View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Appl. Comput. Harmon. Anal. 21 (2006) 349–359 www.elsevier.com/locate/acha Fourier multipliers of classical modulation spaces Hans G. Feichtinger a,∗, Ghassem Narimani b a Fakultät für Mathematik, Universität Wien, Nordbergstrasse 15, 1090 Wien, Austria b Faculty of Mathematics, University of Tabriz, Tabriz, Iran Received 22 September 2005; accepted 5 April 2006 Available online 19 June 2006 Communicated by W.R. Madych Abstract Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131–139]. Finally, we show that any function with ([d/2]+1)-times bounded derivatives is a Fourier multiplier for all modulation spaces Mp,q(Rd ) with p ∈ (1, ∞) and q ∈[1, ∞]. © 2006 Elsevier Inc. All rights reserved. Keywords: Modulation spaces; Wiener amalgam spaces; Fourier multipliers; Pointwise multipliers 1. Introduction Modulation spaces are defined by measuring the time–frequency concentration of functions or distributions in the time–frequency plane. They are the most natural function spaces for studying time–frequency behavior of functions and distributions. An important feature of the “classical” modulation spaces is that they are characterized by a uni- form partition of the frequency plane. In contrast, Besov spaces and similar smoothness spaces are defined by a dyadic decomposition of the frequency plane. Nevertheless, the behavior of modulation spaces concerning duality, interpo- lation, embeddings, or trace operators is similar to Besov spaces and most of the time some properties of modulation spaces are simpler to study than Besov spaces (see [10]). As we shall see, multipliers of modulation spaces can be * Corresponding author. E-mail addresses: [email protected] (H.G. Feichtinger), [email protected] (G. Narimani). URL: http://www.mat.univie.ac.at/NuHAG/fei/ (H.G. Feichtinger). 1063-5203/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.acha.2006.04.010 350 H.G. Feichtinger, G. Narimani / Appl. Comput. Harmon. Anal. 21 (2006) 349–359 better described than multipliers of Besov spaces [25]. Formally, the classical modulation spaces Mp,q, p,q ∈[1, ∞], are defined as the set of all tempered distributions f ∈ S(Rd ) such that q 1 p q p f Mp,q := Vgf(x,w) dx dw < ∞, Rd Rd d where Vgf is the so called short-time Fourier transform of f with respect to a window function g ∈ S(R ), defined by the usual pairing (scalar product) 2πiw· Vgf(x,w)= f,e g(·−x) . Here, S(Rd ) and S(Rd ) denote the Schwartz space of rapidly decreasing smooth functions and the space of tempered distributions, respectively. In time–frequency analysis one is interested to measure, quantitatively, the behavior of functions and distributions d d in the time–frequency plane R × R , and for this purpose ·Mp,q is a natural candidate. That is why the modulation spaces are so useful in time–frequency analysis. Since their definition in early 1980’s [10], modulation spaces have found their way into different areas of mathematical analysis and applications. Just as an example, recent developments in the theory of pseudodifferential operators on modulation spaces or with symbols in modulation spaces, along with improvement of some old results, have significantly simplified the proofs. See [2,14,23,24] for this kind of applications. In this paper we study (Fourier) multipliers of modulation spaces. A bounded function m is called a Fourier multiplier from Mp1,q1 to Mp2,q2 ,ifforsomeC>0 and for all f in a dense subspace of Mp1,q1 one has − F 1 · F p ,q (m f) Mp2,q2 C f M 1 1 . Here F and F −1 denote the Fourier transform and the inverse Fourier transform (if necessary in the sense of distri- butions). The operators defined by f → F −1(m · Ff)for f ∈ S(Rd ) arise naturally in applications and are closely related to bounded translation invariant operators, i.e., the bounded operators T that commute with all translation operators. We will give a characterization of these multipliers using generalized Wiener amalgam spaces. Recall that they constitute a family of (function or) distribution spaces. They allow to quantify the global behavior of a certain local property, they are an indispensable tool when it comes to formulate results in which the global and the local behavior need to be considered separately. As we will demonstrate in this paper, the Fourier multipliers of modula- tion spaces can be completely characterized (in an abstract way) in this setting. Historically, the first use of Wiener amalgam spaces goes back to Norbert Wiener, who used some special cases of this kind of spaces for formulating his famous Tauberian theory. The first systematic study of Wiener amalgams of Lebesgue spaces appeared in [16]. Then, several authors studied properties of these spaces. Here we do not aim to go in more detail, but we mention only that, in [7,8], this concept is generalized, allowing a wide range of Banach spaces of functions and distributions on locally compact groups to be used as local and global components, while still having the expected properties (concerning duality, multipliers, etc.). As an introduction to Wiener amalgam spaces with weighted Lebesgue spaces as local and global components we recommend [15], or the classical papers [7,8]. The organization of this paper is as follows. In Section 2 we provide relevant information about Wiener amal- gam spaces and modulation spaces. Our approach to Wiener amalgam spaces and modulation spaces is based on a special kind of partitions of unity called “bounded uniform partitions of unity” or BUPU for short. In Section 3 we prove that every bounded translation invariant operator on modulation spaces is essentially of convolution type. In Section 4, Theorem 16 we characterize the multipliers of modulation spaces by means of Wiener amalgam spaces. Then we present some relevant applications of Theorem 16 and provide a simple proof using amalgam methods. Subsequently, the usefulness of Theorem 16 is described in two corollaries. In particular, we recapture all known con- volution relations between modulation spaces. Also, a short proof for (a generalization of) the main result of [1] [ d ]+1 is given. Finally, in Theorem 20 we show that any function in C 2 , the space of all functions with bounded d + p,q Rd ∈ ∞ derivatives of order 2 1, is a Fourier multiplier for all modulation spaces M ( ), with p (1, ) and q ∈[1, ∞]. H.G. Feichtinger, G. Narimani / Appl. Comput. Harmon. Anal. 21 (2006) 349–359 351 2. Preliminaries and notations 2.1. General spaces of functions and distributions In this paper → denotes the continuous embeddings of function spaces. I is the reflection operator defined by If(x)= f(−x). An open ball of radius ε with center y is denoted by Bε(y). |A| denotes the cardinality of a finite p d p set A.Forp ∈[1, ∞], L (R ) is the usual Lebesgue space, with norm ·p and is the usual Lebesgue sequence d space. p denotes the conjugate exponent of p. Cb(R ) denotes the space of bounded continuous functions. The Schwartz class of rapidly decreasing smooth functions is denoted by S(Rd ).WeletS(Rd ) be equipped with its usual topology (system of seminorms). For p ∈[1, ∞), S(Rd ) is dense in Lp(Rd ). The continuous linear functionals on S(Rd ) are called tempered distributions, and the space is denoted by S(Rd ). The pairing between S(Rd ) and S(Rd ) is denoted by · , · .Forf ∈ L1(Rd ), its Fourier transform is the bounded and continuous function − · Ff(w)= f(x)e 2πix w dx. Rd If also Ff is in L1(Rd ), the Fourier inversion formula gives − · f(x)= F 1Ff(x)= Ff(w)e2πix w dw. Rd Since L1 is a convolution algebra and F(f ∗ g) = (Ff)(Fg) for f , g in L1(Rd ), the space A(Rd ) = FL1(Rd ) with pointwise product is a Banach algebra with the norm defined by hA =g1, when h = Fg. This is justified because F is injective. The Fourier transformation is also a bijection on S(Rd ), a unitary operator on L2(Rd ), and can be extended to a bijection on S(Rd ) by the relation Fσ,ϕ =σ,Fϕ for σ ∈ S(Rd ) and ϕ ∈ S(Rd ). With this generalized notion of Fourier transformation, for p ∈[1, ∞] we can define FLp(Rd ) as the set of all tempered p p d distributions that are Fourier transform of an L function. By defining the norm of an f ∈ FL (R ) by hFLp = gp, when h = Fg, this space becomes a Banach space, which will be of special interest for us later. The concept of a bounded uniform partition of unity plays an essential role in the theory of Wiener amalgam spaces: d Definition 1 (BUPU). A family Ψ = (ψi)i∈I of functions in A(R ) is called a bounded uniform partition of unity d (BUPU) of size ε>0, if there exists a family of points (yi)i∈I in R , such that := ∞ (1) CΨ supi∈I ψi A < ; (2) supp(ψi) ⊆ Bε(yi) for i ∈ I ; = |{ ∩ =∅}| ∞ (3) NΨ supi∈I j: Bε(yi) Bε(yj ) < ; = ∈ Rd (4) i∈I ψi(x) 1 for all x .