Compactness Properties for Modulation Spaces

Abstract

We prove that if $\omega _1$ and $\omega _2$ are moderate weights and $\mathscr {B}$ is a suitable (quasi-)Banach function space, then a necessary and sufficient condition for the embedding $i\, :\, M (\omega _1,\mathscr {B})\rightarrow M (\omega _2,\mathscr {B})$ between two modulation spaces to be compact is that the quotient $\omega _2/\omega _1$ vanishes at infinity. Moreover we show, that the boundedness of $\omega _2/\omega _1$ is a necessary and sufficient condition for the previous embedding to be continuous.

1 Introduction

A fundamental question in analysis, science and engineering concerns compactness. In the paper we investigate continuity and compactness properties for a broad class of modulation spaces, a family of functions and distribution spaces on the configuration space $\mathbf R^{d}$, introduced by Feichtinger [12]. The basic theory of these spaces was thereafter established by Feichtinger and Gröchenig (see [15, 16, 25] and the references therein).

A modulation space is defined by imposing a certain norm or quasi-norm estimate on the short-time Fourier transform of the involved functions and distributions. (See [28] or Sect. 2 for definitions and notations.) Roughly speaking, this means that each modulation space norm measures in a certain way the phase or time-frequency content. That is, it admits to measure the configuration energy and momentum energy simultaneously, in a certain way.

By the design of these spaces it turns out that they are useful in several fields in analysis, physics and engineering (see [6, 8, 31, 34] and the references therein).

Compactness investigations of embeddings between modulation spaces go in some sense back to [32], where M. Shubin proved that if $t>0$, then the embedding $i\! : \! Q_s\rightarrow Q_{s-t}$ is compact. In the community, the previous compactness property was not obvious because similar facts do not hold for Sobolev spaces of Hilbert type. That is, for $t>0$ it is well-known that the embedding $i\! : \! H_s^2\rightarrow H_{s-t}^2$ is continuous but not compact. Since

$$\begin{aligned} Q_s&= M^{2,2}_{(\omega )}, \quad \omega (X) = (1+|x|+|\xi |)^s \end{aligned}$$

(1.1)

and

$$\begin{aligned} H^2_s&= M^{2,2}_{(\omega )}, \quad \omega (X) = (1+|\xi |)^s, \quad X=(x,\xi ) \end{aligned}$$

(1.2)

the previous compact embedding properties can also be written by means of modulation spaces. A more general situation was considered by M. Dörfler, H. Feichtinger and K. Gröchenig who considered the inclusion map

$$\begin{aligned} i\! : \! M^{p,q}_{(\omega _1)}(\mathbf R^{d})\rightarrow M^{p,q}_{(\omega _2)}(\mathbf R^{d}) \end{aligned}$$

(1.3)

in [9]. It is well-known that this map is well-defined and continuous when $\omega _2\lesssim \omega _1$, see e.g. [25]. In [9, Theorem 5] it was proved that if $p,q\in [1,\infty )$, and $\omega _1$ and $\omega _2$ are certain moderate weights of polynomial type, then (1.3) is compact if and only if $\omega _2/\omega _1$ tends to zero at infinity. By choosing $\omega _j$ in similar ways as in (1.1), the latter compactness result confirms the compactness of the embedding $i\! : \! Q_s\rightarrow Q_{s-t}$ above by Shubin, as well as it confirms the lack of compactness of the embedding $i\! : \! H^2_s\rightarrow H^2_{s-t}$ for Sobolev spaces.

In [3], the compact embedding property [9, Theorem 5] was extended in such a way that all moderate weights $\omega _j$ of polynomial type are included. That is, there are no other restrictions on $\omega _j$ than there should exist constants $N_j>0$ such that

$$\begin{aligned} \omega _j(X+Y)\lesssim \omega _j(X)(1+|Y|)^{N_j},\quad X,Y\in \mathbf R^{2d}, \quad j=1,2. \end{aligned}$$

(1.4)

Moreover, in [3], the Lebesgue exponents p and q in the modulation spaces are allowed to attain $\infty$.

In our main results, Theorems 3.7 and 3.9 in Sect. 3, these continuity and compactness results are extended to involve modulation spaces $M(\omega ,\mathscr {B})$, which are more general in different ways. Firstly, there are no boundedness estimates of polynomial type for the involved weight $\omega$. In most of our considerations, we require that the weights are moderate, which implies that the condition (1.4) is relaxed into

$$\begin{aligned} \omega _j(X+Y)\lesssim \omega _j(X)e^{r_j|Y|},\quad j=1,2, \end{aligned}$$

for some constants $r_1,r_2>0$.

Secondly, $\mathscr {B}$ can be any general translation invariant Banach function space without the restriction that $M(\omega ,\mathscr {B})$ should be of the form $M^{p,q}_{(\omega )}$. We may also have $M(\omega ,\mathscr {B})=M^{p,q}_{(\omega )}$, but in contrast to [3, 9], we here allow p and q to be smaller than 1. Here we notice that if $p<1$ or $q<1$, then $M^{p,q}_{(\omega )}$ fails to be a Banach space because of absence of convex topological structures.

Thirdly, we show that (1.3) is compact when $\omega _2/\omega _1$ tends to zero at infinity, when the assumptions on $\omega _1$ and $\omega _2$ are relaxed into a suitable local moderate condition (cf. Theorem 3.9(1) in Sect. 3). We refer to [40] and to some extent to [42] for a detailed study of modulation spaces with such relaxed conditions on the involved weight functions.

In Sect. 4 we show how Theorem 3.9 has been applied in [1] to deduce index results and lifting properties for certain pseudo-differential operators. Especially it is here indicated in which way our results on compactness are used to show that the operator $\hbox {Op}_A(\omega _0)$ is continuous and bijective from $M^{p,q}_{(\omega )}$ to $M^{p,q}_{(\omega /\omega _0)}$ when

$$\begin{aligned} \omega _0(x,\xi )\equiv p_1(x)^r+p_2(\xi )^\rho , \end{aligned}$$

$r,\rho >0$, and $p_1$ and $p_2$ are positive polynomials on $\mathbf R^{d}$.

In Sect. 4 we also give some links on possible extensions and generalizations.

Finally we remark that for moderate weights, the continuity and compactness properties for (1.3) can also be obtained by Gabor analysis, which transfers (1.3) into

$$\begin{aligned} i\! : \! \ell ^{p,q}_{(\omega _1)}\rightarrow \ell ^{p,q}_{(\omega _2)}, \end{aligned}$$

Since it is clear that the latter inclusion map is compact, if and only if $\omega _2/\omega _1$ tends to 0 at infinity, it follows that the compactness results in [3, 32] as well as some of the results in Sect. 3 can be deduced in such ways. We emphasise however that such technique can not be used in those situations in Sect. 3 when modulation spaces are of the form $M(\omega ,\mathscr {B})$, where either $\mathscr {B}$ is a general BF-space, or $\omega$ fails to be moderate, since it seems that Gabor analysis is not applicable on such modulation spaces.

Parts of the proof of these continuity and compactness results are based on some properties for modulation spaces which might be of independent interests. We prove that if $\mathscr {B}$ is a Banach Function space, then $M(\omega ,\mathscr {B})$ is a Banach space, which is continuously embedded in a weighted modulation space of the form $M^{\infty }_{(1/v)}(\mathbf R^{d})$ for certain weights v. (See Proposition 3.2.) We also need some properties for the Bargmann transform when acting on modulation spaces, essentially deduced in [40, 42], which are presented in Sect. 2.5.

2 Preliminaries

In this section we discuss basic properties for modulation spaces and other related spaces. The proofs are in many cases omitted since they can be found in [10,11,12, 15,16,17, 25, 36,37,39].

2.1 Weight Functions

A weight or weight function$\omega$ on $\mathbf R^{d}$ is a positive function such that $\omega ,1/\omega \in L^\infty _{loc}(\mathbf R^{d})$. Let $\omega$ and v be weights on $\mathbf R^{d}$. Then $\omega$ is called v-moderate or moderate, if

$$\begin{aligned} \omega (x_1+x_2)\lesssim \omega (x_1) v(x_2),\quad x_1,x_2\in \mathbf R^{d}. \end{aligned}$$

(2.1)

Here $f(\theta )\lesssim g(\theta )$ means that $f(\theta )\le cg(\theta )$ for some constant $c>0$ which is independent of $\theta$ in the domain of f and g. If v can be chosen as a polynomial, then $\omega$ is called a weight of polynomial type.

The function v is called submultiplicative, if it is even and (2.1) holds for $\omega =v$. We notice that (2.1) implies that if v is submultiplicative on $\mathbf R^{d}$, then there is a constant $c>0$ such that $v(x)\ge c$ when $x\in \mathbf R^{d}$.

We let $\mathscr {P}_E(\mathbf R^{d})$ be the set of all moderate weights on $\mathbf R^{d}$, and $\mathscr {P}(\mathbf R^{d})$ be the subset of $\mathscr {P}_E(\mathbf R^{d})$ which consists of all polynomially moderate functions on $\mathbf R^{d}$. For $s>0$, also let $\mathscr {P}_{E,s}(\mathbf R^{d})$ ($\mathscr {P}_{E,s}^0(\mathbf R^{d})$) be the set of all weights $\omega$ in $\mathbf R^{d}$ such that

$$\begin{aligned} \omega (x_1+x_2)\lesssim \omega (x_1) e^{r|x_2|^{\frac{1}{s}}},\quad x_1,x_2\in \mathbf R^{d}. \end{aligned}$$

(2.2)

for some $r>0$ (for every $r>0$), and set $\mathscr {P}_E^0=\mathscr {P}_{E,1}^0$. We have

$$\begin{aligned} \mathscr {P}&\subseteq \mathscr {P}_{E,s_1}^0\subseteq \mathscr {P}_{E,s_1}\subseteq \mathscr {P}_{E,s_2}^0\subseteq \mathscr {P}_E \quad \text {when} \,\, s_2<s_1 \end{aligned}$$

and

$$\begin{aligned} \mathscr {P}_{E,s}&= \mathscr {P}_E \quad \text {when} \quad s\le 1, \end{aligned}$$

where the last equality follows from the fact that if $\omega \in \mathscr {P}_E(\mathbf R^{d})$ ($\omega \in \mathscr {P}_E^0(\mathbf R^{d})$), then

$$\begin{aligned} \omega (x+y)\lesssim \omega (x) e^{r|y|^{\frac{1}{s}}} \quad \text {and}\quad e^{-r|x|}\le \omega (x)\lesssim e^{r|x|},\quad x,y\in \mathbf R^{d} \end{aligned}$$

(2.3)

hold true for some $r>0$ (for every $r>0$) in view of [26] when s $\le 1.$

In some situations we shall consider a more general class of weights compared to $\mathscr {P}_E$ given in [40, Definition 1.1].

Definition 2.1

The set $\mathscr {P}_{Q}(\mathbf R^{d})$ consists of all weights $\omega$ on $\mathbf R^{d}$ such that for some constants $R\ge 2$ and $c,r>0$ it holds

$$\begin{aligned} e^{-r|x|^2}\lesssim \omega (x)\lesssim e^{r|x|^2}, \end{aligned}$$

(2.4)

and

$$\begin{aligned} \omega (x)^2\lesssim \omega (x+y)\omega (x-y)\lesssim \omega (x)^2 \quad \text {when}\quad Rc \le |x|\le \frac{c}{|y|}. \end{aligned}$$

(2.5)

2.2 Gelfand–Shilov Spaces

Let $0<h,s\in \mathbf R$. Then we denote the set of all functions $f\in C^\infty (\mathbf R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal S_{s,h}}\equiv \sup \frac{|x^\beta \partial ^\alpha f(x)|}{h^{|\alpha + \beta |}(\alpha !\, \beta !)^s} < \infty \end{aligned}$$

(2.6)

by $\mathcal S_{s,h}(\mathbf R^{d})$. Here the supremum is taken over all $\alpha ,\beta \in \mathbf N^d$ and $x\in \mathbf R^{d}$.

It is clear that $\mathcal S_{s,h}$ is a Banach space which increases with h and s, and that $\mathcal S_{s,h}\hookrightarrow \mathscr {S}$ holds. Here $A\hookrightarrow B$ means that $A\subseteq B$ with continuous embedding. If $s>\frac{1}{2}$, then

$$\begin{aligned} \mathcal S_{s,h} \quad \text {and}\quad \bigcup \limits _{h>0} S_{1/2,h} \end{aligned}$$

contains all finite linear combinations of the Hermite functions. By the density of such linear combinations in $\mathscr {S}$ and in $\mathcal S_{s,h}$, the dual $\mathcal S_{s,h}'(\mathbf R^{d})$ of $\mathcal S_{s,h}(\mathbf R^{d})$ is a Banach space which contains $\mathscr {S}'(\mathbf R^{d})$, for such choices of s.

The inductive and projective limits respectively of $\mathcal S_{s,h}(\mathbf R^{d})$ are called Gelfand–Shilov spaces of Beurling respectively Roumieu type and are denoted by $\mathcal S_s(\mathbf R^{d})$ and $\Sigma _s(\mathbf R^{d})$. Hence

$$\begin{aligned} \mathcal S_s(\mathbf R^{d}) = \bigcup _{h>0}\mathcal S_{s,h}(\mathbf R^{d}) \quad \text {and}\quad \Sigma _s(\mathbf R^{d}) =\bigcap _{h>0} \mathcal S_{s,h}(\mathbf R^{d}), \end{aligned}$$

(2.7)

where the topology for $\mathcal S_s(\mathbf R^{d})$ is the strongest possible one such that the inclusion map from $\mathcal S_{s,h} (\mathbf R^{d})$ to $\mathcal S_s(\mathbf R^{d})$ is continuous, for every choice of $h>0$. Equipped with the seminorms $\Vert \, \cdot \, \Vert _{\mathcal S_{s,h}}$, $h>0$ the space $\Sigma _s(\mathbf R^{d})$ is a Fréchet space. Additionally, $\Sigma _s(\mathbf R^{d})\ne \{ 0\}$, if and only if $s>\frac{1}{2}$, and $\mathcal S_s(\mathbf R^{d})\ne \{ 0\}$, if and only if $s\ge \frac{1}{2}$.

The Gelfand–Shilov distribution spaces$\mathcal S_s'(\mathbf R^{d})$ and $\Sigma _s'(\mathbf R^{d})$ are the projective and inductive limit respectively of $\mathcal S_{s,h}'(\mathbf R^{d})$. This implies that

Note that $\mathcal S_s'(\mathbf R^{d})$ is the dual of $\mathcal S_s(\mathbf R^{d})$, and $\Sigma _s'(\mathbf R^{d})$ is the dual of $\Sigma _s(\mathbf R^{d})$ as proved in [22].

By the definitions we have

$$\begin{aligned}&\mathcal S_{1/2}\hookrightarrow \Sigma _s \hookrightarrow \mathcal S_s \hookrightarrow \Sigma _{s+\varepsilon } \hookrightarrow \mathscr {S}\nonumber \\&\qquad \hookrightarrow \mathscr {S}' \hookrightarrow \Sigma _{s+\varepsilon }' \hookrightarrow \mathcal S_s' \hookrightarrow \Sigma _s' \hookrightarrow \mathcal S_{1/2} ',\quad s>\frac{1}{2},\ \varepsilon >0. \end{aligned}$$

(2.8)

We let $\langle \, \cdot \, ,\, \cdot \, \rangle$ denote the dual form between a topological vector space and its dual. If $z,w\in \mathbf C^{d}$, then $\langle z,w\rangle$ is defined by

$$\begin{aligned} \langle z,w\rangle = \sum _{j=1}^dz_jw_j,\quad z= (z_1,\ldots ,z_d)\in \mathbf C^{d},\ w= (w_1,\ldots ,w_d)\in \mathbf C^{d}. \end{aligned}$$

The Fourier transform of $f\in L^1(\mathbf R^{d})$ is given by

$$\begin{aligned} (\mathscr {F}f)(\xi )= \widehat{f}(\xi ) \equiv (2\pi )^{-\frac{d}{2}}\int _{\mathbf R^{d}} f(x)e^{-i\langle x,\xi \rangle }\, dx. \end{aligned}$$

The map $\mathscr {F}$ extends uniquely to homeomorphisms on $\mathscr {S}'(\mathbf R^{d})$, $\mathcal S_s'(\mathbf R^{d})$ and on $\Sigma _s'(\mathbf R^{d})$. Furthermore, $\mathscr {F}$ restricts to homeomorphisms on $\mathscr {S}(\mathbf R^{d})$, $\mathcal S_s(\mathbf R^{d})$, $\Sigma _s(\mathbf R^{d})$, and to a unitary operator on $L^2(\mathbf R^{d})$. Similar results hold true for partial Fourier transforms.

For a fixed $\phi \in \mathcal S_s (\mathbf R^{d})$ the short-time Fourier transform$V_\phi f$ of $f\in \mathcal S_s ' (\mathbf R^{d})$ with respect to the window function$\phi$ is the Gelfand–Shilov distribution on $\mathbf R^{2d}$, defined by

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-\frac{d}{2}} \langle f,\overline{\phi (\, \cdot \, -x)}e^{-i\langle \, \cdot \, ,\xi \rangle }\rangle \end{aligned}$$

If instead $f\in \Sigma _s'(\mathbf R^{d})$ or $f\in \mathscr {S}'(\mathbf R^{d})$, then $\phi$ should belong to $\Sigma _s(\mathbf R^{d})$ or $\mathscr {S}(\mathbf R^{d})$, respectively. We have

$$\begin{aligned} V_\phi f(x,\xi ) \equiv (\mathscr {F}_2 (U(f\otimes \phi )))(x,\xi ) = \mathscr {F}(f \, \overline{\phi (\, \cdot \, -x)})(\xi ), \end{aligned}$$

(2.9)

where $(UF)(x,y)=F(y,y-x)$. Here $\mathscr {F}_2F$ denotes the partial Fourier transform of $F(x,y)\in \mathcal S_s'(\mathbf R^{2d})$ with respect to the y variable (see (A.1) in [5]).

In the case $f \in \mathcal S_s (\mathbf R^{d})$, $V_\phi f$ can be written as

$$\begin{aligned} V_\phi f(x,\xi ) = (2\pi )^{-\frac{d}{2}}\int f(y)\overline{\phi (y-x)}e^{-i\langle y,\xi \rangle }\, dy. \end{aligned}$$

The next two propositions characterize Gelfand–Shilov functions and their distributions by suitable estimates on the short-time Fourier transforms of the corresponding distributions. The proofs are omitted since the results are special cases of [27, Theorem 2.7]) and [42, Proposition 2.2].

Proposition 2.2

Let $s,s_0\ge \frac{1}{2}$ be such that $s_0\le s$. Also let $\phi \in \mathcal S_{s_0}(\mathbf R^{d}){\setminus } 0$ and $f\in \mathcal S_{s_0}'(\mathbf R^{d})$. Then the following is true:

• (1) $f\in \mathcal S_s(\mathbf R^{d})$, if and only if

  $$\begin{aligned} |V_\phi f(x,\xi )| \lesssim e^{-r (|x|^{\frac{1}{s}}+|\xi |^{\frac{1}{s}})} \end{aligned}$$

  (2.10)

  holds for some $r > 0$;

• (2) if in addition $s_0>\frac{1}{2}$ and $\phi \in \Sigma _{s_0}(\mathbf R^{d})$, then $f\in \Sigma _s(\mathbf R^{d})$, if and only if (2.10) holds for every $r > 0$.

Proposition 2.3

Let $s,s_0\ge \frac{1}{2}$ be such that $s_0\le s$. Also let $\phi \in \mathcal S_{s_0}(\mathbf R^{d}){\setminus } 0$ and $f\in \mathcal S_{s_0}'(\mathbf R^{d})$. Then the following is true:

• (1) $f\in \mathcal S_s'(\mathbf R^{d})$, if and only if

  $$\begin{aligned} |V_\phi f(x,\xi )| \lesssim e^{r(|x|^{\frac{1}{s}}+|\xi |^{\frac{1}{s}})} \end{aligned}$$

  (2.11)

  holds for every $r > 0$;

• (2) if in addition $s_0>\frac{1}{2}$ and $\phi \in \Sigma _{s_0}(\mathbf R^{d})$, then $f\in \Sigma _s'(\mathbf R^{d})$, if and only if (2.11) holds for some $r > 0$.

Remark 2.4

For every $s>0$, the mapping $(f,\phi )\mapsto V_\phi f$ is continuous from $\mathcal S_s(\mathbf R^{d})\times \mathcal S_s(\mathbf R^{d})$ to $\mathcal S_s(\mathbf R^{2d})$ and extends uniquely to continuous mappings from $\mathcal S_s'(\mathbf R^{d})\times \mathcal S_s'(\mathbf R^{d})$ to $\mathcal S_s'(\mathbf R^{2d})$. The same is true if we replace each $\mathcal S_s$ by $\mathscr {S}$ or by $\Sigma _s$. This is admitted by formula (2.9) (cf. e. g. [35, 42]).

2.3 Modulation Spaces

We recall that a quasi-norm $\Vert \, \cdot \, \Vert _{\mathscr {B}}$ of order $r \in (0,1]$ on the vector-space $\mathscr {B}$ is a nonnegative functional on $\mathscr {B}$ which satisfies

$$\begin{aligned} \Vert f+g\Vert _{\mathscr {B}}&\le 2^{\frac{1}{r}-1}(\Vert f\Vert _{\mathscr {B}} + \Vert g\Vert _{\mathscr {B}}), \quad f,g \in \mathscr {B}, \nonumber \\ \Vert \alpha \cdot f \Vert _{\mathscr {B}}&= |\alpha | \cdot \Vert f\Vert _{\mathscr {B}}, \quad \alpha \in \mathbf {C}, \quad f \in \mathscr {B}\end{aligned}$$

(2.12)

and

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}}&= 0\quad \Leftrightarrow \quad f=0. \end{aligned}$$

The vector space $\mathscr {B}$ is called a quasi-Banach space if it is a complete quasi-normed space. If $\mathscr {B}$ is a quasi-Banach space with quasi-norm satisfying (2.12) then by [2, 29] there is an equivalent quasi-norm to $\Vert \, \cdot \, \Vert _{\mathscr {B}}$ which additionally satisfies

$$\begin{aligned} \Vert f+g\Vert _{\mathscr {B}}^r \le \Vert f\Vert _{\mathscr {B}}^r + \Vert g\Vert _{\mathscr {B}}^r, \quad f,g \in \mathscr {B}. \end{aligned}$$

(2.13)

From now on we always assume that the quasi-norm of the quasi-Banach space $\mathscr {B}$ is chosen in such a way that both (2.12) and (2.13) hold.

Let $\phi \in \Sigma _1(\mathbf R^{d}){\setminus } 0$, $p,q\in (0,\infty ]$ and $\omega \in \mathscr {P}_E(\mathbf R^{2d})$. Then the modulation space$M^{p,q}_{(\omega )}(\mathbf R^{d})$ is defined as the set of all $f\in \Sigma _1'(\mathbf R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}}\equiv \Big (\int \Big (\int |V_\phi f(x,\xi )\omega (x,\xi )|^p\, dx\Big )^{q/p}\, d\xi \Big )^{1/q} <\infty \end{aligned}$$

(2.14)

holds. We set $M^p_{(\omega )}=M^{p,p}_{(\omega )}$, and if $\omega =1$, we set $M^{p,q}=M^{p,q}_{(\omega )}$ and $M^{p}=M^{p}_{(\omega )}$.

We summarize some well-known facts about Modulation spaces in the next proposition. See [12, 21, 25, 41] for the proof. Here the conjugate exponent of $p\in (0,\infty ]$ is given by

$$\begin{aligned} p' = {\left\{ \begin{array}{ll} \infty &{} \text {when}\ p\in (0,1], \\ \displaystyle {\frac{p}{p-1}} &{} \text {when}\ p\in (1,\infty ), \\ 1 &{} \text {when}\ p=\infty . \end{array}\right. } \end{aligned}$$

Proposition 2.5

Let $p,q,p_j,q_j,r\in (0,\infty ]$ be such that $r\le \min (1,p,q)$, $j=1,2$, let $\omega ,\omega _1,\omega _2,v\in \mathscr {P}_E(\mathbf R^{2d})$ be such that $\omega$ is v-moderate, $\phi \in M^r_{(v)}(\mathbf R^{d}){\setminus } 0$ and let $f\in \Sigma _1'(\mathbf R^{d})$. Then the following is true:

• (1) $f\in M^{p,q}_{(\omega )}(\mathbf R^{d})$ if and only if (2.14) holds. Moreover, $M^{p,q}_{(\omega )}$ is a quasi-Banach space under the quasi-norm in (2.14) and a Banach space if $p,q \ge 1$. Different choices of $\phi$ give rise to equivalent (quasi-)norms;

• (2) if $p_1\le p_2$, $q_1\le q_2$ and $\omega _2\lesssim \omega _1$, then

  $$\begin{aligned} \Sigma _1(\mathbf R^{d})&\subseteq M^{p_1,q_1}_{(\omega _1)}(\mathbf R^{d}) \subseteq M^{p_2,q_2}_{(\omega _2)}(\mathbf R^{d})\subseteq \Sigma _1'(\mathbf R^{d}); \end{aligned}$$

• (3) if in addition $p,q\ge 1$, then the $L^2$-form on $\Sigma _1(\mathbf R^{d})$ extends uniquely to a continuous sesqui-linear form on $M^{p,q}_{(\omega )}(\mathbf R^{d})\times M^{p',q'}_{(1/\omega )}(\mathbf R^{d})$. Furthermore, if $p,q<\infty$, then the dual of $M^{p,q}_{(\omega )}(\mathbf R^{d})$ can be identified by $M^{p',q'}_{(1/\omega )}(\mathbf R^{d})$ through this form.

2.4 A Broader Family of Modulation Spaces

In this subsection we introduce a broader class of modulation spaces by imposing certain types of translation invariant solid BF-space norms on the short-time Fourier transform, cf. [12,13,14,15,16].

Definition 2.6

Let $\mathscr {B}\subseteq L^r_{loc}(\mathbf R^{d})$ be a quasi-Banach space of order $r\in (0,1]$ which contains $\Sigma _1(\mathbf R^{d})$ with continuous embedding, and let $v _0\in \mathscr {P}_E(\mathbf R^{d})$. Then $\mathscr {B}$ is called a translation invariant Quasi-Banach Function space on$\mathbf R^{d}$ (with respect to $v_0$), or invariant QBF space on$\mathbf R^{d}$ of order r, if there is a constant C such that:

• (1) if $x\in \mathbf R^{d}$ and $f\in \mathscr {B}$, then $f(\, \cdot \, -x)\in \mathscr {B}$, and

  $$\begin{aligned} \Vert f(\, \cdot \, -x)\Vert _{\mathscr {B}}\le Cv_0(x)\Vert f\Vert _{\mathscr {B}}; \end{aligned}$$

  (2.15)

• (2) if $f,g\in L^r_{loc}(\mathbf R^{d})$ satisfy $g\in \mathscr {B}$ and $|f| \le |g|$, then $f\in \mathscr {B}$ and

  $$\begin{aligned} \Vert f\Vert _{\mathscr {B}}\le C\Vert g\Vert _{\mathscr {B}}. \end{aligned}$$

If the weight $v_0$ even is an element of $\mathscr {P}_{E,s}(\mathbf R^{d})$ ($\mathscr {P}_{E,s}^0(\mathbf R^{d})$), then we call $\mathscr {B}$ of Definition 2.6 an invariant QBF-space of Roumieu type (Beurling type) of order r.

By Definition 2.6(2) we have $f\cdot h\in \mathscr {B}$ and

$$\begin{aligned} \Vert f\cdot h\Vert _{\mathscr {B}}\le C\Vert f\Vert _{\mathscr {B}}\Vert h\Vert _{L^\infty }, \end{aligned}$$

(2.16)

when $f\in \mathscr {B}$ and $h\in L^\infty$.

The QBF space $\mathscr {B}$ is called is called an invariant BF-space (with respect to $v_0$), if it is of order 1, is continuously embedded in $\Sigma _1'(\mathbf R^{d})$ and the map $(f,\varphi )\mapsto f*\varphi$ is well-defined and continuous from $\mathscr {B}\times L^1_{(v_0)}(\mathbf R^{d})$ to $\mathscr {B}$. Note here that an invariant QBF space of order $r=1$ is a Banach space. Because of condition (2) an invariant BF-space is a solid BF-space in the sense of (A.3) in [13]. For the invariant BF-space $\mathscr {B}\subseteq L^1_{loc}(\mathbf R^{d})$ with respect to $v_0$ we have Minkowski’s inequality, i. e.

$$\begin{aligned} \Vert f*\varphi \Vert _{\mathscr {B}}\le C \Vert f\Vert _{\mathscr {B}}\Vert \varphi \Vert _{L^1_{(v)}}, \quad f\in \mathscr {B},\ \varphi \in L^1_{(v_0)} (\mathbf R^{d}) \end{aligned}$$

(2.17)

for some $C>0$ which is independent of $f\in \mathscr {B}$ and $\varphi \in L^1_{(v_0)} (\mathbf R^{d})$.

The following result shows that $v_0$ in Definition 2.6 can be replaced by a submultiplicative weight v such that (2.15) is true with v in place of $v_0$ and the constant $C=1$, and such that

$$\begin{aligned} v(x+y)\le v(x)v(y), \quad x,y\in \mathbf R^{d}. \end{aligned}$$

(2.18)

Proposition 2.7

Let $\mathscr {B}$ be an invariant BF-space on $\mathbf R^{d}$ with respect to $v_0\in \mathscr {P}_E(\mathbf R^{d})$. Then there is a submultiplicative $v\in \mathscr {P}_E(\mathbf R^{d})$ which satisfies (2.15) and (2.18) with v in place of $v_0$, and $C=1$.

Proof

Let

$$\begin{aligned} v_1(x) \equiv \sup _{f\in \mathscr {B}} \left( \frac{\Vert f(\, \cdot \, -x)\Vert _{\mathscr {B}}}{\Vert f\Vert _{\mathscr {B}}} \right) . \end{aligned}$$

Then

$$\begin{aligned} v_1(x+y)= & {} \sup _{f\in \mathscr {B}}\left( \frac{\Vert f(\, \cdot \, -x-y)\Vert _{\mathscr {B}}}{\Vert f(\, \cdot \, -y)\Vert _{\mathscr {B}}}\cdot \frac{\Vert f(\, \cdot \, -y)\Vert _{\mathscr {B}}}{\Vert f\Vert _{\mathscr {B}}}\right) \\\le & {} \sup _{f\in \mathscr {B}} \left( \frac{\Vert f(\, \cdot \, -x)\Vert _{\mathscr {B}}}{\Vert f\Vert _{\mathscr {B}}} \right) \cdot \sup _{f\in \mathscr {B}}\left( \frac{\Vert f(\, \cdot \, -y)\Vert _{\mathscr {B}}}{\Vert f\Vert _{\mathscr {B}}}\right) = v_1(x)v_1(y). \end{aligned}$$

The result now follows by letting

$$\begin{aligned} v(x)=\max (v_1(x),v_1(-x)). \end{aligned}$$

$\square$

From now on it is assumed that v and $v_j$ are submultiplicative weights if nothing else is stated.

Example 2.8

For $p,q\in [1,\infty ]$ the space $L^{p,q}_1(\mathbf R^{2d})$ consists of all $f\in L^1_{loc}(\mathbf R^{2d})$ such that

$$\begin{aligned} \Vert f\Vert _{L^{p,q}_1} \equiv \Big ( \int \Big ( \int |f(x,\xi )|^p\, dx\Big )^{q/p}\, d\xi \Big )^{1/q} < \infty . \end{aligned}$$

Additionally $L^{p,q}_2(\mathbf R^{2d})$ is the set of all $f\in L^1_{loc}(\mathbf R^{2d})$ such that

$$\begin{aligned} \Vert f\Vert _{L^{p,q}_2} \equiv \Big ( \int \Big ( \int |f(x,\xi )|^q\, d\xi \Big )^{p/q}\, dx \Big )^{1/p} < \infty . \end{aligned}$$

Then $L^{p,q}_1$ and $L^{p,q}_2$ are translation invariant BF-spaces with respect to $v=1$.

Next we define the extended class of modulation spaces, which are of main interest for us.

Definition 2.9

Let $\mathscr {B}$ be a translation invariant QBF-space on $\mathbf R^{2d}$, $\omega \in \mathscr {P}_E(\mathbf R^{2d})$, and $\phi \in \Sigma _1(\mathbf R^{d}){\setminus } 0$. Then the set $M(\omega ,\mathscr {B})$ consists of all $f\in \Sigma _1'(\mathbf R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{M(\omega ,\mathscr {B})} \equiv \Vert V_\phi f\, \omega \Vert _{\mathscr {B}} <\infty . \end{aligned}$$

Obviously, we have $M^{p,q}_{(\omega )}(\mathbf R^{d})=M(\omega ,\mathscr {B})$ if $\mathscr {B}=L^{p,q}_1(\mathbf R^{2d})$, see e.g. (2.14). We remark that many properties of the classical modulation spaces are also true for $M(\omega ,\mathscr {B})$. For instance, the definition of $M(\omega ,\mathscr {B})$ is independent of the choice of $\phi$ when $\mathscr {B}$ is a Banach space. This statement is formulated in the next proposition. We omit the proof since it can be proved by similar arguments as in Proposition 11.3.2 in [25].

Proposition 2.10

Let $\mathscr {B}$ be an invariant BF-space with respect to $v_0\in \mathscr {P}_E(\mathbf R^{2d})$ and let $\omega ,v\in \mathscr {P}_E(\mathbf R^{2d})$ be such that $\omega$ is v-moderate. Also let $M(\omega ,\mathscr {B})$ be the same as in Definition 2.9, and let $\phi \in M^1_{(v_0v)}(\mathbf R^{d}){\setminus } 0$ and $f\in \Sigma _1'(\mathbf R^{d})$. Then $f\in M(\omega ,\mathscr {B})$ if and only if $V_\phi f\cdot \omega \in \mathscr {B}$, and different choices of $\phi$ gives rise to equivalent norms of $M(\omega ,\mathscr {B})$.

We refer to [12, 15,16,17, 21, 25, 30, 41] for more facts about modulation spaces.

In applications, $\mathscr {B}$ is mostly a mixed quasi-normed Lebesgue space, which is defined next. Let $E= \{ e_1,\ldots ,e_d \}$ be an ordered basis of $\mathbf R^{d}$ and let $E'=\{ e'_1,\ldots ,e'_d \}$ be such that

$$\begin{aligned} \langle e_j,e'_k\rangle = 2\pi \delta _{jk}, \quad j,k =1,\ldots , d. \end{aligned}$$

Then $E'$ is called the dual basis of E. The corresponding lattice and dual lattice are

$$\begin{aligned} \Lambda _E&=\{ \, j_1e_1+\cdots +j_de_d\, ;\, (j_1,\ldots ,j_d)\in \mathbf Z^{d}\, \} \\ \end{aligned}$$

and

$$\begin{aligned} \Lambda '_E&= \Lambda _{E'}=\{ \, \iota _1e'_1+\cdots +\iota _de'_d\, ;\, (\iota _1,\ldots ,\iota _d) \in \mathbf Z^{d}\, \} . \end{aligned}$$

We also let $\kappa (E)$ be the parallelepiped spanned by the basis E.

We define for each ${\varvec{q}}=(q_1,\ldots ,q_d)\in (0,\infty ]^d$,

$$\begin{aligned} \max ({\varvec{q}}) =\max (q_1,\ldots ,q_d) \quad \text {and}\quad \min ({\varvec{q}}) =\min (q_1,\ldots ,q_d). \end{aligned}$$

Definition 2.11

Let $E = \{ e_1,\ldots ,e_d\}$ be an orderd basis of $\mathbf R^{d}$, $\omega$ be a weight on $\mathbf R^{d}$, ${\varvec{p}}=(p_1,\ldots ,p_d)\in (0,\infty ]^{d}$ and $r=\min (1,{\varvec{p}})$. If $f\in L^r_{loc}(\mathbf R^{d})$, then

$$\begin{aligned} \Vert f\Vert _{L^{{\varvec{p}}}_{E,(\omega )}}\equiv \Vert g_{d-1}\Vert _{L^{p_{d}}(\mathbf R)}, \end{aligned}$$

where $g_k(z_k)$, $z_k\in \mathbf R^{d-k}$, $k=0,\ldots ,d-1$, are inductively defined as

$$\begin{aligned} g_0(x_1,\ldots ,x_{d})&\equiv |f(x_1e_1+\cdots +x_{d}e_d) \omega (x_1e_1+\cdots +x_{d}e_d)| \\ \end{aligned}$$

and

$$\begin{aligned} g_k(z_k)&\equiv \Vert g_{k-1}(\, \cdot \, ,z_k)\Vert _{L^{p_k}(\mathbf R)}, \quad k=1,\ldots ,d-1. \end{aligned}$$

The space $L^{{\varvec{p}}}_{E,(\omega )}(\mathbf R^{d})$ consists of all $f\in L^r_{loc}(\mathbf R^{d})$ such that $\Vert f\Vert _{L^{{\varvec{p}}}_{E,(\omega )}}$ is finite, and is called E-split Lebesgue space (with respect to${\varvec{p}}$and$\omega$).

Let E, ${\varvec{p}}$ and $\omega$ be the same as in Definition 2.11. Then the discrete version $\ell ^{{\varvec{p}}}_{E,(\omega )}(\Lambda _E)$ of $L^{{\varvec{p}}}_{E,(\omega )}(\mathbf R^{d})$ is the set of all sequences $a=\{ a(j)\} _{j\in \Lambda _E}$ such that the quasi-norm

$$\begin{aligned} \Vert a\Vert _{\ell ^{{\varvec{p}}}_{E,(\omega )}} \equiv \Vert f_a\Vert _{L^{{\varvec{p}}}_{E,(\omega )}},\quad f_a=\sum _{j\in \Lambda _E} a(j)\chi _j, \end{aligned}$$

is finite. Here $\chi _j$ is the characteristic function of $j+\kappa (E)$. We also set $L^{{\varvec{p}}}_{E} = L^{{\varvec{p}}}_{E,(\omega )}$ and $\ell ^{{\varvec{p}}}_{E} = \ell ^{{\varvec{p}}}_{E,(\omega )}$ when $\omega =1$.

Definition 2.12

Let E be an ordered basis of the phase space $\mathbf R^{2d}$. Then E is called weakly phase split if there is a subset $E_0\subseteq E$ such that the span of $E_0$ equals $\{ \, (x,0)\in \mathbf R^{2d}\, ;\, x\in \mathbf R^{d}\, \}$ and the span of $E{\setminus } E_0$ equals $\{ \, (0,\xi )\in \mathbf R^{2d}\, ;\, \xi \in \mathbf R^{d}\, \}$.

2.5 Pilipović Flat Spaces, Modulation Spaces Outside Time-Frequency Analysis and the Bargmann Transform

Besides the characterization by means of the short-time Fourier transform in Proposition 2.2, Gelfand–Shilov spaces also can be characterized via Hermite function expansions. Recall that the Hermite function of order $\alpha \in {\mathbf N}^{d}$ is given by

$$\begin{aligned} h_\alpha (x) = \pi ^{-\frac{d}{4}}(-1)^{|\alpha |} (2^{|\alpha |}\alpha !)^{-\frac{1}{2}}e^{\frac{|x|^2}{2}} (\partial ^\alpha e^{-|x|^2}). \end{aligned}$$

It is well-known that $\{ h_\alpha \} _{\alpha \in {\mathbf N}^{d}}$ provides an orthonormal basis for $L^2(\mathbf R^{d})$ and a basis for $\mathscr {S}(\mathbf R^{d})$, $\Sigma _s(\mathbf R^{d})$ when $s>\frac{1}{2}$, $\mathcal S_s(\mathbf R^{d})$ when $s\ge \frac{1}{2}$ and their duals.

By [23] we have for $s \ge \frac{1}{2}$ ($s> \frac{1}{2}$) that $f\in \mathcal S_s(\mathbf R^{d})$ ($f\in \Sigma _s(\mathbf R^{d})$), if and only if the coefficients $c_\alpha (f)$ in its Hermite series expansion

$$\begin{aligned} f = \sum _{\alpha \in {\mathbf N}^{d}}c_\alpha (f)h_\alpha ,\quad c_\alpha (f) = (f,h_\alpha )_{L^2(\mathbf R^{d})} \end{aligned}$$

(2.19)

fulfill

$$\begin{aligned} |c_\alpha (f)| \lesssim e^{-r|\alpha |^{\frac{1}{2s}}} \end{aligned}$$

for some $r>0$ (for every $r>0$) with convergence in $\mathcal S_s(\mathbf R^{d})$ ($\Sigma _s(\mathbf R^{d})$). Similarly, $f\in \mathcal S' _s(\mathbf R^{d})$ ($f\in \Sigma ' _s(\mathbf R^{d})$), if and only if

$$\begin{aligned} |c_\alpha (f)| \lesssim e^{r|\alpha |^{\frac{1}{2s}}} \end{aligned}$$

for every $r>0$ (for some $r>0$) with convergence in $\mathcal S_s'(\mathbf R^{d})$ ($\Sigma _s'(\mathbf R^{d})$).

In [19, 42] various kinds of Fourier-invariant function and distribution spaces are obtained by applying suitable topologies on formal power series expansions. In particular, the Pilipović flat space$\mathcal H_\flat (\mathbf R^{d})$, denoted by $\mathcal H_{\flat _1}(\mathbf R^{d})$ in [42], and its dual $\mathcal H_\flat '(\mathbf R^{d})$, are defined by all formal expansions (2.19) such that

$$\begin{aligned} |c_\alpha (f)| \lesssim r^{|\alpha |}\alpha !^{-\frac{1}{2}} \end{aligned}$$

for some $r>0$, respectively

$$\begin{aligned} |c_\alpha (f)| \lesssim r^{|\alpha |}\alpha !^{\frac{1}{2}} \end{aligned}$$

for every $r>0$. For $f\in \mathcal H_\flat '(\mathbf R^{d})$ and $\phi \in \mathcal H_\flat (\mathbf R^{d})$, we define

$$\begin{aligned} (f,\phi )_{L^2(\mathbf R^{d})} \equiv \sum _{\alpha \in {\mathbf N}^{d}} c_\alpha (f)\overline{c_\alpha (\phi )}. \end{aligned}$$

If $\phi , f \in L^2(\mathbf R^{d})$, the pairing $(f,\phi )_{L^2(\mathbf R^{d})}$ agrees with the $L^2(\mathbf R^{d})$ scalar product of those two functions.

We remark that $\mathcal H_\flat '(\mathbf R^{d})$ is larger than any Fourier-invariant Gelfand–Shilov distribution space, and $\mathcal H_\flat (\mathbf R^{d})$ is smaller than any Fourier-invariant Gelfand–Shilov space.

We notice that $\mathcal H_\flat (\mathbf R^{d})$ and $\mathcal H_\flat '(\mathbf R^{d})$ possess interesting mapping properties under the Bargmann transform. In fact, the Bargmann kernel is given by

$$\begin{aligned} \mathfrak A_d(z,y)=\pi ^{-\frac{d}{4}} \exp \Big ( -\frac{1}{2}(\langle z,z\rangle +|y|^2)+2^{\frac{1}{2}}\langle z,y\rangle \Big ), \end{aligned}$$

which is analytic in z. For fixed $z \in \mathbf C^{d}$, the function $y \rightarrow \mathfrak A_d(z,y)$ belongs to $\mathcal H_\flat (\mathbf R^{d})$. The Bargmann transform $(\mathfrak V_df)(z)$ of $f\in \mathcal H_\flat '(\mathbf R^{d})$ is then defined by

$$\begin{aligned} (\mathfrak V_df)(z) =\langle f,\mathfrak A_d(z,\, \cdot \, )\rangle . \end{aligned}$$

Due to [42] we have that $\mathfrak V_d$ is bijective between $\mathcal H_\flat '(\mathbf R^{d})$ and $A(\mathbf C^{d})$, the set of all entire functions on $\mathbf C^{d}$, and restricts to a bijective map from $\mathcal H_\flat (\mathbf R^{d})$ to

$$\begin{aligned} \{ \, F\in A(\mathbf C^{d})\, ;\, |F(z)|\lesssim e^{R|z|},\ \text {for some}\ R>0\, \} . \end{aligned}$$

Later on we need that the Bargmann and the short-time Fourier transform are linked by the formula

$$\begin{aligned} \begin{aligned} (\mathfrak {V} _df)(x+i\xi )&= (2\pi )^{\frac{d}{2}}e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle } (V_\phi f)(2^{\frac{1}{2}}x,-2^{\frac{1}{2}}\xi ), \\ \phi (x)&= \pi ^{-\frac{d}{4}}e^{-\frac{1}{2}|x|^2},\quad x\in \mathbf R^{d}, \end{aligned} \end{aligned}$$

(2.20)

which can be shown by straight-forward computations. By means of the operator

$$\begin{aligned} (U_{\mathfrak V}F)(x,\xi ) = (2\pi )^{\frac{d}{2}}e^{\frac{1}{2}(|x|^2+|\xi |^2)}e^{-i\langle x,\xi \rangle } F(2^{\frac{1}{2}}x,-2^{\frac{1}{2}}\xi ), \end{aligned}$$

(2.21)

where F is a function or a suitable element of $F \in \mathcal H' _\flat (\mathbf R^{d})$ we can write the Bargmann transform as

$$\begin{aligned} (\mathfrak {V} _df)(x+i\xi ) = (U_{\mathfrak V}(V_\phi f))(x,\xi ). \end{aligned}$$

Definition 2.13

Let $\phi$ be as in (2.20), $\omega$ be a weight on $\mathbf R^{2d}$, $\mathscr {B}$ be an invariant QBF-space with respect to $v\in \mathscr {P}_E(\mathbf R^{2d})$ on $\mathbf R^{2d}\simeq \mathbf C^{d}$ of order $r\in (0,1]$, and let $U_{\mathfrak V}$ be given by (2.21). Then

• (1) $B(\omega ,\mathscr {B})$ consists of all $F\in L^r_{loc}(\mathbf R^{2d})= L^r_{loc}(\mathbf C^{d})$ such that

  $$\begin{aligned} \Vert F\Vert _{B(\omega ,\mathscr {B})}\equiv \Vert (U_{\mathfrak V}^{-1}F)\omega \Vert _{\mathscr {B}}<\infty ; \end{aligned}$$

• (2) $A(\omega ,\mathscr {B})$ consists of all $F\in A(\mathbf C^{d})\cap B(\omega ,\mathscr {B})$ with topology inherited from $B(\omega ,\mathscr {B})$;

• (3) $M(\omega ,\mathscr {B})$ consists of all $f\in \mathcal H_\flat ' (\mathbf R^{d})$ such that

  $$\begin{aligned} \Vert f\Vert _{M(\omega ,\mathscr {B})} \equiv \Vert V_\phi f \cdot \omega \Vert _{\mathscr {B}} \end{aligned}$$

  is finite.

We observe the smaller restrictions on $\omega$ compared to what is the main stream. For example, in Definition 2.13 it is not assumed that $\omega$ should belong to $\mathscr {P}_E(\mathbf R^{2d})$ or $\mathscr {P}(\mathbf R^{2d})$. We still call $M(\omega ,\mathscr {B})$ a modulation space. In contrast to earlier situations, it seems that $M(\omega ,\mathscr {B})$ is not invariant under the choice of $\phi$ when $\omega$ fails to belong to $\mathscr {P}_E$. For that reason we always assume that $\phi$ is given by (2.20) for such $\omega$.

We have the following.

Proposition 2.14

Let $\phi$ be as in (2.20), $\omega$ be a weight on $\mathbf R^{2d}$ and let $\mathscr {B}$ be an invariant QBF-space with respect to $v\in \mathscr {P}_E(\mathbf R^{2d})$. Then the following is true:

• (1) the map $\mathfrak V _d$ is an isometric bijection from $M(\omega ,\mathscr {B})$ to $A(\omega ,\mathscr {B})$;

• (2) if in addition $\mathscr {B}$ is a mixed quasi-normed space of Lebesgue type, then $M(\omega ,\mathscr {B})$ and $A(\omega ,\mathscr {B})$ are quasi-Banach spaces, which are Banach spaces when $\mathscr {B}$ is a Banach space.

For moderate weights, Proposition 2.14 is proved in [18, 24] with some completing arguments given in [33]. For the broader weight class $\mathscr {P}_Q$ in Definition 2.1, a proof of Proposition 2.14 is given in [40]. We now present a proof which holds for any weight $\omega$.

Proof

From (2.20), (2.21) and Definition 2.13 it follows that $\mathfrak V _d$ is an isometric injection from $M(\omega ,\mathscr {B})$ to $A(\omega ,\mathscr {B})$. Since any element in $A(\mathbf C^{d})$, and thereby any element in $A(\omega ,\mathscr {B})$ is a Bargmann transform of an element in $\mathcal H' _\flat (\mathbf R^{d})$, it follows that the image of $M(\omega ,\mathscr {B})$ under $\mathfrak V_d$ contains $A(\omega ,\mathscr {B})$. This gives the stated bijectivity in (1).

The completeness of $A(\omega ,\mathscr {B})$, and thereby of $M(\omega ,\mathscr {B})$ follows from [42, Theorem 4.8]. This gives (2).$\square$

3 Compactness Properties for Modulation Spaces

This section is devoted to the questions under which sufficient and necessary conditions the inclusion map

$$\begin{aligned} i: M(\omega _1, \mathscr {B}) \rightarrow M(\omega _2, \mathscr {B}) \end{aligned}$$

is continuous or even compact for suitable invariant QBF-spaces $\mathscr {B}$.

As ingredients for the proof of our main results we need to deduce some properties for moderate weight functions. In what follows let $L^{\infty }_{0,(\omega )}(\mathbf {R} ^{d})$ be the set of all $f \in L^{\infty }_{(\omega )}(\mathbf {R} ^{d})$ with the property

$$\begin{aligned} \lim _{R \rightarrow \infty } \left( \hbox {ess\, sup}_{|x| \ge R} |f(x)\omega (x)| \right) =0, \end{aligned}$$

when $\omega$ is a weight on $\mathbf R^{d}$. We also set $L^{\infty }_{0}=L^{\infty }_{0,(\omega )}$ when $\omega =1$. If $\Lambda$ is a lattice, then the discrete Lebesgue spaces $\ell ^\infty _0(\Lambda )$ and $\ell ^\infty _{0,(\omega )}(\Lambda )$ are defined analogously.

Lemma 3.1

Let E be an ordered basis of $\mathbf R^{d}$ and let $\omega \in \mathscr {P}_E(\mathbf R^{d})$. Then the following is true:

• (1) $\mathscr {P}_E(\mathbf R^{d})$ is a convex cone which is closed under multiplication, division and under compositions with power functions;

• (2) $\mathscr {P}_E(\mathbf R^{d})\cap L^{{\varvec{p}}}_{E,(\omega )}(\mathbf R^{d})$ increases with ${\varvec{p}}\in (0,\infty ]^d$, and

  $$\begin{aligned} \mathscr {P}_E(\mathbf R^{d})\cap L^{{\varvec{p}}}_{E,(\omega )}(\mathbf R^{d}) \subseteq \mathscr {P}_E(\mathbf R^{d})\cap L^\infty _{0,(\omega )}(\mathbf R^{d}), \quad {\varvec{p}}\in (0,\infty )^d. \end{aligned}$$

  (3.1)

Similar properties have been shown in [3, Lemma 2.1] for the subset $\mathscr {P}$ of $\mathscr {P}_E$.

Proof

Claim (1) can easily be verified by means of the definition of moderate weights.

It remains to verify (2). Let $\kappa (E)$ be the (closed) parallelepiped spanned by E and let $\vartheta \in \mathscr {P}_E(\mathbf R^{d})$. By using the map $\vartheta \mapsto \vartheta \cdot \omega$, we reduce ourself to the case when $\omega =1$.

The moderateness of $\vartheta \in \mathscr {P}_E(\mathbf R^{2d})$ implies that

$$\begin{aligned} \vartheta (x_1)\asymp \vartheta (x_1+x_2), \quad \text {when}\quad x_2\in \kappa (E). \end{aligned}$$

(3.2)

Hence, if $\chi _j$ is the characteristic function of $j+\kappa (E)$ and

$$\begin{aligned} \vartheta _0 (x) = \sum _{j\in \Lambda _E}\vartheta (j)\chi _j(x), \end{aligned}$$

then $\vartheta \asymp \vartheta _0$, giving that

$$\begin{aligned} \Vert \vartheta \Vert _{L^{{\varvec{p}}}_E} \asymp \Vert \vartheta _0\Vert _{L^{{\varvec{p}}}_E} \asymp \Vert \vartheta \Vert _{\ell ^{{\varvec{p}}}_E}. \end{aligned}$$

The assertion now follows from the fact that $\ell ^{{\varvec{p}}}_E$ increases with ${\varvec{p}}$ and that if in addition ${\varvec{p}}\in (0,\infty )^d$, then $\ell ^{{\varvec{p}}}_E\subseteq \ell ^\infty _0$.$\square$

We also need the following extension of [43, Theorem 2.5].

Proposition 3.2

Let $v,v_0\in \mathscr {P}_E(\mathbf R^{2d})$ be submultiplicative, $\omega \in \mathscr {P}_E(\mathbf R^{2d})$ be v-moderate, and let $\mathscr {B}$ be an invariant BF-space with respect to $v_0$. Then $M(\omega ,\mathscr {B})$ is a Banach space, and

$$\begin{aligned} M(\omega ,\mathscr {B})\hookrightarrow M^\infty _{(1/(v_0v))}(\mathbf R^{d}). \end{aligned}$$

(3.3)

Remark 3.3

If $\mathscr {B}=L^{{\varvec{p}}}_E(\mathbf R^{2d})$ for some weakly phase split basis E of $\mathbf R^{2d}$, ${\varvec{p}}\in (0,\infty ]^{2d}$ and $\omega \in \mathscr {P}_E(\mathbf R^{2d})$, then $M(\omega ,\mathscr {B})$ is a quasi-Banach space. Moreover $M(\omega , L^{{\varvec{p}}}_E(\mathbf R^{2d}))$ is increasing with ${\varvec{p}}$. In particular, (3.3) is improved in [41] into

$$\begin{aligned} M(\omega , L^{{\varvec{p}}}_E(\mathbf R^{2d}))\hookrightarrow M^\infty _{(\omega )}(\mathbf R^{2d}). \end{aligned}$$

For the proof of Proposition 3.2 we need to consider the twisted convolution, $\widehat{*}$, defined by

$$\begin{aligned} (F{\, \widehat{*}\,}G)(x,\xi ) = (2\pi )^{-\frac{d}{2}} \iint _{\mathbf R^{2d}}F(x-y,\xi -\eta )G(y,\eta ) e^{-i\langle x-y,\eta \rangle }\, dyd\eta \end{aligned}$$

when $F,G\in L^1(\mathbf R^{2d})$. The twisted convolution map $\mathcal T$, which takes suitable pairs of functions and distributions (F, G) into $F{\, \widehat{*}\,}G$, is continuous between several function and distribution spaces, see e. g. [25] or Lemma 3 in [7]. For example the map $\mathcal T$ is continuous from $L^1(\mathbf R^{2d})\times L^1(\mathbf R^{2d})$ to $L^1(\mathbf R^{2d})$. For any $s>0$ it restricts to continuous mappings

$$\begin{aligned} \begin{aligned} \mathcal T\,&:\, \mathcal S_s (\mathbf R^{2d})\times \mathcal S_s (\mathbf R^{2d}) \rightarrow \mathcal S_s(\mathbf R^{2d}), \\ \mathcal T\,&:\, \Sigma _s (\mathbf R^{2d})\times \Sigma _s (\mathbf R^{2d}) \rightarrow \Sigma _s(\mathbf R^{2d}), \end{aligned} \end{aligned}$$

(3.4)

and by duality it follows that these mappings extend to continuous mappings

$$\begin{aligned} \begin{aligned} \mathcal T\,&:\, \mathcal S_s '(\mathbf R^{2d})\times \mathcal S_s (\mathbf R^{2d}) \rightarrow \mathcal S_s'(\mathbf R^{2d}), \\ \mathcal T\,&:\, \Sigma _s '(\mathbf R^{2d})\times \Sigma _s (\mathbf R^{2d}) \rightarrow \Sigma _s'(\mathbf R^{2d}). \end{aligned} \end{aligned}$$

(3.5)

In fact, for some map $A_*$ we have $A_*(F{\, \widehat{*}\,}G) = (A_*F)\circ (A_*G)$. This map $A_*$ is similar to the map A in Sections 1 and 2 in [4], and by identifying operators with their kernels, $A_*$ consists of pullbacks of partial Fourier transforms and non-degenerate linear transformations on the phase space. Since such pullbacks are homeomorphic on any Fourier invariant Gelfand–Shilov space, (3.4) follows from the fact that the sets

$$\begin{aligned} \{ \, T_K\, ;\, K\in \mathcal S_s(\mathbf R^{2d})\, \} \quad \text {and}\quad \{ \, T_K\, ;\, K\in \Sigma _s(\mathbf R^{2d})\, \} \end{aligned}$$

are algebras under compositions. Here $T_K$ denotes the linear operator with distribution kernel K.

The twisted convolution is convenient to use when changing window functions in short-time Fourier transforms. In fact by Fourier’s inversion formula and some straight-forward computations one has

$$\begin{aligned} (\phi _3,\phi _1)_{L^2}\cdot V_{\phi _2}f = (V_{\phi _1}f){\, \widehat{*}\,} (V_{\phi _2}\phi _3) \end{aligned}$$

(3.6)

for every $f\in \Sigma _s'(\mathbf R^{d})$ and $\phi _1,\phi _2,\phi _3 \in \Sigma _s(\mathbf R^{d})$. For any $\phi \in \Sigma _s(\mathbf R^{d}){\setminus } 0$, we are also interested of the operator $P_\phi$, given by

$$\begin{aligned} P_\phi F \equiv \Vert \phi \Vert _{L^2(\mathbf R^{d})}^{-2} F{\, \widehat{*}\,}(V_\phi \phi ), \quad F \in \Sigma _s'(\mathbf R^{2d}). \end{aligned}$$

(3.7)

Since $V_{\phi _2}\phi _3 \in \Sigma _s(\mathbf R^{2d})$ when $\phi _2,\phi _3 \in \Sigma _s(\mathbf R^{d})$, it follows from the continuity properties of the twisted convolution above that $P_\phi$ in (3.7) is continuous on $\Sigma _s(\mathbf R^{2d})$ and on $\Sigma _s'(\mathbf R^{2d})$. Similar facts hold true with $\mathcal S_s$ and $\mathcal S_s'$ in place of $\Sigma _s$ and $\Sigma _s'$, respectively, at each place.

The operator $P_\phi$ has the following properties.

Lemma 3.4

Let $s>0$, $\phi \in \Sigma _s(\mathbf R^{d}){\setminus } 0$. Then the following is true:

• (1) $P_\phi$ in (3.7) is a continuous projection from $\Sigma _s'(\mathbf R^{2d})$ to

  $$\begin{aligned} V_\phi (\Sigma _s'(\mathbf R^{d})) \equiv \{ \, V_\phi f\, ;\, f\in \Sigma _s'(\mathbf R^{d})\, \} \subseteq \Sigma _s'(\mathbf R^{2d})\bigcap C^\infty (\mathbf R^{2d}); \end{aligned}$$

• (2) $P_\phi$ in (3.7) restricts to a continuous projection from $\Sigma _s(\mathbf R^{2d})$ to

  $$\begin{aligned} V_\phi (\Sigma _s(\mathbf R^{d})) \equiv \{ \, V_\phi f\, ;\, f\in \Sigma _s(\mathbf R^{d})\, \} \subseteq \Sigma _s(\mathbf R^{2d}); \end{aligned}$$

• (3) if $\mathscr {B}$ is an invariant BF-space on $\mathbf R^{2d}$, then $P_\phi$ is continuous on $\mathscr {B}$.

Similar facts hold true with $\mathcal S_s$ and $\mathcal S_s'$ in place of $\Sigma _s$ and $\Sigma _s'$, respectively, at each place.

Related results can essentially be found in e. g. [20, 24]. In order to be self-contained, we here give a short proof.

Proof

We only prove the result in the Beurling case. The Roumieu case follows by similar arguments and is left for the reader.

By (3.6) it is clear that $P_\phi$ is the identity map on $V_\phi (\Sigma _s'(\mathbf R^{d}))$ and thereby on $V_\phi (\Sigma _s(\mathbf R^{d}))$.

Let $V_\phi ^*$ be the $L^2$-adjoint of $V_\phi$. That is, $V_\phi ^*F$ satisfies

$$\begin{aligned} (V_\phi ^*F,\psi )_{L^2(\mathbf R^{d})} = (F,V_\phi \psi )_{L^2(\mathbf R^{2d})}, \quad F\in \Sigma _s'(\mathbf R^{2d}),\ \psi \in \Sigma _s(\mathbf R^{d}). \end{aligned}$$

By the continuity properties of $V_\phi$ on $\Sigma _s$ and $\Sigma _s'$, it follows that $V_\phi ^*$ is continuous from $\Sigma _s'(\mathbf R^{2d})$ to $\Sigma _s'(\mathbf R^{d})$ and restricts to a continuous map from $\Sigma _s(\mathbf R^{2d})$ to $\Sigma _s(\mathbf R^{d})$.

By a straight-forward application of Fourier’s inversion formula it follows that

$$\begin{aligned} P_\phi F = V_\phi f \quad \text {when}\quad f= \Vert \phi \Vert _{L^2}^{-2}V_\phi ^*F, \end{aligned}$$

which shows that the images of $\Sigma _s'(\mathbf R^{2d})$ and $\Sigma _s(\mathbf R^{2d})$ under $P_\phi$ equal $V_\phi (\Sigma _s'(\mathbf R^{d}))$ and $V_\phi (\Sigma _s(\mathbf R^{d}))$, respectively. This gives (1) and (2).

If $\mathscr {B}$ is an invariant BF-space on $\mathbf R^{2d}$ and $F\in \mathscr {B}$, then it follows from the definitions that

$$\begin{aligned} |P_\phi F |\lesssim |F|*\Phi , \end{aligned}$$

where $\Phi = |V_\phi \phi |$ belongs to $L^1_{(v)}(\mathbf R^{2d})$ for every choice of $v\in \mathscr {P}_E(\mathbf R^{2d})$. Hence, a combination of (2) in Definition 2.6 and (2.17) gives $P_\phi F \in \mathscr {B}$ and

$$\begin{aligned} \Vert P_\phi F\Vert _{\mathscr {B}} \lesssim \Vert F\Vert _{\mathscr {B}}\Vert \Phi \Vert _{L^1_{(v)}} \end{aligned}$$

for some $v\in \mathscr {P}_E(\mathbf R^{2d})$, and the continuity of $P_\phi$ on $\mathscr {B}$ follows. This gives (3).$\square$

Lemma 3.5

If $\mathscr {B}$ is an invariant BF-space of $\mathbf R^{d}$, then

$$\begin{aligned} L^\infty _{(v)}(\mathbf R^{d})\hookrightarrow \mathscr {B}\end{aligned}$$

for some $v\in \mathscr {P}_E(\mathbf R^{d})$.

Proof

Since $\Sigma _1(\mathbf R^{d})$ is continuously embedded in $\mathscr {B}$ we have

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}} \lesssim \sup _{\beta \in {\mathbf N}^{d}} \left( \frac{\Vert (D^\beta f) \cdot e^{|\, \cdot \, |/h_0}\Vert _{L^\infty }}{h^{|\beta |}_0\beta !} \right) \end{aligned}$$

for some $h_0>0$. Let $\omega = e^{-2|\, \cdot \, |/h_0}$, $\omega _0 =\omega *e^{-|\, \cdot \, |^2/2}$ and let $v=1/\omega _0$. Then $\omega \in \mathscr {P}_E(\mathbf R^{d})$, and [1, Proposition 1.6] shows that $\omega _0 \in \mathscr {P}_E(\mathbf R^{d})$ and that

$$\begin{aligned} |D^\beta \omega _0| \lesssim h^{|\beta |}\beta !\, e^{-2|\, \cdot \, |/h_0} \end{aligned}$$

holds for every $h>0$. By choosing $h<h_0$ we get

$$\begin{aligned} \Vert \omega _0\Vert _{\mathscr {B}} \lesssim \sup _{\beta \in {\mathbf N}^{d}} \left( \frac{\Vert (D^\beta \omega _0) \cdot e^{|\, \cdot \, |/h_0}\Vert _{L^\infty }}{h_0^{|\beta |}\beta !} \right) \lesssim \sup _{\beta \in {\mathbf N}^{d}} \left( \frac{h^{|\beta |}\beta !\Vert \omega _0 \cdot e^{|\, \cdot \, |/h_0}\Vert _{L^\infty }}{h_0^{|\beta |}\beta !} \right) \\ \lesssim \Vert e^{-2|\, \cdot \, |/h_0}\cdot e^{|\, \cdot \, |/h_0}\Vert _{L^\infty } <\infty . \end{aligned}$$

Hence, if $f\in L^\infty _{(v)}(\mathbf R^{d})$, then $|f|\lesssim \omega _0$ which implies that $f\in \mathscr {B}$ in view of Definition 2.6(2). Furthermore,

$$\begin{aligned} \Vert f\Vert _{\mathscr {B}} \lesssim \Vert \omega _0\Vert _{\mathscr {B}}\Vert f\cdot v\Vert _{L^\infty } \asymp \Vert f\Vert _{L^\infty _{(v)}} \end{aligned}$$

and the result follows.$\square$

In what follows we let

$$\begin{aligned} \mathscr {B}_{(\omega )} \equiv \{ \, f\in L^1_{loc}(\mathbf R^{d})\, ;\, f\cdot \omega \in \mathscr {B}\, \} , \end{aligned}$$

where $\mathscr {B}$ be an invariant BF-space on $\mathbf R^{d}$.

Lemma 3.6

Let $\mathscr {B}$ be an invariant BF-space on $\mathbf R^{d}$ and $\omega \in \mathscr {P}_E(\mathbf R^{d})$. Then $\mathscr {B}_{(\omega )}$ is an invariant BF-space under the norm

$$\begin{aligned} f\mapsto \Vert f\Vert _{\mathscr {B}_{(\omega )}} \equiv \Vert f\cdot \omega \Vert _{\mathscr {B}}. \end{aligned}$$

Proof

It is obvious that $\mathscr {B}_{(\omega )}$ is complete. Let v be as in Lemma 3.5. By Lemma 3.5, $L^\infty _{(\omega \cdot v)} \hookrightarrow \mathscr {B}_{(\omega )}$. Since

$$\begin{aligned} \Sigma _1(\mathbf R^{d})\hookrightarrow L^\infty _{(\omega \cdot v)}(\mathbf R^{d}), \end{aligned}$$

we obtain that $\Sigma _1(\mathbf R^{d})$ is continuously embedded in $\mathscr {B}_{(\omega )}$.

By straight-forward computations it follows that both (1) and (2) in Definition 2.6 are fulfilled with $\mathscr {B}_{(\omega )}$ in place of $\mathscr {B}$ provided v in that definition has been modified in suitable ways.$\square$

Proof of Proposition 3.2

Let $\phi \in \Sigma _1(\mathbf R^{d}){\setminus } 0$ be fixed. Since $\omega$ is a moderate function, it follows by the previous lemma that $\mathscr {B}_{(\omega )}$ is an invariant BF-space.

Let $\{ f_j\} _{j=1}^\infty$ be a Cauchy sequence in $M(\omega ,\mathscr {B})$. Then $\{ V_\phi f_j\} _{j=1}^\infty$ is a Cauchy sequence in $\mathscr {B}_{(\omega )}$. Since $\mathscr {B}_{(\omega )}$ is a Banach space, there is a unique $F\in \mathscr {B}_{(\omega )}$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty }\Vert V_\phi f_j -F\Vert _{\mathscr {B}_{(\omega )}} =0. \end{aligned}$$

Let $f=\Vert \phi \Vert _{L^2}^{-2}V_\phi ^*F$. Then $V_\phi f=P_\phi F$ belongs to $\mathscr {B}_{(\omega )}$ in view of Lemma 3.4 (3). Since $P_\phi$ is continuous on $\mathscr {B}_{(\omega )}$ and satisfies the mapping properties given in Lemma 3.4, we get

$$\begin{aligned}&\lim _{j\rightarrow \infty } \Vert f_j-f\Vert _{M(\omega ,\mathscr {B})}=\lim _{j\rightarrow \infty } \Vert V_\phi (f_j-f)\Vert _{\mathscr {B}_{(\omega )}}\\&\quad =\lim _{j\rightarrow \infty } \Vert P_\phi (V_\phi f_j - F)\Vert _{\mathscr {B}_{(\omega )}}\lesssim \lim _{j\rightarrow \infty } \Vert V_\phi f_j - F\Vert _{\mathscr {B}_{(\omega )}} =0. \end{aligned}$$

Hence, $f_j\rightarrow f$ in $M(\omega ,\mathscr {B})$, and the completeness of $M(\omega ,\mathscr {B})$ follows. Consequently, $M(\omega ,\mathscr {B})$ is a Banach space.

The embedding (3.3) is an immediate consequence of [43, Theorem 2.5] and the fact that $M(\omega ,\mathscr {B})$ is a Banach space.$\square$

If we assume that $\mathscr {B}$ is an invariant QBF-space (instead of invariant BF-space) with respect of $v_0$, then it seems to be an open question wether (3.3) might be violated or not.

Before studying compactness of embeddings between modulation spaces, we first consider related continuity questions.

Theorem 3.7

Let $\omega _1$ and $\omega _2$ be weights on $\mathbf R^{2d}$, $\mathscr {B}$ be an invariant BF-space on $\mathbf R^{2d}$ with respect to $v \in \mathscr {P}_E(\mathbf R^{2d})$ or a mixed quasi-normed space of Lebesgue type. Then the following is true:

• (1) if ${\omega _2}/{\omega _1}\in L^\infty (\mathbf R^{2d})$, then $M(\omega _1,\mathscr {B}) \subseteq M(\omega _2, \mathscr {B})$ and the injection

  $$\begin{aligned} i\! : M(\omega _1,\mathscr {B})\rightarrow M(\omega _2,\mathscr {B}). \end{aligned}$$

  (3.8)

  is continuous;

• (2) if in addition $\omega _1,\omega _2\in \mathscr {P}_E(\mathbf R^{2d})$ and v is bounded, then the map (3.8) is a continuous injection, if and only if ${\omega _2}/{\omega _1}\in L^\infty (\mathbf R^{2d})$.

The next lemma is related to Remark 3.3 and is needed to verify the previous theorem.

Lemma 3.8

Let v be submultiplicative and bounded on $\mathbf R^{2d}$, $\mathscr {B}$ be an invariant BF-space and let $\omega \in \mathscr {P}_E(\mathbf R^{2d})$. Then $M(\omega ,\mathscr {B}) \hookrightarrow M^\infty _{(\omega )} (\mathbf R^{d})$.

Proof

Let $\mathscr {B}'$ be the $L^2$-dual of $\mathscr {B}$ and let $f \in \Sigma _1'(\mathbf R^{d})$. Then it follows by straight-forward computations that both $\mathscr {B}$ and $\mathscr {B}'$ are translation invariant Banach spaces of order 1 which contain $\Sigma _1(\mathbf R^{2d})$. Let $\phi \in \Sigma _1(\mathbf R^{d})$ be such that $\Vert \phi \Vert _{L^2}=1$ and let

$$\begin{aligned} \Omega = \{ \, g\in \Sigma _1 (\mathbf R^{d})\, ;\, \Vert g\Vert _{M^1_{(1/\omega )}}\le 1\, \} . \end{aligned}$$

By Feichtinger’s minimality principle (cf. the extension [43, Theorem 2.4] of [24, Theorem 12.1.9]) one gets

$$\begin{aligned} \Vert V_\phi g/\omega \Vert _{\mathscr {B}'} \asymp \Vert g\Vert _{M(1/\omega ,\mathscr {B}' \, )} \lesssim \Vert g\Vert _{M^1_{(1/\omega )}}<\infty . \end{aligned}$$

Together with Proposition 2.5 we get

$$\begin{aligned} \Vert f\Vert _{M^\infty _{(\omega )}} \asymp \sup _{g\in \Omega } |(f,g)_{L^2(\mathbf R^{d})}| = \sup _{g\in \Omega } |(V_\phi f\cdot \omega ,V_\phi g/\omega )_{L^2(\mathbf R^{2d})}| \\ \le \sup _{g\in \Omega } \Vert V_\phi f\cdot \omega \Vert _{\mathscr {B}} \Vert V_\phi g/\omega \Vert _{\mathscr {B}'} \lesssim \Vert V_\phi f\cdot \omega \Vert _{\mathscr {B}} \asymp \Vert f\Vert _{M(\omega ,\mathscr {B})}. \end{aligned}$$

$\square$

Proof of Theorem 3.7

Claim (1) is an immediate consequence of the boundedness of ${\omega _2}/{\omega _1}$ and of $\mathscr {B}$ being an invariant BF-space.

Assume instead that the embedding i in (3.8) is continuous and the assumptions of the second claim hold. Claim (2) follows if we prove the boundedness of $\omega _2/\omega _1$, which we aim to deduce by contradiction. Therefore suppose that $\omega _2/\omega _1$ is unbounded and $M(\omega _1, \mathcal B)$ is continuously embedded in $M(\omega _2, \mathcal B)$. Then there is a sequence $(x_k,\xi _k) \in \mathbf R^{2d}$ with $|(x_k,\xi _k)| \rightarrow \infty$ when $k \rightarrow \infty$ and such that

$$\begin{aligned} \frac{\omega _2(x_k,\xi _k)}{\omega _1(x_k,\xi _k)} \ge k \quad \text {for all } k \in \mathbf {N}. \end{aligned}$$

(3.9)

Let $\phi$ be as in (2.20) and set

$$\begin{aligned} f_k=\frac{1}{\omega _1(X_k)} e^{i\langle \, \cdot \, ,\xi _k\rangle }\phi (\, \cdot \, -x_k),\quad X_k=(x_k,\xi _k). \end{aligned}$$

(3.10)

Also let $v_0\in \mathscr {P}_E (\mathbf R^{2d})$ be submultiplicative and such that $\omega _1$ is $v_0$-moderate and that $v_0\ge 1$.

By

$$\begin{aligned} V_{\phi }(e^{i \langle \, \cdot \, ,\xi \rangle }f(\, \cdot \, -x))(y,\eta ) = e^{i\langle x,\eta -\xi \rangle } (V_{\phi }f)(y-x, \eta -\xi ), \end{aligned}$$

which follows by straight-forward computations, see e. g. [25], we get

$$\begin{aligned} \Vert e^{i\langle \, \cdot \, ,\xi \rangle } f(\, \cdot \, -x)\Vert _{M(\omega _1, \mathscr {B})} \le C \omega _1(x,\xi ) \Vert f\Vert _{M(v_0,\mathscr {B})},\quad f\in M{(v_0, \mathscr {B})}. \end{aligned}$$

This gives

$$\begin{aligned} \Vert f_k \Vert _{M(\omega _1, \mathscr {B})} =\frac{1}{\omega _1(X_k)}\Vert e^{i\langle \, \cdot \, ,\xi _k\rangle }\phi (\, \cdot \, -x_k)\Vert _{M(\omega _1, \mathscr {B})} \le C \Vert \phi \Vert _{M(v_0, \mathscr {B})}< \infty , \end{aligned}$$

where C is independent of $k \in \mathbb {N}$. Then the hypothesis provides the boundedness of the sequence $\{ f_k\}$ in ${M(\omega _2, \mathscr {B})}$.

Since $M(\omega _2, \mathscr {B}) \hookrightarrow M^\infty _{(\omega _2)}$ due to Lemma 3.8 we have

$$\begin{aligned} \sup _{X\in \mathbf R^{2d}}\omega _2(X)|(V_\phi f_k)(X)|\le C\Vert f_k\Vert _{M(\omega _2, \mathscr {B})}\le C \quad \text {for all } k \in \mathbb {N} \end{aligned}$$

(3.11)

for some $C>0$. In particular by letting $X=X_k$ in (3.11) we obtain

$$\begin{aligned}&\frac{\omega _2(X_k)}{\omega _1(X_k)}=(2\pi )^{\frac{d}{2}}\,\frac{\omega _2(X_k)}{\omega _1(X_k)}|(V_\phi \phi )(0)|\nonumber \\&\quad =(2\pi )^{\frac{d}{2}}\,\frac{\omega _2(X_k)}{\omega _1(X_k)}|(V_\phi (e^{i\langle \, \cdot \, ,\xi _k\rangle }\phi (\, \cdot \, -x_k))(X_k)|\nonumber \\&\quad =(2\pi )^{\frac{d}{2}}\,\omega _2(X_k)|(V_\phi f_k)(X_k)| \le C, \end{aligned}$$

(3.12)

which contradicts (3.9) and proves the result.$\square$

We have now the following extension of [3, Theorem 1.2], which is our main result.

Theorem 3.9

Let $\omega _1,\omega _2\in \mathscr {P}_Q(\mathbf R^{2d})$, $v \in \mathscr {P}_E(\mathbf R^{2d})$ be submultiplicative, $\mathscr {B}$ be an invariant BF-space on $\mathbf R^{2d}$ with respect to v or a mixed quasi-normed space of Lebesgue type. Then the following is true:

• (1) if ${\omega _2}/{\omega _1}\in L^\infty _0(\mathbf R^{2d})$, then the injection (3.8) is compact;

• (2) if in addition $\omega _1,\omega _2\in \mathscr {P}_E(\mathbf R^{2d})$ and v is bounded, then the injection (3.8) is compact, if and only if ${\omega _2}/{\omega _1}\in L^\infty _0(\mathbf R^{2d})$.

We need the following lemma for the proof.

Lemma 3.10

Let $\mathscr {B}$ be an invariant BF space on $\mathbf R^{2d}$, $\phi (x)=\pi ^{-\frac{d}{4}}e^{-\frac{1}{2}\cdot |x|^2}$, $x\in \mathbf R^{d}$, $\omega \in \mathscr {P}_Q(\mathbf R^{2d})$ and let $\{ f_j\} _{j=1}^\infty \subseteq \Sigma _1'(\mathbf R^{d})$ be a bounded set in $M(\omega ,\mathscr {B})$. Then there is a subsequence $\{ f_{j_k}\} _{k=1}^\infty$ of $\{ f_j\} _{j=1}^\infty$ such that $\{ V_\phi f_{j_k}\} _{k=1}^\infty$ is locally uniformly convergent.

Proof

By the link (2.20) between the Bargmann transform and the Gaussian windowed short-time Fourier transform, the result follows if we prove the assertion with $F_j=\mathfrak V_df_j$ in place of $V_\phi f_j$. For any $R>0$, let $D_R$ be the poly-disc

$$\begin{aligned} D_R \equiv&\{ \, (x,\xi )\in \mathbf R^{2d}\, ;\, x_j^2+\xi _j^2<R^2,\ j=1,\ldots ,d\, \} \end{aligned}$$

in $\mathbf R^{2d}$ which we identify with

$$\begin{aligned}&\{ \, x+i\xi \in \mathbf C^{d}\, ;\, x_j^2+\xi _j^2<R^2,\ j=1,\ldots ,d\, \} \end{aligned}$$

in $\mathbf C^{d}$. By Cantor’s diagonalization principle the result follows if we prove that for each $R>0$, there is a subsequence $\{ f_{j_k}\} _{k=1}^\infty$ of $\{ f_j\} _{j=1}^\infty$ such that $\{ F_{j_k}\} _{k=1}^\infty$ is uniformly convergent on $D_R$.

By [40, Theorem 3.2], we get the boundedness of $\{ f_j\} _{j=1}^\infty$ in $M^{\infty }_{(\omega _0)}(\mathbf R^{d})$ for some choice of $\omega _0\in \mathscr {P}_Q(\mathbf R^{2d})$. Hence, $\{ V_\phi f_{j}\} _{j=1}^\infty$ and thereby $\{ F_{j}\} _{j=1}^\infty$ are locally uniformly bounded on $\mathbf R^{2d}$. In particular,

$$\begin{aligned} C_R \equiv \sup _{j\ge 1}\Vert F_j\Vert _{L^\infty (D_{2R})} \quad \text {and}\quad C_{R,\omega _0} \equiv \sup _{j\ge 1}\Vert F_j\omega _0\Vert _{L^\infty (D_{2R})} \end{aligned}$$

(3.13)

are finite for every weight $\omega _0$ on $\mathbf C^{d}\simeq \mathbf R^{2d}$.

By Cauchy’s and Taylor’s formulae we have

$$\begin{aligned} F_j(z)&= \sum _{\alpha \in {\mathbf N}^{d}} a_j(\alpha )z^\alpha ,\quad z\in D_R, \end{aligned}$$

(3.14)

where

$$\begin{aligned} |a_j(\alpha )|&\le C_R(2R)^{-|\alpha |}, \quad \alpha \in {\mathbf N}^{d}. \end{aligned}$$

(3.15)

In particular, if $\{ \beta _l\} _{l=1}^\infty$ is an enumeration of ${\mathbf N}^{d}$, then for each $l\ge 1$, $\{ a_j(\beta _l )\} _{j=1}^\infty$ is a bounded set in $\mathbf C$. Hence, for a subsequence $I_1=\{ k_{1,1},k_{1,2},\ldots \}$ of $\mathbf Z_+=\{ 1,2,\ldots \}$, the limit

$$\begin{aligned} \lim _{m\rightarrow \infty } a_{k_{1,m}}(\beta _1) \end{aligned}$$

exists. By induction it follows that for some family of subsequences

$$\begin{aligned} I_N=\{ k_{N,1},k_{N,2},\ldots \} \subseteq \mathbf Z _+, \end{aligned}$$

which decreases with N, the limit

$$\begin{aligned} \lim _{m\rightarrow \infty } a_{k_{N,m}}(\beta _n) \end{aligned}$$

exists for every $n\le N$.

By Cantor’s diagonal principle, there is a subsequence $\{ {j_k}\} _{k=1}^\infty$ of $\mathbf Z_+$ and sequence $\{ b(\alpha )\} _{\alpha \in {\mathbf N}^{d}}$ such that

$$\begin{aligned} \lim _{k\rightarrow \infty } a_{j_k}(\alpha ) = b(\alpha ). \end{aligned}$$

By (3.15) we get

$$\begin{aligned} |b(\alpha )| \le C_R(2R)^{-|\alpha |}. \end{aligned}$$

This in turn gives

$$\begin{aligned} \sup _{j\ge 1}\Vert a_j(\alpha )z^\alpha \Vert _{L^\infty (D_R)} \le C_R2^{-|\alpha |} \quad \text {and}\quad \Vert b(\alpha )z^\alpha \Vert _{L^\infty (D_R)} \le C_R2^{-|\alpha |}. \end{aligned}$$

(3.16)

Hence, (3.14) and the Taylor series

$$\begin{aligned} F(z) \equiv \sum _{\alpha \in {\mathbf N}^{d}}b(\alpha )z^\alpha \end{aligned}$$

are uniformly convergent on $D_R$, and by using (3.16), we obtain by straight-forward computations that $F_{j_k}$ tends to F uniformly on $D_R$ when k tends to infinity.$\square$

Proof of Theorem 3.9

In order to verify (1) we need to show that a bounded sequence $\{f_j\}_{j=1}^{\infty }$ in $M(\omega _1,\mathscr {B})$ has a convergent subsequence in $M(\omega _2,\mathscr {B})$. By means of the assumptions there is a sequence of increasing balls $B_k$, $k\in \mathbf Z_+$, centered at the origin with radius tending to $+\infty$ as $k\rightarrow \infty$ such that

$$\begin{aligned} \frac{\omega _2(x,\xi )}{\omega _1(x,\xi )}\le \frac{1}{k},\quad \text {when}\quad (x,\xi )\in \mathbf R^{2d}{\setminus } B_k. \end{aligned}$$

(3.17)

By Lemma 3.10 it follows that if $\phi (x)=\pi ^{-\frac{d}{4}}e^{-\frac{1}{2}\cdot |x|^2}$, $x\in \mathbf R^{d}$, then there is a subsequence $\{ h_j\}_{j=1}^\infty$ of $\{ f_{j}\}_{j=1}^\infty$ such that $\{ V_\phi h_j\}_{j=1}^\infty$ converges uniformly on any $B_k$, and converges on the whole $\mathbf R^{2d}$.

Claim (1) follows if we prove $\Vert h_{m_1} - h_{m_2}\Vert _{M(\omega _2,\mathscr {B})}\rightarrow 0$ as $m_1,m_2\rightarrow \infty$. Let $\chi _k$ be the characteristic function of $B_k$, $k\ge 1$. From the fact that $C=C_R$ in (3.13) is bounded we have

$$\begin{aligned}&\Vert h_{m_1} - h_{m_2} \Vert _{M(\omega _2,\mathscr {B})} = \Vert V_\phi h_{m_1}-V_\phi h_{m_2}\Vert _{\mathscr {B}_{(\omega _2)}}\nonumber \\&\quad \lesssim \Vert (V_\phi h_{m_1}-V_\phi h_{m_2})\chi _{k}\Vert _{\mathscr {B}_{(\omega _2)}}+ {\Vert V_\phi h_{m_1}-V_\phi h_{m_2}\Vert _{\mathscr {B}_{(\omega _1)}}}/{k}\nonumber \\&\quad \le \Vert (V_\phi h_{m_1}-V_\phi h_{m_2})\chi _{k}\Vert _{\mathscr {B}_{(\omega _2)}}+ {2C}/{k}. \end{aligned}$$

(3.18)

In order to make the right-hand side arbitrarily small, k is first chosen large enough. Note that $V_\phi h_1, V_\phi h_2,\ldots$ is a sequence of bounded continuous functions converging uniformly on the compact set $\overline{B}_k$. Since $\omega _2$ is a weight and $\mathscr {B}$ is an invariant BF-space we obtain that

$$\begin{aligned}&\Vert \left( V_{\phi }h_{m_1}-V_{\phi }h_{m_2} \right) \chi _{k} \Vert _{\mathscr {B}_{(\omega _2)}} =\Vert \left( V_{\phi }h_{m_1}-V_{\phi }h_{m_2} \right) \omega _2 \chi _{k} \Vert _{\mathscr {B}} \\&\qquad \lesssim \left( \sup _{(x,\xi ) \in B_k} |\left( V_{\phi }h_{m_1}(x,\xi )-V_{\phi }h_{m_2}(x,\xi ) \right) \omega _2(x, \xi )| \right) \Vert \chi _{k}\Vert _{\mathscr {B}} \\&\qquad \lesssim \sup _{(x,\xi ) \in B_k} |V_{\phi }h_{m_1}(x,\xi )-V_{\phi }h_{m_2}(x,\xi )| \end{aligned}$$

tends to zero as $m_1$ and $m_2$ tend to infinity. This proves (1).

In order to verify (2) we suppose that the embedding i in (3.8) is compact and all assumptions of the second claim hold. From the first part of the proof, the result follows if we prove that $\omega _2/\omega _1$ turns to zero at infinity. We prove this claim by contradiction.

Suppose there is a sequence $(x_k,\xi _k) \in \mathbf R^{2d}$ with $|(x_k,\xi _k)| \rightarrow \infty$ if $k \rightarrow \infty$ and a $C>0$ fulfilling

$$\begin{aligned} \frac{\omega _2(x_k,\xi _k)}{\omega _1(x_k,\xi _k)} \ge C \quad \text {for all } k \in \mathbf {N}. \end{aligned}$$

(3.19)

Let $\psi \in \Sigma _1 (\mathbf R^{d}), \phi$ be as in (2.20) and let $\{f_k\}_{k =1}^{\infty }$ be as in (3.10).

By the proof of Theorem 3.7, we get the boundedness of $\{ f_k \} _{k=1}^\infty$ in $M(\omega _1,\mathscr {B})$, and by the assumptions, $\{ f_k \} _{k=1}^\infty$ is precompact in ${M(\omega _2, \mathscr {B})}$.

Since $\phi \in \Sigma _1 (\mathbf R^{d})$ we have $|V_{\phi }\psi | \lesssim e^{-r|\, \cdot \, |}$ for every $r>0$ by Proposition 2.2(2). From the fact $\omega _1\gtrsim e^{-r_0|\, \cdot \, |}$ for some $r_0>0$ we get

$$\begin{aligned} \int \psi (x)\overline{f_k(x)}\, dx = \frac{1}{\omega _1(x_k,\xi _k)}(V_{\phi }\psi )(x_k,\xi _k)\rightarrow 0, \end{aligned}$$

as $k \rightarrow \infty$, which implies that $f_k$ tends to zero in $\Sigma _1 '(\mathbf {R} ^{d})$. Hence the only possible limit point in $M(\omega _2, \mathscr {B})$ of $\{ f_k \} _{k=1}^\infty$ is zero.

As $\{ f_k \} _{k=1}^\infty$ is precompact in $M(\omega _2, \mathscr {B})$, we can extract a subsequence $\{ f_{k_j}\} _{j=1}^\infty$ which converges to zero in $M(\omega _2, \mathscr {B})$.

Since $M(\omega _2, \mathscr {B}) \hookrightarrow M^\infty _{(\omega _2)}$ due to Lemma 3.8 we have

$$\begin{aligned} \sup _{X\in \mathbf R^{2d}}\omega _2(X)|(V_\phi (f_{k_j}))(X)|\le C\Vert f_{k_j}\Vert _{M(\omega _2, \mathscr {B})}\rightarrow 0 \end{aligned}$$

(3.20)

as $j\rightarrow \infty$. Taking $X=X_{k_j}$ in the previous inequality provides

$$\begin{aligned}&\frac{\omega _2(X_{k_j})}{\omega _1(X_{k_j})} =(2\pi )^{\frac{d}{2}}\, \frac{\omega _2(X_{k_j})}{\omega _1(X_{k_j})}|(V_\phi \phi )(0)|\nonumber \\&\quad =(2\pi )^{\frac{d}{2}}\, \frac{\omega _2(X_{k_j})}{\omega _1(X_{k_j})}|(V_\phi (e^{i\langle \, \cdot \, ,\xi _{k_j}\rangle }\phi (\, \cdot \, -x_{k_j}))(X_{k_j})|\nonumber \\&\quad =(2\pi )^{\frac{d}{2}}\, \omega _2(X_{k_j})|(V_\phi (f_{k_j}))(X_{k_j})| \rightarrow 0, \end{aligned}$$

(3.21)

which contradicts (3.19) and proves (2).$\square$

As an immediate consequence of Lemma 3.1 and Theorem 3.9 we get the following.

Corollary 3.11

Assume that $\omega _1,\omega _2\in \mathscr {P}_E (\mathbf R^{2n})$, and that $p,p_0,q,q_0\in (0,\infty ]$ are such that $p_0,q_0<\infty$. Also assume that ${\omega _2}/{\omega _1}\in L^{p_0,q_0} (\mathbf R^{2d})$. Then the embedding (3.8) is compact.

4 Applications and Open Questions

Compactness is a fundamental property in analysis, science and engineering, as remarked in the introduction. In this section we make a review on how the compactness results from the previous section are applied in [1] to deduce index and lifting results for pseudo-differential operators and Toeplitz operators. Thereafter we give some links on some open questions and further developments.

4.1 Applications to Index Results for Pseudo-Differential and Toeplitz Operators

Let A be a real $d\times d$ matrix and $\phi \in \Sigma _1(\mathbf R^{d}){\setminus } 0$ be fixed, and let $a\in \Sigma _1 (\mathbf R^{2d})$. We recall that the pseudo-differential operator $\hbox {Op}_A(a)$ and the Toeplitz operator $\hbox {Tp}_\phi (a)$ are the linear and continuous operators from $\Sigma _1(\mathbf R^{d})$ to $\Sigma _1'(\mathbf R^{d})$, defined by the formulae

$$\begin{aligned} (\hbox {Op}_A(a)f)(x)&= (2\pi )^{-d}\iint _{\mathbf R^{2d}}a(x-A(x-y),\xi )f(y) e^{i\langle x-y,\xi \rangle }\, dyd\xi \end{aligned}$$

and

$$\begin{aligned} (\hbox {Tp}_\phi (a)f,g)_{L^2(\mathbf R^{d})}&= (a \cdot V_\phi f,V_\phi g)_{L^2(\mathbf R^{2d})}, \quad f,g \in \Sigma _1(\mathbf R^{d}). \end{aligned}$$

(Cf. [1, 5, 7, 25, 28, 32, 34,35,36,37,38,39,40, 43, 44].) The definitions of pseudo-differential and Toeplitz operators extend in different ways. For example, let $p_1$ and $p_2$ be positive polynomials on $\mathbf R^{d}$ of degrees $n_1$ and $n_2$, and let

$$\begin{aligned} \omega _0(x,\xi )&= p_1(x)^{r_1}+p_2(\xi )^{\rho _1} \end{aligned}$$

(4.1)

or

$$\begin{aligned} \omega _0(x,\xi )&= \exp \big ( (p_1(x)^{r_2}+p_2(\xi )^{\rho _2})^r \big ) \end{aligned}$$

(4.2)

for some $r,r_j,\rho _j>0$ for $j=1,2$, which satisfy

$$\begin{aligned} r\cdot \max (r_2n_1,\rho _2n_2) <1. \end{aligned}$$

If $\omega \in \mathscr {P}_E^0(\mathbf R^{2d})$ and $p,q\in (0,\infty ]$, then it is proved in [44] that the definition of $\hbox {Op}_A(a)$ above is uniquely extended in such ways that

$$\begin{aligned} \hbox {Op}_A(\omega _0) \, : \, M^{p,q}_{(\omega )}(\mathbf R^{d})&\rightarrow M^{p,q}_{(\omega /\omega _0)}(\mathbf R^{d}) \end{aligned}$$

(4.3)

is continuous. In [1, Section 6] it is also proved that

$$\begin{aligned} \hbox {Op}_A(\omega _0) -\hbox {Tp}_\phi (\omega _0) \, : \, M^{p,q}_{(\omega )}(\mathbf R^{d})&\rightarrow M^{p,q}_{(\omega /\omega _1)}(\mathbf R^{d}) \end{aligned}$$

(4.4)

is continuous for some $\omega _1$ such that $\omega _1/\omega _0$ tends to zero at infinity. Finally, in [1, Section 5] it is also proved that

$$\begin{aligned} \hbox {Tp}_\phi (\omega _0) \, : \, M^{p,q}_{(\omega )}(\mathbf R^{d})&\rightarrow M^{p,q}_{(\omega /\omega _0)}(\mathbf R^{d}) \end{aligned}$$

(4.5)

is a continuous bijection with continuous inverse. A combination of (4.4), the fact that $\omega _1/\omega _0$ tends to zero at infinity and Theorem 3.9 then shows that

$$\begin{aligned} \hbox {Op}_A(\omega _0) -\hbox {Tp}_\phi (\omega _0) \, : \, M^{p,q}_{(\omega )}(\mathbf R^{d})&\rightarrow M^{p,q}_{(\omega /\omega _0)}(\mathbf R^{d}) \end{aligned}$$

(4.6)

is compact.

In particular, if in addition $p,q\ge 1$, then the involved modulation spaces are Banach spaces. Hence, Fredholm’s theorem shows that the indices of the operators in (4.3) and (4.5) satisfy

$$\begin{aligned} \hbox {Ind}(\hbox {Op}_A(\omega _0)) =\hbox {Ind}(\hbox {Tp}_\phi (\omega _0)) =0. \end{aligned}$$

Here the last equality follows from the fact that (4.5) is a continuous bijection.

If $\omega _0$ is given by (4.1), then it follows by straight-forward computations that $\hbox {Op}_A(\omega _0)$ is injective. Since $\hbox {Ind}(\hbox {Op}_A(\omega _0))=0$, it follows that the map (4.3) in this case is bijective.

4.2 Open Questions and Further Developments

The main objective in the paper is Theorem 3.9 which completely characterizes compactness for the injection map (3.8) when $\mathscr {B}$ is either an invariant BF-space or a mixed quasi-normed space of Lebesgue type, and $\omega _1,\omega _2\in \mathscr {P}_E(\mathbf R^{2d})$. An open question here concerns wether such characterizations can be deduced when $\omega _1$ and $\omega _2$ are allowed to belong to a broader weight class than $\mathscr {P}_E(\mathbf R^{2d})$.

In fact, for general weights, Theorem 3.9 gives some sufficient but no necessary conditions for the map (3.8) to be compact.

An other open question concerns wether Theorem 3.9 holds for any QBF-space $\mathscr {B}$ with respect to $v=1$ and not only when $\mathscr {B}$ is either an invariant BF-space or a mixed quasi-normed space of Lebesgue type.