Kernel Theorems for Modulation Spaces

Abstract

We deal with kernel theorems for modulation spaces. We completely characterize the continuity of a linear operator on the modulation spaces $M^p$ for every $1\le p\le \infty$, by the membership of its kernel in (mixed) modulation spaces. Whereas Feichtinger’s kernel theorem (which we recapture as a special case) is the modulation space counterpart of Schwartz’ kernel theorem for tempered distributions, our results do not have a counterpart in distribution theory. This reveals the superiority, in some respects, of the modulation space formalism over distribution theory, as already emphasized in Feichtinger’s manifesto for a post-modern harmonic analysis, tailored to the needs of mathematical signal processing. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. For operators acting on $M^{p,q}$ a similar characterization is not expected, but sufficient conditions for boundedness can be stated in the same spirit.

1 Introduction

Schwartz’ kernel theorem certainly is one of the most important results in modern functional analysis. It states, in the framework of tempered distributions, that every linear continuous operator $A:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ can be regarded as an integral operator in a generalized sense, namely

$$\begin{aligned} \langle Af,g\rangle =\langle K,g\otimes \overline{f}\rangle ,\qquad f,g\in \mathcal {S}(\mathbb {R}^d), \end{aligned}$$

in the sense of distributions, for some kernel $K\in \mathcal {S}'({\mathbb {R}^{2d}})$, and vice versa [21]. We write, formally,

$$\begin{aligned} A f(x)=\int _{\mathbb {R}^d}K(x,y)\,f(y)dy,\quad f\in \mathcal {S}(\mathbb {R}^d). \end{aligned}$$

(1)

This result provides a general framework that includes most of the operators of interest in harmonic analysis. However, for many applications in time-frequency analysis and mathematical signal processing the spaces $\mathcal {S}(\mathbb {R}^d)$ and $\mathcal {S}'(\mathbb {R}^d)$ can be fruitfully replaced by the modulation spaces $M^1(\mathbb {R}^d)$ and $M^\infty (\mathbb {R}^d)$, respectively, introduced by H. Feichtinger in [14] and nowadays widely used as a fundamental tool in time-frequency analysis (see [19] and also Feichtinger’s manifesto for a postmodern harmonic analysis [15]). One of the obvious advantages is that $M^1(\mathbb {R}^d)$ and $M^\infty (\mathbb {R}^d)$ are Banach spaces; secondly their norm gives a direct and transparent information on the time-frequency concentration of a function or tempered distribution. The same holds, more generally, for a whole scale of intermediate spaces $M^p(\mathbb {R}^d)$, $1\le \ p\le \infty$, as well as the more general mixed norm spaces $M^{p,q}(\mathbb {R}^d)$, $1\le p,q\le \infty$.

In short, for $(x,\xi )\in \mathbb {R}^d\times \mathbb {R}^d$, define the time-frequency shifts $\pi (x,\xi )$ by

$$\begin{aligned} \pi (x,\xi )f(t)=e^{2\pi i t \xi } f(t-x), \quad f\in \mathcal {S}'(\mathbb {R}^d). \end{aligned}$$

Then $M^{p,q}(\mathbb {R}^d)$ is the space of tempered distributions f in $\mathbb {R}^d$ such that the function

$$\begin{aligned} (x,\xi )\mapsto \langle f,\pi (x,\xi )g\rangle , \end{aligned}$$

for some (and therefore any) window $g\in \mathcal {S}(\mathbb {R}^d){\setminus }\{0\}$, is in $L^{p,q}({\mathbb {R}^{2d}})$, endowed with the obvious norm (see Sect. 2 below for details). We also set $M^p(\mathbb {R}^d)=M^{p,p}(\mathbb {R}^d)$. Weighted versions are also used in the literature but here we limit ourselves, for simplicity, to the unweighted case.

Now, a kernel theorem in the framework of modulation spaces was announced in [11] and proved in [16]; see also [19, Theorem 14.4.1]. It states that linear continuous operators $A:M^1(\mathbb {R}^d)\rightarrow M^\infty (\mathbb {R}^d)$ are characterized by the membership of their distributional kernel K to $M^\infty ({\mathbb {R}^{2d}})$.

In this note we provide a similar characterization of linear continuous operators acting on the following spaces:

• $M^1(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$, for a fixed $1\le p\le \infty$

• $M^p(\mathbb {R}^d)\rightarrow M^\infty (\mathbb {R}^d)$, for a fixed $1\le p\le \infty$

• $M^p(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$, for every $1\le p\le \infty$.

In all cases the characterization is given in terms of the membership of their distributional kernels to certain mixed modulation spaces, first introduced and studied in [4]. The proof uses in an essential way a discretization of the problem by means of Gabor frames. We also show the equivalence of the operator norm and the modulation space norm of the corresponding kernel. We address to [1, Theorem 6] for a related result.

These results seem remarkable; no such a characterization is known, for example, for linear continuous operators from $\mathcal {S}(\mathbb {R}^d)$ into itself and similarly from $\mathcal {S}'(\mathbb {R}^d)$ into itself. This again shows the superiority, in some respects, of the modulation space framework over distribution theory.

For operators acting on mixed-norm modulation spaces $M^{p,q}(\mathbb {R}^d)\rightarrow M^{p,q}(\mathbb {R}^d)$ such a characterization is not expected, as we will see in Sect. 3 below. We give however sufficient conditions for boundedness in the same spirit.

Let us also observe that in the literature there are plenty of sufficient conditions for operators in certain classes (Fourier multipliers, localization operators, pseudodifferential operators, Fourier integral operators) to be bounded $M^p(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$ for every $1\le p\le \infty$, or $M^{p,q}(\mathbb {R}^d)\rightarrow M^{p,q}(\mathbb {R}^d)$ for every $1\le p,q\le \infty$. For example it is known that pseudodifferential operators with Weyl symbol in $M^{\infty ,1}$ are in fact bounded $M^{p,q}(\mathbb {R}^d)\rightarrow M^{p,q}(\mathbb {R}^d)$ for every $1\le p,q\le \infty$ [6, 19, 20, 27, 28] (see also [25]). On the other hand, the Fourier multiplier with symbol $e^{i|\xi |^2}$ is still bounded on these spaces [3], although that symbol does not belong to $M^{\infty ,1}({\mathbb {R}^{2d}})$. Similarly, it is easy to see that any linear continuous operator $A:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ whose Gabor matrix enjoys decay estimates such as

$$\begin{aligned} |\langle A\pi (z)g,\pi (w)g\rangle |\le C(1+|w-\chi (z)|)^{-s}\qquad z,w\in {\mathbb {R}^{2d}}\end{aligned}$$

for some $s>2d$, $g\in \mathcal {S}(\mathbb {R}^d){\setminus }\{0\}$, and bi-Lipschitz mapping $\chi :{\mathbb {R}^{2d}}\rightarrow {\mathbb {R}^{2d}}$, is bounded $M^p(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$ for every $1\le p\le \infty$ [9] (see also [7,8,9]). But once again this condition is not necessary at all.

The very simple characterization provided in the present paper when $p=q$ therefore includes, at least implicitly, all these continuity results on $M^p$.

2 Backgrounds on Time-Frequency Analysis

2.1 Mixed Norm Spaces

We summarize the main properties of mixed norm spaces, we refer to [2] for a full treatment of this subject.

Consider measure spaces $(X_i,\mu _i)$, indices $p_i\in [1,\infty ]$, with $i=1,\dots ,d$. Then the mixed norm space $L^{p_1,p_2,\dots , p_d}(X_1\times X_2\times \dots \times X_d;\mu _1\times \mu _2\times \dots \times \mu _d)$ consists of the measurable functions $F:X_1\times X_2\times \dots \times X_d\rightarrow \mathbb {C}$ such that the following norm is finite:

$$\begin{aligned} \Vert F\Vert _{L^{p_1,p_2,\dots , p_d}}=\left( \int _{X_d}\cdots \left( \int _{X_1}|F(x_1,\dots ,x_d)|^{p_1 }\,d\mu _{1}(x_1)\right) ^{\frac{p_2}{p_1}}\cdots \,d\mu _{d}(x_d)\right) ^{\frac{1}{p_d}}\nonumber \\ \end{aligned}$$

(2)

with standard modification when $p_i=\infty$ for some i.

If $X_i=\mathbb {R}$ and $\mu _i$ is the Lebesgue measure on $\mathbb {R}$ for every $i=1,\dots ,d$, we simply write $L^{p_1,\dots , p_d}$. If each $X_i$ is a countable set and the corresponding measure $\mu _i$ is the counting measure, we use the notation $\ell ^{p_1,\dots , p_d}(X_1\times X_2\times \dots \times X_d)$.

The mixed norm spaces $L^{p_1,p_2,\dots , p_d}(X_1\times X_2\times \dots \times X_d;\mu _1\times \mu _2\times \dots \times \mu _d)$ are Banach spaces, generalizations of the classical $L^p$ and $\ell ^p$ spaces, cf. [2].

2.2 Modulation Spaces

For $x,\xi \in \mathbb {R}^d$, define the translation operator $T_x$ and modulation operator $M_\xi$ by

$$\begin{aligned} T_x f(t)=f(t-x),\quad M_\xi f(t)= e^{2\pi i tx} f(t). \end{aligned}$$

For $z=(x,\xi )$, the composition of operators $\pi (z)=M_\xi T_x$ is called a time-frequency  shift of the phase space ${\mathbb {R}^{2d}}$.

For a fixed non-zero $g \in \mathcal {S}(\mathbb {R}^d)$, the short-time Fourier transform (STFT) of $f \in \mathcal {S}' (\mathbb {R}^d)$ with respect to the window g is given by

$$\begin{aligned} V_gf(x,\xi )=\int _{\mathbb {R}^d} f(t)\, {\overline{g(t-x)}} \, e^{-2\pi i\xi t}\,dt\, =\langle f, \pi (z)g\rangle \quad z=(x,\xi ), \end{aligned}$$

(3)

where the integral represents the (anti-)duality between $\mathcal {S}'$ and $\mathcal {S}$. The short-time Fourier transform  gives information about the time-frequency  content of the signal f: indeed, roughly speaking, it can be seen as a localized Fourier transform  of f in a neighbourhood of x.

The STFT is the basic tool in the definition of modulation spaces. Indeed, modulation space norms are a measure of the joint time-frequency distribution of $f\in \mathcal {S}'$. For their basic properties we refer [10, 12, 13] (see also [4, 22,23,24]) and the textbook [19].

Definition 2.1

Fix $g\in \mathcal {S}(\mathbb {R}^d)$, $p=(p_1,\dots ,p_d)$, $q=(q_1,\dots ,q_d)$, $p_i,q_i\in [1,\infty ]$, $i=1,\dots ,d$. Then

$$\begin{aligned} M^{p,q}(\mathbb {R}^d)=\{f\in \mathcal {S}'(\mathbb {R}^d):\, \Vert f\Vert _{M^{p,q}(\mathbb {R}^d)}<\infty \}, \end{aligned}$$

where

$$\begin{aligned} \Vert f\Vert _{M^{p,q}(\mathbb {R}^d)}=\Vert V_g f\Vert _{L^{p_1,\dots ,p_d,q_1,\dots ,q_d}}. \end{aligned}$$

If $p=p_1,\dots ,p_d,\quad q=q_1,\dots ,q_d$, we come back to the classical modulation spaces of distributions $f\in \mathcal {S}'(\mathbb {R}^d)$ such that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}(\mathbb {R}^d)}=\left( \int _{\mathbb {R}^d}\left( \int _{\mathbb {R}^d}|V_g f (x,\xi )|^p\,dx\right) ^{\frac{q}{p}} d\xi \right) ^\frac{1}{q} <\infty \end{aligned}$$

(cf. [14]). The extension of the original modulation spaces in Definition 2.1 was widely studied and applied to the investigation of boundedness and sampling properties for pseudodifferential operators in [22,23,24].

To state our kernel theorems we need a further extension of Definition 2.1, that was introduced by Bishop in [4], in her study of Schatten p-class properties for integral operators. This is a natural generalization of the classical modulation spaces, as will be clear in the sequel.

We shall mainly use the notation from [4]. From now on we assume that c is a permutation of the set $\{1,\dots ,2d\}$. We identify c with the linear bijection

$$\begin{aligned} \tilde{c}:{\mathbb {R}^{2d}}\rightarrow {\mathbb {R}^{2d}},\quad \tilde{c}(x_1,\dots ,x_{2d})=(x_{c(1)},\dots , x_{c(2d)}). \end{aligned}$$

Definition 2.2

Consider $g\in \mathcal {S}(\mathbb {R}^d)$ and let c be a permutation corresponding to the map $\tilde{c}$ as above. Then $M(c)^{p_1,p_2,\dots ,p_{2d}}$ is the mixed modulation space of tempered distributions $f\in \mathcal {S}'(\mathbb {R}^d)$ for which

$$\begin{aligned} \Vert f\Vert _{M(c)^{p_1,p_2,\dots ,p_{2d}}} =\Vert V_g f\circ \tilde{c}\Vert _{L^{p_1,p_2,\dots ,p_{2d}}} <\infty . \end{aligned}$$

(4)

If $p=p_1=p_2=\dots =p_d,\quad q=p_{d+1}=p_{d+2}=\dots =p_{2d}$ we simply write $M(c)^{p,q}$; if $p=p_i$, $i=1,\dots ,2d$, we shorten to $M(c)^{p}$.

Remark 2.3

• (i) If c is the identity permutation and the indices satisfy

  $$\begin{aligned} p=p_1=p_2=\dots =p_d,\quad q=p_{d+1}=p_{d+2}=\dots =p_{2d} \end{aligned}$$

  then $M(c)^{p_1,p_2,\dots ,p_{2d}}=M^{p,q}(\mathbb {R}^d)$ is the original modulation space introduced by Feichtinger in [14]. If the indices satisfy

  $$\begin{aligned} p=p_i,\quad i=1,\dots ,2d, \end{aligned}$$

  given any permutation c, a simple change of variables yields

  $$\begin{aligned} M(c)^{p}=M^p(\mathbb {R}^d). \end{aligned}$$

• (ii) If c is the identity permutation and $p=(p_1,\dots ,p_d),\, q=(q_1,\dots ,q_d)$, we come back to the modulation spaces in Definition 2.1.

• (iii) In general the mixed modulation spaces differ from the classical ones of Definition 2.1. A simple example is provided by the permutation

  $$\begin{aligned} c_0(1,2,\dots ,d,d+1,\dots ,2d)=(d+1,d+2,\dots ,2d,1,2,\dots , d). \end{aligned}$$

  (5)

  If we choose

  $$\begin{aligned} p=p_1=p_2=\dots =p_d,\quad q=p_{d+1}=p_{d+2}=\dots =p_{2d} \end{aligned}$$

  then it is easy to see that

  $$\begin{aligned} M^{p,q}(c_0)=W(\mathcal {F}L^p,L^q)(\mathbb {R}^d), \end{aligned}$$

  the Wiener amalgam spaces of distributions $f\in \mathcal {S}'(\mathbb {R}^d)$ such that

  $$\begin{aligned} \Vert f\Vert _{W(\mathcal {F}L^p,L^q)(\mathbb {R}^d)}=\left( \int _{\mathbb {R}^d}\left( \int _{\mathbb {R}^d}|V_g f (x,\xi )|^p\,d\xi \right) ^{\frac{q}{p}} dx \right) ^\frac{1}{q} <\infty \end{aligned}$$

  for some window function $g\in \mathcal {S}(\mathbb {R}^d){\setminus }\{0\}$, cf. [13].

It can be shown (cf. [4]) that the definition of mixed modulation spaces is independent of the choice of the window g in $\mathcal {S}(\mathbb {R}^d)$, with different windows giving equivalent norms. Furthermore, this fact also holds for g in the larger space $M^1(\mathbb {R}^d)$, as follows by the subsequent Theorem 2.4 (cf. [4]), which gives the inversion formula on $M(c)^{p_1,\dots , p_{2d}}$.

Theorem 2.4

Suppose c is a permutation of $\{1,\dots ,2d\}$, and for a measurable function $\psi :\mathbb {R}^d\rightarrow \mathbb {C}$ define formally the operator $\Upsilon _\psi$ by

$$\begin{aligned} \Upsilon _\psi F(t)=\int _{{\mathbb {R}^{2d}}}F(x) \pi (\tilde{c}(x))\psi (t)\, dx. \end{aligned}$$

(6)

• (i) Given $\psi , g\in M^1(\mathbb {R}^d)$, $p_1,\dots , p_{2d}\in [1,\infty ]$, and $f\in M(c)^{p_1,\dots , p_{2d}}$, then

  $$\begin{aligned} \Upsilon _\psi (V_g f\circ \tilde{c})=\langle \psi , g\rangle f. \end{aligned}$$

• (ii) Different windows in $M^1(\mathbb {R}^d)$ define equivalent norms on $M(c)^{p_1,\dots , p_{2d}}$.

Observe that the operator $\Upsilon _\psi$ in (6) can be seen as a permuted synthesis operator.

Corollary 2.5

For any $p_1,\dots , p_{2d}\in [1,\infty ]$, $M(c)^{p_1,\dots , p_{2d}}$ is a Banach space.

We also have the expected duality result.

Theorem 2.6

If $p_1,\dots , p_{2d}\in [1,\infty )$, $M(c)^{p'_1,\dots , p'_{2d}}$ is the dual space of $M(c)^{p_1,\dots , p_{2d}}$, where $p'_i\in (1,\infty ]$ satisfies $1/p_i+1/p_i'=1$.

2.3 Gabor Frames

Given a lattice $\Lambda$ in ${\mathbb {R}^{2d}}$ and a non-zero square integrable function g (called window) on $\mathbb {R}^d$ the system

$$\begin{aligned} \mathcal {G}(g,\Lambda )=\{\pi (\lambda )g\,:\,\lambda \in \Lambda \} \end{aligned}$$

is called a Gabor frame if it is a frame for $L^2(\mathbb {R}^d)$, that is, there exist constants $0<A\le B$ such that

$$\begin{aligned} A\Vert f\Vert _2^2\le \sum _{\lambda \in \Lambda }|\langle f,\pi (\lambda ) g\rangle |^2\le B\Vert f\Vert _2^2,\quad \forall f \in L^2(\mathbb {R}^d). \end{aligned}$$

(7)

If (7) holds, then there exists $\gamma \in L^2(\mathbb {R}^d)$ (the so-called dual window), such that $\mathcal {G}(\gamma ,\Lambda )$ is a frame for $L^2(\mathbb {R}^d)$ and every $f\in L^2(\mathbb {R}^d)$ can be expanded as

$$\begin{aligned} f=\sum _{\lambda \in \Lambda }\langle f, \pi (\lambda )g \rangle \pi (\lambda )\gamma = \sum _{\lambda \in \Lambda }\langle f,\pi (\lambda )\gamma \rangle \pi (\lambda ) g, \end{aligned}$$

(8)

with unconditional convergence in $L^2(\mathbb {R}^d)$.

From now on we shall work with the quadratic lattice $\Lambda =\alpha \mathbb {Z}^{2d}$, for suitable $\alpha >0$.

If the window function belongs to the Feichtinger’s algebra $M^1(\mathbb {R}^d)$, then a Gabor frame for $L^2(\mathbb {R}^d)$ is a frame also for mixed modulation spaces (cf. [4, Theorem 4.6]).

Theorem 2.7

Let g be a window function in $M^1(\mathbb {R}^d)$ such that $\mathcal {G}(g,\Lambda )$ is a frame for $L^2(\mathbb {R}^d)$ with a dual frame $\mathcal {G}(\gamma ,\Lambda )$. Consider $p_1,\dots ,p_{2d} \in [1,\infty ]$.

• (i) There exist constants $0<A\le B$ independent of $p_1,\dots ,p_{2d}$ such that

  $$\begin{aligned} A\Vert f\Vert _{M(c)^{p_1,\dots , p_{2d}}}\le \Vert V_g f\circ {\tilde{c}}|_{\Lambda }\Vert _{\ell ^{p_1,\dots ,p_{2d}}}\le B\Vert f\Vert _{M(c)^{p_1,\dots , p_{2d}}},\quad \forall f \in {M(c)^{p_1,\dots , p_{2d}}}. \end{aligned}$$

• (ii) We have

  $$\begin{aligned} f=\sum _{\lambda \in \Lambda }\langle f, \pi (\lambda )g \rangle \pi (\lambda )\gamma = \sum _{\lambda \in \Lambda }\langle f,\pi (\lambda )\gamma \rangle \pi (\lambda ) g, \end{aligned}$$

  (9)

  with unconditional convergence in $M(c)^{p_1,\dots , p_{2d}}$ if $p_1,\dots ,p_{2d}\in [1,\infty )$, and weak* convergence in $M^\infty (\mathbb {R}^d)$ if $p_1,\dots ,p_{2d}\in [1,\infty ]$.

This generalizes the basic fact, valid for the identity permutation, that the coefficient operator $C_g$ and the reconstruction operator $D_\gamma$, defined by

$$\begin{aligned} (C_g f)_\lambda :=\langle f, \pi (\lambda )g\rangle , \qquad D_\gamma c:=\sum _{\lambda \in \Lambda } c_\lambda \pi (\lambda )\gamma , \end{aligned}$$

(10)

satisfy

$$\begin{aligned} C_g\,: M^{p,q}(\mathbb {R}^d)\, \rightarrow \ell ^{p,q}(\Lambda ), \quad D_\gamma :\, \ell ^{p,q}(\Lambda ) \rightarrow M^{p,q}(\mathbb {R}^d), \end{aligned}$$

(11)

with $D_\gamma C_g=I$, the identity on $M^{p,q}(\mathbb {R}^d)$, cf. (9), and similarly if we interchange the windows g and $\gamma$.

3 Kernel Theorems

In this section we will present the characterization of continuity properties on modulation spaces of integral operators given by (1).

We need the following result (see e.g. Tao’s Lecture Notes [26, Proposition 5.2]).

Proposition 3.1

Assume that $(X,\nu _X)$, $(Y,\mu _Y)$ are measure spaces and $p\in [1,\infty ]$. Suppose that $K:X\times Y\rightarrow \mathbb {C}$, and $\Vert K(\cdot ,y)\Vert _{L^p(X)}$ is uniformly bounded on Y. Then the operator A with kernel K maps $L^1(X)$ into $L^p(Y)$ and

$$\begin{aligned} \Vert A\Vert _{L^1(X)\rightarrow L^p(Y)}={{\mathrm{ess\,sup\,}}}_{y\in Y} \Vert K(\cdot ,y)\Vert _{L^p(X)}. \end{aligned}$$

(12)

Similarly, we have a dual statement (see e.g. [26, Proposition 5.4]).

Proposition 3.2

Assume that $(X,\nu _X)$, $(Y,\mu _Y)$ are measure spaces and $p\in [1,\infty ]$. Suppose that $K:X\times Y\rightarrow \mathbb {C}$, and $\Vert K(x,\cdot )\Vert _{L^{p'}(Y)}$ is uniformly bounded on Y. Then the operator A with kernel K maps $L^p(X)$ into $L^\infty (Y)$ and

$$\begin{aligned} \Vert A\Vert _{L^p(X)\rightarrow L^\infty (Y)}={{\mathrm{ess\,sup\,}}}_{x\in X} \Vert K(x,\cdot )\Vert _{L^{p'}(Y)},\quad \frac{1}{p}+\frac{1}{p'}=1. \end{aligned}$$

(13)

Let us introduce the permutation $c_1$ of the set $\{1,\dots , 4d\}$ with related $\tilde{c}_1$ given by, for $x_i\in \mathbb {R}^d$, $i=1,\dots ,4$,

$$\begin{aligned} \tilde{c}_1(x_1,x_2, x_3,x_{4})= \begin{pmatrix} I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\\ 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\end{pmatrix}\,\begin{pmatrix} x_1\\ x_{2}\\ x_{3}\\ x_{4} \end{pmatrix}=(x_1,x_3,x_2,x_4). \end{aligned}$$

(14)

Recall that we only consider lattices of the form $\Lambda =\alpha {\mathbb {Z}^{2d}}$.

Theorem 3.3

Suppose $p\in [1,\infty ]$. Let g be a window function in $\mathcal {S}(\mathbb {R}^d)$ such that $\mathcal {G}(g,\Lambda )$ is a frame for $L^2(\mathbb {R}^d)$ with a dual frame $\mathcal {G}(\gamma ,\Lambda )$. A linear continuous operator $A:\mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is bounded $M^1(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$ if and only if its distributional kernel K satisfies

$$\begin{aligned} V_G K\circ \tilde{c_1}\in \ell ^{p,\infty }(\Lambda \times \Lambda ), \end{aligned}$$

(15)

where $G=g\otimes \bar{\gamma }$.

Moreover

$$\begin{aligned} \Vert A\Vert _{M^1\rightarrow M^p}\asymp \Vert V_G K\circ \tilde{c_1}\Vert _{\ell ^{p,\infty }(\Lambda \times \Lambda )}. \end{aligned}$$

Proof

Assume first that (15) holds. For $\lambda =(\lambda _1,\lambda _2), \mu =(\mu _1,\mu _2)\in \Lambda$, observe that the Gabor matrix of the operator A is given by

$$\begin{aligned} K_{\lambda ,\mu }&:= \langle A \pi (\mu )\gamma , \pi (\lambda )g\rangle \nonumber \\&=\langle K, \pi (\lambda _1,\lambda _2)g\otimes \pi (\mu _1,-\mu _2)\bar{\gamma }\rangle \nonumber \\&=V_G K\circ \tilde{c}_1(\lambda _1,\lambda _2,\mu _1,-\mu _2). \end{aligned}$$

(16)

This matrix turns out to be the kernel of the operator $\tilde{A}:=C_g A D_\gamma$, acting on sequences on $\Lambda$, where the coefficient operator $C_g$ and synthesis operator $D_\gamma$ are defined in (10). By (15) and Proposition 3.1, $\tilde{A}$ is a bounded operator $\ell ^1(\Lambda )\rightarrow \ell ^p(\Lambda )$. Since $C_g: M^p(\mathbb {R}^d)\rightarrow \ell ^p (\Lambda )$ and $D_\gamma : \ell ^p(\Lambda )\rightarrow M^{p}(\mathbb {R}^d)$ are continuous, we have $A=D_\gamma \tilde{A} C_g: M^1(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$, with

$$\begin{aligned} \Vert A\Vert _{M^1\rightarrow M^p}\le C \sup _{\mu \in \Lambda } \Vert K_{\cdot , \mu }\Vert _{\ell ^p(\Lambda )}. \end{aligned}$$

Vice versa, consider the finite dimensional space

$$\begin{aligned} E_N =\{(a_\lambda )_{\lambda \in \Lambda }\,:\, a_\lambda =0\,\,\text{ for }\,\, |\lambda |>N\} \end{aligned}$$

and the natural projection $P_N:\,\ell ^\infty (\Lambda )\rightarrow E_N$. Consider the composition operator

$$\begin{aligned} P_N C_gAD_\gamma |_{E_N}: (E_N,\Vert \cdot \Vert _{\ell ^1})\rightarrow (E_N,\Vert \cdot \Vert _{\ell ^p}). \end{aligned}$$

(17)

This operator is represented by the (finite) matrix

$$\begin{aligned} \{K_{\lambda ,\mu }\}_{\lambda ,\mu \in \Lambda , |\lambda |\le N, |\mu |\le N}. \end{aligned}$$

Using formula (12) in Proposition 3.1 (we work on finite sets, so that the assumptions are trivially satisfied) we have

$$\begin{aligned} \sup _{\mu \in \Lambda , |\mu |\le N} \Vert K_{\lambda ,\mu }\Vert _{\ell ^p(\Lambda , |\lambda |\le N)}=\Vert P_N C_gAD_\gamma |_{E_N}\Vert _{\ell ^1\rightarrow \ell ^p}\le \Vert C_g\Vert \Vert D_\gamma \Vert \Vert A\Vert _{M^1\rightarrow M^p}. \end{aligned}$$

Since the right-hand side is independent of N, taking the supremum with respect to N, we obtain

$$\begin{aligned} \sup _{\mu \in \Lambda } \Vert K_{\cdot , \mu }\Vert _{\ell ^p(\Lambda )}\le C \Vert A\Vert _{M^1\rightarrow M^p}. \end{aligned}$$

This concludes the proof. $\square$

We observe that, given a frame $\mathcal {G}(g,\Lambda )$ for $L^2(\mathbb {R}^d)$, $\Lambda =\alpha \mathbb {Z}^{2d}$, with dual frame $\mathcal {G}(\gamma ,\Lambda )$, then $\mathcal {G}(\bar{\gamma },\Lambda )$ is still a frame for $L^2(\mathbb {R}^d)$ and $\mathcal {G}(g\otimes \bar{\gamma },\Lambda \times \Lambda )$ is a frame for $L^2({\mathbb {R}^{2d}})$. Using Theorem 2.7 we can therefore reformulate the previous result in terms of mixed modulation spaces.

Corollary 3.4

Suppose $p\in [1,\infty ]$. A linear continuous operator $A: \mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is bounded from $M^1(\mathbb {R}^d)$ into $M^p(\mathbb {R}^d)$ if and only if its distributional kernel K satisfies

$$\begin{aligned} K\in M(c_1)^{p,\infty }, \end{aligned}$$

(18)

where $c_1$ is identified with (14).

Moreover

$$\begin{aligned} \Vert A\Vert _{M^1\rightarrow M^p}\asymp \Vert K\Vert _{M(c_1)^{p,\infty }}. \end{aligned}$$

Remark 3.5

When $p=\infty$, condition (18) becomes $K\in M^\infty ({\mathbb {R}^{2d}})$ and we recapture Feichtinger’s kernel Theorem [16].

A similar characterization of bounded operators $A: M^p(\mathbb {R}^d) \rightarrow M^{\infty }(\mathbb {R}^d)$ can be obtained arguing as above, using Proposition 3.2, (16) and Theorem 2.7. Precisely, define the permutation $c_2$ of the set $\{1,\dots , 4d\}$ with related $\tilde{c}_2$ by

$$\begin{aligned} \tilde{c}_2(x_1,x_2, x_3,x_{4})= \begin{pmatrix} 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\\ I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\\ 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\end{pmatrix}\,\begin{pmatrix} x_1\\ x_{2}\\ x_{3}\\ x_{4} \end{pmatrix}=(x_3,x_1,x_4,x_2), \end{aligned}$$

(19)

for $x_i\in \mathbb {R}^d$, $i=1,\dots ,4$.

Theorem 3.6

Suppose $p\in [1,\infty ]$. A linear continuous operator $A: \mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is bounded from $M^p(\mathbb {R}^d)$ into $M^\infty (\mathbb {R}^d)$ if and only if its distributional kernel K satisfies

$$\begin{aligned} K\in M(c_2)^{p',\infty }, \end{aligned}$$

(20)

where $1/p+1/p'=1$.

Moreover

$$\begin{aligned} \Vert A\Vert _{M^p\rightarrow M^\infty }\asymp \Vert K\Vert _{M(c_2)^{p',\infty }}. \end{aligned}$$

As a consequence of Theorems 3.3 and 3.6 we have the following result.

Corollary 3.7

Let $A:\mathcal {S}(\mathbb {R}^d) \rightarrow \mathcal {S}'(\mathbb {R}^d)$ be an operator with distributional kernel $K\in \mathcal {S}'({\mathbb {R}^{2d}})$. The following conditions are equivalent:

• (i) $A: M^p(\mathbb {R}^d)\rightarrow M^p(\mathbb {R}^d)$, bounded for every $1\le p\le \infty$.

• (ii) $A: M^1(\mathbb {R}^d)\rightarrow M^1(\mathbb {R}^d)$ and $A: M^\infty (\mathbb {R}^d)\rightarrow M^\infty (\mathbb {R}^d)$ bounded.

• (iii) $K\in M(c_1)^{1,\infty }\cap M(c_2)^{1,\infty }$.

Proof

All the equivalences follow trivially from Theorems 3.3 and 3.6 but $(ii) \Rightarrow (i)$, which follows by complex interpolation for modulation spaces [14]. $\square$

4 Operators Acting on $M^{p,q}$

Let us now consider the problem of boundedness of an integral operator on $M^{p,q}(\mathbb {R}^d)$, $1\le p,q\le \infty$. We no longer expect a characterization as in the case $p=q$, studied in the previous section (see Remark 4.2 below). However sufficient conditions in the same spirit can be stated.

Consider the permutations $c_3$ and $c_4$ of the set $\{1,\dots , 4d\}$ with related $\tilde{c}_3$, $\tilde{c}_4$ defined by

$$\begin{aligned} \tilde{c}_3(x_1,x_2, x_3,x_{4})= \begin{pmatrix} 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\\ I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\end{pmatrix}\,\begin{pmatrix} x_1\\ x_{2}\\ x_{3}\\ x_{4} \end{pmatrix}=(x_2,x_3,x_1,x_4), \end{aligned}$$

(21)

and

$$\begin{aligned} \tilde{c}_4(x_1,x_2, x_3,x_{4})= \begin{pmatrix} I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d \\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\\ 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\end{pmatrix}\,\begin{pmatrix} x_1\\ x_{2}\\ x_{3}\\ x_{4} \end{pmatrix}=(x_1,x_4,x_2,x_3), \end{aligned}$$

(22)

for $x_i\in \mathbb {R}^d$, $i=1,\dots ,4$.

The following result was proved in [8, Proposition 5.1] (a continuous version is contained in [5, Proposition 2.4]).

Proposition 4.1

Consider an at most countable index set J and the operator defined on sequences $(a_{\lambda _1,\lambda _2})$ on the set $\Lambda = J\times J$ by

$$\begin{aligned} (Aa)_{\lambda _1,\lambda _2}=\sum _{\mu _1,\mu _2\in J} {K_{\lambda _1,\lambda _2,\mu _1,\mu _2}}a_{\mu _1,\mu _2}. \end{aligned}$$

• (i) If $K\circ \tilde{c_3}\in \ell ^{1,\infty ,1,\infty }(J^4)$, then A is continuous on $\ell ^{\infty ,1}(\Lambda )$. Moreover,

  $$\begin{aligned} \Vert A\Vert _{\ell ^{\infty ,1}\rightarrow \ell ^{\infty ,1}}\le \Vert K\circ \tilde{c_3}\Vert _{\ell ^{1,\infty ,1,\infty }}. \end{aligned}$$

  (23)

• (ii) If $K\circ \tilde{c_4}\in \ell ^{1,\infty ,1,\infty }(J^4)$, then A is continuous on $\ell ^{1,\infty }(\Lambda )$, with

  $$\begin{aligned} \Vert A\Vert _{\ell ^{1,\infty }\rightarrow \ell ^{1,\infty }}\le \Vert K\circ \tilde{c_4}\Vert _{\ell ^{1,\infty ,1,\infty }}. \end{aligned}$$

  (24)

• (iii) If K satisfies assumptions (i) and (ii) and moreover $K\circ c_0, K\in \ell ^{1,\infty }(\Lambda \times \Lambda )$, where the permutation $c_0$ is defined in (5), then the operator A is continuous on $\ell ^{p,q}(\Lambda )$, for every $1\le p,q\le \infty$.

Remark 4.2

Observe that the reverse inequalities in (23), (24) do not hold, even if one allows a multiplicative constant. Consider, for example, the estimate (23), with $J=\{0,1,\ldots , N-1\}$, $N\in \mathbb {N}$, where

$$\begin{aligned} K_{\lambda _1,\lambda _2,\mu _1,\mu _2}=F_{\lambda _2,\mu _1}=\frac{1}{\sqrt{N}}e^{-2\pi i \lambda _2\mu _1} \end{aligned}$$

is the Fourier matrix. We have

$$\begin{aligned} (K\circ \tilde{c_3})_{\mu _1,\lambda _1,\lambda _2,\mu _2} =K_{\lambda _1,\lambda _2,\mu _1,\mu _2} \end{aligned}$$

so that

$$\begin{aligned} \Vert K\circ \tilde{c_3}\Vert _{\ell ^{1,\infty ,1,\infty }}=\Vert F\Vert _{\ell ^1}=N^{3/2} \end{aligned}$$

but

$$\begin{aligned} \Vert A\Vert _{\ell ^{\infty ,1}\rightarrow \ell ^{\infty ,1}}\le N, \end{aligned}$$

(25)

which blows up, as $N\rightarrow +\infty$, at a lower rate.

Let us verify (25). We have

$$\begin{aligned} \left\| \left\| \sum _{\mu _1,\mu _2\in J} {K_{\lambda _1,\lambda _2,\mu _1,\mu _2}}a_{\mu _1,\mu _2}\right\| _{\ell ^\infty _{\lambda _1}}\right\| _{\ell ^1_{\lambda _2}}&=\left\| \sum _{\mu _1\in J} {F_{\lambda _2,\mu _1}}\sum _{\mu _2\in J}a_{\mu _1,\mu _2}\right\| _{\ell ^1_{\lambda _2}}\\&=\left\| \sum _{\mu _1\in J} {F_{\lambda _2,\mu _1}}\tilde{a}_{\mu _1}\right\| _{\ell ^1_{\lambda _2}} \end{aligned}$$

with $\tilde{a}_{\mu _1}:=\sum _{\mu _2\in J}a_{\mu _1,\mu _2}$. Now, using the embeddings

$$\begin{aligned} \Vert b\Vert _{\ell ^1(J)}\le \sqrt{N} \Vert b\Vert _{\ell ^2(J)},\qquad \Vert b\Vert _{\ell ^2(J)}\le \sqrt{N} \Vert b\Vert _{\ell ^\infty (J)} \end{aligned}$$

and the fact that F is a unitary transformation of $\ell ^2(J)$ we get (25).

As a consequence, we obtain the following boundedness result.

Consider the permutations $c_5$ and $c_6$ of the set $\{1,\dots , 4d\}$ with related $\tilde{c}_5$, $\tilde{c}_6$ defined by

$$\begin{aligned} \tilde{c}_5= \tilde{c}_1 \tilde{c}_3= \begin{pmatrix} 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\\ I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\end{pmatrix} \end{aligned}$$

(26)

and

$$\begin{aligned} \tilde{c}_6= \tilde{c}_1 \tilde{c}_4= \begin{pmatrix} I_d&{}\quad 0_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad I_d&{}\quad 0_d&{}\quad 0_d\\ 0_d&{}\quad 0_d&{}\quad 0_d&{}\quad I_d\\ 0_d&{}\quad 0_d&{}\quad I_d&{}\quad 0_d\end{pmatrix}. \end{aligned}$$

(27)

Theorem 4.3

• (i) A linear continuous operator $A: \mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is bounded on $M^{\infty ,1}(\mathbb {R}^d)$ if its distributional kernel K satisfies

  $$\begin{aligned} K\in M(c_5)^{1,\infty ,1,\infty }. \end{aligned}$$

  (28)

• (ii) A linear continuous operator $A: \mathcal {S}(\mathbb {R}^d)\rightarrow \mathcal {S}'(\mathbb {R}^d)$ is bounded on $M^{1,\infty }(\mathbb {R}^d)$ if its distributional kernel K satisfies

  $$\begin{aligned} K\in M(c_6)^{1,\infty ,1,\infty }. \end{aligned}$$

  (29)

Proof

The result is a consequence of (16), Proposition 4.1 and Theorem 2.7, by using the same arguments as in the first part of the proof of Theorem 3.3. $\square$

By interpolation we also obtain the following result.

Corollary 4.4

Let $A:\mathcal {S}(\mathbb {R}^d) \rightarrow \mathcal {S}'(\mathbb {R}^d)$ be an operator with distributional kernel

$$\begin{aligned} K\in M(c_5)^{1,\infty ,1,\infty }\cap M(c_6)^{1,\infty ,1,\infty }. \end{aligned}$$

Then $A: M^{p',p}(\mathbb {R}^d)\rightarrow M^{p',p}(\mathbb {R}^d)$ continuously, for every $1\le p\le \infty$, with $1/p+1/p'=1$.

By combining Corollaries 3.7 and 4.4, we finally obtain the following result.

Corollary 4.5

Let $A:\mathcal {S}(\mathbb {R}^d) \rightarrow \mathcal {S}'(\mathbb {R}^d)$ be an operator with distributional kernel

$$\begin{aligned} K\in M(c_1)^{1,\infty }\cap M(c_2)^{1,\infty }\cap M(c_5)^{1,\infty ,1,\infty }\cap M(c_6)^{1,\infty ,1,\infty }. \end{aligned}$$

Then $A: M^{p,q}(\mathbb {R}^d)\rightarrow M^{p,q}(\mathbb {R}^d)$ continuously, for every $1\le p,q\le \infty$.