One objective of this thesis is to extend the theory of Gabor frames to other. Banach spaces ...... mate a local criteria for membership with a global criteria. We will ...
Characterization of function spaces and boundedness of bilinear pseudodifferential operators through Gabor frames
A Thesis Presented to The Academic Faculty by
Kasso Akochay´ e Okoudjou
In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
School of Mathematics Georgia 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 Yang Wang
Professor Jeffrey Geronimo
Professor Anthony Yezzi ECE
Professor Gerd Mockenhaupt
Date Approved
To my family.
iii
ACKNOWLEDGEMENTS
First and foremost, I would like to thank my advisor, Professor Chris Heil for introducing 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 Gr¨ochenig 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 D¨orfler, Norbert Kaiblinger, Tobias Werther, as well as to Erik Alap¨ aa¨ 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 K´elom`e, 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 I
II
viii
PRELIMINARIES
1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
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 3.1 Weighted amalgam spaces . . . . . . . . . . . . . . . . . . . . . . . .
22 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 on amalgam 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 INTO MODULATION SPACES 51
V
4.1 The Besov and Triebel-Lizorkin Spaces . . . . . . . . . . . . . . . .
52
4.2 Embedding of Besov, Triebel-Lizorkin spaces into modulation spaces
58
BILINEAR PSEUDODIFFERENTIAL OPERATORS ON MODULATION 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 {fn }n∈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, exponential frames, or Gabor frames—have proven very useful in numerous applications. Gabor frames, also known as Weyl-Heisenberg frames, are generated by timefrequency shifts of a single function which is called the window function or the generator. 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 functions 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 modulation 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 simultaneously 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 signal. He used building blocks generated by time-frequency shifts of a Gaussian, i.e., 2
building blocks of the form gm,n = e2πim· g(· − n) where g(x) = e−πx , 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 reconstruction, 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 {fn }n∈I for which there exist constants 0 < A, B < ∞ —called frame bounds— such that A kf k2H ≤
X
|hf, fn i|2 ≤ Bkf k2H
∀f ∈ H.
n∈I
It is remarkable that the above inequalities imply the existence of a (canonical) dual
1
frame {f˜n }n∈I , such that the following reconstruction formula holds for every f ∈ H: f=
X
hf, f˜n i fn .
n∈I
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 redundancy 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. Gr¨ochenig [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 0 (R2d ) —the symbol of the operator— a linear operator Tσ : S(Rd ) → S 0 (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 pseudodifferential 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. Symbols in the so-called H¨ormander class are known to yield bounded pseudodifferential operators on various Banach spaces, cf. [26, Chapter 2]. In particular, Calder´onVaillancourt [10] proved that if σ is smooth enough and has enough decay, then Tσ can be extended to a bounded operator on L2 . Gr¨ochenig 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. Gr¨ochenig 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 H¨ olderLipschitz, 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 pseudodifferential 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 0 (Rd ), and are not just generalizations of their linear counterparts, but are important 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. B´enyi [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 products 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 characterization of the weighted amalgam spaces by Gabor frames. To obtain this characterization we study the behavior of the various operators connected with (Gabor) frames theory. 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 = x1 y1 + x2 y2 + . . . + xn yn . 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). R 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 Z ˆ F f (ω) = f (ω) = f (t) e−2πit·ω dt for ω ∈ Rd . Rd
Similarly, the inverse Fourier transform of f is defined formally by Z −1 ˇ f (ω) e2πiω·t dω for t ∈ Rd . F f (t) = f (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 kukX or simply kuk when the appropriate space is clear from the context. Moreover, if two norms k · k1 and k · k2 , are equivalent on a Banach space X we will write kuk1 kuk2 to mean the existence of two positive constant C1 , C2 such that C1 kuk1 ≤ kuk2 ≤ C2 kuk1
∀u ∈ X.
The dual of a Banach space X is denoted X ∗ . We write hf, gi for the action of f ∈ X 0 on g ∈ X. The adjoint of an operator T is denoted by T ∗ . For 1 ≤ p ≤ ∞, p0 will denote the conjugate of p, i.e.,
1 p
+
1 p0
= 1.
Lp (Rd ) is the Banach space of complex-valued functions f on Rd with norm Z 1/p p |f (x)| dx , kf kp = kf kLp = Rd
6
for 1 ≤ p < ∞. If p = ∞, the norm is given by kf k∞ = ess sup |f (x)|. x∈Rd
Similarly, `p (Zd ) is the Banach space of complex-valued sequences c on Zd with norm kckp = kck`p =
X
1/p |cn |
p
,
n∈Zd
for 1 ≤ p < ∞. If p = ∞, the norm is given by kck∞ = sup |cn |. n∈Zd
2d We will also consider weighted mixed-norm spaces Lp,q ν (R ), which are Banach
spaces of complex-valued functions f on R2d with norm Z Z kf kLp,q = ν
q/p 1/q |f (x, y)| ν(x, y) dx dy , p
Rd
Rd
p
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,q ν˜ as the Banach space of complex-valued sequences c on Z2d with norm kck`p,q = ν
X X l∈Zd
q/p 1/q |ck,l | ν˜(k, l) p
p
,
k∈Zd
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 kf kp,E = kf χE kp 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 kT kX→Y , or simply by kT k if there is no confusion. P For a multi-index α = (α1 , . . . , αd ), we write |α| = di=1 αi . The differentiation operator Dα and the multiplication operator X β are defined by d Y (∂xi )αi f (x), D f (x) = α
and
β
X f (x) =
i=1
d Y
xβi i f (x).
i=1
S(Rd ) is the Schwartz space of all infinitely differentiable functions f for which the seminorms kf k(M,N ) =
X X
kD α X β f k∞
|α|≤M |β|≤N
are finite for all non-negative integers M, N . Its topological dual, S 0 (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 Mνp,q known as the modulation spaces. The frame expansions converge unconditionally in the norm of those spaces, and membership of a tempered distribution in Mνp,q is characterized by membership of its sequence of Gabor coefficients in a weighted sequence space `p,q ν˜ . 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 function 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 1 ≤ Cν ω(x) , ν(x + y) 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 Mb f (t) = e2πit·b f (t)
Ta f (t) = f (t − a), 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 Dc f (t) = |c|−d/2 f (t/c). It is easily seen that the translation, modulation, and dilation operators are unitary on L2 , and that they map S and S 0 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. Ta Mb f = e−2πia·b Mb Ta f. b. Dc Ta f = Tca Dc f. c. Dc Mb f = M b Dc f. c
ˆ ˆ [ d. Td a f = M−a f, and Mb f = Tb f . 11
d ˆ e. D cf = D 1 f . c
2πia·b ˆ \ M−a Tb f. f. M b Ta f = e
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) = {Mβn Tαk g}k,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 kf k2L2 ≤
X
|hf, Mβn Tαk gi|2 ≤ B kf k2L2 .
(2)
k,n∈Zd
We should point out that due to the commutation relations between the translation 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(ˆ g , β, α). Definition 2.2.4. Consider the collection of time-frequency shifts G(g, α, β) generated 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 Cg f = hf, Mβn Tαk gi k,n∈Zd , for f ∈ L2 . b. The synthesis operator associated with G(g, α, β) is the operator Rg : `2 (Z2d ) → P L2 (Rd ) defined formally by Rg c = k,n∈Zd ckn Mβn Tαk g, 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 Cg f = hf, Mβn Tαk gi
k,n∈Zd
is a bounded mapping from
L2 into `2 , and we have the norm equivalence kf k2 kCg f k`2 . b. The synthesis operator Rg c =
P k,n∈Zd
ckn Mβn Tαk g is a bounded mapping from
`2 into L2 . The series defining Rg c converges unconditionally in L2 for every c ∈ `2 . c. Rg = Cg∗ , and the frame operator Sg = Rg Cg : L2 → L2 is strictly positive. d. The dual window γ = Sg−1 g generates a Gabor frame G(γ, α, β) for L2 (Rd ) with frame bounds 1/B, 1/A. e. Rγ Cg = I on L2 (Rd ), i.e., we have the Gabor expansions X
f = Rγ Cg f =
hf, Mβn Tαk γi Mβn Tαk g
(3)
k,n∈Zd
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 kf k2 kCg f k`2 . b. Let c = (ckn )k,n∈F , where F is a finite subset of Z2d . For f ∈ L2 , we have |hRg c, f i| = |
X
ckn hMβn Tαk g, f i|
k,n∈F
≤
X
|ckn ||hf, Mβn Tαk gi|
k,n∈F
≤
X
|ckn |
kn∈F
≤
√
B kf kL2
2
1/2 X
X kn∈F
13
1/2 |hf, Mβn Tαk gi|
kn∈Zd
1/2
|ckn |
2
,
2
where we have used the second inequality in (2). By duality we obtain kRg ckL2 = sup |hRg c, f i| kf kL2 =1
≤
√
B
X
1/2 |ckn |
2
kn∈F
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 unconditionally. c. For f ∈ L2 and c = (ckn )k,n∈Zd , we have hRg c, f i =
X
ckn hf, Mβn Tαk gihc, Cg f i.
k,n∈Zd
Hence, Rg = Cg∗ . The frame operator Sg = Rg Cg 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 kf k2L2 ≤ hSg f, f i ≤ B kf k2L2 for all f ∈ L2 , or equivalently in operator notation as A I ≤ 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 ≤ Sg−1 ≤ 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 Sg−1 . Hence, Sg−1 = Sγ , which together with the last operator inequality concludes the proof of this part. e. Follows from the fact that f = Sg Sγ f , 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 (hf, Mβn Tαk gi)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 = Rγ Cg . In all these statements, and throughout this thesis, the roles of g and γ may be interchanged.
2.3 2.3.1
Modulation spaces 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 Z Vg f (x, y) = hf, My Tx gi =
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 0 and g ∈ S. Moreover, analogously to the Fourier transform, the STFT extends in a distributional sense to f , g in the space of tempered distributions S 0 , 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: Vg f (x, y) = hf, My Tx gi = f · Tx g
∧
(y)
= e2πix·y f ∗ My g ∗ (x) ˆ Ty M−x gˆi = hf, = e−2πix·y hfˆ, M−x Ty gˆi = e−2πix·y Vgˆfˆ(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 kVg f kL2 = kf kL2 kgkL2 . That is, Vg is a multiple of an isometry from L2 (Rd ) into L2 (R2d ). Proof. First assume that f ∈ S then, f · Tx g ∈ L2 (Rd ) for almost all x ∈ Rd . Therefore, we have kVg f k2L2
ZZ = R2d
ZZ =
R2d
ZZ =
R2d
ZZ =
R2d
|Vg f (x, ω)|2 dx dω |(f\ · Tx g)(ω)|2 dω dx |(f · Tx g)(t)|2 dt dx |f (t)|2 |g(t − x)|2 dt dx
= kf k2L2 kgk2L2 . Thus kVg f kL2 = kf kL2 kgkL2 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 Mνp,q (Rd ) is the space of all distributions f ∈ S 0 for which the following norm is finite: Z Z kf kMνp,q =
q/p 1/q |Vg f (x, ξ)| ν(x, ξ) dx dξ = kVg f kLp,q , ν p
Rd
Rd
p
(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 kf kgMi νp,q denote the norm of f in the modulation space Mνp,q as measured by the window gi when i = 1, 2. Then there exist two constants C1 , C2 > 0 such that C1
1 kf kgM2 νp,q ≤ kf kgM1 νp,q ≤ C2 kVg1 g2 kL1ω kf kgM2 νp,q . kVg2 g1 kL1ω
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 ≤ ∞, Mνp,q is a Banach space.
17
b. If p, q < ∞, S is a dense subspace of Mνp,q . Moreover, the dual of Mνp,q is the 0
0
p ,q modulation space M1/ν . More precisely we have that
kf kMνp,q =
sup kgk
p0 ,q 0 =1 M 1/ν
|hf, gi|.
c. If p1 ≤ p2 , and q1 ≤ q2 , then Mνp1 ,q1 ⊂ Mνp2 ,q2 . d. If p, q < ∞, then Mω1 is a dense subspace of Mνp,q . Remark 2.3.8.
a. If p = q we denote the modulation space Mνp,p simply by Mνp .
Moreover, if ν = 1 we simply denote Mνp,q by M p,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 M 2 = L2 , • if ν(x, ξ) = (1 + |x|2 )s/2 where s > 0, and p = q = 2, then Mν2 = L2s , a weighted-L2 space, • if ν(x, ξ) = (1 + |ξ|2)s/2 where s > 0, and p = q = 2, then Mν2 = Hs2 , the standard Sobolev space. c. Lp for p 6= 2 does not coincide with any modulation space [25]. d. The modulation M 1 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 0 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 ) =
\
Mv∞s
S 0 (Rd ) =
and
s≥0
[
∞ M1/v . s
s≥0
The next proposition, due to Feichtinger [21], on complex interpolation of modulation 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
1 p
=
1−θ p0
+
θ , p1
and
1 q
=
1−θ q0
+
θ q1
M p0 ,q0 , M p1 ,q1
2.3.2
then
[θ]
= M p,q .
(5)
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 modulation 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 ∈ Mω1 , and does not require that g generate a frame for L2 . Recall that the mixed-norm sequence space `p,q ν˜ consists of all sequences c = (ckn )k,n∈Zd such that kck`p,q = ν ˜
XX n∈Zd
q/p 1/q |ckn | ν˜(k, n) p
p
< ∞,
k∈Zd
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 ∈ Mω1 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 Cg f = hf, Mβn Tαk gi
k,n∈Zd
is a bounded map-
ping from Mνp,q to `p,q ν˜ , and we have the norm equivalence kf kMνp,q kCg f k`p,q . ν ˜ b. The synthesis operator defined by Rg c =
P
k,n∈Zd ckn
Mβn Tαk g is a bounded map-
p,q ping from `p,q ν˜ to Mν . The series defining Rg c converges unconditionally in the ∞,∞ norm of Mνp,q for every c ∈ `p,q if p = ∞ or ν˜ (weak* unconditionally in M1/ω
q = ∞). c. The frame operator Sg = Rg Cg is a continuously invertible mapping of Mνp,q onto itself. d. The dual window γ = Sg−1 g lies in Mω1 . e. Rγ Cg = I on Mνp,q , i.e., we have the Gabor expansions f = Rγ Cg f =
X
hf, Mβn Tαk γi Mβn Tαk g
(6)
k,n∈Zd
for f ∈ Mνp,q , with unconditional convergence of the series if p, q < ∞, and with unconditional weak* convergence otherwise. f. A distribution f ∈ Mν∞,∞ belongs to Mνp,q if and only if Cg f ∈ `p,q ν˜ . If g ∈ S, then a tempered distribution f ∈ S 0 (Rd ) belongs to Mνp,q if and only if Cg f ∈ `p,q ν˜ . In brief, the `p,q ν˜ norm of the Gabor coefficients (hf, Mβn Tαk gi)k,n∈Zd is an equivalent norm for Mνp,q , and the Gabor expansions (3) are valid in Mνp,q with unconditional convergence of that series in the norm of Mνp,q . 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 kCg f k`p,q is not only an ν ˜ equivalent norm for Mνp,q , but membership of f in the modulation space is characterized by membership of its sequence of Gabor coefficients Cg f in `p,q ν˜ . In particular, 20
only the magnitude of these coefficients is important in determining whether a given distribution lies in Mνp,q . The proof of Theorem 2.3.11 requires deep analysis. In particular, the invertibility of Sg on Mω1 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 Mνp,q factorizes as I = Rγ Cg .
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 representation 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 amalgamate 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, membership 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 expansions 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 3.1.1
Weighted amalgam spaces 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 kf k
W (Lp ,Lqν )
=
X
1/q kf · Tαk χQα kqp ν(αk)q
(7)
k∈Zd
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 sometimes used to denote F L1 ), cf. [51]. It was shown in [59] that W (L∞ , L1 ) is a convenient 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 relations: if p1 ≥ p2 , and q1 ≤ q2 , then 2 W (Lp1 , Lqω1 ) ⊂ W (Lp1 , Lqν1 ) ⊂ W (Lp2 , Lqν2 ) ⊂ W (Lp2 , Lq1/ω ).
23
In particular, the inclusions W (L∞ , L1ω ) ⊂ W (Lp , Lqν ) ⊂ 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 , Lqω1 ) ⊂ W (Lp1 , Lqν1 ) follows from the (7). Now let f ∈ W (Lp1 , Lqν1 ). Then kf kW (Lp2 ,Lqν2 ) =
X
kf ·
Tαk χQα kqp22
1/q2 ν(αk)
q2
k∈Zd
≤C
X
kf ·
Tαk χQα kqp21
1/q2 ν(αk)
q2
k∈Zd
≤C
X
kf · Tαk χQα kqp11 ν(αk)q1
1/q1 ,
k∈Zd
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 , Lqν1 ) ⊂ W (Lp2 , Lqν2 ). 2 ) follows again from the fact that ν The last inclusion, W (Lp2 , Lqν2 ) ⊂ W (Lp2 , Lq1/ω
is ω-moderate, and so is 1/ν, and so
1 ν(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.
P 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 ,
X
fk
f − k∈F
< ε.
X
c. For every ε > 0, there exists a finite F0 ⊂ J such that ∀ finite F ⊃ F0 ,
sup
X
∗
|hfk , hi| : h ∈ X , khkX ∗ = 1
< ε.
k ∈F /
Now let X be a Banach function space in the sense of [5]. In particular, this includes the amalgam spaces W (Lp , Lqν ). The K¨ othe dual of X (or the associated ˜ consisting of all measurable functions space, as it is called in [5]), is the space X ˜ is a closed, normh such that f h ∈ L1 for each f ∈ X. By [5, Thm. 1.2.9], X fundamental subspace of X ∗ , so in particular, ∀ f ∈ X,
˜ khk ˜ = 1 . kf kX = sup |hf, hi| : h ∈ X, X
˜ topology, i.e., the weak topology on X By [5, Cor. 1.5.3], X is complete in the σ(X, X) P ˜ In particular, a series ˜ generated by X. k∈J fk converges in the σ(X, X) topology if P ˜ k∈J hfk , hi 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 hfk , hi are scalars, this occurs if and only if ˜ ∀ h ∈ X,
X
|hfk , hi| < ∞.
k∈J
Remark 3.1.4.
a. If X1 and X2 are two Banach function spaces such that X1 ⊂
X2 then X˜2 ⊂ X˜1 . Indeed, let f ∈ X˜2 and g ∈ X1 with kgkX1 = 1, then by the definition of the K¨othe dual, we have that f g¯ ∈ L1 , which implies that f ∈ X˜1 , and, moreover, Z |hf, gi| =
Rd
f (x) g(x) dx
≤ kf kX˜2 kgkX1 ≤ kf kX˜2 . Thus, kf kX˜1 ≤ kf kX˜2 .
(8)
˜˜ = X, cf. [5, Theorem 2.7]. b. For every Banach function space X we have X The dual and K¨othe dual of the amalgam spaces are given in the next lemma. Lemma 3.1.5. Let ν be an ω-moderate weight. 0
0
a. For 1 ≤ p, q < ∞, the dual space of W (Lp , Lqν ) is W (Lp , Lq1/ν ). 0
0
b. For 1 ≤ p, q ≤ ∞, the K¨othe dual of W (Lp , Lqν ) is W (Lp , Lq1/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
0
Case I: 1 ≤ p < ∞ and q = ∞. Let f ∈ W (Lp , L11/ν ), and g ∈ W (Lp , L∞ ν ) with kgkW (Lp ,L∞ = 1. Then we have ν ) Z |hf, gi| = f (x) g(x) dx Rd Z |f (x)| |g(x)| dx ≤ Rd
=
XZ k∈Zd
≤
X
|f (x)| |g(x)| dx αk+Qα
kf · Tαk χQα kp0 kg · Tαk χQα kp
k∈Zd
=
X
kf · Tαk χQα kp0
k∈Zd
1 ν(αk) kg · Tαk χQα kp ν(αk)
≤ sup kg · Tαk χQα kp ν(αk) k∈Zd
X
kf · Tαk χQα kp0
k∈Zd
= kgkW (Lp ,L∞ kf kW (Lp0 ,L1 ν )
1/ν
= kf kW (Lp0 ,L1
1/ν
1 ν(αk)
)
).
Thus, kf kW˜ (Lp ,L∞ ≤ kf kW (Lp0 ,L1 ν )
1/ν
),
0 ˜ (Lp , L∞ ). From part a and (8), as well as and consequently, W (Lp , L11/ν ) ⊂ W ν
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 fˆ defined by Z fˆ(n) = F f (n) = β
f (t) e−2πiβn·t dt,
d
n ∈ Zd .
Q1/β
For 1 ≤ p ≤ ∞, let F Lp (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 ∈ F Lp (Q1/β ) then there exists a unique function m ∈ Lp (Q1/β ) such that m(n) ˆ = cn for every n, and the norm on F Lp (Q1/β ) is defined by kckFLp (Q1/β ) = kmkp,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 Sν˜p,q = `qν˜ (F Lp (Q1/β )) will denote the space of all F Lp (Q1/β )-valued sequences which are `qν˜ -summable. That is, a doubly-indexed sequence c = (ckn )k,n∈Zd lies in Sν˜p,q if for each k ∈ Zd there exists mk ∈ Lp (Q1/β ) such that m ˆ k (n) = ckn , and such that kck
Sνp,q ˜
=
X
k, n ∈ Zd , 1/q
kmk kqp,Q1/β
ν˜(k)
q
< ∞,
k∈Zd
with the usual change if q = ∞. When 1 < p < ∞, we can write mk as a Fourier series mk (x) =
X n∈Zd
28
ckn e2πiβn·x ,
(10)
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 Sν˜p,q as kckSνp,q = ˜
X Z k∈Zd
Q1/β
p q/p 1/q X 2πiβn·x q dx ckn e ν˜(k) . n∈Zd
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, kgk ≤ kf k. Note that for p = 2, we have via the Plancherel theorem that Sν˜2,q = `2,q ν˜ , thus is a solid space. However, for general p 6= 2, Sν˜p,q 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 translates of Q1/β required to cover any αZd -translate of Qα , i.e., Kαβ = max # ` ∈ Zd : ( β` + Q1/β ) ∩ (αk + Qα ) > 0 . k∈Zd
29
1 d Z β
Then given 1 ≤ p ≤ ∞, we have for any 1/β-periodic function m ∈ Lp (Q1/β ) and any k ∈ Zd that 1/p
kmkp,αk+Qα ≤ Kαβ kmkp,Q1/β , 1/∞
where Kαβ = 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 : |( βl + Q1/β ) ∩ (αk + Qα )| > 0}. Then we have: Z kmkpp,αk+Qα
|m(x)|p dx
= αk+Qα
=
XZ l∈Zd
= ≤ =
l +Q1/β β
XZ
l∈Ak
l +Q1/β β
l∈Ak
l +Q1/β β
XZ
XZ
l∈Ak
|m(x)|p Tαk χQα (x) dx |m(x)|p Tαk χQα (x) dx |m(x)|p dx
|m(x)|p dx Q1/β
Z
|m(x)|p dx
≤ Kα,β Q1/β
= Kα,β kmkpp,Q1/β . If p = ∞, then it is easily seen that kmk∞,αk+Qα ≤ kmk∞,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 X
kGn k∞ ω( nβ ) ≤ C kgkW (L∞ ,L1ω ) kγkW (L∞ ,L1ω ) .
n∈Zd
Proof. It follows from the fact that ω is ω-moderate that kf ωkW (L∞,L1 ) is an equivalent norm for W (L∞ , L1ω ). In particular, we have gω, γω ∈ W (L∞ , L1 ), so by [41, Lemma 6.3.1], X
˜ n k∞ ≤ kG
1 α
+1
d
(2β + 1)d kgωkW (L∞,L1 ) kγωkW (L∞ ,L1 ) ,
n∈Zd
˜ n is the analogue of Gn with g replaced by |g|ω and γ replaced by |γ|ω. where G Hence, X
kGn k∞ ω( nβ ) =
n∈Zd
X n∈Zd
X ess sup g(x − d x∈R
k∈Zd
ω (x − αk) − (x − ≤
X n∈Zd
ess sup x∈Rd
X
n β
|g(x −
n β
− αk) γ(x − αk)×
− αk) n β
− αk)| ω(x −
n β
− αk)×
k∈Zd
|γ(x − αk)| ω(x − αk) =
X
˜ n k∞ kG
n∈Zd
≤ C kgkW (L∞,L1ω ) kγkW (L∞ ,L1ω ) .
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 kTα` f kW (Lp ,Lqν ) ≤ Cν ω(α`) kf kW (Lp,Lqν ) . 31
Proof. Let f ∈ W (Lp , Lqν ), and l ∈ Zd . Then using the fact that ν is ω-moderate, we obtain kTαl f kW (Lp ,Lqν ) =
X
1/q kTαl f ·
Tαk χQα kqp
ν(αk)
q
k∈Zd
=
X
1/q kf ·
Tα(k−l) χQα kqp
kf ·
Tαk χQα kqp
ν(αk)
q
k∈Zd
=
X
1/q
k∈Zd
≤ C ω(αl)
X
ν(α(k + l))
q
1/q kf ·
Tαk χQα kqp
ν(αk)
q
k∈Zd
= C ω(αl) kf kW (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∞ , L1ω ). Then the following statement is true. Given c ∈ Sν˜p,q , let mk ∈ Lp (Qα ) be the unique functions satisfying m ˆ k (n) = ckn for all k, n ∈ Zd . Then the series Rg c =
X
mk · Tαk g
(11)
k∈Zd
converges unconditionally in W (Lp , Lqν ) (unconditionally in the 0
0
σ(W (Lp , Lqν ), W (Lp , Lq1/ν )) topology if p = ∞ or q = ∞), and Rg is a bounded mapping from Sν˜p,q into W (Lp , Lqν ). Proof. We divide the proof into cases. First, we consider the case 1 ≤ p, q < ∞. We are given c ∈ Sν˜p,q , and we must prove that the series (11) defining Rg c converges unconditionally in the norm of W (Lp , Lqν ), and that Rg so defined is a bounded
32
mapping from Sν˜p,q into W (Lp , Lqν ) . To show the convergence we will make use of Lemma 3.1.3. Fix ε > 0. Then, by definition of the norm in Sν˜p,q , we have that X
kmk kqp,Q1/β ν˜(k)q < ∞.
Hence there exists a finite set F0 such that X
∀ finite F ⊃ F0 ,
k ∈F /
kmk kqp,Q1/β ν˜(k)q < εq .
(12)
Recall that 1/ν is an ω-moderate weight, and let Kαβ be the constant appearing in 0
0
Lemma 3.2.1. Fix any h ∈ W (Lp , Lq1/ν ). Then Z X
X mk · Tαk g, h ≤ |mk (x) Tαk g(x) h(x)| dx k ∈F /
Rd
k ∈F /
=
XXZ k ∈F / n∈Zd
≤
XX k ∈F /
|mk (x) Tαk g(x) h(x)| Tαn+αk χQα (x) dx Qα
kTαk g · Tαn+αk χQα k∞ kmk kp,αn+αk+Qα ×
n∈Zd
kh · Tαn+αk χQα kp0 ≤
X
ν(αk) ν(αn + αk − αn)
kg · Tαn χQα k∞ ×
n∈Zd
X
1/p
Kαβ kmk kp,Q1/β kh · Tαn+αk χQα kp0
k ∈F / 1/p
≤ Cν Kαβ
X
Cν ν(αk) ω(αn) ν(αn + αk)
kg · Tαn χQα k∞ ω(αn)×
n∈Zd
X k ∈F /
1/q kmk kqp,Q1/β
X
ν(αk)
q
0
kh · Tαn+αk χQα kqp0
k∈Zd
× 1 ν(αn + αk)q0
1/q0 .
Combining (12) and (13), we have that X
mk · Tαk g, h ≤ εCν K 1/p kgkW (L∞ ,L1 ) khk 0 . αβ ω W (Lp0 ,Lq ) 1/ν
k ∈F /
33
(13)
Therefore, taking the supremum over all h of unit norm and appealing to Lemma 3.1.3, P we see that Rg c = mk · Tαk g converges unconditionally. Further, replacing F by Zd in the calculation in (13) yields |hRg c, hi| ≤
X
mk · Tαk g, h k∈Zd 1/p
khkW (Lp0 ,Lq0 ) . ≤ Cν Kαβ kgkW (L∞,L1ω ) kckSνp,q ˜ 1/ν
0
(14)
0
Since W (Lp , Lq1/ν ) is the dual space of W (Lp , Lqν ), taking the suprema over all h of unit norm in (14) shows that kRg ckW (Lp ,Lqν ) = sup |hRg c, hi| : khkW (Lp0 ,Lq0
1/ν
)
=1
1/p
≤ Cν Kαβ kgkW (L∞,L1ω ) kckSνp,q , ˜
(15)
so Rg is bounded. This completes the proof for the case 1 ≤ p, q < ∞. 0
0
When p = ∞ or q = ∞, we make use of the fact that W (Lp , Lq1/ν ) is the K¨othe dual of W (Lp , Lqν ). The fact that the series defining Rg c converges in the weak topology is given by the same calculations as in (13), (14), and the fact that the K¨othe dual is a norm-fundamental subspace of the dual space means that we can again estimate kRg ckSνp,q 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 Rg c to be written as the iterated sum XX 2πiβn·x Tαk g(x), Rg c(x) = ckn e k∈Zd
(16)
n∈Zd
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 Rg c 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∞ , L1ω ). Then the analysis operator defined by Cg f = hf, Mβn Tαk gi k,n∈Zd is a bounded mapping from W (Lp , Lqν ) into Sν˜p,q , Moreover, there exist unique functions mk ∈ Lp (Q1/β ) which satisfy m ˆ k (n) = Cg f (k, n) for all k, n ∈ Zd , and these are given explicitly by mk (x) = β −d
X
f · Tαk g¯ (x − nβ )
n∈Zd
= β −d
X
T nβ f · Tαk+ nβ g¯ (x).
(17)
n∈Zd
The series on the right side of (17) converges unconditionally in Lp (Q1/β ) (unconditionally 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 , Lqν ), 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 · Tαk g, 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 0
a useful estimate, fix any 1/β-periodic function h ∈ Lp (Q1/β ). Then for each fixed k, we have Z
X
Q1/β
=
Rd
|f (x) Tαk g(x) h(x)| dx
XZ n∈Zd
≤
−
n ) h(x) dx β
n∈Zd
Z ≤
f (x −
n ) Tαk g(x β
X
|f (x) Tαk g(x) h(x)| Tαk+αn χQα (x) dx Qα
kTαk g · Tαk+αn χQα k∞ kf · Tαk+αn χQα kp ×
n∈Zd
35
khkp0,αk+αn+Qα ≤
X
ν(αk + αn − αn) ν(αk) 1/p0
kg · Tαn χQα k∞ kf · Tαk+αn χQα kp Kαβ ×
n∈Zd
Cν ν(αk + αn) ω(αn) ν(αk) 1 X kg · Tαn χQα k∞ ω(αn) × ν(αk) d
khkp0,Q1/β 1/p0
= Cν Kαβ khkp0 ,Q1/β
n∈Z
kf · Tαk+αn χQα kp ν(αk + αn).
(18)
This yields the desired convergence, and taking the suprema in (18) over h with unit norm implies the estimate 1/p0
kmk kp,Q1/β ≤ β −d Cν Kαβ
1 ν(αk)
P n∈Zd
kg · Tαn χQα k∞ ω(αn) ×
kf · Tαk+αn χQα kp ν(αk + αn).
(19) 0
Second, we show that m ˆ k (n) has the correct form. Since e2πiβn·x ∈ Lp (Q1/β ), we have by the weak convergence of the series defining mk that
m ˆ k (n) = β d mk , e2πiβn·x XZ
= T ` f · Tαk+ ` g¯, e2πiβn·x `∈Zd
=
XZ
`∈Zd
Z
= Rd
β
Q1/β
Q1/β
β
f (x − β` ) Tαk g¯(x − β` ) e−2πiβn·(x−`/β) dx
f · Tαk g¯ (x) e−2πiβn·x dx
= f, Mβn Tαk g = Cg f (k, n). Finally, we must show that Cg is a bounded mapping of W (Lp , Lqν ) into Sν˜p,q . Given f ∈ W (Lp , Lqν ), to show that Cg f ∈ Sν˜p,q we must show that the sequence r given by r(k) = kmk kp,Q1/β , 36
k ∈ Zd ,
0
lies in `qν˜ . To do this, fix any sequence a ∈ `q1/˜ν . Then, using (19), we have |hr, ai| ≤
X
kmk kp,Q1/β |a(k)|
k∈Zd 1/p0
≤ β −d Cν Kαβ
X
kg · Tαn χQα k∞ ω(αn) ×
n∈Zd
X
kf · Tαk+αn χQα kp ν(αk + αn) |a(k)|
k∈Zd 1/p0
≤ β −d Cν Kαβ X
X
kg · Tαn χQα k∞ ω(αn) ×
n∈Zd
kf ·
1/q
Tαk+αn χQα kqp
k∈Zd
X
1 ν(αk)
1 |a(k)| ν(αk)q0 d q0
ν(αk + αn)
q
×
1/q0
k∈Z
1/p0
≤ β −d Cν Kαβ kgkW (L∞ ,L1ω ) kf kW (Lp ,Lqν ) kak`q0 .
(20)
1/˜ ν
0
Since `q1/˜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 1/p0
kCg f kSνp,q = krk`pν˜ ≤ β −d Cν Kαβ kgkW (L∞ ,L1ω ) kf kW (Lp ,Lqν ) . ˜ Hence Cg is a bounded mapping of W (Lp , Lqν ) into Sν˜p,q . Remark 3.2.7. For the case 1 < p, q < ∞, the boundedness of Cg could also be shown 0
0
0
0
p ,q q p by proving that Cg : W (Lp , Lqν ) → Sν˜p,q is the adjoint of Rg : S1/˜ ν → W (L , L1/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 amalgam 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∞ , L1ω ). Then the Walnut representation X
Rγ Cg f = β −d
Gn · T n f
n∈Zd
(21)
β
holds for f ∈ W (Lp , Lqν ), with the series on the right of (21) converging absolutely in W (Lp , Lqν ), and where Gn (x) =
X k∈Zd
=
X
g(x −
n β
− αk) γ(x − αk)
Tαk+ n g¯ · Tαk γ (x).
(22)
β
k∈Zd
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, kT nβ f kW (Lp ,Lqν ) ≤ Cν ω( nβ ) kf kW (Lp ,Lqν ) . Therefore, using the autocorrelation functions Gn defined in (22), we have for f ∈ W (Lp , Lqν ) that X
kGn · T nβ f kW (Lp ,Lqν ) ≤
n∈Zd
X
kGn k∞ kT nβ f kW (Lp ,Lqν )
n∈Zd
≤ Cν kf kW (Lp ,Lqν )
X
kGn k∞ ω( nβ )
n∈Zd
≤ Cν kf kW (Lp ,Lqν ) kgkW (L∞ ,L1ω ) kγkW (L∞ ,L1ω ) , the last inequality following from Lemma 3.2.2. Hence the series
P
Gn ·T nβ f converges
absolutely in W (Lp , Lqν ). Now fix f ∈ W (Lp , Lqν ). Then Cg f ∈ Sν˜p,q by Theorem 3.2.6. Letting mk be P ˆ k (n). Further, Rγ Cg f = mk · Tαk γ, this defined by (17), we have Cg f (k, n) = m 38
series converging unconditionally if p, q < ∞, or unconditionally in the weak topology 0
0
otherwise. In any case, for h ∈ W (Lp , Lq1/˜ν ) we have
X
Rγ Cg f, h = mk · Tαk γ, h k∈Zd
=
XZ
k∈Zd
=β
−d
Rd
¯ mk (x) Tαk γ(x) h(x) dx
XZ Rd
k∈Zd
= β −d = β −d = β −d
X
XZ
n∈Zd
Rd
n∈Zd
Rd
XZ
¯ T nβ f (x) Tαk+ nβ g¯(x) Tαk γ(x) h(x) dx
n∈Zd
X
¯ T nβ f (x) Tαk+ nβ g¯(x) Tαk γ(x) h(x) dx
k∈Zd
¯ T nβ f (x) Gn (x) h(x) dx.
X
Gn · T nβ f, h ,
n∈Zd
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∞ , L1ω ) 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. Rγ Cg = I on W (Lp , Lqν ). b. We have the norm equivalence kf kW (Lp ,Lqν ) kCg f kSνp,q . ˜ p,q p q c. A function f ∈ W (L1 , L∞ 1/ω ) belongs to W (L , Lν ) if and only if Cg f ∈ Sν˜ .
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 ) → Sν˜p,q and Rg , Rγ : Sν˜p,q → 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 Rγ Cg = 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 Rγ Cg = I holds on L2 implies by [41, Thm. 7.3.1] that β −d G0 = 1 a.e.
Gn = 0 a.e. for n 6= 0.
and
Consequently, using the Walnut representation (21) of Rγ Cg on the space W (Lp , Lqν ), we have for f ∈ W (Lp , Lqν ) that Rγ Cg f = β −d
X
Gn · T nβ f = f.
n∈Zd
Hence Rγ Cg = 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 kf kW (Lp ,Lqν ) = kRγ Cg f kW (Lp ,Lqν ) ≤ kRγ k kCg f kSνp,q ˜ ≤ kRγ k kCg k kf kW (Lp ,Lqν ) . 40
Consequently, kCg f kSνp,q kf kW (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 Cg f ∈ Sν˜p,q . The forward direction, that if f ∈ W (Lp , Lqν ) then Cg f ∈ Sν˜p,q , is simply Theorem 3.2.6. For the reverse direction, assume that Cg f ∈ Sν˜p,q . Then by Theorem 3.2.6, the function f˜ = Rγ (Cg f ) lies in W (Lp , Lqν ). However, the factorization Rγ Cg = I holds on every amalgam space, including W (L1 , L∞ 1/ω ) in particular, so we also know that f = Rγ Cg f . 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 = Rγ Cg . The specific form of these expansions P is that given f , there exist functions mk such that f = Rγ Cg f = mk · Tαk g. When 1 < p < ∞, the functions mk can be realized as Fourier series, leading to an expansion of the form f (x) = Rγ Cg f (x) =
X X k∈Zd
2πiβn·x
hf, Mβn Tαk γi e
Tαk g(x).
(23)
n∈Zd
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 characterized 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 0 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 Mω1 , then the dual window γ will lie in Mω1 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 ∈ Sν˜p,q , then the partial sums SK,N c =
X X
ckn Mβn Tαk g,
K, N > 0,
|k|≤K |n|≤N
converge to Rg c in the norm of W (Lp , Lqν ), i.e., for each ε > 0 there exist K0 , N0 > 0 such that ∀ K ≥ K0 ,
∀ N ≥ N0 , 42
kRg c − SK,N ckW (Lp ,Lqν ) < ε.
b. If f ∈ W (Lp , Lqν ), then the partial sums of the Gabor expansion of f , SK,N (Cg f ) =
X X
hf, Mβn Tαk gi Mβn Tαk γ,
|k|≤K |n|≤N
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 ∈ Sν˜p,q , and let mk be defined by (17). For N > 0, write SN mk (x) =
X
ckn e2πiβn·x
|n|≤N
for the partial sums of the Fourier series of mk . The exponentials {e2πiβn·x }n∈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 lim kmk − SN mk kp,Q1/β = 0
(24)
sup kSN mk kp,Q1/β ≤ C1 kmk kp,Q1/β .
(25)
N →∞
and N >0
Since c ∈ Sν˜p,q , given ε > 0, we can find K0 > 0 such that ∀ K ≥ K0 ,
X |k|≥K
1/q kmk kqp,Q1/β
ν˜(k)
q
< ε.
(26)
Because of (24) and the fact that K0 is finite, we can find an N0 > 0 such that ∀ N ≥ N0 ,
sup kmk − SN mk kp,Q1/β ν˜(k)
K0
≤ kRg kε.
(29)
For the second term, define skn = ckn for |k| ≤ K0 and |n| ≤ N, and skn = 0 otherwise. Then SK0 ,N c = Rg s, so using (27), we have kSK0 ,∞ c − SK0 ,N ckW (Lp ,Lqν ) ≤ kRg k kr − skSνp,q ˜ 1/q X q q = kRg k kmk − SN mk kp,Q1/β ν˜(k) |k|≤K0
≤ kRg kε.
(30)
For the third term, define tkn = ckn for |k| ≤ K and |n| ≤ N, and tkn = 0 otherwise. Then SK0 ,N c = Rg t, so using (25) and (26), we have kSK0 ,N c − SK,N ckW (Lp ,Lqν ) ≤ kRg k ks − tkSνp,q ˜ 1/q X q q = kRg k kSN mk kp,Q1/β ν˜(k) K0 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
ei arg g χJ . Using the equivalent norm for W (Lp , L∞ ) obtained by
replacing α in (7) by 1/β, we have that kf kW (Lp ,L∞ ) ≤ 1. By hypothesis, Cg f ∈ ˆ k (n) = Cg f (k, n). Since S p,∞ , so there exist 1/β-periodic functions mk such that m 45
f · Tαk g¯ ∈ L1 , it is easy to see that mk is given by (17). In particular, considering k = 0 we have km0 kpp,Q1/β
=β
−pd
Z Q1/β
β −pd = |J|
p X n n dx f (x − ) g ¯ (x − ) β β n∈Zd
Z ` +Q1/β β
χJ (x) |g(x)|p dx
≥ β −pdD p . Hence D ≤ β d sup kmk kp,Q1/β k∈Zd
= β d kCg f kS p,∞ ≤ β d kCg k kf kW (Lp ,L∞ ) ≤ β d kCg k. 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)| ≥ 12 kgk∞, βn +Q1/β .
Then set Jn0 = Jn if |Jn | ≤ ε, otherwise let Jn0 be a subset of Jn of measure ε. Let Nε = sup{N ∈ N : |Jn0 | ≥
ε 2
for all |n| ≤ N}.
Note that Nε → ∞ as ε → 0 (and may even be ∞ for some ε). Define f = P ei arg g |n|≤Nε χJn0 , and note that kf kW (Lp ,L∞ ) ≤ ε1/p . Therefore Cg f ∈ S p,∞ , and
46
letting mk be defined by (17), we have km0 kpp,Q1/β
=β
−pd
X Z n +Q1/β β
|n|≤Nε
≥β
−pd
|g(x)|p χJn0 (x) dx
X kg · T nβ χQ1/β k∞ p 2
|n|≤Nε
X
≥ β −pd 2−p−1ε
|n|≤Nε
|Jn0 |
kg · T nβ χQ1/β kp∞ .
Hence X |n|≤Nε
kg · T nβ χQ1/β kp∞ ≤
β pd 2p+1 sup kmk kpp,Q1/β ε k∈Zd
=
β pd2p+1 kCg f kpS p,∞ ε
≤
β pd 2p+1 kCg kp kf kpW (Lp ,L∞ ) ε
≤ β pd 2p+1 kCg kp . Since Nε → ∞ as ε → 0, this implies that g ∈ W (L∞ , Lp ). 0
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. 0
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 0
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
P
k,n hf, Mβn Tαk gi Mβn Tαk γ
converges un-
conditionally in Lploc , b. the frame operators Sg = Rg Cg and Sγ = Rγ Cγ 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→
P
k,n hf, Mβn Tαk giMβn Tαk γ
is a bounded mapping from W (Lp , L∞ ) into LpLoc , and the series converges uncondiP tionally in Lploc . We first show that f 7→ Sk f = n∈Zd hf, Mβn Tαk giMβn Tαk γ is uniformly bounded on W (Lp , L∞ ), with respect to k ∈ Zd . Fix k ∈ Zd and any f ∈ W (Lp , L∞ ). Then the sequence X
hf, Mβn Tαk giMβn Tαk γ
|n|≤N
converges in Lp (F ) as N → ∞, hence is a bounded sequence in Lp (F ). Moreover, because the operators SN,k (·) : =
X
h·, Mβn Tαk giMβn Tαk γ,
|n|≤N
are bounded, we conclude from the Uniform Boundedness Principle that for each fixed k the sequence of operators {SN,k }N ≥0 , are uniformly bounded in N, i.e., for each k 48
there is a constant Ck > 0 such that kSN,k kW (Lp ,L∞ )→Lp (F ) ≤ Ck ∀N. Next, given > 0 and f ∈ W (Lp , L∞ ) with kf kW (Lp ,L∞ ) = 1, there exists N0 > 0 P such that k |n|≥N0 hf, Mβn Tαk giMβn Tαk γkp,F ≤ . Thus, kSk (f )kp,F
X
≤ hf, M T giM T γ βn αk βn αk
+
p,F
|n|≥N0
X
hf, M T giM T γ βn αk βn αk
|n|≤N0
p,F
≤ Ck + . Letting → 0, we conclude that kSk (f )kp,F ≤ Ck . It then follows that Sk is a bounded P 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 kSk kW (Lp ,L∞ )→Lp (F ) ≤ C(F ) for all k ∈ Zd . Moreover, it is easy to see that kSk kW (Lp ,L∞ )→Lp (F +α) = kSk−1 kW (Lp ,L∞ )→Lp (F ) ≤ C(F ). Thus if we let F = Qα then kSk (f )kp,F ≤ C(F )kf kW (Lp,L∞ ) . On the other hand, taking F and δ > 0 as in the definition of persistency, for any k ∈ Zd we have
X
hf, Mβn Tαk giMβn Tαk γ
n∈Zd
p,F +αk
= kSk (f )kp,F +αk
X
2πiβ(·+αk)
= γ(·) hf, M T gie βn αk
p,F
n∈Zd
≥ δβ d kmk kp,Q1/β . Therefore, kmk kp,Q1/β ≤ δ −1 β d kSk (f )kp,F +αk ≤ Cδ −1 β d kf kW (Lp ,L∞ ) 49
∀k ∈ Zd .
Hence, kCg f kS p,∞ = sup kmk kp,Q1/β ≤ C δ −1 β d kf kW (Lp ,L∞ ) . k
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,q norm of the ν 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, Gr¨ochenig in [40], Galperin and Gr¨ochenig in [32], and Hogan and Lakey in [47] derived sufficient conditions for membership in the modulation space M 1 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 M 1 in the special case of dimension d = 1.
51
Other embeddings results between modulation spaces and Besov spaces were arrived at independently and by different techniques by P. Gr¨obner [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, ψ(x) = 0,
if |x| ≤ 1, if |x| ≥ 3/2.
φ0 (x) = ψ(x), φ1 (x) = ψ( x2 ) − ψ(x), φk (x) = φ1 (2−k+1x), k = 2, 3, ....
Define
Then {φk }∞ k=0 is a partition of unity, and satisfies supp(φk ) ⊂ {x ∈ Rd : 2k−1 ≤ |x| ≤ 3 · 2k−1 }. 0
Definition 4.1.1. Let s ∈ R, 1 ≤ q ≤ ∞, and f ∈ S . s (i) For 1 ≤ p < ∞ the Triebel-Lizorkin space Fp,q is defined by:
f∈
s Fp,q
s ⇐⇒ kf kFp,q =
Z X ∞ Rd
p/q 2
skq
|F
−1
q ˆ (φk f)(x)|
1/p dx < ∞.
(32)
k=0
s is defined by: (ii) For 1 ≤ p ≤ ∞, the Besov space Bp,q
f∈
s Bp,q
s ⇐⇒ kf kBp,q =
X ∞
Z 2
skq Rd
k=0
|F
−1
q/p < ∞. (φk fˆ)(x)| dx p
(33)
s is defined by: (iii) For p = ∞, the Triebel-Lizorkin space F∞,q
f∈
s F∞,q
⇐⇒
∃{fk }∞ k=0 ,
f=
∞ X
F
−1
(φk fˆk ),
k=0
sup Rd
X ∞
1/q 2
ksq
|fk (x)|
q
< ∞,
k=0
(34) 52
with norm
s kf kF∞,q = inf sup
X ∞
Rd
2
ksq
1/q , |fk (x)| q
k=0
the infimum being taken over all admissible representations. (iv) For 1 ≤ p ≤ ∞, the fractional Sobolev space Hps is defined by: Z f∈
Hps
⇐⇒ kf kHps =
Rd
|F
−1
2 s/2
((1 + |x| )
1/p p ˆ < ∞. f)(x)| dx
(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]. s s a. If 1 ≤ p = q ≤ ∞ and s ∈ R, then Bp,p = Fp,p , this follows from the definition. 0 = Lp . b. If 1 < p < ∞, then Fp,2 s = Hps . c. If 1 < p < ∞ and s ∈ R, then Fp,2 s ⊂ Lp , additionally if Moreover, for 1 ≤ p, q ≤ ∞ and s > 0 we have that Bp,q s p < ∞ we also have Fp,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 s s and 4.1.4 give equivalent definitions of Fp,q and Bp,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
0
s Proposition 4.1.3. Let s ∈ R, 1 < p ≤ ∞, and 1 < q ≤ ∞. If f ∈ S then f ∈ Fp,q
if and only if ∃ {fk }∞ k=0
⊂L ,f = p
∞ X
F
−1
Z X ∞
(φk fˆk ),
Rd
k=0
Furthermore, inf
Z X ∞ Rd
is an equivalent norm on
s Fp,q ,
p/q 2
ksq
2
1/p dx < ∞. (36)
k=0
p/q ksq
|fk (x)|
q
|fk (x)|
q
1/p dx
k=0
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. 0
s Proposition 4.1.4. Let 1 ≤ p, q ≤ ∞ and s > 0. If f ∈ S then f ∈ Bp,q if and only
if ∃
{bk }∞ =0
⊂L , p
f=
∞ X
bk ,
X ∞
k=0
Furthermore, inf
X ∞
Z 2
kqs Rd
k=0
Z 2
kqs
(37)
q/p 1/q |bk (x)| dx p
Rd
k=0
q/p 1/q |bk (x)| dx < ∞. p
s is an equivalent norm on Bp,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 interpolation 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 ,
1 p
=
1−θ p0
+
θ , p1
and
1 q
Bps00,q0 , Bps11,q1
=
1−θ q0
[θ]
+
θ q1
s = Bp,q .
then (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 in M 1 for s > d. Because M 1 is invariant under Fourier transforms we then conclude that the Bessel potential m−s itself is in M 1 for s > d. 2
Proposition 4.1.6. Define g(x) = e−πx and ga (x) = e− 1 1 Gs (x) = (4π)s/2 Γ(s/2)
Z
∞
t
−d+s 2
e−(
πx2 a
. Let
πx2 t + 4π ) t
0
dt t
for s, a > 0 and x ∈ Rd , and where Γ refers to the Gamma-function. Then the following hold. (a) Vga g(x, ω) =
d/2 2πi x·ω a e a+1 a+1
(b) Vg m ˇ −s = (2π)
d/2
ga+1 (x) g a+1 (ω). a
1
Vg (D 1 Gs ) ∈ L for s > d, where Da is the unitary dilation 2π
operator defined by Da g(x) = |a|−d/2 g(x/a). (c) m−s ∈ M 1 for s > d. Proof. (a) First note that from Lemma 2.2.2 the operator Da f (t) = |a|−d/2 f (t/a) d d ˆ where a > 0 is unitary on L2 , and D a f = D1/a f . Now for x, ω ∈ R we have:
Z Vga g(x, ω) =
t2
2
e−π a e−2πit·ω e−π(t−x) dt
Rd
Z
−π (a+1)t2 −2at·x+ax2 a
e
=
e−2πit·ω
Rd
Z =
a a −π a+1 (t− a+1 x)2 + a+1 x2 a
e Rd
π − a+1 x2
Z
e−π
=e
a+1 a (t− a+1 x)2 a
Rd 2
∧ a a ) (ω) = e− a+1 x (T a+1 x g a+1 π
55
e−2πt·ω dt e−2πt·ω dt
a a (ω) = ga+1 (x) M− a+1 x gd a+1
=
a d/2 a ga+1 (x) M− a+1 x g a+1 (ω) a a+1
=
a d/2 −2πi a x·ω a+1 e ga+1 (x) g a+1 (ω), a a+1
where we have used the fact that the Fourier transform of the Gaussian g(x) = e−πx is itself, i.e., gˆ = g. (b) For s > 0 it is shown in [55, Proposition 3.1.2] that ˆ s (ω) = (1 + 4π 2 |x|2 )−s/2 . G Notice for future references that Gs ∈ L1 , see [55, Proposition 3.1.2]. Thus, m−s (x) = (1 + |x|2 )−s/2 ˆ s (x/2π) =G ˆ s (x) = (2π)d/2 D2π G = (2π)d/2 D\ 1/2π Gs (x). Consequently, we have that m ˇ −s (x) = (2π)d/2 (D 1 Gs )(x). Therefore: 2π
Vg m ˇ −s (x, ω) = (2π)d/2 Vg (D 1 Gs )(x, ω) 2π Z = (2π)d/2 D 1 Gs (t) e−2πit·ω g(t − x) dt, Rd
2π
= (2π)d/2 D 1 Gs , Mω Tx g 2π
= (2π)d/2 Gs , D2π Mω Tx g
ω T = Gs , M 2π 2πx g2π Z Z ∞ 2 s−d u 1 −( πtu + 4π ) −2πit·ω/2π 2 e = u e × s/2 (4π) Γ(s/2) Rd 0 du 2 dt e−π(t−2πx) /2π u 56
2
1 = s/2 (4π) Γ(s/2)
Z
∞
−u/4π
e
u
s−d −1 2
Z
0
e−
πt2 u
Rd
×
(t−2πx)2
=
1 (4π)s/2 Γ(s/2)
e−2πit·ω/2π e−π 2π dt du Z ∞ Z 2 −u/4π 2s −1 e u e−πt × 0
Rd
−2πit·ω
e 1 = s/2 (4π) Γ(s/2)
Z
∞
−u/4π
e
√
u 2π
u
−πu
e
s −1 2
0
√ )2 (t− 2πx u 2π
dt du
√ 2πx uω ) du. Vg 2π g( √ , u u 2π
The last equality follows from some changes of variable and by using part a. Therefore, we have that: π (d−s)/2 Vg m ˇ −s (x, ω) = s−d/2 2 Γ(s/2)
Z
∞
0
4π 2 x2
2
e−u/4π e−π u+2π e−π 2π(2π+u) ω × u
1 du, (2π + u)d/2
u 2 −1 s
where we have used Lemma 2.2.2 to obtain the last equation. By changing the R 2 variables and using the fact that Rd e−πx dx = 1, we have: ZZ kVg m ˇ −s kL1 =
R2d
|Vg m ˇ −s (x, ω)| dx dω
π (d−s)/2 ≤ s−d/2 2
Z
∞ 0
e−u/4π u 2 −1 s
Z
2 2
x −π 4π 2π+u
Z
1 × (2π + u)d/2
e π (d−s)/2 = s−d/2 2
Z
Rd ∞
u −π 2π(2π+u) ω2
e
dω dx du
Rd
e−u/4π u
−d+s −1 2
0
and the last expression is finite if s > d. (c) Follows from (b) and the comments above.
57
(u + 2π)d/2 du,
(39)
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. 0
s Proposition 4.2.1. (a) Bp,q ⊂ Lp ⊂ M p,p when s > 0, 1 ≤ p ≤ 2, and 1 ≤ q ≤ ∞. s (b) Bp,q ⊂ Lp ⊂ M p 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 , Vg f (x, ω) = f\ · Tx g(ω). Note that the f · Tx g ∈ Lp since f ∈ Lp and g ∈ S. Moreover, since 1 ≤ p ≤ 2 ≤ p0 , we have that p0 /p ≥ 1, and thus using Hausdorff-Young’s inequality and Minkowski’s inequality (for integrals) we have: Z Z kVg f kpLp,p0
p0 /p p/p0 |Vg f (x, ω)| dx dω p
= Rd
Rd
Z Z ≤
Rd
Z Z
p0
Rd
Z ≤ Z
Rd
Z Z
Rd
Rd
= Rd
Rd
|Vg f (x, ω)| dω |f\ · Tx g(ω)| dω p0
= Rd
p/p0 dx p/p0 dx
|f · Tx g(t)|p dt dx |f (t)|p |g(t − x)|p dt dx
= kf kpLp kgkpLp . Hence, kVg f kLp,p0 = kf kM p,p0 ≤ kf kLp kgkp , 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 ≤ p0 ≤ 2. Moreover, choosing g ∈ S we see using H¨older’s inequality 0
with p/p0 ≥ 1, that f · Tx g ∗ ∈ Lp for almost all x ∈ Rd . By using Hausdorff-Young’s inequality as well as Young’s inequality we have that Z Z 1/p p kVg f kLp = |Vg f (x, ω)| dx dω Z Z
R2d
1/p |f\ · Tx g(ω)| dω dx p
= R2d
Z Z ≤
Rd
Rd
Z
p/p0 |(f · Tx g)(t)| dt , dx p0
Z
= Rd
Z
1/p p/p0 |f (t)| |g (x − t)| dt dx p0
Rd
p0
= Rd
∗ p0
p0
∗
p/p0
|f | ∗ |g | (x)
1/p dx
0 0 1/p0 = |f |p ∗ |g ∗ |p Lp/p0 . 0
0
Now notice that p ≥ p0 , and that f ∈ Lp ⇐⇒ |f |p ∈ Lp/p . Thus, applying Young’s inequality with parameter p/p0 we have that
0 0 0 kVg f kpLp ≤ |f |p ∗ |g ∗|p Lp/p0
0 0 ≤ k|f |p kLp/p0 |g ∗|p L1 0
0
= kf kpLp kgkpLp0 . Consequently, kf kM p,p0 ≤ kgkLp0 kf kLp , 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, Gr¨ochenig in [40] and Hogan and Lakey in [47] derived sufficient conditions for membership in the modulation 59
space M 1 from certain uncertainty principles related to the STFT. Precisely, it was shown that Lpa ∩ F Lqb ⊂ M 1 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 F Lqb is the image of Lqb under the Fourier transform. Somewhat more general embeddings involving weighted Lp spaces were given by Galperin and Gr¨ochenig in [32]. Another interesting example appears in [45]. There, Heil, Ramanathan and Topiwala proved, in our notation, that C s (R2d ) ⊂ M ∞,1 (R2d ) for s > 2d, while working on a timefrequency 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: Vg f (x, ω) = f · Tx g
∧
(ω)
= e−2πixω Vgˆfˆ(ω, −x) = e−2πixω F −1(fˆ · Tω gˆ)(x) = e−2πixω f ∗ (Mω g)(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 0
s ⊂ M p ,p . Bp,q
s Proof. Let f ∈ Bp,q , and use (37) to write f =
supp(ˆbk ) ⊂ {|x| ≤ 2k },
P
(41)
bk where bk ∈ Lp , and
s kf kBp,q ∼ inf
and
∞ X
2ksq kbk kqLp
1/q
,
k=0
where the infimum is over all possible such representations of f . Given g ∈ S, we have using (40) that 60
Vg f (x, ω) =
∞ X
e−2πixω F −1 (ˆbk · Tω gˆ)(x).
k=0
Hence by the Hausdorff-Young inequality, kVg f (·, ω)kLp0 ≤
∞ X
kF −1 (ˆbk · Tω gˆ)kLp0
k=0
≤
∞ X
kˆbk · Tω gˆkLp .
k=0
Therefore, by Minkowski’s inequality, kVg fkLp0 ,p ≤
∞ Z X
1/p
Rd
k=0
= kˆ g kLp
kˆbk · Tω gˆkpLp dω
∞ X
kˆbk kLp .
(42)
k=0
Now, ˆbk has compact support, which is contained in Ek = {|x| ≤ 2k }. Since 1 ≤ p ≤ 2 ≤ p0 , we have that p0 /p ≥ 1, and using H¨older’s inequality and HausdorffYoung’s inequality we obtain the following estimates kˆbk kpLp = kˆbk kpp,Ek Z = |ˆbk (ω)|p dω Ek
≤ |Ek | ≤C2
1− pp0
kˆbk kpEk ,p0
kd(1− pp0 )
kˆbk kpp0 .
where C is the volume of the ball of center 0 and radius 1. Thus, 1
p
kd( − ) kˆbk kLp ≤ C 1/p 2 p p0 kˆbk kp0
≤ C0 2
kd( p1 − p10 )
61
kbk kLp .
(43)
Substituting (43) into (42) and applying H¨older’s inequality yields 0
g kp kVg f kLp0 ,p ≤ C kˆ
∞ X
2
kd( p1 − p10 )
kbk kp
k=0 0
= C kˆ g kp ≤C
0
X ∞
∞ X
2
kd( p1 − p10 − ds )
k=0
2
k=0 s ≤ C kf kBp,q
ksq
kbk kqLp
X ∞
2ks kbk kp
1/q X ∞
2
kq 0 d( p1 − p10 − ds )
1/q0
k=0
2
kq 0 d( 1p − p10 − ds )
1/q0 .
(44)
k=0
The last term in (44) is finite if and only if s > d( 1p −
1 ). p0
The next result recovers and extends the embedding in [45], and follows by idens s tifying Fp,p with Bp,p (see Remark 4.1.2) and by using Proposition 4.1.3 as the appros s priate definition of Bp,p . However, it does not include the fact that B2,2 ⊂ L2 = M 2
for s ≥ 0. This last embedding is obtained as corollary by using complex interpolation methods. Theorem 4.2.3. Let 1 ≤ p ≤ ∞. If s >
d p0
then 0
s ⊂ M p,p . Bp,p
(45)
Proof. We divide the proof in two cases. P 2 s −1 , if f = ∞ (φk fˆ), then letting g(x) = e−πx , we Case 1: p = 1. Let f ∈ B1,1 k=0 F have Vg f (x, ω) =
∞ X
ˆ k ) ∗ F −1 (Tω gˆ)(x). F −1 (fφ
k=0
Therefore we have the following estimates kVg f (·, ω)kL1 ≤
∞ X
kF −1(φk fˆ)k1 kF −1(δˆ · Tω gˆ)k1
k=0
kVg f kL1,∞ ≤
∞ X
kF
−1
Z (φk fˆ)k1 sup ω∈Rd
k=0
62
Rd
|F −1(δˆ · Tω gˆ)(x)| dx
≤
∞ X
kF
−1
Z (φk fˆ)k1 sup
Rd
ω∈Rd
k=0
|Vg δ(x, ω)| dx.
However, Z
Z
sup ω∈Rd
Rd
|Vg δ(x, ω)| dx = sup
Z
Rd
ω∈Rd
|Mω Tx g(0)| dx =
Rd
|g(x)| dx < ∞.
Thus, ∞ X
kVg f kL1,∞ ≤ kgk1
kF −1 (φk fˆ)k1
k=0 ∞ X
≤ kgk1
2−ks kF −1 (φk fˆ)k1 ,
k=0
for all s > 0. Hence, using (33) we have that s , kf kM 1,∞ ≤ kgk1 kf kB1,1
which concludes the proof for this case. Case 2: Assume 1 < p ≤ ∞. By (36) with p = q we obtain an equivalent norm P 2 s s s −1 for Fp,p = Bp,p . Let f ∈ Bp,p , then f = ∞ (φk fˆk ). Let g(x) = e−πx . Then, k=0 F using (40), Vg f (x, ω) =
∞ X
fk ∗ F −1(φk · Tω gˆ)(x),
k=0
so by Young’s convolution inequality, kVg f (., ω)kLp ≤
∞ X
kfk kLp kF −1 (φk · Tω gˆ)kL1 .
k=0
Hence, by Minkowski’s inequality and H¨older’s inequality, kVg f kLp,p0 ≤
∞ X
Z kfk kLp
k=0
=
∞ X
Rd
2 kfk kLp 2 sk
−sk
k=0
≤
X ∞ k=0
2
ksp
kF
kfk kpLp
−1
(φk · Tω gˆ)kL1
Z Rd
kF
1/p X ∞ k=0
63
2
−1
p0
1/p0 dω
(φk · Tω gˆ)kL1
−ksp0
Z Rd
kF
−1
p0
1/p0 dω
(φk ·
0 Tω gˆ)kpL1
1/p0 dω
,
and therefore s kf kM p,p0 ≤ kf kBp,p
X ∞
2
−ksp0
Z Rd
k=0
kF
−1
(φk ·
0 Tω gˆ)kpL1
1/p0 dω
.
(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 = 2 Z Rd
kF
−1
(φk ·
k−1 2
D2k−1 g, we have:
0 Tω gˆ)kpL1
1/p0 dω
= kVg φˇk kL1,p0 .
But, using Lemma 2.2.2, we obtain:
Vg φˇk (x, ω) = φˇk , Mω Tx g
= φk , Tω M−x gˆ
= 2(k−1)/2 D2k−1 φ1 , Tω M−x gˆ
\ = 2(k−1)/2 φ1 , T21−k ω M−2k−1 x D 2k−1 g
= 2(k−1)/2 φˇ1 , M21−k ω T2k−1 x D2k−1 g
= φˇ1 , M21−k ω T2k−1 x gk = Vgk φˇ1 (2k−1x, 21−k ω). Therefore, Z Rd
kF
−1
(φk ·
0 Tω gˆ)kpL1
1/p0 dω
= kVg φˇk kL1,p0 = kVgk φˇ1 (2k−1 ·, 2−k+1·)kL1,p0 0
= 2d(−k+1) 2d(k−1)/p kVgk φˇ1 kL1,p0 ≤ C1 2d/p 2−kd/p kVgk gkL1,1 kVg φˇ1 kL1,p0 ,
(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 kVgk gkL1 = kVga gkL1 = (1 + 22k−2 )d/2 .
(48)
Combining (46), (47) and (48) yields: s kf kM p,p0 ≤ C2 kf kBp,p
X ∞
2
−kp0 (s−d+d/p)
1/p0 .
(49)
k=0
The last term in the right-hand side of (49) is finite if and only if s > d(1− p1 ) =
d . p0
Corollary 4.2.4. If 2 ≤ p ≤ ∞ and s > d(1 − 2p ) then 0
s Bp,p ⊂ M p,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 s0 s0 B2,2 = F2,2 = H2s0 ⊂ L2 = M 2
for s0 ≥ 0,
(51)
and applying Theorem 4.2.3 to p = ∞ yields: s1 B∞,∞ ⊂ M ∞,1
for s1 > d.
(52)
By [57, Remark 4, Sect. 2.4.1] we have that s0 s1 B2,2 , B∞,∞
[θ]
⊂ M 2,2 , 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 and
1 q
=
1−θ 2
− θ. Consequently,
1 p
+
1 q
=
1−θ , 2
= 1. It follows from the above referenced
proposition that M 2,2 , M ∞,1
1 p
0
[θ]
= M p,q = M p,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 ,
1 p
=
1−θ 2
= 1q , i.e., p = q. It
follows that 0 s1 B2,2 , B∞,∞
[θ]
s = Bp,p ,
and moreover, because 0 < θ = 1 − 2/p < 1, we have that p ≥ 2, hence, s = s1 θ > 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 s s of the Triebel-Lizorkin space Fp,2 into a modulation space, since Fp,2 = Hps when
1 < p < ∞ [57, 58, 53]. Theorem 4.2.5. If 1 ≤ p ≤ ∞, then Hps ⊂ M p,1
for s > d.
(53)
2
Proof. Let m−s (x) = (1 + |x|2 )−s/2 and g(x) = e−πx . Then for f ∈ Hps we have: Vg f (x, ω) = e−2πixω F −1 (fˆ · Tω gˆ)(x) = e−2πixω F −1 (fˆms · m−s Tω gˆ)(x) ˆ s ) ∗ F −1(m−s · Tω gˆ)(x) . = e−2πixω F −1 (fm Hence, by Young’s convolution inequality, kVg f (·, ω)kLp ≤ kF −1 (fˆms )kLp kF −1(m−s · Tω gˆ)kL1 , and so Z kVg f kLp,1 ≤ kf kHps Z = kf kHps
Rd
Rd
Z Z
Rd
Rd
|F −1 (m−s · Tω gˆ)(x)|dx dω |Vgˆm−s (ω, −x)|dx dω
ˇ −s kL1 . = kf kHps kVg m 66
(54)
Using (39), we have that kVg m ˇ −s kL1 < ∞ for s > d, which concludes the proof. The next corollary holds only in dimension one, and gives a useful sufficient condition on a function to be in M 1 . In particular, (55) below gives a new proof of a conjecture of Feichtinger, that W 2,1 ⊂ M 1 when d = 1. We point out that another (unpublished) proof of this conjecture was obtained by Gr¨ochenig. The corollary follows from the identification of the Bessel potential space Hp2 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 Hp2 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
0
00
(R) = {f ∈ L (R) : f , f ∈ L (R), kf kW 2,p = p
p
2 X
kf (k) kp < ∞}.
k=0
Then Hp2 = W 2,p Proof. Case I: Assume p = 1. Let f ∈ W 2,1 (R), then f, f 0 , f 00 ∈ L1 (R). Moreover, fˆ00 (ω) = −4π 2 fˆ(ω). Thus, ˆ = F −1 (fˆ) + F −1 (ω 2fˆ) F −1 ((1 + |ω|2 )f) =f−
1 00 f . 4π 2
Consequently, F −1((1 + |ω|2)fˆ) ∈ L1 (R),
67
and moreover, ˆ L1 ≤ kf kL1 + kF −1 ((1 + |ω|2 )f)k ≤C
2 X
1 kf 00 kL1 2 4π
kf (k) kL1 .
k=0
Thus, W 2,1 ⊂ H12 . For the converse, let h be the function defined on R, such that e−x : x ≥ 0, h(x) = 0 : x < 0. ˆ We easily obtain that for ω ∈ R, h(ω) =
1 . 1+2πiω
Now let f ∈ H12 (R), then f ∈ L1 (R) and F −1 ((1 + ω 2)fˆ) ∈ L1 . Thus, g = F −1 (ω 2 fˆ) ∈ L1 . Moreover, f ∈ L1 (R) implies that f ∈ S 0 (R), and so f 00 exists as an element of S 0 (R), hence fˆ00 (ω) = −4π 2 ω 2 fˆ, where the equality holds in a distributional sense. Thus, fˆ00 = gˆ, and g ∈ L1 (R), therefore the uniquness of the Fourier transform implies that f 00 = g ∈ L1 (R). We now show that f, f 00 ∈ L1 (R) implies that f 0 which exists as a distribution is in L1 (R). To that end, note that from the fact that f, f 00 , h ∈ L1 (R), we obtain that f ∗ h, and f 00 ∗ h are both L1 functions, and we have the following equalities: (−2πiω)2 ˆ f (ω) h\ ∗ f 00 (ω) = 1 + 2πiω = (−1 + 2πiω +
1 ) hatf (ω) 1 + 2πiω
= −fˆ(ω) + 2πiω fˆ(ω) +
ˆ f(ω) . 1 + 2πiω
Consequently, h ∗ f 00 = −f + f ∗ h + k,
68
ˆ The last equation also implies that k ∈ L1 (R), and morewhere k = F −1(2πiω, f). over, ˆ ˆ k(ω) = 2πiω f(ω) = fˆ0 (ω). Using again the uniqueness of the Fourier transform, we conclude that f 0 = k ∈ L1 (R), and moreover, kf 0 kL1 ≤ 2kf kL1 + kf 00 kL1 , where we have used the fact that khkL1 = 1. Additionally, using the fact that H12 ⊂ L1 , we conclude that 2 X
kf (k) kL1 ≤ C kf kH12 ,
k=0
thus, H12 ⊂ 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) ⊂ M 1 (R).
(55)
Proof. If d = 1 and p ∈ {1, ∞}, then from Lemma 4.2.6 we have that Hp2 = 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 particular, 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 KohnNirenberg correspondences are well-known examples of pseudodifferential operators, which can be expressed as a superposition of time-frequency shifts. In particular, if σ ∈ S 0 (R2d ) the Weyl correspondence associate to it the operator Tσ : S(Rd ) −→ S 0 (Rd ), such that
ZZ Tσ f =
R2d
σ ˆ (ξ, u) e−πiξ·u T−u Mξ du dξ,
for f ∈ S(Rd ). Thus, because the operator can be realized as superposition of timefrequency 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 5.1.1
Bilinear operators on modulation spaces Definition and background
A bilinear pseudodifferential operator Tσ is ´a priori defined through its (distributional) symbol σ ∈ S 0 (R3d ) as a mapping from S(Rd ) × S(Rd ) into S 0 (Rd ) by: Z Z ˆ gˆ(η) e2πix·(ξ+η) dξ dη, σ(x, ξ, η) f(ξ) Tσ (f, g)(x) = Rd
(56)
Rd
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 |∂xα ∂ξβ ∂ηγ σ(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
1 p
+
1 q
=
1 r
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 modulation 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 A we denote Ω˜s the weight function defined by
˜ A (X, Y ) = Ωs (A(X, Y )), Ω s where X, Y ∈ R3d . Additionally, we define ω ˜ s on Z2d by ω ˜ s (l, k) = ωs (αl, βk) for fs is defined similarly. α, β > 0, and Ω Before considering general bilinear integral operators, we state a result which characterizes the modulation space MΩ1 s (R3d ) in terms of Gabor frames using standard tensor product arguments; see [41, p. 272] for further details. Proposition 5.1.1. Let φ ∈ Mω1s (Rd ) be such that {Mβn Tαk φ}k,n∈Zd is a Gabor frame for L2 (Rd ) with (canonical) dual γ ∈ Mω1s (Rd ). Then K ∈ MΩ1 s if and only if K=
X
hK, Mβn Tαm γ ⊗ Mβl Tαk γ ⊗ Mβj Tαi γi Mβn Tαm φ ⊗ Mβl Tαk φ ⊗ Mβj Tαi φ
k,m,i,l,n,j∈Zd
with unconditional convergence of the series in MΩ1 s (R3d ). Moreover, the norm of K in MΩ1 s is equivalent to the norm of its sequence of Gabor coefficients hK, Mβn Tαm γ ⊗ Mβl Tαk γ ⊗ Mβj Tαi γi k,m,i,l,n,j∈Zd in `1Ω˜ s (Z6d ).
72
5.1.2
Bilinear operators
Definition 5.1.2. A bilinear operator associated with a kernel K ∈ S 0 (R3d ), is a mapping BK defined ´a priori from S(Rd ) × S(Rd ) into S 0 (Rd ) by Z
Z BK (f, g)(x) =
K(x, y, z) f (y) g(z) dy dz, Rd
(58)
Rd
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 0 . Moreover, U ∗ f (x, y, z) = U −1 f (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 0 (R3d ) defined by (56). Then Tσ is a bilinear integral operator BK with kernel K(x, y, z) = UF1−1 σˆ (x, y, z), where F1−1 denotes the inverse Fourier transform in the first variable, and U is the operator defined above. Proof. For f, g ∈ S we have: Z Tσ (f, g)(x) =
Rd
Z
ˆ gˆ(η) e2πix·(ξ+η) dξ dη σ(x, ξ, η) f(ξ) Rd
ZZZZ =
σ(x, ξ, η) f (y) g(z) e−2πiξ·y e−2πiη·z e2πix·(ξ+η) dξ dη dy dz
ZZ =
K(x, y, z) f (y) g(z) dy dz = BK (f, g)(x),
73
where Z Z K(x, y, z) =
σ(x, ξ, η)e−2πiξ·(y−x) e−2πiη·(z−x) dξ dη
= F2 F3 σ(x, y − x, z − x) = UF1−1 σ ˆ (x, y, z). Here, Fj denotes the Fourier transform in the j th variable. We show in the next proposition that the symbol of the bilinear pseudodifferential operator is in MΩ1 Bs if and only if the corresponding integral kernel as defined in Proposition 5.1.3 is in MΩ1 s , 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 ∈ MΩ1 Bs (R3d ) if and only if K = UF1−1 σ ˆ ∈ MΩ1 s (R3d ). Proof. Let G ∈ S(R3d ). For u = (u1 , u2 , u3), and v = (v1 , v2 , v3 ) ∈ R3d we have VG σ(u, v) = hσ, Mv Tu Gi = hF −1 F1−1 U ∗ K, Mv Tu Gi = hK, UF1−1F Mv Tu Gi. Hence, VG σ(u, v) = e−2πi(u2 ·v2 +u3 ·v3 ) hK, M(v1 +u2 +u3 ,−u2 ,−u3 ) T(u1 ,v2 +u1 ,v3 +u1 ) Hi = e−2πi(u2 ·v2 +u3 ·v3 ) VH K(u1 , v2 + u1 , v3 + u1 , v1 + u2 + u3 , −u2 , −u3 ) = e−2πi(u2 ·v2 +u3 ·v3 ) VH K(B(u, v)),
(59)
ˆ Consequently, we have where H = UF1−1 G. |VG σ(u, v)| = |VH K(B(u, v))|. Therefore, Z Z R3d
R3d
Z |VH K(u, v)|Ωs (u, v) du dv =
R3d
Z
Z Z
R3d
= R3d
74
R3d
|VG σ(B −1 (u, v))|Ωs (u, v) du dv |VG σ(u, v))|ΩB s (u, v) du dv.
Thus kKkMΩ1 = kσkM 1 B , Ωs
s
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: ZZZ hBK (f, g), hi =
K(x, y, z) R3d
X
X
hf, Mβl Tαk γi Mβl Tαk φ(y)×
k,l∈Zd
hg, Mβn Tαm γi Mβn Tαm φ(z)
m,n∈Zd
=
hh, Mβj Tαi γi Mβj Tαi φ(x) dx dy dz
i,j∈Zd
XXX hf, Mβl Tαk γi hg, MβnTαm γi hh, Mβj Tαi γi× k,l m,n
i,j
Z
Z Rd
=
X
Z Rd
Rd
K(x, y, z) Mβj Tαi φ(x)Mβl Tαk φ(y) Mβn Tαm φ(z) dx dy dz
XXX hf, Mβl Tαk γi hg, MβnTαm γi hh, Mβj Tαi γi× i,j
k,n l,m
hBK (Mβl Tαk φ, Mβn Tαm φ), Mβj Tαi φi.
(60)
The exchange of the integrals and summations above is justified since f, g, h ∈ S have S ∞ absolutely summable Gabor coefficients. Moreover, K ∈ S 0 (R3d ) = s≥0 M1/ ( see Ωs 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) = Mβj Tαi φ(x) Mβl Tαk φ(y) MβnTαm φ(z). ∞ where Clearly, Mβ(j,l,n) Tα(i,k,m) Φ ∈ S(R3d ) ⊂ MΩ1 s for all s ≥ 0. Moreover, K ∈ M1/Ω s
s > 0. Thus, using the fact that the time-frequency shift operator acts isometrically 75
on Mω1s , we have Z Z Z
R3d
K(x, y, z) Mβj Tαi φ(x)Mβl Tαk φ(y) MβnTαm φ(z) dx dy dz
ZZZ ≤
R3d
|K(x, y, z)| |Mβ(j,l,n)Tα(i,k,m) Φ(x, y, z)| dx dy dz
∞ ≤ kKkM1/Ω kMβ(j,l,n)Tα(i,k,m) ΦkMΩ1 s
s
∞ = kKkM1/Ω kφk3Mω1 . s s
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 = hBK (Mβl Tαk φ, Mβn Tαm φ), Mβj Tαi φi
(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 modulation spaces. For an infinite matrix (amn,ij,kl ), let O denote the bilinear operator associated to it, i.e., (O(fij ), (gkl ))mn =
X
amn,ij,kl fij gkl ,
ij,kl
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
1 r1
=
1 p1
+ q11 . If (amn,ij,kl ) ∈ `1Ω˜ s (Z6d ), then O is a bounded operator from
`pν˜1 ,p2 (Z2d ) × `qν˜1 ,q2 (Z2d ) into `rν˜1 ,r2 (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 ). r 0 ,r 0
2d 0 0 1 2 Proof. Let (fij ) ∈ `pν˜1 ,p2 (Z2d ), (gkl ) ∈ `qν˜1 ,q2 (Z2d ) and (hmn ) ∈ `1/˜ ν (Z ) where r1 , r2
76
are the dual indices of r1 , respectively r2 . We have |hO((fij ), (gkl)), (hmn )i| ≤
X
|amn,kl,ij | |fij | |gkl| |hmn |
m,n,i,j,k,l
=
X
1
|amn,kl,ij | p1 |fij |
m,n,i,j,k,l 1
0
|amn,kl,ij | r1 |hmn | X
≤ C3
1 ν˜(i, j) ν˜(k, l) |amn,kl,ij | q1 |gkl | × ν˜(i, j) ν˜(k, l)
ν˜(m, n) ν˜(m, n)
1
|amn,kl,ij | p1 ν˜(i, j) |fij | ω˜s (i, j)×
m,n,i,j,k,l 1
|amn,ij,kl| q1 ν˜(k, l) |gkl | ω˜s(k, l)× |amn,ij,kl X
= C3
1 0 r | 1
1 |hmn | ω˜s (m, n) ν˜(m, n)
|˜ amn,kl,ij |
1/p1
|fij | ν˜(i, j)×
m,n,i,j,k,l
|˜ amn,kl,ij | |˜ amn,kl,ij |
1/q1 1/r10
|gkl | ν˜(k, l)× |hmn |
1 , ν˜(m, n)
where ˜ s (m, n, k, l, i, j) = amn,kl,ij ω a ˜mn,kl,ij = amn,kl,ij Ω ˜ s (m, n) ω ˜ s (k, l) ω ˜ s (i, j). 1 ν
We have used the fact that ν, and 1 p1
+
1 q1
=
1 , r1
or equivalently
1 p1
+
1 q1
are s-moderate with the same constant C. Since +
1 , r10
we can apply H¨older’s inequality to obtain
the following: |hO((fij ), (gkl )), (hmn )i| ≤ C
3
X
|˜ amn,ij,kl| |fij | ν˜(i, j) p1
p1
1
p1
×
m,n,i,j,k,l
X m,n,i,j,k,l
77
|˜ amn,ij,kl| |gkl |q1 ν˜(k, l)q1
1
q1
×
X
0 1 r1 |˜ amn,ij,kl| |hmn | 0 r 1 ν˜(m, n) m,n,i,j,k,l
≤C
3
1
r10
sup |fij | ν˜(i, j) sup |gkl | ν˜(k, l) × i,j
k,l
sup |hmn | m,n
3
≤ C kamn,ij,kl k`Ω˜ 1
1 ν˜(m, n)
XX
s
i
XX k
X X
|˜ amn,ij,kl |
m,i,k n,j,l
|fij | ν˜(i, j) p1
p1
p2 1 p1
p2
×
j
|gkl | ν˜(k, l) q1
q1
q2 1 q1
q2
×
l
XX m
n
r20
0 0 1 r1 r2 |hmn | 0 ν˜(m, n)r1 r10
1
≤ C 3 kamn,ij,kl k`1˜ kfi,j k`pν˜1 ,p2 kgk,lk`qν˜1 ,q2 khm,n k r10 ,r20 , Ωs
`1/˜ ν
where we have used the fact that `p,q (Z2d ) ⊂ `∞ (Z2d ), i.e., X X
sup |xm,n | ≤ m,n
n∈Zd
|xm,n |p
q/p 1/q
.
m∈Zd
Moreover, using the duality of the `p,q ν˜ -spaces i.e., kak`rν˜1 ,r2 =
X
sup kbk
r 0 ,r 0 =1 ` 1 2 1/˜ ν
|am,n | |bm,n |,
m,n∈Zd
we get that kO((fij ), (gkl ))k`rν˜1 ,r2 ≤ C 3 kamn,ij,klk`1˜ k(fij )k`pν˜1 ,p2 k(gkl )k`qν˜1 ,q2 . Ωs
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 MΩ1 s — 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
1 p1
+
1 q1
=
1 . r1
If K ∈ MΩ1 s (R3d ), then the bilinear integral operator BK
defined by (58) can be extended as a bounded operator from Mνp1 ,p2 (Rd ) × Mνq1 ,q2 (Rd ) into Mνr1 ,r2 (Rd ). Proof. Let f, g, h ∈ S(Rd ) and expand each of these functions into their Gabor seP P ries, i.e., f = hf, M T φi M T γ, g = βj αi βj αi i,j k,l hg, Mβl Tαk φi Mβl Tαk γ, and h = P m,n hh, Mβn Tαm φi Mβn Tαm γ, where φ and γ are dual Gabor frames as in Theorem 2.3.11. By Proposition 5.1.1, the matrix defined by (61) belongs to `1Ω˜s since K ∈ MΩ1 s . Therefore, using Theorem 5.1.5 we have the following estimates: |hBK (f, g), hi| = |
XXX mn
ij
amn,ij,kl hf, Mβj Tαi φi hg, MβlTαk φi hh, Mβn Tαm φi|
kl
≤ C kamn,ij,kl k`1˜ khf, Mβj Tαi φik`pν˜1 ,p2 × Ωs
khg, Mβl Tαk φik`qν˜1 ,q2 khh, Mβn Tαm φik r10 ,r20 `1/˜ ν
≤ C kKkMΩ1 kf kMνp1,p2 kgkMνq1,q2 khk s
0 r 0 ,r2
1 M1/ν
,
by duality we obtain kBK (f, g)kMνr1 ,r2 ≤ C kKkMΩ1 kf kMνp1 ,p2 kgkMνq1 ,q2 . s
The result then follows by standard density arguments, using the fact that S(Rd ) is dense in Mνp,q 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
1 p1
+
1 q1
=
1 . r1
If σ ∈ MΩ1 B (R3d ), then the bilinear pseudodifferential s
operator Tσ defined by (56) can be extended to a bounded operator from Mνp1 ,p2 (Rd ) × Mνq1 ,q2 (Rd ) into Mνr1 ,r2 (Rd ). Proof. By Proposition 5.1.4, σ ∈ MΩ1 B if and only if K ∈ MΩ1 s , where K is the kernel s
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
1 p
+
1 q
=
1 r
. If
σ ∈ M 1 (R3d ), then Tσ can be extended to a bounded operator from Lp (Rd ) × Lq (Rd ) into Lr (Rd ). In particular, if σ ∈ M 1 (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 ⊂ M p ,
and
Lq ⊂ M q .
Thus, Lp ×Lq ⊂ M p ×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 ∈ M 1 (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 ωs1 be the restriction of ωs to Rd × {0}. Then the following holds. Corollary 5.1.10. If σ ∈ MΩ1 B then Tσ can be extended as a bounded bilinear pseus
dodifferential operator from Mω2s × Mω2s into L1ωs1 . 80
Proof. Notice that Mω1s is continuously embedded in L1ωs1 , cf. [41, Prop. 12.1.4]. So, we only need to prove that under the hypotheses of the corollary, the bilinear pseudodifferential operator can be extended to a bounded operator from Mω2s × Mω2s into Mω1s . This follows from Theorem 5.1.7 by taking ν = ωs . Remark 5.1.11.
a. If the symbol σ satisfies the estimates |∂ξβ ∂ηγ σ(x, ξ, η)| ≤ Cβ,γ (1 + |ξ| + |η|)−d−
(62)
for all (x, ξ, η) ∈ R3d , all multi-indices β and γ, and some > 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 M 1 (R3d ). Therefore, our corollary does not apply. On the other hand, functions in M 1 must be continuous, but there are non-differentiable functions in M 1 , 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 81
the modulation spaces. The Hilbert transform of a function f ∈ S(R) is defined by Hf (x) =
1 lim π →0
Z |t|>
f (x−t) t
1 dt = lim π →0
Z |x−t|>
f (t) x−t
dt.
(63)
The boundedness of the Hilbert transform on the Lp spaces (1 < p < ∞) was established by M. Riesz using complex variable methods. The real variable method, initiated by Besicovitch and Titchmarsh, and further developed by Calder´on 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: C |{x ∈ R : (Hf (x)) > α}| ≤ α
Z 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, d(ω) = −i sign(ω) f(ω), ˆ it is known that Hf where sign denotes the sign function. Thus, using Plancherel’s theorem we obtain d L2 kHf kL2 = kHfk ˆ L2 = kfk = kf kL2 , and this last equality proves the boundedness of H 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 M p,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
82
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(ˆ γ ) ⊂ (0, β) and γˆ ≥ 0. Assume, moreover, that {Mβn Tαk φ}k,n , and {Mβn Tαk γ}k,n are dual Gabor frames for L2 (R). Then {Mβn Tαk φ}k,n and {Mβn Tαk γ}k,n are also dual Gabor frames for all the modulation spaces M p,q (R) (see Theorem 2.3.11). Proposition 5.2.1. Let f, g ∈ S(R), then X X
hHf, gi =
hf, Mβn Tαm φi hg, MβlTαk φi hHMβn Tαm γ, Mβl Tαk γi.
(64)
m,n∈Z k,l∈Z
Proof. If f, g ∈ S(R), then we can expand them into their Gabor expansions, i.e., f=
X
hf, Mβn Tαm φi Mβn Tαm γ,
and g =
m,n∈Z
X
hg, Mβl Tαk φi Mβl Tαk γ,
k,l∈Z
with unconditional convergence in any modulation space. In particular, the assumptions on f, g, φ, and γ imply that the Gabor coefficients of f and g are absolutely summable. Then we have: Z hHf, gi = =
R
Hf (x) · g(x) dx
Z X R k,l∈Z
hg, Mβl Tαk φi Hf (x) · Mβl Tαk γ(x) dx.
Because H is bounded on L2 , we have that for each k, n ∈ Z, Hf · Mβl Tαk γ ∈ L1 (R). Moreover, since hg, Mβl Tαk φi ∈ `1 (Z × Z), we can apply Fubini’s Theorem to obtain:
hHf, gi =
X
Z hg, Mβl Tαk φi
k,l∈Z
R
Hf (x) · Mβl Tαk γ(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: Z R
Hf (x) Mβl Tαk γ(x) dx = hHf, Mβl Tαk γi = hf, H ∗ Mβl Tαk γi = −hf, HMβl Tαk γi =
X
hf, Mβn Tαm φi hHMβnTαm γ, Mβl Tαk γi.
(66)
m,n∈Z
We can now use (65) and (66) to obtain; hHf, gi =
XX
hf, Mβn Tαm φi hg, MβlTαk φi hHMβn Tαm γ, Mβl Tαk γi.
(67)
k,l m,n
We denote Aα,β,γ the sequence defined for (k, l) ∈ Z2 by Aα,β,γ (k, l) = |Vγˆ γˆ (βl, αk)|. The choice of the window γ implies in particular that γ ∈ M 1 , hence X
Aα,β,γ (k, l) < ∞.
l,k∈Z
We can use the above proposition to prove the following. Proposition 5.2.2. If f, g ∈ S(R), then |hHf, gi| ≤
X X
Aα,β,γ (m − k, l − n) |hf, Mβn Tαm φi| |hg, MβlTαk φi|.
m,n∈Z k,l∈Z
Proof. Let S be the sign function defined by −1 : x < 0, S(x) = sign(x) = 0 : x = 0, +1 : x > 0.
84
(68)
We use again the L2 -theory of the Hilbert transform. In particular, using the Fourier transform we have hHMβn Tαm γ, Mβl Tαk γi = h(HMβn Tαm γ)∧ , (Mβl Tαk γ)∧ i = −ihS · (Mβn Tαm γ)∧ , (Mβl Tαk γ)∧ i = −ie2πiαβ(kl−mn) hS · M−αm Tβn γˆ , M−αk Tβl γˆ i = −ie2πiαβk(n−l) Vγˆ ((T−βn S) · γˆ )(β(l − n), α(m − k)).
(69)
With that choice of γ it is easy to see that for all n ∈ Z, T−βn S · γˆ = ±ˆ γ.
(70)
hHMβn Tαm γ, Mβl Tαk γi = ±ie2πiαβk(n−l) Vγˆ γˆ (β(l − n), α(m − k)).
(71)
Hence, (69) becomes
By putting all the above together into (65) we have hHf, gi = ±i
X X
e2πiαβk(n−l) Vγˆ γˆ (β(l −n), α(m−k)) hf, Mβn Tαm φi hg, MβlTαk φi.
m,n∈Z k,l∈Z
(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 M p,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 M p,q . In particular, for any f ∈ M p,q we have the following estimate: kHfkM p,q ≤ C kf kM p,q , for some positive constant C independent of f .
85
Proof. Let f, g ∈ S(R), then, by Proposition 68, we have that |hHf, gi| ≤
X
|hf, Mβn Tαm φi|
m,n∈Z
=
X
X
Aα,β,γ (m − k, l − n) |hg, MβlTαk φi|
k,l∈Z
|hf, Mβn Tαm φi| Aα,β,γ ∗ |hg, Mβ· Tα· φi|(m, n)
n,m
≤ khf, Mβ· Tα· φik`p,q kAα,β,γ ∗ |hg, Mβ· Tα· φi|k`p0 ,q0 ≤ kAα,β,γ k`1 khf, Mβ· Tα· φik`p,q khg, Mβ· Tα· φik`p0 ,q0 ≤ C kγkM 1 kf kM p,q kgkM p0,q0 . We have used Young’s inequality to obtain the fourth inequality. By duality we then obtain kHf kM p,q ≤ C kγkM 1 kf kM p,q . The result then follows since S(R) is a dense subspace of each of the modulation spaces M p,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 M 1 . To see this notice that M 1 is a dense subspace of L2 , and that functions in M 1 are continuous. Since M 1 is invariant under the Fourier transform, it is easy to see that for f ∈ M 1 , d ∈ M 1. Hf ∈ M 1 ⇐⇒ Hf d = −S · fˆ, and this function cannot belong to M 1 since it has a disconHowever, Hf tinuity at the origin.
86
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VITA
Kasso Akochay´e Okoudjou was born on March 30, 1973, in Parakou, B´enin. He attended both Lyc´ee Toffa and Lyc´ee B´ehanzin high schools in Porto-Novo, B´enin and graduated from the latter in 1991. Subsequently, he entered the Universit´e Nationale du B´enin in Abomey-Calavi, B´enin and completed a Maitrise `es Sciences Math´ematiques in 1996. After teaching mathematics at Complexe Scolaire William Ponty high school in Porto-Novo, B´enin from 1996 to 1998, he was admitted to the Ph.D. program in the School of Mathematics at the Georgia Institute of Technology in the fall of 1998. He graduated from Georgia Tech with an M. S. in Electrical and Computer Engineering in May 2003, and with a Ph. D. in Mathematics in August 2003. His thesis was written under the supervision of Professor Christopher Heil. He then moved to Ithaca, NY, to join the Mathematics Department of Cornell University as an H. C. Wang Assistant Professor.
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