. Let m be a 1/β-periodic function in Lp(Q1/β), where 1 ≤ p < ∞. For any

k ∈ Zd define Ak = l ∈ Zd : |( lβ

+Q1/β) ∩ (αk +Qα)| > 0. Then we have:

‖m‖pp,αk+Qα

=

∫αk+Qα

|m(x)|p dx

=∑l∈Zd

∫lβ+Q1/β

|m(x)|p TαkχQα(x) dx

=∑l∈Ak

∫lβ+Q1/β

|m(x)|p TαkχQα(x) dx

≤∑l∈Ak

∫lβ+Q1/β

|m(x)|p dx

=∑l∈Ak

∫Q1/β

|m(x)|p dx

≤ Kα,β

∫Q1/β

|m(x)|p dx

= Kα,β ‖m‖pp,Q1/β

.

If p = ∞, then it is easily seen that

‖m‖∞,αk+Qα ≤ ‖m‖∞,Q1/β.

The second lemma is a weighted version of an estimate that is useful in the Walnut

representation of the Gabor frame operator on L2, see [59, Lemma 2.2].

Lemma 3.2.2. Let ω be a submultiplicative weight, and let α, β > 0 be given. Then

there exists a constant C = C(α, β, ω) > 0 such that if g, γ ∈ W (L∞, L1ω) and the

30

functions Gn are defined by (22), then

∑n∈Zd

‖Gn‖∞ ω(nβ) ≤ C ‖g‖W (L∞,L1

ω) ‖γ‖W (L∞,L1ω).

Proof. It follows from the fact that ω is ω-moderate that ‖fω‖W (L∞,L1) is an equiv-

alent norm for W (L∞, L1ω). In particular, we have gω, γω ∈ W (L∞, L1), so by [41,

Lemma 6.3.1],

∑n∈Zd

‖Gn‖∞ ≤ (1α

+ 1)d

(2β + 1)d ‖gω‖W (L∞,L1) ‖γω‖W (L∞,L1),

where Gn is the analogue of Gn with g replaced by |g|ω and γ replaced by |γ|ω.

Hence,

∑n∈Zd

‖Gn‖∞ ω(nβ) =

∑n∈Zd

ess supx∈Rd

∣∣∣∣∑k∈Zd

g(x− nβ− αk) γ(x− αk)×

ω((x− αk) − (x− n

β− αk)

)∣∣∣∣≤

∑n∈Zd

ess supx∈Rd

∑k∈Zd

|g(x− nβ− αk)|ω(x− n

β− αk)×

|γ(x− αk)|ω(x− αk)

=∑n∈Zd

‖Gn‖∞

≤ C ‖g‖W (L∞,L1ω) ‖γ‖W (L∞,L1

ω).

Finally, we need an estimate on the effect of translations on the amalgam space

norm.

Lemma 3.2.3. Let ν be an ω-moderate weight. Then for 1 ≤ p, q ≤ ∞, we have for

each f ∈W (Lp, Lqν) and ` ∈ Zd that

‖Tα`f‖W (Lp,Lqν) ≤ Cν ω(α`) ‖f‖W (Lp,Lq

ν).

31

Proof. Let f ∈W (Lp, Lqν), and l ∈ Zd. Then using the fact that ν is ω-moderate, we

obtain

‖Tαlf‖W (Lp,Lqν) =

(∑k∈Zd

‖Tαlf · TαkχQα‖qp ν(αk)

q

)1/q

=

(∑k∈Zd

‖f · Tα(k−l)χQα‖qp ν(αk)

q

)1/q

=

(∑k∈Zd

‖f · TαkχQα‖qp ν(α(k + l))q

)1/q

≤ C ω(αl)

(∑k∈Zd

‖f · TαkχQα‖qp ν(αk)

q

)1/q

= C ω(αl) ‖f‖W (Lp,Lqν).

3.2.2 Boundedness of the synthesis operator

Theorem 3.2.4. Let ν be an ω-moderate weight on Rd. Let α, β > 0 and 1 ≤ p, q ≤∞ be given. Fix g, γ ∈ W (L∞, L1

ω). Then the following statement is true. Given

c ∈ Sp,qν , let mk ∈ Lp(Qα) be the unique functions satisfying mk(n) = ckn for all k,

n ∈ Zd. Then the series

Rgc =∑k∈Zd

mk · Tαkg (11)

converges unconditionally in W (Lp, Lqν) (unconditionally in the

σ(W (Lp, Lqν), W (Lp′, Lq′

1/ν)) topology if p = ∞ or q = ∞), and Rg is a bounded

mapping from Sp,qν into W (Lp, Lq

ν).

Proof. We divide the proof into cases. First, we consider the case 1 ≤ p, q < ∞. We

are given c ∈ Sp,qν , and we must prove that the series (11) defining Rgc converges

unconditionally in the norm of W (Lp, Lqν), and that Rg so defined is a bounded

32

mapping from Sp,qν into W (Lp, Lq

ν) . To show the convergence we will make use of

Lemma 3.1.3.

Fix ε > 0. Then, by definition of the norm in Sp,qν , we have that

∑‖mk‖q

p,Q1/βν(k)q <∞.

Hence there exists a finite set F0 such that

∀ finite F ⊃ F0,∑k/∈F

‖mk‖qp,Q1/β

ν(k)q < εq. (12)

Recall that 1/ν is an ω-moderate weight, and let Kαβ be the constant appearing in

Lemma 3.2.1. Fix any h ∈W (Lp′, Lq′1/ν). Then

∑k/∈F

∣∣⟨mk · Tαkg, h⟩∣∣ ≤ ∑

k/∈F

∫Rd

|mk(x)Tαkg(x) h(x)| dx

=∑k/∈F

∑n∈Zd

∫Qα

|mk(x)Tαkg(x) h(x)| Tαn+αkχQα(x) dx

≤∑k/∈F

∑n∈Zd

‖Tαkg · Tαn+αkχQα‖∞ ‖mk‖p,αn+αk+Qα ×

‖h · Tαn+αkχQα‖p′ν(αk)

ν(αn+ αk − αn)

≤∑n∈Zd

‖g · TαnχQα‖∞×

∑k/∈F

K1/pαβ ‖mk‖p,Q1/β

‖h · Tαn+αkχQα‖p′Cν ν(αk)ω(αn)

ν(αn+ αk)

≤ CνK1/pαβ

∑n∈Zd

‖g · TαnχQα‖∞ ω(αn)×

(∑k/∈F

‖mk‖qp,Q1/β

ν(αk)q

)1/q

×(∑

k∈Zd

‖h · Tαn+αkχQα‖q′p′

1

ν(αn+ αk)q′

)1/q′

. (13)

Combining (12) and (13), we have that

∑k/∈F

∣∣⟨mk · Tαkg, h⟩∣∣ ≤ εCνK

1/pαβ ‖g‖W (L∞,L1

ω) ‖h‖W (Lp′ ,Lq′1/ν

).

33

Therefore, taking the supremum over all h of unit norm and appealing to Lemma 3.1.3,

we see that Rgc =∑mk · Tαkg converges unconditionally. Further, replacing F by

Zd in the calculation in (13) yields

|〈Rgc, h〉| ≤∑k∈Zd

∣∣⟨mk · Tαkg, h⟩∣∣

≤ CνK1/pαβ ‖g‖W (L∞,L1

ω) ‖c‖Sp,qν

‖h‖W (Lp′ ,Lq′

1/ν). (14)

Since W (Lp′, Lq′1/ν) is the dual space of W (Lp, Lq

ν), taking the suprema over all h of

unit norm in (14) shows that

‖Rgc‖W (Lp,Lqν) = sup

|〈Rgc, h〉| : ‖h‖W (Lp′ ,Lq′

1/ν)= 1

≤ CνK

1/pαβ ‖g‖W (L∞,L1

ω) ‖c‖Sp,qν, (15)

so Rg is bounded. This completes the proof for the case 1 ≤ p, q <∞.

When p = ∞ or q = ∞, we make use of the fact thatW (Lp′, Lq′1/ν) is the Kothe dual

of W (Lp, Lqν). The fact that the series defining Rgc converges in the weak topology

is given by the same calculations as in (13), (14), and the fact that the Kothe dual

is a norm-fundamental subspace of the dual space means that we can again estimate

‖Rgc‖Sp,qν

by using (15). Hence Rg is bounded, and the proof is complete.

Remark 3.2.5. When 1 < p <∞, the functions mk appearing in (11) can be written

as Fourier series, allowing Rgc to be written as the iterated sum

Rgc(x) =∑k∈Zd

(∑n∈Zd

ckn e2πiβn·x

)Tαkg(x), (16)

i.e., the same series as appears in the Gabor expansions in (3), or more generally the

Gabor expansions in modulation spaces (6). When p = 1 or p = ∞, this is not the

case. The functions mk are still uniquely determined by c, but cannot be written as

Fourier series. When p = q = 2, both the inner and outer sums in the iterated series

in (16) converge unconditionally, and then Rgc can also be written as the double sum

given by (6), with unconditional convergence of that series.

34

3.2.3 Boundedness of the analysis operator

Theorem 3.2.6. Let ν be an ω-moderate weight on Rd. Let α, β > 0 and 1 ≤ p, q ≤∞ be given. Fix g, γ ∈ W (L∞, L1

ω). Then the analysis operator defined by Cgf =(〈f,MβnTαkg〉)

k,n∈Zd is a bounded mapping from W (Lp, Lqν) into Sp,q

ν , Moreover, there

exist unique functions mk ∈ Lp(Q1/β) which satisfy mk(n) = Cgf(k, n) for all k, n ∈Zd, and these are given explicitly by

mk(x) = β−d∑n∈Zd

(f · Tαkg

)(x− n

β)

= β−d∑n∈Zd

(Tn

βf · Tαk+ n

βg)(x). (17)

The series on the right side of (17) converges unconditionally in Lp(Q1/β) (uncondi-

tionally in the σ(L∞(Q1/β), L1(Q1/β) topology if p = ∞).

Proof. We are given that g ∈W (L∞, L1ω) and that 1 ≤ p, q ≤ ∞. Let f ∈W (Lp, Lq

ν),

which is a subspace of W (L1, L∞1/ω). First we must show that the functions mk given

by (17) are well-defined. Since mk is the 1/β-periodization of the integrable function

f · Tαkg, the series defining mk converges at least in L1(Q1/β). To show that the

periodization converges unconditionally in Lp(Q1/β) (weakly if p = ∞) and to derive

a useful estimate, fix any 1/β-periodic function h ∈ Lp′(Q1/β). Then for each fixed

k, we have

∣∣∣∫Q1/β

∑n∈Zd

f(x− nβ)Tαkg(x− n

β) h(x) dx

∣∣∣≤

∫Rd

|f(x)Tαkg(x) h(x)| dx

=∑n∈Zd

∫Qα

|f(x)Tαkg(x) h(x)| Tαk+αnχQα(x) dx

≤∑n∈Zd

‖Tαkg · Tαk+αnχQα‖∞ ‖f · Tαk+αnχQα‖p ×

35

‖h‖p′,αk+αn+Qα

ν(αk + αn− αn)

ν(αk)

≤∑n∈Zd

‖g · TαnχQα‖∞ ‖f · Tαk+αnχQα‖pK1/p′αβ ×

‖h‖p′,Q1/β

Cνν(αk + αn)ω(αn)

ν(αk)

= CνK1/p′αβ ‖h‖p′,Q1/β

1

ν(αk)

∑n∈Zd

‖g · TαnχQα‖∞ ω(αn) ×

‖f · Tαk+αnχQα‖p ν(αk + αn). (18)

This yields the desired convergence, and taking the suprema in (18) over h with

unit norm implies the estimate

‖mk‖p,Q1/β≤ β−dCνK

1/p′αβ

1ν(αk)

∑n∈Zd ‖g · TαnχQα‖∞ ω(αn) ×

‖f · Tαk+αnχQα‖p ν(αk + αn). (19)

Second, we show that mk(n) has the correct form. Since e2πiβn·x ∈ Lp′(Q1/β), we

have by the weak convergence of the series defining mk that

mk(n) = βd⟨mk, e

2πiβn·x⟩=

∑`∈Zd

∫Q1/β

⟨T `

βf · Tαk+ `

βg, e2πiβn·x⟩

=∑`∈Zd

∫Q1/β

f(x− `β)Tαkg(x− `

β) e−2πiβn·(x−`/β) dx

=

∫Rd

(f · Tαkg

)(x) e−2πiβn·x dx

=⟨f, MβnTαkg

⟩= Cgf(k, n).

Finally, we must show that Cg is a bounded mapping of W (Lp, Lqν) into Sp,q

ν .

Given f ∈ W (Lp, Lqν), to show that Cgf ∈ Sp,q

ν we must show that the sequence r

given by

r(k) = ‖mk‖p,Q1/β, k ∈ Zd,

36

lies in `qν . To do this, fix any sequence a ∈ `q′

1/ν . Then, using (19), we have

|〈r, a〉| ≤∑k∈Zd

‖mk‖p,Q1/β|a(k)|

≤ β−dCνK1/p′αβ

∑n∈Zd

‖g · TαnχQα‖∞ ω(αn)×

∑k∈Zd

‖f · Tαk+αnχQα‖p ν(αk + αn) |a(k)| 1

ν(αk)

≤ β−dCνK1/p′αβ

∑n∈Zd

‖g · TαnχQα‖∞ ω(αn)×

(∑k∈Zd

‖f · Tαk+αnχQα‖qp ν(αk + αn)q

)1/q

×

(∑k∈Zd

|a(k)|q′ 1

ν(αk)q′

)1/q′

≤ β−dCνK1/p′αβ ‖g‖W (L∞,L1

ω) ‖f‖W (Lp,Lqν) ‖a‖`q′

1/ν

. (20)

Since `q′

1/ν equals (`qν)∗ when q <∞ and is a norm-fundamental subspace when q = ∞,

taking the suprema in (20) over sequences a with unit norm yields the estimate

‖Cgf‖Sp,qν

= ‖r‖`pν≤ β−dCνK

1/p′αβ ‖g‖W (L∞,L1

ω) ‖f‖W (Lp,Lqν).

Hence Cg is a bounded mapping of W (Lp, Lqν) into Sp,q

ν .

Remark 3.2.7. For the case 1 < p, q <∞, the boundedness of Cg could also be shown

by proving that Cg : W (Lp, Lqν) → Sp,q

ν is the adjoint of Rg : Sp′,q′1/ν → W (Lp′, Lq′

1/ν),

and then using the reflexivity of the space W (Lp, Lqν) and the fact that 1/ν is also

ω-moderate.

37

3.2.4 The Walnut representation of the Gabor frame operator on amal-gam spaces

Theorem 3.2.8. Let ν be an ω-moderate weight on Rd. Let α, β > 0 and 1 ≤ p, q ≤∞ be given. Fix g, γ ∈W (L∞, L1

ω). Then the Walnut representation

RγCgf = β−d∑n∈Zd

Gn · Tnβf (21)

holds for f ∈W (Lp, Lqν), with the series on the right of (21) converging absolutely in

W (Lp, Lqν), and where

Gn(x) =∑k∈Zd

g(x− nβ− αk) γ(x− αk)

=∑k∈Zd

(T

αk+nβg · Tαkγ

)(x). (22)

Proof. We are given g, γ ∈ W (L∞, L1ω) and 1 ≤ p, q ≤ ∞. For this proof, let us use

the equivalent norm for W (Lp, Lqν) obtained by replacing α in (7) by 1/β. Then by

Lemma 3.2.3,

‖Tnβf‖W (Lp,Lq

ν) ≤ Cν ω(nβ) ‖f‖W (Lp,Lq

ν).

Therefore, using the autocorrelation functions Gn defined in (22), we have for f ∈W (Lp, Lq

ν) that

∑n∈Zd

‖Gn · Tnβf‖W (Lp,Lq

ν) ≤∑n∈Zd

‖Gn‖∞ ‖Tnβf‖W (Lp,Lq

ν)

≤ Cν ‖f‖W (Lp,Lqν)

∑n∈Zd

‖Gn‖∞ ω(nβ)

≤ Cν ‖f‖W (Lp,Lqν) ‖g‖W (L∞,L1

ω) ‖γ‖W (L∞,L1ω),

the last inequality following from Lemma 3.2.2. Hence the series∑Gn ·Tn

βf converges

absolutely in W (Lp, Lqν).

Now fix f ∈ W (Lp, Lqν). Then Cgf ∈ Sp,q

ν by Theorem 3.2.6. Letting mk be

defined by (17), we have Cgf(k, n) = mk(n). Further, RγCgf =∑mk · Tαkγ, this

38

series converging unconditionally if p, q <∞, or unconditionally in the weak topology

otherwise. In any case, for h ∈W (Lp′, Lq′1/ν) we have

⟨RγCgf, h

⟩=

∑k∈Zd

⟨mk · Tαkγ, h

⟩

=∑k∈Zd

∫Rd

mk(x)Tαkγ(x) h(x) dx

= β−d∑k∈Zd

∫Rd

∑n∈Zd

Tnβf(x)Tαk+ n

βg(x)Tαkγ(x) h(x) dx

= β−d∑n∈Zd

∫Rd

∑k∈Zd

Tnβf(x)Tαk+ n

βg(x)Tαkγ(x) h(x) dx

= β−d∑n∈Zd

∫Rd

Tnβf(x)Gn(x) h(x) dx.

= β−d∑n∈Zd

⟨Gn · Tn

βf, h

⟩,

from which (21) follows. The interchanges of integration and summation can be

justified by Lemma 3.2.2 and Fubini’s Theorem.

3.3 Gabor expansions in the amalgam spaces

Under the assumption that G(g, α, β) is a frame for L2(Rd), we obtain the following

result, which makes precise the characterization of the amalgam spaces in terms of

Gabor frames. In particular, we show in this section that there is an analogue for

the amalgam spaces of the Gabor expansions of functions in the modulation spaces

(see Theorem 2.3.11). This is surprising, because the modulation spaces are the

natural setting for Gabor analysis. And indeed, while Gabor expansions converge

unconditionally in the modulation spaces, the convergence in the amalgam spaces is

conditional in general and even the meaning of the term “expansion” must be handled

appropriately. Throughout, we will use the notation ν(k) = ν(αk).

39

Theorem 3.3.1. Let ν be an ω-moderate weight on Rd, and let α, β > 0 and 1 ≤p, q ≤ ∞ be given. Assume that g, γ ∈W (L∞, L1

ω) are such that G(g, α, β) is a Gabor

frame for L2 with dual frame G(γ, α, β). Then the following statements hold.

a. RγCg = I on W (Lp, Lqν).

b. We have the norm equivalence ‖f‖W (Lp,Lqν) ‖Cgf‖Sp,q

ν.

c. A function f ∈W (L1, L∞1/ω) belongs to W (Lp, Lq

ν) if and only if Cgf ∈ Sp,qν .

Proof. We are given g, γ ∈ W (L∞, L1ω) such that G(g, α, β) is a Gabor frame for L2

and γ is the dual window to g. By Theorem 3.2.6, we have that Cg, Cγ : W (Lp, Lqν) →

Sp,qν and Rg, Rγ : Sp,q

ν → W (Lp, Lqν) are bounded mappings for each 1 ≤ p, q ≤ ∞

and each ω-moderate weight ν. Further, for the case p = q = 2 and ν = 1, the

frame hypothesis implies that the identity RγCg = I holds on L2, and the definition

of Rγ given in Chapter 2 coincides in this case with the definition of Rγ given in

Theorem 3.2.4. Letting Gn be the autocorrelation functions defined in (22), the fact

that RγCg = I holds on L2 implies by [41, Thm. 7.3.1] that

β−dG0 = 1 a.e. and Gn = 0 a.e. for n 6= 0.

Consequently, using the Walnut representation (21) of RγCg on the space W (Lp, Lqν),

we have for f ∈W (Lp, Lqν) that

RγCgf = β−d∑n∈Zd

Gn · Tnβf = f.

Hence RγCg = I holds on W (Lp, Lqν) as well. This proves part a of Theorem 3.3.1.

Next, given f ∈W (Lp, Lqν), we have

‖f‖W (Lp,Lqν) = ‖RγCgf‖W (Lp,Lq

ν)

≤ ‖Rγ‖ ‖Cgf‖Sp,qν

≤ ‖Rγ‖ ‖Cg‖ ‖f‖W (Lp,Lqν).

40

Consequently, ‖Cgf‖Sp,qν

‖f‖W (Lp,Lqν), which proves part b of Theorem 3.3.1.

Finally, we prove part c of Theorem 3.3.1. Let f ∈ W (L1, L∞1/ω) be given. We

must show that f ∈ W (Lp, Lqν) if and only if Cgf ∈ Sp,q

ν . The forward direction,

that if f ∈ W (Lp, Lqν) then Cgf ∈ Sp,q

ν , is simply Theorem 3.2.6. For the reverse

direction, assume that Cgf ∈ Sp,qν . Then by Theorem 3.2.6, the function f = Rγ(Cgf)

lies in W (Lp, Lqν). However, the factorization RγCg = I holds on every amalgam

space, including W (L1, L∞1/ω) in particular, so we also know that f = RγCgf . Thus

f = f ∈W (Lp, Lqν), which completes the proof.

Remark 3.3.2. a. Theorem 3.3.1 says that, given an appropriate condition on the

window g and its dual window γ, a Gabor frame for L2 extends to the amalgam spaces

and provides “Gabor expansions” for the amalgam spaces in the sense that we have

the factorization of the identity as I = RγCg. The specific form of these expansions

is that given f , there exist functions mk such that f = RγCgf =∑mk · Tαkg. When

1 < p <∞, the functions mk can be realized as Fourier series, leading to an expansion

of the form

f(x) = RγCgf(x) =∑k∈Zd

(∑n∈Zd

〈f,MβnTαkγ〉 e2πiβn·x)Tαkg(x). (23)

The inner sum defining mk converges conditionally in general, while the outer sum

converges unconditionally.

b. For the case p = 1, the functions mk cannot be written as Fourier series, so

we do not have a series expansion of the form (23). A different approach to the case

p = q = 1 and ν = 1, based on Littlewood–Paley theory, is developed by Gilbert

and Lakey in [33], where they show that Gabor frames can be used to characterize a

Hardy-type space on the line.

c. Theorem 3.3.1c says that if we use the “largest” amalgam space W (L1, L∞1/ω) as

our “universe,” then membership of a function in an amalgam W (Lp, Lqν) is character-

ized by membership of its sequence of Gabor coefficients in an appropriate sequence

41

space. By imposing additional restrictions on g, γ, we could enlarge the universe on

which this characterization is valid. In particular, if we required g, γ to lie in the

Schwartz class S, then the universe on which this characterization was valid would

be the space S ′ of tempered distributions.

d. For the case of the modulation spaces, there is a deep result that states that

if g lies in the Feichtinger algebra M1ω, then the dual window γ will lie in M1

ω as

well, [43]. For the case of the amalgam spaces, we do not know if the assumption

g ∈ W (L∞, L1ω) implies that the dual window γ also lies in that space. This is an

interesting and possibly difficult open question.

3.4 Convergence of Gabor expansions

As pointed out above, when 1 < p < ∞, the synthesis operator Rg can be written

as the iterated sum (16). The inner series in this sum converges conditionally in

general, while the outer series converges unconditionally. Our next result shows that

this series can also be written as a double sum as in Theorem 2.3.11, but because the

proof relies on the convergence of Fourier series in Lp, the convergence is conditional

in general. In dealing with Fourier series in higher dimensions, it is important to use

the maximum norm |x| = max|x1|, . . . , |xd| on Rd.

Theorem 3.4.1. Let ν be an ω-moderate weight. Let α, β > 0 and 1 < p < ∞,

1 ≤ q < ∞ be given. Assume that g, γ ∈ W (L∞, L1ω) are such that G(g, α, β) is a

Gabor frame for L2 with dual window γ. Then the following statements hold.

a. If c ∈ Sp,qν , then the partial sums

SK,Nc =∑|k|≤K

∑|n|≤N

cknMβnTαkg, K,N > 0,

converge to Rgc in the norm of W (Lp, Lqν), i.e., for each ε > 0 there exist K0,

N0 > 0 such that

∀K ≥ K0, ∀N ≥ N0, ‖Rgc− SK,Nc‖W (Lp,Lqν) < ε.

42

b. If f ∈W (Lp, Lqν), then the partial sums of the Gabor expansion of f ,

SK,N(Cgf) =∑|k|≤K

∑|n|≤N

〈f,MβnTαkg〉MβnTαkγ,

converge to f in the norm of W (Lp, Lqν).

Proof. We are given g, γ ∈ W (L∞, L1ω) such that G(g, α, β) is a Gabor frame for L2

and γ is the dual window to g, and we fix 1 < p <∞ and 1 ≤ q <∞.

Assume that c ∈ Sp,qν , and let mk be defined by (17). For N > 0, write

SNmk(x) =∑|n|≤N

ckn e2πiβn·x

for the partial sums of the Fourier series of mk. The exponentials e2πiβn·xn∈Zd form

a basis for Lp(Q1/β) [48], [63], so, letting C1 denote the basis constant for this system,

we have for each k ∈ Zd that

limN→∞

‖mk − SNmk‖p,Q1/β= 0 (24)

and

supN>0

‖SNmk‖p,Q1/β≤ C1 ‖mk‖p,Q1/β

. (25)

Since c ∈ Sp,qν , given ε > 0, we can find K0 > 0 such that

∀K ≥ K0,

( ∑|k|≥K

‖mk‖qp,Q1/β

ν(k)q

)1/q

< ε. (26)

Because of (24) and the fact that K0 is finite, we can find an N0 > 0 such that

∀N ≥ N0, sup|k|≤K0

‖mk − SNmk‖p,Q1/βν(k) <

ε

(2K0 + 1)d/q. (27)

Now, since c ∈ Sp,qν and 1 < p <∞, we know that Rgc can be written as the iterated

series (16). Write the partial sums of the outer series as

SK,∞c =∑|k|≤K

(∑n∈Zd

ckn e2πiβn·x

)Tαkg =

∑|k|≤K

mk · Tαkg.

43

Given K ≥ K0 and N ≥ N0, write

Rgc− SK,Nc = (Rgc− SK0,∞c) + (SK0,∞c− SK0,Nc) + (SK0,Nc− SK,Nc). (28)

We will calculate the W (Lp, Lqν) norm of each of these terms separately.

For the first term, define a sequence r by rkn = ckn for |k| ≤ K0 and n ∈ Zd,

and rkn = 0 otherwise. Then SK0,∞c = Rgr, and Rg is a bounded mapping of

Sp,qν →W (Lp, Lq

ν), so using (26) we have

‖Rgc− SK0,∞c‖W (Lp,Lqν) = ‖Rg(c− r)‖W (Lp,Lq

ν)

≤ ‖Rg‖ ‖c− r‖Sp,qν

= ‖Rg‖( ∑

|k|>K0

‖mk‖qp,Q1/β

ν(k)q

)1/q

≤ ‖Rg‖ε. (29)

For the second term, define skn = ckn for |k| ≤ K0 and |n| ≤ N , and skn = 0

otherwise. Then SK0,Nc = Rgs, so using (27), we have

‖SK0,∞c− SK0,Nc‖W (Lp,Lqν)

≤ ‖Rg‖ ‖r − s‖Sp,qν

= ‖Rg‖( ∑

|k|≤K0

‖mk − SNmk‖qp,Q1/β

ν(k)q

)1/q

≤ ‖Rg‖ε. (30)

For the third term, define tkn = ckn for |k| ≤ K and |n| ≤ N , and tkn = 0

otherwise. Then SK0,Nc = Rgt, so using (25) and (26), we have

‖SK0,Nc− SK,Nc‖W (Lp,Lqν) ≤ ‖Rg‖ ‖s− t‖Sp,q

ν

= ‖Rg‖( ∑

K0<|k|≤K

‖SNmk‖qp,Q1/β

ν(k)q

)1/q

≤ C1‖Rg‖( ∑

K0<|k|≤K

‖mk‖qp,Q1/β

ν(k)q

)1/q

44

‖SK0,Nc− SK,Nc‖W (Lp,Lqν) ≤ C1‖Rg‖ε. (31)

Applying (29)–(31) to (28), we see that ‖Rgc− SK,Nc‖W (Lp,Lqν) ≤ (2 +C1) ‖Rg‖ ε,

which completes the proof.

3.5 Necessary conditions on the window

In this section we prove a partial converse to Theorem 3.2.6. In particular, Theo-

rem 3.2.6 implies that if g ∈ W (L∞, L1ω), then Cg is bounded on each W (Lp, Lq

ν).

In the converse direction, if g is a measurable function and 1 ≤ p, q ≤ ∞ are

given, then in order for Cg to be well-defined on W (Lp, Lqν), we must at least have

Cgf(0, 0) = 〈f, g〉 =∫f g defined for each f ∈W (Lp, Lq

ν). Hence f g ∈ L1 for all such

f , so we immediately have that g must lie in the Kothe dual of W (Lp, Lqν), which is

W (Lp′, Lq′1/ν).

For the unweighted case, we obtain the following further necessary condition in

order that Cg be bounded on W (Lp, L∞). For the case p = 2, this result was obtained

by Balan in [2] and published in [4], [3].

Theorem 3.5.1. Let α, β > 0 and 1 < p <∞ be given. If g ∈ W (Lp′, L1) and Cg is

a bounded map from W (Lp, L∞) to Sp,∞, then g ∈W (L∞, Lp).

Proof. We assume that g ∈ W (Lp′, L1) is such that Cg is a bounded map from

W (Lp, L∞) to Sp,∞, where 1 < p < ∞, and we wish to show that g ∈ W (L∞, Lp).

Let us show first that g ∈ L∞. If not, then given any D > 0 there would exist a set

J contained in some cube `β

+Q1/β and with positive measure such that |g(x)| > D

on J .

Set f = 1|J |1/p e

i arg g χJ . Using the equivalent norm for W (Lp, L∞) obtained by

replacing α in (7) by 1/β, we have that ‖f‖W (Lp,L∞) ≤ 1. By hypothesis, Cgf ∈Sp,∞, so there exist 1/β-periodic functions mk such that mk(n) = Cgf(k, n). Since

45

f · Tαkg ∈ L1, it is easy to see that mk is given by (17). In particular, considering

k = 0 we have

‖m0‖pp,Q1/β

= β−pd

∫Q1/β

∣∣∣∣∑n∈Zd

f(x− nβ) g(x− n

β)

∣∣∣∣p dx=β−pd

|J |∫

`β+Q1/β

χJ(x) |g(x)|p dx

≥ β−pdDp.

Hence

D ≤ βd supk∈Zd

‖mk‖p,Q1/β

= βd ‖Cgf‖Sp,∞

≤ βd ‖Cg‖ ‖f‖W (Lp,L∞) ≤ βd ‖Cg‖.

But since D is arbitrary, this contradicts the fact that Cg is a bounded mapping.

Hence g must be in L∞.

Now we show that g ∈W (L∞, Lp). Fix ε > 0, and for each n ∈ Zd define

Jn =x ∈ n

β+Q1/β : |g(x)| ≥ 1

2‖g‖∞, n

β+Q1/β

.

Then set J ′n = Jn if |Jn| ≤ ε, otherwise let J ′

n be a subset of Jn of measure ε. Let

Nε = supN ∈ N : |J ′n| ≥ ε

2for all |n| ≤ N.

Note that Nε → ∞ as ε → 0 (and may even be ∞ for some ε). Define f =

ei arg g∑

|n|≤NεχJ ′

n, and note that ‖f‖W (Lp,L∞) ≤ ε1/p. Therefore Cgf ∈ Sp,∞, and

46

letting mk be defined by (17), we have

‖m0‖pp,Q1/β

= β−pd∑

|n|≤Nε

∫nβ+Q1/β

|g(x)|p χJ ′n(x) dx

≥ β−pd∑

|n|≤Nε

(‖g · TnβχQ1/β

‖∞2

)p

|J ′n|

≥ β−pd2−p−1ε∑

|n|≤Nε

‖g · TnβχQ1/β

‖p∞.

Hence

∑|n|≤Nε

‖g · TnβχQ1/β

‖p∞ ≤ βpd2p+1

εsupk∈Zd

‖mk‖pp,Q1/β

=βpd2p+1

ε‖Cgf‖p

Sp,∞

≤ βpd2p+1

ε‖Cg‖p ‖f‖p

W (Lp,L∞)

≤ βpd2p+1 ‖Cg‖p.

Since Nε → ∞ as ε → 0, this implies that g ∈W (L∞, Lp).

Remark 3.5.2. As noted above, the hypothesis g ∈ W (Lp′, L1) is not a limitation

on the generality of the result, as it is necessary in order that Cg can even be defined.

Furthermore, if 1 < p < ∞ then W (L∞, Lp) is not contained in W (Lp′, L1) nor

conversely, so Theorem 3.5.1 is not a trivial consequence of embeddings of amalgam

spaces. The result is also true if p = 1, but in this case W (L∞, Lp) = W (Lp′, L1) and

there is no new information gained.

We now show that, with a mild hypothesis, we obtain a necessary condition on the

analysis window. This hypothesis is formulated in terms of the following condition;

we refer to [14], [2] for examples.

47

Definition 3.5.3. A function f : Rd → C has persistency length a if there exists a

δ > 0 and a compact set K congruent to Qa mod a such that |f(x)| ≥ δ for every

x ∈ K.

Theorem 3.5.4. Let α, β > 0 and 1 ≤ p < ∞ be given. Let g, γ be measurable

functions on Rd. Suppose the following:

a. for each f ∈ W (Lp, L∞), the series∑

k,n〈f,MβnTαkg〉MβnTαkγ converges un-

conditionally in Lploc,

b. the frame operators Sg = RgCg and Sγ = RγCγ are bounded mappings of

W (Lp, L∞) onto itself,

c. γ has persistency length 1/β.

Then g ∈W (L∞, Lp).

Proof. Let F ⊂ Rd be compact. Then by hypothesis, f 7→ ∑k,n〈f,MβnTαkg〉MβnTαkγ

is a bounded mapping from W (Lp, L∞) into LpLoc, and the series converges uncondi-

tionally in Lploc. We first show that f 7→ Skf =

∑n∈Zd〈f,MβnTαkg〉MβnTαkγ is

uniformly bounded on W (Lp, L∞), with respect to k ∈ Zd.

Fix k ∈ Zd and any f ∈W (Lp, L∞). Then the sequence

∑|n|≤N

〈f,MβnTαkg〉MβnTαkγ

converges in Lp(F ) as N → ∞, hence is a bounded sequence in Lp(F ). Moreover,

because the operators

SN,k(·) : =∑|n|≤N

〈·,MβnTαkg〉MβnTαkγ,

are bounded, we conclude from the Uniform Boundedness Principle that for each fixed

k the sequence of operators SN,kN≥0, are uniformly bounded in N , i.e., for each k

48

there is a constant Ck > 0 such that

‖SN,k‖W (Lp,L∞)→Lp(F ) ≤ Ck∀N.

Next, given ε > 0 and f ∈W (Lp, L∞) with ‖f‖W (Lp,L∞) = 1, there exists N0 > 0

such that ‖∑|n|≥N0

〈f,MβnTαkg〉MβnTαkγ‖p,F ≤ ε. Thus,

‖Sk(f)‖p,F ≤∥∥∥∥ ∑|n|≥N0

〈f,MβnTαkg〉MβnTαkγ

∥∥∥∥p,F

+

∥∥∥∥ ∑|n|≤N0

〈f,MβnTαkg〉MβnTαkγ

∥∥∥∥p,F

≤ Ck + ε.

Letting ε→ 0, we conclude that ‖Sk(f)‖p,F ≤ Ck. It then follows that Sk is a bounded

operator from W (Lp, L∞) into Lp(F ). Now because∑

k∈Zd Sk(f) converges in Lp(F ),

we know that Sk(f) is bounded independently of k, so by the Uniform Boundedness

principle, we conclude that ‖Sk‖W (Lp,L∞)→Lp(F ) ≤ C(F ) for all k ∈ Zd. Moreover, it

is easy to see that

‖Sk‖W (Lp,L∞)→Lp(F+α) = ‖Sk−1‖W (Lp,L∞)→Lp(F ) ≤ C(F ).

Thus if we let F = Qα then ‖Sk(f)‖p,F ≤ C(F )‖f‖W (Lp,L∞). On the other hand,

taking F and δ > 0 as in the definition of persistency, for any k ∈ Zd we have∥∥∥∥∑n∈Zd

〈f,MβnTαkg〉MβnTαkγ

∥∥∥∥p,F+αk

= ‖Sk(f)‖p,F+αk

=

∥∥∥∥γ(·) ∑n∈Zd

〈f,MβnTαkg〉e2πiβ(·+αk)

∥∥∥∥p,F

≥ δβd ‖mk‖p,Q1/β.

Therefore,

‖mk‖p,Q1/β≤ δ−1βd ‖Sk(f)‖p,F+αk ≤ Cδ−1βd‖f‖W (Lp,L∞) ∀k ∈ Zd.

49

Hence,

‖Cgf‖Sp,∞ = supk

‖mk‖p,Q1/β≤ C δ−1 βd ‖f‖W (Lp,L∞).

Thus, we conclude by Theorem 3.5.1 that g ∈W (Lp, L∞), which concludes the proof.

50

CHAPTER IV

EMBEDDINGS OF BESOV,

TRIEBEL-LIZORKIN SPACES INTO

MODULATION SPACES

The apparently simple definition of the modulation spaces (see Chapter 2) hides

the practical problem of how to decide whether or not a distribution belongs to

a given modulation space. In principle one has to estimate the Lp,qν norm of the

STFT, which can be a non-trivial task. Therefore it is important to understand the

relationship between time-frequency content and other properties of distributions,

e.g., smoothness properties. Such relationships may appear in the form of embeddings

of certain spaces that measure smoothness and/or decay into modulation spaces.

For example, Grochenig in [40], Galperin and Grochenig in [32], and Hogan and

Lakey in [47] derived sufficient conditions for membership in the modulation space M1

from certain uncertainty principles related to the STFT. Another interesting example

appears in [45], where Heil, Ramanathan and Topiwala obtained an embedding that

is particularly important in relation to pseudodifferential operator theory.

In the present chapter we prove sufficient conditions for a tempered distribution

to belong to certain (unweighted) modulation spaces by proving some embeddings of

classical Banach spaces such as the Besov, Triebel-Lizorkin, or Sobolev spaces into the

modulation spaces. As corollaries, we obtain some embeddings which generalize the

embedding from [45] mentioned above, and, moreover, we will give an easy sufficient

condition for membership of a distribution in M1 in the special case of dimension

d = 1.

51

Other embeddings results between modulation spaces and Besov spaces were ar-

rived at independently and by different techniques by P. Grobner [37], and J. Toft

[56].

4.1 The Besov and Triebel-Lizorkin Spaces

Let ψ ∈ S be a function such that

0 ≤ ψ(x) ≤ 1,

ψ(x) = 1, if |x| ≤ 1,

ψ(x) = 0, if |x| ≥ 3/2.

Define

φ0(x) = ψ(x),

φ1(x) = ψ(x2) − ψ(x),

φk(x) = φ1(2−k+1x), k = 2, 3, ....

Then φk∞k=0 is a partition of unity, and satisfies

supp(φk) ⊂ x ∈ Rd : 2k−1 ≤ |x| ≤ 3 · 2k−1.

Definition 4.1.1. Let s ∈ R, 1 ≤ q ≤ ∞, and f ∈ S ′.

(i) For 1 ≤ p <∞ the Triebel-Lizorkin space F sp,q is defined by:

f ∈ F sp,q ⇐⇒ ‖f‖F s

p,q=

(∫Rd

( ∞∑k=0

2skq |F−1(φkf)(x)|q)p/q

dx

)1/p

<∞. (32)

(ii) For 1 ≤ p ≤ ∞, the Besov space Bsp,q is defined by:

f ∈ Bsp,q ⇐⇒ ‖f‖Bs

p,q=

( ∞∑k=0

2skq

(∫Rd

|F−1(φkf)(x)|p dx)q/p)

<∞. (33)

(iii) For p = ∞, the Triebel-Lizorkin space F s∞,q is defined by:

f ∈ F s∞,q ⇐⇒ ∃fk∞k=0, f =

∞∑k=0

F−1(φkfk), supRd

( ∞∑k=0

2ksq |fk(x)|q)1/q

<∞,

(34)

52

with norm

‖f‖F s∞,q= inf

(supRd

( ∞∑k=0

2ksq|fk(x)|q)1/q)

,

the infimum being taken over all admissible representations.

(iv) For 1 ≤ p ≤ ∞, the fractional Sobolev space Hsp is defined by:

f ∈ Hsp ⇐⇒ ‖f‖Hs

p=

(∫Rd

|F−1((1 + |x|2)s/2f)(x)|p dx)1/p

<∞. (35)

Remark 4.1.2. The classes of Besov and Triebel-Lizorkin spaces comprise many of

the spaces encounter in analysis, e.g., we have the following identifications whose

proofs may be found in [57, Sect. 2.3.5].

a. If 1 ≤ p = q ≤ ∞ and s ∈ R, then Bsp,p = F s

p,p, this follows from the definition.

b. If 1 < p <∞, then F 0p,2 = Lp.

c. If 1 < p <∞ and s ∈ R, then F sp,2 = Hs

p .

Moreover, for 1 ≤ p, q ≤ ∞ and s > 0 we have that Bsp,q ⊂ Lp, additionally if

p <∞ we also have F sp,q ⊂ Lp. We refer to [58, Sect. 2.3.2] for the proof of these last

assertions.

More generally, we refer to [57], [58], [53] and [55] for background and information

about the Triebel-Lizorkin, Besov, and Sobolev spaces.

Because the Besov and the Tribel-Lizorkin spaces have been rediscovered (under

different names) by various authors, they have a number of equivalent definitions. We

collect here some of those results that will be needed in the sequel: Propositions 4.1.3

and 4.1.4 give equivalent definitions of F sp,q and Bs

p,q, respectively, while Proposition

4.1.5 is a result on interpolation of Besov spaces.

The following result is proved in [57, Proposition 1, 2.3.4].

53

Proposition 4.1.3. Let s ∈ R, 1 < p ≤ ∞, and 1 < q ≤ ∞. If f ∈ S ′then f ∈ F s

p,q

if and only if

∃ fk∞k=0 ⊂ Lp, f =∞∑

k=0

F−1(φkfk),

(∫Rd

( ∞∑k=0

2ksq|fk(x)|q)p/q

dx

)1/p

<∞. (36)

Furthermore,

inf

(∫Rd

( ∞∑k=0

2ksq|fk(x)|q)p/q

dx

)1/p

is an equivalent norm on F sp,q, where the infimum is taken over all admissible repre-

sentations of f .

See [53, Theorem 2, 2.3.2] for a proof of the following result.

Proposition 4.1.4. Let 1 ≤ p, q ≤ ∞ and s > 0. If f ∈ S ′then f ∈ Bs

p,q if and only

if

∃ bk∞=0 ⊂ Lp, f =∞∑

k=0

bk,

( ∞∑k=0

2kqs

(∫Rd

|bk(x)|p dx)q/p)1/q

<∞. (37)

Furthermore,

inf

( ∞∑k=0

2kqs

(∫Rd

|bk(x)|p dx)q/p)1/q

is an equivalent norm on Bsp,q, where the infimum is taken over all admissible repre-

sentations of f .

See [57, Theorem 2.4.7] for a proof of the following result about complex interpo-

lation of Besov spaces.

Proposition 4.1.5. Let s0, s1 ∈ R, 1 ≤ p0, q0, p1, q1 ≤ ∞, and 0 < θ < 1.

If s = (1 − θ)s0 + θs1,1p

= 1−θp0

+ θp1

, and 1q

= 1−θq0

+ θq1

then

(Bs0

p0,q0, Bs1

p1,q1

)[θ]

= Bsp,q. (38)

The next proposition collects some of the computations involved in the proofs of

our results. Part (a) computes the STFT of a Gaussian with respect to a dilated

54

Gaussian. The result is essentially the product of two Gaussians (one in time and the

other in frequency). Part (b) shows that the inverse Fourier transform of the Bessel

potential

m−s(x) = (1 + |x|2)−s/2

is inM1 for s > d. BecauseM1 is invariant under Fourier transforms we then conclude

that the Bessel potential m−s itself is in M1 for s > d.

Proposition 4.1.6. Define g(x) = e−πx2and ga(x) = e−

πx2

a . Let

Gs(x) =1

(4π)s/2

1

Γ(s/2)

∫ ∞

0

t−d+s

2 e−(πx2

t+ t

4π)dt

t

for s, a > 0 and x ∈ Rd, and where Γ refers to the Gamma-function. Then the

following hold.

(a) Vgag(x, ω) =(

aa+1

)d/2e2πi x·ω

a+1 ga+1(x) g a+1a

(ω).

(b) Vgm−s = (2π)d/2 Vg(D 12πGs) ∈ L1 for s > d, where Da is the unitary dilation

operator defined by Dag(x) = |a|−d/2g(x/a).

(c) m−s ∈M1 for s > d.

Proof. (a) First note that from Lemma 2.2.2 the operator Daf(t) = |a|−d/2 f(t/a)

where a > 0 is unitary on L2, and Daf = D1/af . Now for x, ω ∈ Rd we have:

Vgag(x, ω) =

∫Rd

e−π t2

a e−2πit·ω e−π(t−x)2 dt

=

∫Rd

e−πa

((a+1)t2−2at·x+ax2

)e−2πit·ω

=

∫Rd

e−π a+1a

((t− a

a+1x)2+ a

a+1x2

)e−2πt·ω dt

= e−π

a+1x2

∫Rd

e−π a+1a

(t− aa+1

x)2 e−2πt·ω dt

= e−π

a+1x2

(T aa+1

x g aa+1

)∧(ω)

55

= ga+1(x)M− aa+1

x g aa+1

(ω)

=( a

a+ 1

)d/2ga+1(x)M− a

a+1x g a+1

a(ω)

=( a

a+ 1

)d/2e−2πi a

a+1x·ω ga+1(x) g a+1

a(ω),

where we have used the fact that the Fourier transform of the Gaussian g(x) = e−πx2

is itself, i.e., g = g.

(b) For s > 0 it is shown in [55, Proposition 3.1.2] that

Gs(ω) = (1 + 4π2|x|2)−s/2.

Notice for future references that Gs ∈ L1, see [55, Proposition 3.1.2]. Thus,

m−s(x) = (1 + |x|2)−s/2

= Gs(x/2π)

= (2π)d/2D2πGs(x)

= (2π)d/2 D1/2πGs(x).

Consequently, we have that m−s(x) = (2π)d/2 (D 12πGs)(x). Therefore:

Vgm−s(x, ω) = (2π)d/2 Vg(D 12πGs)(x, ω)

= (2π)d/2

∫Rd

D 12πGs(t) e

−2πit·ω g(t− x) dt,

= (2π)d/2⟨D 1

2πGs, MωTxg

⟩= (2π)d/2

⟨Gs, D2πMωTxg

⟩=

⟨Gs, M ω

2πT2πxg2π

⟩=

1

(4π)s/2 Γ(s/2)

∫Rd

∫ ∞

0

us−d2 e−(πt2

u+ u

4π) e−2πit·ω/2π×

e−π(t−2πx)2/2π du

udt

56

=1

(4π)s/2 Γ(s/2)

∫ ∞

0

e−u/4π us−d2

−1

∫Rd

e−πt2

u ×

e−2πit·ω/2π e−π (t−2πx)2

2π dt du

=1

(4π)s/2 Γ(s/2)

∫ ∞

0

e−u/4π us2−1

∫Rd

e−πt2×

e−2πit·ω√

u2π e−πu

(t− 2πx√u

)2

2π dt du

=1

(4π)s/2 Γ(s/2)

∫ ∞

0

e−u/4π us2−1 Vg 2π

u

g(2πx√u,

√uω

2π) du.

The last equality follows from some changes of variable and by using part a. Therefore,

we have that:

Vgm−s(x, ω) =π(d−s)/2

2s−d/2 Γ(s/2)

∫ ∞

0

e−u/4π e−π 4π2x2

u+2π e−π u2π(2π+u)

ω2 ×

us2−1 1

(2π + u)d/2du,

where we have used Lemma 2.2.2 to obtain the last equation. By changing the

variables and using the fact that∫

Rd e−πx2

dx = 1, we have:

‖Vgm−s‖L1 =

∫∫R2d

|Vgm−s(x, ω)| dx dω

≤ π(d−s)/2

2s−d/2

∫ ∞

0

e−u/4π us2−1 1

(2π + u)d/2×

∫Rd

e−π 4π2x2

2π+u

∫Rd

e−π u

2π(2π+u)ω2

dω dx du

=π(d−s)/2

2s−d/2

∫ ∞

0

e−u/4π u−d+s

2−1 (u+ 2π)d/2 du, (39)

and the last expression is finite if s > d.

(c) Follows from (b) and the comments above.

57

4.2 Embedding of Besov, Triebel-Lizorkin spaces

into modulation spaces

There are several embeddings between the Besov or Triebel-Lizorkin and modulation

spaces that can easily be derived. Some of these embeddings are summarized in the

following result.

Proposition 4.2.1. (a) Bsp,q ⊂ Lp ⊂Mp,p′ when s > 0, 1 ≤ p ≤ 2, and 1 ≤ q ≤ ∞.

(b) Bsp,q ⊂ Lp ⊂Mp when s > 0, 2 ≤ p ≤ ∞, and 1 ≤ q ≤ ∞.

Proof. (a) The first of these embeddings was mentioned in Remark 4.1.2, and its proof

can be found in [58, Remark 3, Sect. 2.3.2]. To prove the second one, let f ∈ Lp, and

let g ∈ S. Then for x, ω ∈ Rd, Vgf(x, ω) = f · Txg(ω). Note that the f · Txg ∈ Lp

since f ∈ Lp and g ∈ S. Moreover, since 1 ≤ p ≤ 2 ≤ p′, we have that p′/p ≥ 1, and

thus using Hausdorff-Young’s inequality and Minkowski’s inequality (for integrals) we

have:

‖Vgf‖p

Lp,p′ =

(∫Rd

(∫Rd

|Vgf(x, ω)|p dx)p′/p

dω

)p/p′

≤∫

Rd

(∫Rd

|Vgf(x, ω)|p′dω)p/p′

dx

=

∫Rd

(∫Rd

|f · Txg(ω)|p′dω)p/p′

dx

≤∫

Rd

∫Rd

|f · Txg(t)|pdt dx

=

∫Rd

∫Rd

|f(t)|p |g(t− x)|p dt dx

= ‖f‖pLp ‖g‖p

Lp.

Hence,

‖Vgf‖Lp,p′ = ‖f‖Mp,p′ ≤ ‖f‖Lp ‖g‖p,

which concludes the proof the second inclusion.

58

(b) We now prove the second of these inclusions. Let f ∈ Lp, since 2 ≤ p ≤ ∞,

we have that 1 ≤ p′ ≤ 2. Moreover, choosing g ∈ S we see using Holder’s inequality

with p/p′ ≥ 1, that f · Txg∗ ∈ Lp′ for almost all x ∈ Rd. By using Hausdorff-Young’s

inequality as well as Young’s inequality we have that

‖Vgf‖Lp =

(∫∫R2d

|Vgf(x, ω)|p dx dω)1/p

=

(∫∫R2d

|f · Txg(ω)|p dω dx)1/p

≤∫

Rd

(∫Rd

|(f · Txg)(t)|p′ dt)p/p′

, dx

=

(∫Rd

(∫Rd

|f(t)|p′ |g∗(x− t)|p′ dt)p/p′dx

)1/p

=

(∫Rd

(|f |p′ ∗ |g∗|p′(x))p/p′dx

)1/p

=∥∥|f |p′ ∗ |g∗|p′∥∥1/p′

Lp/p′ .

Now notice that p ≥ p′, and that f ∈ Lp ⇐⇒ |f |p′ ∈ Lp/p′. Thus, applying Young’s

inequality with parameter p/p′ we have that

‖Vgf‖p′Lp ≤ ∥∥|f |p′ ∗ |g∗|p′∥∥

Lp/p′

≤ ‖|f |p′‖Lp/p′∥∥|g∗|p′∥∥

L1

= ‖f‖p′Lp ‖g‖p′

Lp′ .

Consequently,

‖f‖Mp,p′ ≤ ‖g‖Lp′ ‖f‖Lp,

which concludes the proof in this case.

Other more subtle embeddings of classical spaces into modulation spaces were

obtained as byproducts of other results. For example, Grochenig in [40] and Hogan

and Lakey in [47] derived sufficient conditions for membership in the modulation

59

space M1 from certain uncertainty principles related to the STFT. Precisely, it was

shown that Lpa ∩ FLq

b ⊂ M1 under appropriate conditions on p, q and the weight

parameters a, b, where Lpa is a weighted Lp space (with weight (1 + |x|)p), and FLq

b

is the image of Lqb under the Fourier transform. Somewhat more general embeddings

involving weighted Lp spaces were given by Galperin and Grochenig in [32]. Another

interesting example appears in [45]. There, Heil, Ramanathan and Topiwala proved,

in our notation, that Cs(R2d) ⊂ M∞,1(R2d) for s > 2d, while working on a time-

frequency approach to pseudodifferential operators.

The embeddings we will prove are more difficult, and require an appropriate norm

on the Besov or Triebel-Lizorkin space in consideration, along with a correct choice

of the form of the STFT. In particular, the following equivalent forms of the STFT

(see Proposition 2.3.3) will be useful:

Vgf(x, ω) =(f · Txg

)∧(ω)

= e−2πixω Vgf(ω,−x)

= e−2πixωF−1(f · Tω g)(x)

= e−2πixω(f ∗ (Mωg)(x)

). (40)

Our first main embedding result involves the Besov spaces and is as follows.

Theorem 4.2.2. Let 1 ≤ p ≤ 2 and 1 ≤ q ≤ ∞. If s > d(2p− 1) then

Bsp,q ⊂ Mp′,p. (41)

Proof. Let f ∈ Bsp,q, and use (37) to write f =

∑bk where bk ∈ Lp, and

supp(bk) ⊂ |x| ≤ 2k, and ‖f‖Bsp,q

∼ inf( ∞∑

k=0

2ksq‖bk‖qLp

)1/q,

where the infimum is over all possible such representations of f . Given g ∈ S, we

have using (40) that

60

Vgf(x, ω) =∞∑

k=0

e−2πixω F−1(bk · Tωg)(x).

Hence by the Hausdorff-Young inequality,

‖Vgf(·, ω)‖Lp′ ≤∞∑

k=0

‖F−1(bk · Tω g)‖Lp′

≤∞∑

k=0

‖bk · Tω g‖Lp.

Therefore, by Minkowski’s inequality,

‖Vgf‖Lp′,p ≤∞∑

k=0

(∫Rd

‖bk · Tω g‖pLp dω

)1/p

= ‖g‖Lp

∞∑k=0

‖bk‖Lp. (42)

Now, bk has compact support, which is contained in Ek = |x| ≤ 2k. Since

1 ≤ p ≤ 2 ≤ p′, we have that p′/p ≥ 1, and using Holder’s inequality and Hausdorff-

Young’s inequality we obtain the following estimates

‖bk‖pLp = ‖bk‖p

p,Ek

=

∫Ek

|bk(ω)|p dω

≤ |Ek|1−pp′ ‖bk‖p

Ek,p′

≤ C 2kd(1− p

p′ ) ‖bk‖pp′.

where C is the volume of the ball of center 0 and radius 1. Thus,

‖bk‖Lp ≤ C1/p 2kd( 1

p− p

p′ ) ‖bk‖p′

≤ C ′ 2kd( 1p− 1

p′ ) ‖bk‖Lp. (43)

61

Substituting (43) into (42) and applying Holder’s inequality yields

‖Vgf‖Lp′,p ≤ C ′ ‖g‖p

∞∑k=0

2kd( 1

p− 1

p′ ) ‖bk‖p

= C ′ ‖g‖p

∞∑k=0

2kd( 1

p− 1

p′− sd)2ks‖bk‖p

≤ C ′( ∞∑

k=0

2ksq ‖bk‖qLp

)1/q ( ∞∑k=0

2kq′d( 1

p− 1

p′ − sd)

)1/q′

≤ C ‖f‖Bsp,q

( ∞∑k=0

2kq′d( 1

p− 1

p′− sd)

)1/q′

. (44)

The last term in (44) is finite if and only if s > d(1p− 1

p′ ).

The next result recovers and extends the embedding in [45], and follows by iden-

tifying F sp,p with Bs

p,p (see Remark 4.1.2) and by using Proposition 4.1.3 as the appro-

priate definition of Bsp,p. However, it does not include the fact that Bs

2,2 ⊂ L2 = M2

for s ≥ 0. This last embedding is obtained as corollary by using complex interpolation

methods.

Theorem 4.2.3. Let 1 ≤ p ≤ ∞. If s > dp′ then

Bsp,p ⊂Mp,p′. (45)

Proof. We divide the proof in two cases.

Case 1: p = 1. Let f ∈ Bs1,1, if f =

∑∞k=0 F−1(φkf), then letting g(x) = e−πx2

, we

have

Vgf(x, ω) =

∞∑k=0

F−1(fφk) ∗ F−1(Tωg)(x).

Therefore we have the following estimates

‖Vgf(·, ω)‖L1 ≤∞∑

k=0

‖F−1(φk f)‖1 ‖F−1(δ · Tω g)‖1

‖Vgf‖L1,∞ ≤∞∑

k=0

‖F−1(φk f)‖1 supω∈Rd

∫Rd

|F−1(δ · Tωg)(x)| dx

62

≤∞∑

k=0

‖F−1(φk f)‖1 supω∈Rd

∫Rd

|Vgδ(x, ω)| dx.

However,

supω∈Rd

∫Rd

|Vgδ(x, ω)| dx = supω∈Rd

∫Rd

|Mω Tx g(0)| dx =

∫Rd

|g(x)| dx <∞.

Thus,

‖Vgf‖L1,∞ ≤ ‖g‖1

∞∑k=0

‖F−1(φk f)‖1

≤ ‖g‖1

∞∑k=0

2−ks ‖F−1(φk f)‖1,

for all s > 0. Hence, using (33) we have that

‖f‖M1,∞ ≤ ‖g‖1 ‖f‖Bs1,1,

which concludes the proof for this case.

Case 2: Assume 1 < p ≤ ∞. By (36) with p = q we obtain an equivalent norm

for F sp,p = Bs

p,p. Let f ∈ Bsp,p, then f =

∑∞k=0 F−1(φkfk). Let g(x) = e−πx2

. Then,

using (40),

Vgf(x, ω) =∞∑

k=0

fk ∗ F−1(φk · Tωg)(x),

so by Young’s convolution inequality,

‖Vgf(., ω)‖Lp ≤∞∑

k=0

‖fk‖Lp ‖F−1(φk · Tωg)‖L1.

Hence, by Minkowski’s inequality and Holder’s inequality,

‖Vgf‖Lp,p′ ≤∞∑

k=0

‖fk‖Lp

(∫Rd

(‖F−1(φk · Tω g)‖L1

)p′dω

)1/p′

=

∞∑k=0

2sk ‖fk‖Lp 2−sk

(∫Rd

(‖F−1(φk · Tω g)‖L1

)p′dω

)1/p′

≤( ∞∑

k=0

2ksp ‖fk‖pLp

)1/p ( ∞∑k=0

2−ksp′∫

Rd

‖F−1(φk · Tωg)‖p′L1 dω

)1/p′

,

63

and therefore

‖f‖Mp,p′ ≤ ‖f‖Bsp,p

( ∞∑k=0

2−ksp′∫

Rd

‖F−1(φk · Tωg)‖p′L1 dω

)1/p′

. (46)

Now we will estimate the terms in the summation on the right-hand side of (46).

Setting gk(x) = g(2−k+1x), i.e., gk = 2k−12 D2k−1g, we have:

(∫Rd

‖F−1(φk · Tω g)‖p′L1 dω

)1/p′

= ‖Vgφk‖L1,p′ .

But, using Lemma 2.2.2, we obtain:

Vgφk(x, ω) =⟨φk, MωTxg

⟩=

⟨φk, TωM−xg

⟩= 2(k−1)/2

⟨D2k−1φ1, TωM−xg

⟩= 2(k−1)/2

⟨φ1, T21−kωM−2k−1xD2k−1g

⟩= 2(k−1)/2

⟨φ1, M21−kωT2k−1xD2k−1g

⟩=

⟨φ1, M21−kωT2k−1xgk

⟩= Vgk

φ1(2k−1x, 21−kω).

Therefore,

(∫Rd

‖F−1(φk · Tω g)‖p′L1 dω

)1/p′

= ‖Vgφk‖L1,p′

= ‖Vgkφ1(2

k−1·, 2−k+1·)‖L1,p′

= 2d(−k+1) 2d(k−1)/p′ ‖Vgkφ1‖L1,p′

≤ C1 2d/p 2−kd/p ‖Vgkg‖L1,1 ‖Vgφ1‖L1,p′ , (47)

the last inequality following from the independence of the definition of the modulation

space with respect to the window used to compute the STFT (see Proposition 2.3.6).

64

Using Proposition 4.1.6 with a = 2k−1, we have that

‖Vgkg‖L1 = ‖Vgag‖L1 = (1 + 22k−2)d/2. (48)

Combining (46), (47) and (48) yields:

‖f‖Mp,p′ ≤ C2 ‖f‖Bsp,p

( ∞∑k=0

2−kp′(s−d+d/p)

)1/p′

. (49)

The last term in the right-hand side of (49) is finite if and only if s > d(1− 1p) = d

p′ .

Corollary 4.2.4. If 2 ≤ p ≤ ∞ and s > d(1 − 2p) then

Bsp,p ⊂Mp,p′. (50)

Proof. We will prove this part by interpolating between the cases p = 2, s0 ≥ 0 and

p = ∞, s1 > d. In particular, we trivially have

Bs02,2 = F s0

2,2 = Hs02 ⊂ L2 = M2 for s0 ≥ 0, (51)

and applying Theorem 4.2.3 to p = ∞ yields:

Bs1∞,∞ ⊂M∞,1 for s1 > d. (52)

By [57, Remark 4, Sect. 2.4.1] we have that

(Bs0

2,2, Bs1∞,∞

)[θ]

⊂ (M2,2,M∞,1

)[θ]

for appropriate values of s and θ. We now apply the interpolation result of modulation

spaces Proposition 2.3.10, with p0 = q0 = 2 and p1 = ∞, q1 = 1. Hence 1p

= 1−θ2

,

and 1q

= 1−θ2

− θ. Consequently, 1p

+ 1q

= 1. It follows from the above referenced

proposition that (M2,2,M∞,1

)[θ]

= Mp,q = Mp,p′.

65

Similarly, if we apply the interpolation of Besov spaces Proposition 4.1.5 with s0 = 0,

s1 > d, p0 = q0 = 2, and p1 = q1 = ∞, hence s = θs1,1p

= 1−θ2

= 1q, i.e., p = q. It

follows that (B0

2,2, Bs1∞,∞

)[θ]

= Bsp,p,

and moreover, because 0 < θ = 1 − 2/p < 1, we have that p ≥ 2, hence, s = s1θ >

d(1 − 2p), which concludes the proof.

Our next result yields an embedding of a fractional Sobolev space (or Bessel

potential space) into a modulation space. This can also be seen as an embedding

of the Triebel-Lizorkin space F sp,2 into a modulation space, since F s

p,2 = Hsp when

1 < p <∞ [57, 58, 53].

Theorem 4.2.5. If 1 ≤ p ≤ ∞, then

Hsp ⊂Mp,1 for s > d. (53)

Proof. Let m−s(x) = (1 + |x|2)−s/2 and g(x) = e−πx2. Then for f ∈ Hs

p we have:

Vgf(x, ω) = e−2πixω F−1(f · Tωg)(x)

= e−2πixω F−1(fms ·m−sTωg)(x)

= e−2πixω(F−1(fms) ∗ F−1(m−s · Tω g)(x)

).

Hence, by Young’s convolution inequality,

‖Vgf(·, ω)‖Lp ≤ ‖F−1(fms)‖Lp ‖F−1(m−s · Tω g)‖L1,

and so

‖Vgf‖Lp,1 ≤ ‖f‖Hsp

∫Rd

∫Rd

|F−1(m−s · Tω g)(x)|dx dω

= ‖f‖Hsp

∫Rd

∫Rd

|Vgm−s(ω,−x)|dx dω

= ‖f‖Hsp‖Vgm−s‖L1. (54)

66

Using (39), we have that ‖Vgm−s‖L1 <∞ for s > d, which concludes the proof.

The next corollary holds only in dimension one, and gives a useful sufficient con-

dition on a function to be in M1. In particular, (55) below gives a new proof of a

conjecture of Feichtinger, that W 2,1 ⊂ M1 when d = 1. We point out that another

(unpublished) proof of this conjecture was obtained by Grochenig. The corollary fol-

lows from the identification of the Bessel potential space H2p with the Sobolev space

W 2,p obtained by imposing that f and its first two (distributional) derivatives belong

to Lp (p = 1 or p = ∞).

Before proving this corollary, we present the proof of the identification of H2p and

W 2,p when p = 1 or p = 2 and d = 1. We refer to [55, Sect. 6.6], and [9, Theorem

16] for more details on these identifications.

Lemma 4.2.6. If p = 1 or p = ∞, let

W 2,p(R) = f ∈ Lp(R) : f ′, f ′′ ∈ Lp(R), ‖f‖W 2,p =

2∑k=0

‖f (k)‖p <∞.

Then

H2p = W 2,p

Proof. Case I: Assume p = 1. Let f ∈ W 2,1(R), then f, f ′, f ′′ ∈ L1(R). Moreover,

f ′′(ω) = −4π2f(ω). Thus,

F−1((1 + |ω|2)f) = F−1(f) + F−1(ω2f)

= f − 1

4π2f ′′.

Consequently,

F−1((1 + |ω|2)f) ∈ L1(R),

67

and moreover,

‖F−1((1 + |ω|2)f)‖L1 ≤ ‖f‖L1 +1

4π2‖f ′′‖L1

≤ C

2∑k=0

‖f (k)‖L1.

Thus,

W 2,1 ⊂ H21 .

For the converse, let h be the function defined on R, such that

h(x) =

e−x : x ≥ 0,

0 : x < 0.

We easily obtain that for ω ∈ R, h(ω) = 11+2πiω

.

Now let f ∈ H21 (R), then f ∈ L1(R) and F−1((1 + ω2)f) ∈ L1. Thus, g =

F−1(ω2 f) ∈ L1. Moreover, f ∈ L1(R) implies that f ∈ S ′(R), and so f ′′ exists

as an element of S ′(R), hence f ′′(ω) = −4π2 ω2 f , where the equality holds in a

distributional sense. Thus, f ′′ = g, and g ∈ L1(R), therefore the uniquness of the

Fourier transform implies that f ′′ = g ∈ L1(R).

We now show that f, f ′′ ∈ L1(R) implies that f ′ which exists as a distribution is

in L1(R). To that end, note that from the fact that f, f ′′, h ∈ L1(R), we obtain that

f ∗ h, and f ′′ ∗ h are both L1 functions, and we have the following equalities:

h ∗ f ′′(ω) =(−2πiω)2

1 + 2πiωf(ω)

= (−1 + 2πiω +1

1 + 2πiω) hatf(ω)

= −f (ω) + 2πiω f(ω) +f(ω)

1 + 2πiω.

Consequently,

h ∗ f ′′ = −f + f ∗ h+ k,

68

where k = F−1(2πiω, f). The last equation also implies that k ∈ L1(R), and more-

over,

k(ω) = 2πiωf(ω) = f ′(ω).

Using again the uniqueness of the Fourier transform, we conclude that f ′ = k ∈ L1(R),

and moreover,

‖f ′‖L1 ≤ 2‖f‖L1 + ‖f ′′‖L1 ,

where we have used the fact that ‖h‖L1 = 1. Additionally, using the fact that

H21 ⊂ L1, we conclude that

2∑k=0

‖f (k)‖L1 ≤ C ‖f‖H21,

thus, H21 ⊂W 2,1, and the proof in this case is concluded.

The case p = ∞, is identical and so we omit it.

Using the above lemma and Theorem 4.2.5 we have the following result in the

case the dimension is d = 1.

Corollary 4.2.7. If d = 1 and p ∈ 1,∞, then

W 2,p(R) ⊂M1(R). (55)

Proof. If d = 1 and p ∈ 1,∞, then from Lemma 4.2.6 we have that H2p = W 2,p and

so the proof follows from Theorem 4.2.5.

69

CHAPTER V

BILINEAR PSEUDODIFFERENTIAL

OPERATORS ON MODULATION SPACES

In this chapter we present some applications of the modulation spaces. In particu-

lar, we study the boundedness of bilinear pseudodifferential operators on modulation

spaces, as well as the boundedness of the linear Hilbert transform.

Modulation spaces have recently been used to formulate and prove boundedness

results of linear pseudodifferential operators, which are formalisms that assign to a

distribution a linear operator in such a way that properties of the distribution can

be inferred from properties of the corresponding operator. The Weyl and the Kohn-

Nirenberg correspondences are well-known examples of pseudodifferential operators,

which can be expressed as a superposition of time-frequency shifts. In particular, if

σ ∈ S ′(R2d) the Weyl correspondence associate to it the operator Tσ : S(Rd) −→S ′(Rd), such that

Tσf =

∫∫R2d

σ(ξ, u) e−πiξ·u T−uMξ du dξ,

for f ∈ S(Rd). Thus, because the operator can be realized as superposition of time-

frequency shifts, the modulation spaces appear to be natural spaces in which to

formulate and prove boundedness results of such operators. We refer to [45, 42, 41],

for more details on the recent developments of pseudodifferential operators in the

realm of the modulation spaces.

In the first section of the present chapter, we deal with bilinear integral operators

(defined by a non-smooth kernel) on modulation spaces. This class of operators is

large enough to include the bilinear pseudodifferential operators with non-smooth

70

symbols. In particular, we prove that symbols in the Feichtinger algebra give rise

to bounded bilinear pseudodifferential operators. We refer to [11, 13, 12, 49] for

background and more detail about these operators.

The second section is devoted to the boundedness of the linear Hilbert transform

on the modulation spaces defined on the real line. We use a discrete approach to

study the Hilbert transform, and rely on its L2 theory to some extent.

5.1 Bilinear operators on modulation spaces

5.1.1 Definition and background

A bilinear pseudodifferential operator Tσ is a priori defined through its (distributional)

symbol σ ∈ S ′(R3d) as a mapping from S(Rd) × S(Rd) into S ′(Rd) by:

Tσ(f, g)(x) =

∫Rd

∫Rd

σ(x, ξ, η) f(ξ) g(η) e2πix·(ξ+η) dξ dη, (56)

for f, g ∈ S(Rd). A natural problem then is to find sufficient (nontrivial) conditions

on the symbol that ensure the boundedness of the operator on products of certain

Banach spaces such as Lebesgue, Sobolev, or Besov spaces [11, 13, 12, 35, 36]. For

instance, it is known that the condition

|∂αx ∂

βξ ∂

γησ(x, ξ, η)| ≤ Cα,β,γ (1 + |ξ| + |η|)−|β|−|γ| (57)

for (x, ξ, η) ∈ R3d and all multi-indices α, β, γ is enough to prove the boundedness

of the operator defined by (56) from Lp(Rd) × Lq(Rd) into Lr(Rd) when 1p

+ 1q

= 1r

and p, q > 1. This result was first obtained by Coifman and Meyer [11], [13], [12],

who noticed that, in general, if the symbol is smooth and has certain decay, then the

boundedness of the corresponding operator can be studied through its decomposition

into elementary operators via techniques related to Littlewood-Paley theory. Grafakos

and Torres [35] used the wavelet expansions of the Triebel-Lizorkin spaces that were

proved by Frazier and Jawerth [29, 30] to decompose instead the function on which the

operator acts, and thereby converting the boundedness question into the boundedness

71

of an infinite matrix. By imposing some convenient decay conditions on the entries

of the corresponding matrix they obtain some boundedness results on the operator

side on products Triebel-Lizorkin spaces. Here again, the symbols of the operators

are assumed to be sufficiently smooth and to have decay at infinity.

In this section, we use Gabor expansions of tempered distributions in the modula-

tion spaces to prove the boundedness of bilinear integral operators with non-smooth

kernels, of which (56) will be shown to be a particular case.

Throughout this chapter, ωs will denote the submultiplicative weight function

defined on R2d by ωs(x, y) = (1 + |x|2 + |y|2)s/2. Moreover, we let Ωs denote the

extension of ωs on R6d given by

Ωs(X, Y ) = (ωs ⊗ ωs ⊗ ωs)(X, Y ) = ωs(x1, x2)ωs(x3, y1)ωs(y2, y3),

where X = (x1, x2, x3), Y = (y1, y2, y3) ∈ R3d. If A is an invertible operator on R6d

we denote ΩsA

the weight function defined by

ΩAs (X, Y ) = Ωs(A(X, Y )),

where X, Y ∈ R3d. Additionally, we define ωs on Z2d by ωs(l, k) = ωs(αl, βk) for

α, β > 0, and Ωs is defined similarly.

Before considering general bilinear integral operators, we state a result which

characterizes the modulation space M1Ωs

(R3d) in terms of Gabor frames using standard

tensor product arguments; see [41, p. 272] for further details.

Proposition 5.1.1. Let φ ∈M1ωs

(Rd) be such that MβnTαkφk,n∈Zd is a Gabor frame

for L2(Rd) with (canonical) dual γ ∈M1ωs

(Rd). Then K ∈M1Ωs

if and only if

K =∑

k,m,i,l,n,j∈Zd

〈K,MβnTαmγ ⊗MβlTαkγ ⊗MβjTαiγ〉MβnTαmφ⊗MβlTαkφ⊗MβjTαiφ

with unconditional convergence of the series in M1Ωs

(R3d). Moreover, the norm of Kin M1

Ωsis equivalent to the norm of its sequence of Gabor coefficients

(〈K,MβnTαmγ⊗MβlTαkγ ⊗MβjTαiγ〉

)k,m,i,l,n,j∈Zd in `1

Ωs(Z6d).

72

5.1.2 Bilinear operators

Definition 5.1.2. A bilinear operator associated with a kernel K ∈ S ′(R3d), is a

mapping BK defined a priori from S(Rd) × S(Rd) into S ′(Rd) by

BK(f, g)(x) =

∫Rd

∫Rd

K(x, y, z) f(y) g(z) dy dz, (58)

for f, g ∈ S(Rd).

One of our objectives in this section is to study the boundedness of (58) on

products of modulation spaces, and to derive from such results the boundedness of

(56). The next proposition establishes the relationship between a bilinear integral

operator and a bilinear pseudodifferential operator defined by (56). We define an

operator U acting on functions defined on R3d by

Uf(x, y, z) = f(x, y − x, z − x).

It easy to check that U is a unitary operator on L2, is an isomorphism on S, and

extends to an isomorphism on S ′. Moreover,

U∗f(x, y, z) = U−1f(x, y, z) = f(x, y + x, z + x).

Proposition 5.1.3. Let Tσ be a bilinear pseudodifferential operator associated to a

symbol σ ∈ S ′(R3d) defined by (56). Then Tσ is a bilinear integral operator BK with

kernel K(x, y, z) = UF−11 σ(x, y, z), where F−1

1 denotes the inverse Fourier transform

in the first variable, and U is the operator defined above.

Proof. For f, g ∈ S we have:

Tσ(f, g)(x) =

∫Rd

∫Rd

σ(x, ξ, η) f(ξ) g(η) e2πix·(ξ+η) dξ dη

=

∫ ∫ ∫ ∫σ(x, ξ, η) f(y) g(z) e−2πiξ·y e−2πiη·z e2πix·(ξ+η) dξ dη dy dz

=

∫ ∫K(x, y, z) f(y) g(z) dy dz = BK(f, g)(x),

73

where

K(x, y, z) =

∫ ∫σ(x, ξ, η)e−2πiξ·(y−x) e−2πiη·(z−x) dξ dη

= F2F3σ(x, y − x, z − x)

= UF−11 σ(x, y, z).

Here, Fj denotes the Fourier transform in the jth variable.

We show in the next proposition that the symbol of the bilinear pseudodifferential

operator is in M1ΩB

sif and only if the corresponding integral kernel as defined in

Proposition 5.1.3 is in M1Ωs

, where B is the invertible transformation defined on R6d

defined by

B(X, Y ) = (x1, x1 + y2, x1 + y3, x2 + x3 + y1 + y3,−x2,−x3),

for X = (x1, x2, x3), Y = (y1, y2, y3) ∈ R3d.

Proposition 5.1.4. σ ∈M1ΩB

s(R3d) if and only if K = UF−1

1 σ ∈M1Ωs

(R3d).

Proof. Let G ∈ S(R3d). For u = (u1, u2, u3), and v = (v1, v2, v3) ∈ R3d we have

VGσ(u, v) = 〈σ,MvTuG〉 = 〈F−1F−11 U∗K,MvTuG〉 = 〈K,UF−1

1 FMvTuG〉. Hence,

VGσ(u, v) = e−2πi(u2·v2+u3·v3) 〈K,M(v1+u2+u3,−u2,−u3)T(u1,v2+u1,v3+u1)H〉

= e−2πi(u2·v2+u3·v3) VHK(u1, v2 + u1, v3 + u1, v1 + u2 + u3,−u2,−u3)

= e−2πi(u2·v2+u3·v3) VHK(B(u, v)), (59)

where H = UF−11 G. Consequently, we have

|VGσ(u, v)| = |VHK(B(u, v))|.

Therefore,∫R3d

∫R3d

|VHK(u, v)|Ωs(u, v) du dv =

∫R3d

∫R3d

|VGσ(B−1(u, v))|Ωs(u, v) du dv

=

∫R3d

∫R3d

|VGσ(u, v))|ΩBs (u, v) du dv.

74

Thus

‖K‖M1Ωs

= ‖σ‖M1

ΩBs

,

and the proof is complete.

5.1.3 A discrete model

Consider φ ∈ S(Rd) that generates a Gabor frame for L2 with (canonical) dual

γ ∈ S(Rd). We can then expand f, g and h in S(Rd) as in Theorem 3.8, where the

series converge unconditionally in every modulation space norm as long as p, q 6= ∞.

Then using (58), we obtain:

〈BK(f, g), h〉 =

∫∫∫R3d

K(x, y, z)∑

k,l∈Zd

〈f,MβlTαkγ〉MβlTαkφ(y)×

∑m,n∈Zd

〈g,MβnTαmγ〉MβnTαmφ(z)∑

i,j∈Zd

〈h,MβjTαiγ〉MβjTαiφ(x) dx dy dz

=∑i,j

∑k,l

∑m,n

〈f,MβlTαkγ〉 〈g,MβnTαmγ〉 〈h,MβjTαiγ〉×

∫Rd

∫Rd

∫Rd

K(x, y, z)MβjTαiφ(x)MβlTαkφ(y)MβnTαmφ(z) dx dy dz

=∑i,j

∑k,n

∑l,m

〈f,MβlTαkγ〉 〈g,MβnTαmγ〉 〈h,MβjTαiγ〉×

〈BK(MβlTαkφ,MβnTαmφ),MβjTαiφ〉. (60)

The exchange of the integrals and summations above is justified since f, g, h ∈ S have

absolutely summable Gabor coefficients. Moreover, K ∈ S ′(R3d) =⋃

s≥0M∞1/Ωs

( see

Proposition 2.3.9), and φ ∈ S implies that the triple integral in the second equality

is uniformly bounded with respect to i, j, k, l,m, n ∈ Zd. More precisely, define

Mβ(j,l,n)Tα(i,k,m)Φ(x, y, z) = MβjTαiφ(x)MβlTαkφ(y)MβnTαmφ(z).

Clearly, Mβ(j,l,n)Tα(i,k,m)Φ ∈ S(R3d) ⊂M1Ωs

for all s ≥ 0. Moreover, K ∈M∞1/Ωs

where

s > 0. Thus, using the fact that the time-frequency shift operator acts isometrically

75

on M1ωs

, we have

∣∣∣∫∫∫R3d

K(x, y, z)MβjTαiφ(x)MβlTαkφ(y)MβnTαmφ(z) dx dy dz∣∣∣

≤∫∫∫

R3d

|K(x, y, z)| |Mβ(j,l,n)Tα(i,k,m)Φ(x, y, z)| dx dy dz

≤ ‖K‖M∞1/Ωs

‖Mβ(j,l,n)Tα(i,k,m)Φ‖M1Ωs

= ‖K‖M∞1/Ωs

‖φ‖3M1

ωs.

Therefore, to study the boundedness of BK on products of modulation spaces, it

suffices to analyze the boundedness of the matrix B = (bij,kl,mn) defined by

bij,kl,mn = 〈BK(MβlTαkφ,MβnTαmφ),MβjTαiφ〉 (61)

on products of appropriate sequence spaces.

The next theorem will be of special importance in proving our main results. In

particular, it shows that, under some mild condition on its entries, an infinite matrix

yields a bounded operator on products of sequence spaces associated with the mod-

ulation spaces. For an infinite matrix (amn,ij,kl), let O denote the bilinear operator

associated to it, i.e.,

(O(fij), (gkl))mn =∑ij,kl

amn,ij,kl fij gkl,

where (fij) and (gkl) are sequences defined on Z2d.

Theorem 5.1.5. Let ν be an s-moderate weight, and let 1 ≤ pi, qi, ri <∞ for i = 1, 2

be such that 1r1

= 1p1

+ 1q1

. If (amn,ij,kl) ∈ `1Ωs

(Z6d), then O is a bounded operator from

`p1,p2

ν (Z2d)× `q1,q2

ν (Z2d) into `r1,r2

ν (Z2d). In particular, if (amn,ij,kl) ∈ `1(Z6d) then O is

a bounded operator from `p1,p2(Z2d) × `q1,q2(Z2d) into `r1,r2(Z2d).

Proof. Let (fij) ∈ `p1,p2

ν (Z2d), (gkl) ∈ `q1,q2

ν (Z2d) and (hmn) ∈ `r′1,r′21/ν (Z2d) where r′1, r

′2

76

are the dual indices of r1, respectively r2. We have

|〈O((fij), (gkl)), (hmn)〉| ≤∑

m,n,i,j,k,l

|amn,kl,ij| |fij| |gkl| |hmn|

=∑

m,n,i,j,k,l

|amn,kl,ij|1p1 |fij | ν(i, j)

ν(i, j)|amn,kl,ij|

1q1 |gkl| ν(k, l)

ν(k, l)×

|amn,kl,ij|1r′1 |hmn| ν(m,n)

ν(m,n)

≤ C3∑

m,n,i,j,k,l

|amn,kl,ij|1p1 ν(i, j) |fij| ωs(i, j)×

|amn,ij,kl|1q1 ν(k, l) |gkl| ωs(k, l)×

|amn,ij,kl|1r′1

1

ν(m,n)|hmn| ωs(m,n)

= C3∑

m,n,i,j,k,l

(|amn,kl,ij|)1/p1 |fij| ν(i, j)×

(|amn,kl,ij|)1/q1 |gkl| ν(k, l)×(|amn,kl,ij|)1/r′1 |hmn| 1

ν(m,n),

where

amn,kl,ij = amn,kl,ij Ωs(m,n, k, l, i, j) = amn,kl,ij ωs(m,n) ωs(k, l) ωs(i, j).

We have used the fact that ν, and 1ν

are s-moderate with the same constant C. Since

1p1

+ 1q1

= 1r1

, or equivalently 1p1

+ 1q1

+ 1r′1

, we can apply Holder’s inequality to obtain

the following:

|〈O((fij), (gkl)), (hmn)〉| ≤ C3( ∑

m,n,i,j,k,l

|amn,ij,kl| |fij|p1 ν(i, j)p1

) 1p1×

( ∑m,n,i,j,k,l

|amn,ij,kl| |gkl|q1 ν(k, l)q1

) 1q1×

77

( ∑m,n,i,j,k,l

|amn,ij,kl| |hmn|r′1 1

ν(m,n)r′1

) 1r′1

≤ C3

(supi,j

|fij | ν(i, j)) (

supk,l

|gkl| ν(k, l))×

(supm,n

|hmn| 1

ν(m,n)

) (∑m,i,k

∑n,j,l

|amn,ij,kl|)

≤ C3 ‖amn,ij,kl‖`Ω1

s

(∑i

(∑j

|fij |p1 ν(i, j)p1

)p2

p1

) 1p2×

(∑k

(∑l

|gkl|q1 ν(k, l)q1

) q2

q1

) 1q2 ×

(∑m

(∑n

|hmn|r′1 1

ν(m,n)r′1

) r′2r′1

) 1r′2

≤ C3 ‖amn,ij,kl‖`1Ωs

‖fi,j‖`p1,p2ν

‖gk,l‖`q1,q2ν

‖hm,n‖`r′1

,r′2

1/ν

,

where we have used the fact that `p,q(Z2d) ⊂ `∞(Z2d), i.e.,

supm,n

|xm,n| ≤(∑

n∈Zd

( ∑m∈Zd

|xm,n|p)q/p)1/q

.

Moreover, using the duality of the `p,qν -spaces i.e.,

‖a‖`r1,r2ν

= sup‖b‖

`r′1,r′21/ν

=1

∑m,n∈Zd

|am,n| |bm,n|,

we get that

‖O((fij), (gkl))‖`r1,r2ν

≤ C3 ‖amn,ij,kl‖`1Ωs

‖(fij)‖`p1,p2ν

‖(gkl)‖`q1,q2ν

.

The second part of the theorem follows by choosing ν = ω0 ≡ 1.

5.1.4 Boundedness of bilinear pseudodifferential operators

Our first main result of this section shows that a bilinear integral operator with kernel

in the modulation space M1Ωs

— in particular, in the Feichtinger algebra — gives rise

to a bounded operator.

78

Theorem 5.1.6. Let ν be an s-moderate weight, and let 1 ≤ pi, qi, ri <∞ for i = 1, 2

be such that 1p1

+ 1q1

= 1r1

. If K ∈ M1Ωs

(R3d), then the bilinear integral operator BK

defined by (58) can be extended as a bounded operator from Mp1,p2ν (Rd)×M q1,q2

ν (Rd)

into M r1,r2ν (Rd).

Proof. Let f, g, h ∈ S(Rd) and expand each of these functions into their Gabor se-

ries, i.e., f =∑

i,j〈f,MβjTαiφ〉MβjTαiγ, g =∑

k,l〈g,MβlTαkφ〉MβlTαkγ, and h =∑m,n〈h,MβnTαmφ〉MβnTαmγ, where φ and γ are dual Gabor frames as in Theo-

rem 2.3.11. By Proposition 5.1.1, the matrix defined by (61) belongs to `1Ωs

since

K ∈M1Ωs

. Therefore, using Theorem 5.1.5 we have the following estimates:

|〈BK(f, g), h〉| = |∑mn

∑ij

∑kl

amn,ij,kl 〈f,MβjTαiφ〉 〈g,MβlTαkφ〉 〈h,MβnTαmφ〉|

≤ C ‖amn,ij,kl‖`1Ωs

‖〈f,MβjTαiφ〉‖`p1,p2ν

×

‖〈g,MβlTαkφ〉‖`q1,q2ν

‖〈h,MβnTαmφ〉‖`r′1,r′21/ν

≤ C ‖K‖M1Ωs

‖f‖Mp1,p2ν

‖g‖Mq1,q2ν

‖h‖M

r′1,r′21/ν

,

by duality we obtain

‖BK(f, g)‖Mr1,r2ν

≤ C ‖K‖M1Ωs

‖f‖Mp1,p2ν

‖g‖Mq1,q2ν

.

The result then follows by standard density arguments, using the fact that S(Rd) is

dense in Mp,qν for 1 ≤ p, q <∞.

The previous result together with Propositions 5.1.3 and 5.1.4 yields our second

main result of this chapter, which provides a sufficient condition on the symbol so

that the operator (56) is bounded on products of modulation spaces. Recall that the

invertible transformation B was defined on R6d by

B(X, Y ) = (x1, x1 + y2, x1 + y3, x2 + x3 + y1 + y3,−x2,−x3).

79

Theorem 5.1.7. Let ν be an s-moderate weight, and let 1 ≤ pi, qi, ri <∞ for i = 1, 2

be such that 1p1

+ 1q1

= 1r1

. If σ ∈ M1ΩB

s(R3d), then the bilinear pseudodifferential

operator Tσ defined by (56) can be extended to a bounded operator from Mp1,p2ν (Rd)×

M q1,q2ν (Rd) into M r1,r2

ν (Rd).

Proof. By Proposition 5.1.4, σ ∈ M1ΩB

sif and only if K ∈M1

Ωs, where K is the kernel

of the corresponding integral operator, and the result follows from Theorem 5.1.6.

If we assume that ν = ω0 ≡ 1, and that p1 = p2 = p and q1 = q2 = q (hence

r1 = r2 = r), we obtain the following.

Corollary 5.1.8. Let 2 ≤ p, q < ∞ and 1 ≤ r ≤ 2 be such that 1p

+ 1q

= 1r

. If

σ ∈ M1(R3d), then Tσ can be extended to a bounded operator from Lp(Rd) × Lq(Rd)

into Lr(Rd). In particular, if σ ∈ M1(R3d), then Tσ has a bounded extension from

L2(Rd) × L2(Rd) into L1(Rd).

Proof. If of ≤ p, q <∞ by Proposition 4.2.1 we have the following embeddings:

Lp ⊂Mp, and Lq ⊂M q.

Thus, Lp×Lq ⊂ Mp×M q. Moreover, since 1 ≤ r ≤ 2, we have by the same proposition

that M r ⊂ Lr. These continuous embeddings combined with Theorem 5.1.7 imply

then the result.

Remark 5.1.9. It is remarkable that the condition σ ∈M1(R3d) does not necessarily

imply any smoothness nor decay on the symbol. In particular, Coifman-Meyer-type

conditions (57) are not necessarily satisfied by the symbols we consider.

Assume that ν(x, y) = ωs(x, y) = (1 + |x|2 + |y|2)s/2 for some s > 0, and that

pi = qi = 2. Let ω1s be the restriction of ωs to Rd × 0. Then the following holds.

Corollary 5.1.10. If σ ∈ M1ΩB

sthen Tσ can be extended as a bounded bilinear pseu-

dodifferential operator from M2ωs

×M2ωs

into L1ω1

s.

80

Proof. Notice that M1ωs

is continuously embedded in L1ω1

s, cf. [41, Prop. 12.1.4]. So,

we only need to prove that under the hypotheses of the corollary, the bilinear pseu-

dodifferential operator can be extended to a bounded operator from M2ωs

×M2ωs

into

M1ωs

. This follows from Theorem 5.1.7 by taking ν = ωs.

Remark 5.1.11. a. If the symbol σ satisfies the estimates

|∂βξ ∂

γησ(x, ξ, η)| ≤ Cβ,γ (1 + |ξ| + |η|)−d−ε (62)

for all (x, ξ, η) ∈ R3d, all multi-indices β and γ, and some ε > 0, then it

follows from [7, Theorem 1] that the corresponding bilinear pseudodifferential

operator is bounded from L2 × L2 into L1. We wish to point out that, in

general, neither that result nor Corollary 5.1.8 in this section imply each other.

On one hand, if g ∈ S(R2d) then σ1(x, ξ, η) = χ[0,1[d(x)g(ξ, η), where χ[0,1[d

is the characteristic function of the unit cube in Rd, satisfies (62) and hence

it yields a bounded operator from L2 × L2 into L1. However, because σ1 is

not a continuous function, it is not in M1(R3d). Therefore, our corollary does

not apply. On the other hand, functions in M1 must be continuous, but there

are non-differentiable functions in M1, hence they do not satisfy (62), thus [7,

Theorem 1] does not apply here.

b. Notice that (62) requires smoothness of the symbols only in the ξ and η variables

whereas (57) imposes smoothness on all the variables x, ξ and η. Thus the two

conditions are different.

5.2 Linear Hilbert transform on the modulation

spaces

In this section, we consider the boundedness of the one-dimensional (linear) Hilbert

transform — which is a prototypical example of a singular integral operator — on

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the modulation spaces. The Hilbert transform of a function f ∈ S(R) is defined by

Hf(x) = 1π

limε→0

∫|t|>ε

f(x−t)t

dt =1

πlimε→0

∫|x−t|>ε

f(t)x−t

dt. (63)

The boundedness of the Hilbert transform on the Lp spaces (1 < p < ∞) was es-

tablished by M. Riesz using complex variable methods. The real variable method,

initiated by Besicovitch and Titchmarsh, and further developed by Calderon and

Zygmund, establishes that the Hilbert transform is of weak-type (1, 1) (in fact, their

theory applies to more general operators). More precisely, the following estimate

holds:

|x ∈ R : (Hf(x)) > α| ≤ C

α

∫R

|f(x)| dx,

where C is a constant independent of f and α > 0. Moreover, using a Fourier

approach, it is easy to prove the boundedness of the Hilbert transform on L2. Indeed,

it is known that Hf(ω) = −i sign(ω) f(ω), where sign denotes the sign function.

Thus, using Plancherel’s theorem we obtain

‖Hf‖L2 = ‖Hf‖L2

= ‖f‖L2

= ‖f‖L2,

and this last equality proves the boundedness ofH on L2. The weak-type (1, 1) result,

and the L2 boundedness together with duality and interpolation methods can be used

to prove that H is bounded on Lp for 1 < p < ∞. We refer to [55, Sect. 2] for more

background on the Hilbert transform, and more generally, on the theory of singular

integrals.

We prove the boundedness of H on the modulation spaces Mp,q, for 1 < p, q <∞,

by converting the boundedness question into the boundedness of an infinite matrix

acting on appropriate sequence spaces. We achieve this goal by expanding the function

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on which H acts into their Gabor expansions, and thus we are reduced to studying

the boundedness of an associated infinite matrix on `p,q(Z × Z).

Let α, β > 0 be given, and let φ, γ ∈ S(R) be such that supp(γ) ⊂ (0, β) and

γ ≥ 0. Assume, moreover, that MβnTαkφk,n, and MβnTαkγk,n are dual Gabor

frames for L2(R). Then MβnTαkφk,n and MβnTαkγk,n are also dual Gabor frames

for all the modulation spaces Mp,q(R) (see Theorem 2.3.11).

Proposition 5.2.1. Let f, g ∈ S(R), then

〈Hf, g〉 =∑

m,n∈Z

∑k,l∈Z

〈f,MβnTαmφ〉 〈g,MβlTαkφ〉 〈HMβnTαmγ,MβlTαkγ〉. (64)

Proof. If f, g ∈ S(R), then we can expand them into their Gabor expansions, i.e.,

f =∑

m,n∈Z

〈f,MβnTαmφ〉MβnTαmγ, and g =∑k,l∈Z

〈g,MβlTαkφ〉MβlTαkγ,

with unconditional convergence in any modulation space. In particular, the assump-

tions on f, g, φ, and γ imply that the Gabor coefficients of f and g are absolutely

summable. Then we have:

〈Hf, g〉 =

∫R

Hf(x) · g(x) dx

=

∫R

∑k,l∈Z

〈g,MβlTαkφ〉Hf(x) ·MβlTαkγ(x) dx.

Because H is bounded on L2, we have that for each k, n ∈ Z, Hf ·MβlTαkγ ∈ L1(R).

Moreover, since 〈g,MβlTαkφ〉 ∈ `1(Z×Z), we can apply Fubini’s Theorem to obtain:

〈Hf, g〉 =∑k,l∈Z

〈g,MβlTαkφ〉∫

R

Hf(x) ·MβlTαkγ(x) dx. (65)

Further, the adjoint H∗ of H is bounded on L2 (in fact, one can show that H∗ =

83

−H), so we have the following:

∫R

Hf(x)MβlTαkγ(x) dx = 〈Hf,MβlTαkγ〉

= 〈f,H∗MβlTαkγ〉

= −〈f,HMβlTαkγ〉

=∑

m,n∈Z

〈f,MβnTαmφ〉 〈HMβnTαmγ,MβlTαkγ〉. (66)

We can now use (65) and (66) to obtain;

〈Hf, g〉 =∑k,l

∑m,n

〈f,MβnTαmφ〉 〈g,MβlTαkφ〉 〈HMβnTαmγ,MβlTαkγ〉. (67)

We denote Aα,β,γ the sequence defined for (k, l) ∈ Z2 by

Aα,β,γ(k, l) = |Vγ γ(βl, αk)|.

The choice of the window γ implies in particular that γ ∈M1, hence

∑l,k∈Z

Aα,β,γ(k, l) <∞.

We can use the above proposition to prove the following.

Proposition 5.2.2. If f, g ∈ S(R), then

|〈Hf, g〉| ≤∑

m,n∈Z

∑k,l∈Z

Aα,β,γ(m− k, l − n) |〈f,MβnTαmφ〉| |〈g,MβlTαkφ〉|. (68)

Proof. Let S be the sign function defined by

S(x) = sign(x) =

−1 : x < 0,

0 : x = 0,

+1 : x > 0.

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We use again the L2-theory of the Hilbert transform. In particular, using the Fourier

transform we have

〈HMβnTαmγ,MβlTαkγ〉 = 〈(HMβnTαmγ)∧, (MβlTαkγ)

∧〉

= −i〈S · (MβnTαmγ)∧, (MβlTαkγ)

∧〉

= −ie2πiαβ(kl−mn) 〈S ·M−αmTβnγ,M−αkTβlγ〉

= −ie2πiαβk(n−l)Vγ((T−βnS) · γ)(β(l − n), α(m− k)). (69)

With that choice of γ it is easy to see that for all n ∈ Z,

T−βnS · γ = ±γ. (70)

Hence, (69) becomes

〈HMβnTαmγ,MβlTαkγ〉 = ±ie2πiαβk(n−l)Vγ γ(β(l − n), α(m− k)). (71)

By putting all the above together into (65) we have

〈Hf, g〉 = ±i∑

m,n∈Z

∑k,l∈Z

e2πiαβk(n−l)Vγ γ(β(l−n), α(m−k)) 〈f,MβnTαmφ〉 〈g,MβlTαkφ〉.

(72)

Taking the magnitude of both sides then yields the desired result.

We are now ready to prove the boundedness of the Hilbert transform on all the

modulation spaces Mp,q with 1 < p, q <∞.

Theorem 5.2.3. Let 1 < p, q < ∞, then the Hilbert transform H extends to a

bounded linear operator on Mp,q. In particular, for any f ∈Mp,q we have the following

estimate:

‖Hf‖Mp,q ≤ C ‖f‖Mp,q ,

for some positive constant C independent of f .

85

Proof. Let f, g ∈ S(R), then, by Proposition 68, we have that

|〈Hf, g〉| ≤∑

m,n∈Z

|〈f,MβnTαmφ〉|∑k,l∈Z

Aα,β,γ(m− k, l − n) |〈g,MβlTαkφ〉|

=∑n,m

|〈f,MβnTαmφ〉|(Aα,β,γ ∗ |〈g,Mβ·Tα·φ〉|(m,n)

)

≤ ‖〈f,Mβ·Tα·φ〉‖`p,q ‖Aα,β,γ ∗ |〈g,Mβ·Tα·φ〉|‖`p′,q′

≤ ‖Aα,β,γ‖`1 ‖〈f,Mβ·Tα·φ〉‖`p,q ‖〈g,Mβ·Tα·φ〉‖`p′,q′

≤ C ‖γ‖M1 ‖f‖Mp,q ‖g‖Mp′,q′ .

We have used Young’s inequality to obtain the fourth inequality. By duality we then

obtain

‖Hf‖Mp,q ≤ C ‖γ‖M1 ‖f‖Mp,q .

The result then follows since S(R) is a dense subspace of each of the modulation

spaces Mp,q for 1 < p, q <∞.

Remark 5.2.4. The technique that we use to prove the boundedness of H on the

modulation spaces is different from the one used in the Lp theory, but relies heavily

on the L2 theory.

We remark that the H cannot be bounded on the Feichtinger algebra M1. To

see this notice that M1 is a dense subspace of L2, and that functions in M1 are

continuous. Since M1 is invariant under the Fourier transform, it is easy to see that

for f ∈ M1,

Hf ∈ M1 ⇐⇒ Hf ∈M1.

However, Hf = −S · f , and this function cannot belong to M1 since it has a discon-

tinuity at the origin.

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VITA

Kasso Akochaye Okoudjou was born on March 30, 1973, in Parakou, Benin. Heattended both Lycee Toffa and Lycee Behanzin high schools in Porto-Novo, Beninand graduated from the latter in 1991. Subsequently, he entered the Universite Na-tionale du Benin in Abomey-Calavi, Benin and completed a Maitrise es SciencesMathematiques in 1996. After teaching mathematics at Complexe Scolaire WilliamPonty high school in Porto-Novo, Benin from 1996 to 1998, he was admitted to thePh.D. program in the School of Mathematics at the Georgia Institute of Technologyin the fall of 1998. He graduated from Georgia Tech with an M. S. in Electrical andComputer Engineering in May 2003, and with a Ph. D. in Mathematics in August2003. His thesis was written under the supervision of Professor Christopher Heil. Hethen moved to Ithaca, NY, to join the Mathematics Department of Cornell Universityas an H. C. Wang Assistant Professor.

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