Decay and Smoothness for Eigenfunctions of Localization Operators

TL;DR

This paper investigates the decay and smoothness of eigenfunctions of localization operators, showing they are highly localized or Schwartz functions depending on the symbol class, with new convolution and multiplication relations in quasi-Banach spaces.

Contribution

It establishes decay and smoothness properties of eigenfunctions for localization operators with symbols in modulation spaces, introducing new convolution and multiplication relations in quasi-Banach spaces.

Findings

Eigenfunctions are highly localized onto few Gabor atoms.

Eigenfunctions are Schwartz functions for certain symbol classes.

New convolution and multiplication relations for modulation and Wiener amalgam spaces.

Abstract

We study decay and smoothness properties for eigenfunctions of compact localization operators. Operators with symbols a in the wide modulation space M^{p,\infty} (containing the Lebesgue space L^p), p<\infty, and windows \f_1,\f_2 in the Schwartz class are known to be compact. We show that their L^2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces M^{\infty}_{v_s\otimes 1} (\rdd), s>0 (subspaces of M^{p,\infty}(\rdd), p>2d/s) the L^2-eigenfunctions of the localization operator are actually Schwartz functions. An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.

Equations

$$M_{\omega }f\left(t\right)=e^{2\pi it\omega }f\left(t\right),T_{x}f\left(t\right)=f\left(t-x\right),\omega ,x\in {\mathbb{R}}^d.$$

$$V_{\phi }f(x,\omega )=⟨f,\pi (x,\omega )\phi ⟩=\mathcal{F}(f\overline{T_x\phi })(\omega )={\int }_{{\mathbb{R}}^d}f(y)\overline{\phi (y-x)}{e}^{-2\pi iy\omega }dy.$$

$$A_a^{{\phi }_1,{\phi }_2}f(t)={\int }_{{\mathbb{R}}^{2d}}a(x,\omega )V_{{\phi }_1}f(x,\omega )M_{\omega }{T}_x{\phi }_2(t)dxd\omega .$$

$$(A_a^{{\phi }_1,{\phi }_2}{)}^{\ast }=A_{\bar{a}}^{{\phi }_2,{\phi }_1};$$

$${\sigma }_P(T)=\{\lambda \in \mathbb{C}:\exists \text{mbox}anon-zerof\in L^2({\mathbb{R}}^d)\text{mbox}suchthatTf=\lambda f\}.$$

$$\sigma (A_a^{{\phi }_1,{\phi }_2})\setminus \{0\}={\sigma }_P(A_a^{{\phi }_1,{\phi }_2})\setminus \{0\}.$$

$$A_a^{{\phi }_1,{\phi }_2}f=\lambda f.$$

$$\Vert f{\Vert }_{M^{p,q}}:=\Vert V_{g}f{\Vert }_{L^{p,q}}={\left({\int }_{{\mathbb{R}}^d}{\left({\int }_{{\mathbb{R}}^d}|V_{g}f(x,\omega ){|}^{p}dx\right)}^{\frac{q}{p}}d\omega \right)}^{\frac{1}{q}}<\infty$$

$$A\Vert f{\Vert }_2^2\le \sum _{\lambda \in \Lambda }|⟨f,\pi (\lambda )g⟩{|}^2\le B\Vert f{\Vert }_2^2,\forall f\in L^2({\mathbb{R}}^d).$$

$$\Vert f{\Vert }_2^2=\sum _{\lambda \in \Lambda }|⟨f,\pi (\lambda )g⟩{|}^2,\forall f\in L^2({\mathbb{R}}^d).$$

$${\Sigma }_N=\{p=\sum _{k,n\in F}{c}_{k,n}\pi (\alpha k,\beta n)g:c_{k,n}\in \mathbb{C},F\subset {\mathbb{Z}}^d\times {\mathbb{Z}}^d,\text{mbox}cardF\le N\}$$

$${\sigma }_N(f)=\underset{p\in {\Sigma }_N}{\inf }\Vert f-p{\Vert }_2.$$

$${\sigma }_N(f)\le C{N}^{-r}.$$

$$\mathcal{F}f(\omega )={\int }_{{\mathbb{R}}^d}{e}^{-2\pi ix\omega }f(x)dx.$$

$$\Vert f{\Vert }_{L_m^p}={\left({\int }_{{\mathbb{R}}^d}|f(x){|}^{p}m(x{)}^{p}dx\right)}^{1/p},$$

$$\pi (z)\pi (w)=e^{-2\pi i{z}_1{w}_2}\pi (w)\pi (z).$$

$$W(f,g)(x,\omega )={\int }_{{\mathbb{R}}^d}f(x+\frac{t}{2})\overline{g(x-\frac{t}{2})}{e}^{-2\pi it\omega }dt.$$

$$v_s(z)=⟨z{⟩}^s=(1+|z{|}^2{)}^{s/2},s\in \mathbb{R},z\in {\mathbb{R}}^d(\text{mbox}or{\mathbb{Z}}^d).$$

$$(m_1\otimes m_2)(x,\omega )=m_1(x)m_2(\omega ),x,\omega \in {\mathbb{R}}^d(\text{mbox}or{\mathbb{Z}}^d).$$

$$\Vert a{\Vert }_{{\ell }_m^{p,q}}={\left(\sum _{n\in {\mathbb{Z}}^d}{\left(\sum _{k\in {\mathbb{Z}}^d}|a_{k,n}{|}^{p}m(k,n{)}^p\right)}^{\frac{q}{p}}\right)}^{\frac{1}{q}}$$

$$\Vert a{\Vert }_{{\ell }_m^p}={\left(\sum _{k\in {\mathbb{Z}}^d}|a_k{|}^{p}m(k{)}^p\right)}^{\frac{1}{p}}<\infty .$$

$$\frac{1}{p}+\frac{1}{q}=1+\frac{1}{r},\text{mbox}for1\le r\le \infty$$

$$p=q=r,\text{mbox}for0<r<1.$$

$$\Vert a\ast b{\Vert }_{{\ell }_m^r}\le C\Vert a{\Vert }_{{\ell }_m^p}\Vert b{\Vert }_{{\ell }_v^q},$$

$${\ell }_m^p({\mathbb{Z}}^d)\cdot {\ell }_{1/m}^q({\mathbb{Z}}^d)\hookrightarrow {\ell }^r({\mathbb{Z}}^d),$$

$$F_f^Q\left(k\right):={\Vert f\cdot T_k{\chi }_Q\Vert }_{L^p}\in {\ell }_m^q\left({\mathbb{Z}}^d\right),k\in {\mathbb{Z}}^d.$$

$${\Vert f\Vert }_{W\left(L^p,L_m^q\right)}$$

$$={\Vert {\Vert f\cdot T_k{\chi }_Q\Vert }_{L^p}\Vert }_{{\ell }_m^q}$$

$$={\left(\sum _{k\in {\mathbb{Z}}^d}{\left({\int }_{{\mathbb{R}}^d}{\left|f\left(t\right)\right|}^p{\chi }_Q\left(t-k\right)\mathrm{d}t\right)}^{\frac{q}{p}}{m}^q\left(k\right)\right)}^{\frac{1}{q}},$$

$$W(L^{p_2},L_m^{q_2})({\mathbb{R}}^d)\hookrightarrow W(L^{p_1},L_m^{q_1})({\mathbb{R}}^d).$$

$$W(L^{p_1},L_{m_1}^{q_1})\ast W(L^{p_2},L_{m_2}^{q_2})\hookrightarrow W(L^{p_3},L_{m_1}^{q_3}).$$

Full text

An important role is played by quasi-Banach Wiener amalgam and modulation spaces. As a tool, new convolution relations for modulation spaces and multiplication relations for Wiener amalgam spaces in the quasi-Banach setting are exhibited.

1. Introduction

The study of localization operators has a long-standing tradition. They have become popular with the papers by I. Daubechies [11, 12] and from then widely investigated by several authors in different fields of mathematics: from signal analysis to pseudodifferential calculus, see, for instance [1, 2, 4, 7, 8, 10, 31, 32, 43, 44, 45, 50]. In quantum mechanics they were already known as Anty-Wick operators, cf. [41] and the references therein.

Localization operators can be introduced via the time-frequency representation known as short-time Fourier transform (STFT). Let us though introduce the STFT. Recall first the modulation $M_{\omega}$ and translation $T_{x}$ operators of a function $f$ on $\mathbb{R}^{d}$:

[TABLE]

For $z=(z_{1},z_{2})\in{\mathbb{R}^{2d}}$, we define the time-frequency  shift $\pi(z)=M_{z_{2}}T_{z_{1}}$. Fix $\varphi\in\mathcal{S}(\mathbb{R}^{d})\setminus\{0\}$. We define the short-time Fourier transform of a tempered distribution $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ as

[TABLE]

The localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ with symbol $a$ and windows $\varphi_{1},\varphi_{2}$ is formally defined to be

[TABLE]

If $\varphi_{1}(t)=\varphi_{2}(t)=e^{-\pi t^{2}}$, then $A_{a}=A_{a}^{\varphi_{1},\varphi_{2}}$ is the classical Anti-Wick operator and the mapping $a\mapsto A_{a}^{\varphi_{1},\varphi_{2}}$ is a quantization rule in quantum mechanics [5, 13, 41, 50].

If one considers a symbol $a$ in the Lebesgue space $L^{q}({\mathbb{R}^{2d}})$ ($1\leq q<\infty$) and window functions $\varphi_{1},\varphi_{2}$ in the Feichtinger’s algebra $M^{1}(\mathbb{R}^{d})$ (see below for its definition) then the localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ is in the Schatten class $S_{q}$ (see [7]). This implies, in particular, that $A_{a}^{\varphi_{1},\varphi_{2}}$ is a bounded and compact operator on $L^{2}(\mathbb{R}^{d})$.

Sharp results for localization operators on modulation spaces (Banach case) were obtained in [9], thus concluding the open issues related to this problem.

The focus of this paper is the properties of eigenfunctions of compact localization operators.

The study of eigenvalues and eigenfunctions of a restrict class of compact and self-adjoint localization operators, namely of the type $A_{\chi_{\Omega}}^{\varphi,\varphi}$, where $\Omega$ is a compact domain of the time-frequency  plane and the window $\varphi$ is in $L^{2}(\mathbb{R}^{d})$, was pursued in [1, 2, 3]. The focus of the previous papers, as well as new recent contributions [37, 38], extending the previous results, is the asymptotic behavior of the eigenvalues, depending on the domain $\Omega$.

Our perspective is different: properties of eigenfunctions of a compact localization operator having a general symbol $a$, without any requirement on the geometry of the function $a$. In particular, the symbol $a$ does not need to have a compact support. Besides, the related localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ is not necessarily a self-adjoint operator (but $A_{a}^{\varphi_{1},\varphi_{2}}$ is compact). It is easy to check that the adjoint of a localization operator is given by

[TABLE]

hence the self-adjointness property forces the choice $\varphi_{1}=\varphi_{2}$ and the symbol $a$ real valued, as for the case $A_{\chi_{\Omega}}^{\varphi,\varphi}$ mentioned above. Our framework can allow the use of two different windows $\varphi_{1}$ and $\varphi_{2}$ to analyze and synthesize the signal $f$, respectively. Moreover, the symbol $a$ can be a complex-valued function.

For a linear bounded operator $T$ on $L^{2}(\mathbb{R}^{d})$ we denote by $\sigma(T)$ the spectrum of $T$, that is the set $\{\lambda\in\mathbb{C}:\,T-\lambda I\,\mbox{is\,not\,invertible}\}$; in particular, the set $\sigma_{P}(T)$ denotes the point spectrum of $T$, that is

[TABLE]

Under our assumptions, the operator $A_{a}^{\varphi_{1},\varphi_{2}}$ is a compact mapping on $L^{2}(\mathbb{R}^{d})$. Hence, the spectral theory for compact operators yields

[TABLE]

For sake of completeness, we recall that for compact operators on $L^{2}(\mathbb{R}^{d})$ the number zero is always in the spectrum $\sigma(A_{a}^{\varphi_{1},\varphi_{2}})$ and that the point spectrum $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$ (possibly empty) is at most countably infinite.

A function $f\in L^{2}(\mathbb{R}^{d})$ is called an eigenfunction of the operator $A_{a}^{\varphi_{1},\varphi_{2}}$ if there exists $\lambda\in\mathbb{C}$ such that

[TABLE]

We are interested in the properties of eigenfuctions of $A_{a}^{\varphi_{1},\varphi_{2}}$ related to eigenvalues $\lambda\in\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$, whenever $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}\not=\emptyset$.

To chase our goal, the new idea is to use properties of quasi-Banach modulation and Wiener amalgam spaces. Some of these issues are investigated for the first time in this paper.

To give a flavour of these results, we recall the definition of modulation spaces (un-weighted case: for the more general cases see below).

Fix a non-zero window function $g\in\mathcal{S}(\mathbb{R}^{d})$ and $p,q\in(0,\infty)$. The modulation space $M^{p,q}(\mathbb{R}^{d})$ consists of all tempered distributions $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that

[TABLE]

(with natural modifications when $p=\infty$ or $q=\infty$). For $p,q\geq 1$ the function $\|\cdot\|_{M^{p,q}}$ is a norm, and different window functions $g\in\mathcal{S}(\mathbb{R}^{d})$ yield equivalent norms, thus the same space. Modulation spaces $M^{p,q}(\mathbb{R}^{d})$, $p,q\geq 1$, are Banach spaces, invented by H. Feichtinger in [19], where many of their properties were already investigated.

We remark that a localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ with symbol $a\in M^{p,\infty}$, $1\leq p<\infty$, and windows $\varphi_{1},\varphi_{2}\in M^{1}$, belongs to the Schatten class $S_{p}$ and, in particular, it is a compact operator on $L^{2}(\mathbb{R}^{d})$ (cf. [8, Theorem 1]).

The (quasi-)Banach space $M^{p,q}$, $p,q>0$, were first introduced and studied by Y.V. Galperin and S. Samarah in [27], see the next section for more details. Roughly speaking, the mapping $f$ is in $M^{p,q}(\mathbb{R}^{d})$ if it locally behaves like a function in $\mathcal{F}L^{q}(\mathbb{R}^{d})$ and “decays” as a function in $L^{p}(\mathbb{R}^{d})$ at infinity.

In this paper we extend convolution relations for Banach modulation spaces exhibited in [7, 47] to the quasi-Banach ones, see Proposition 3.1 below. This result seems remarkable by itself.

Such convolution relations will be crucial for our main result, that can be simplified as follows (cf. Theorem 3.7).

Theorem 1.1**.**

Consider a symbol $a\in M^{p,\infty}({\mathbb{R}^{2d}})$, $0<p<\infty$, non-zero windows $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$ and assume that $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}\not=\emptyset$. If $\lambda\in\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$, any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ satisfies $f\in\bigcap_{\gamma>0}M^{\gamma}(\mathbb{R}^{d})$.

Roughly speaking, this means that $L^{2}$ eigenfunctions of such compact localization operators reveal to be extremely well-localized. To make this statement more precise, we use Gabor frames. Let $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$ be a lattice of the time-frequency plane. The set of time-frequency shifts $\mathcal{G}(g,\Lambda)=\{\pi(\lambda)g:\ \lambda\in\Lambda\}$, for a non-zero $g\in L^{2}(\mathbb{R}^{d})$, is called a Gabor system. The set $\mathcal{G}(g,\Lambda)$ is a Gabor frame if there exist constants $A,B>0$ such that

[TABLE]

If $A=B=1$ we call $\mathcal{G}(g,\Lambda)$ a Parseval Gabor frame. In this case (3) reduces to

[TABLE]

From now on we restrict to Parseval Gabor frames. If the eigenfunction $f$ above satisfies $f\in\bigcap_{\gamma>0}M^{\gamma}(\mathbb{R}^{d})$, then $f$ is highly compressed onto a few Gabor atoms $\pi(\lambda)g$. Indeed, for $N\in\mathbb{N}_{+}$, its $N$-term approximation error $\sigma_{N}(f)$ presents super-polynomial decay. We define

[TABLE]

(the set of all linear combinations of Gabor atoms consisting of at most $N$ terms). Note that $\Sigma_{N}$ is not a linear subspace since $\Sigma_{N}+\Sigma_{N}=\Sigma_{2N}$. That is why the approximation of a signal $f$ by elements of $\Sigma_{N}$ is often referred to as non-linear approximation. Given a function $f\in L^{2}(\mathbb{R}^{d})$, the $N$-term approximation error in $L^{2}(\mathbb{R}^{d})$ is

[TABLE]

That is, $\sigma_{N}(f)$ is the error produced when $f$ is approximated optimally by a linear combination of $N$ Gabor atoms.

We shall show in Corollary 3.9 that, if $f\in\bigcap_{\gamma>0}M^{\gamma}(\mathbb{R}^{d})$, then, for every $r>0$ there exists a positive constant $C=C(r,f)$ such that

[TABLE]

Since $\sigma_{N}(f)$ is the error produced when $f$ is approximated optimally by a linear combination of $N$ Gabor atoms, the decay above shows the high compression of the eigenfunction onto such atoms, cf. [28, Subsection 12.4].

Another main result (Theorem 3.10) states, in the same spirit, that for symbols in the weighted modulation space $M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})$, $s>0$ (see Definition 2.4 below), the corresponding $L^{2}$ eigenfunctions of $A_{a}^{\varphi_{1},\varphi_{2}}$, with eigenvalues $\lambda\not=0$, are actually in $\mathcal{S}(\mathbb{R}^{d})$. Also in this case $A_{a}^{\varphi_{1},\varphi_{2}}$ is a compact operator, since $M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})\subset M^{p,\infty}({\mathbb{R}^{2d}})$, whenever $p>2d/s$, cf. the subsequent Remark 2.7. These symbol classes include certain measures and the result applies, in particular, to Gabor multipliers (see e.g., [21]). We leave the precise statement to the interested reader.

In short, the paper is organized as follows. Section 2 is devoted to the function spaces involved in our study. In particular, we show new multiplication relations for Wiener amalgam spaces in the quasi-Banach setting and we prove convolution relations for quasi-Banach modulation spaces. Section 3 represents the core of the paper. We first exhibit continuity results for Weyl operators on modulation spaces (involving the quasi-Banach setting), cf. Theorem 3.3. Then we study the eigenfunctions’ properties for Weyl operators (Propositions 3.5 and 3.6 below). Next we show our main result on the eigenfunctions’ regularity and smoothness of localization operators: Theorem 3.7 and related consequences in terms of the $N$-term approximation (Proposition 3.8 and Corollary 3.9). The last Section $4$ is devoted to the study of eigenfunctions of localization operators with symbols in the weighted Lebesgue spaces $L^{q}_{m}(\mathbb{R}^{d})$, $1\leq q<\infty$. This section suggests a wider study of the topic for localization operators on groups [50]. We shall pursue this issue in a subsequent paper. Let us conclude this introduction with another open problem, that can be worth investigating. Namely, in this paper we work with compact localization operators having symbols $a$ in $M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})$, for $s>0$, or in the larger space $M^{p,\infty}({\mathbb{R}^{2d}})$, $0<p<\infty$. Notice that the symbol classes $M^{p,\infty}({\mathbb{R}^{2d}})$ do not exhaust the class of compact localization operators $A_{a}^{\varphi_{1},\varphi_{2}}$, characterized in [22, Theorem 3.15] (see also the contributions [23, 24]); in fact, the localization operators object of study belong to the Schatten class $S_{p}$, $p<\infty$ (see the characterization in [8, Theorem 1]), that is a proper linear subspace of the space of compact operators. More comments on this topic are contained in the subsequent Remark 3.11.

2. Preliminaries

Notation. We define $xy=x\cdot y$, the scalar product on $\mathbb{R}^{d}$. The Schwartz class is denoted by $\mathcal{S}(\mathbb{R}^{d})$, the space of tempered distributions by $\mathcal{S}^{\prime}(\mathbb{R}^{d})$. We use the brackets $\langle f,g\rangle$ to denote the extension to $\mathcal{S}^{\prime}(\mathbb{R}^{d})\times\mathcal{S}(\mathbb{R}^{d})$ of the inner product $\langle f,g\rangle=\int f(t){\overline{g(t)}}dt$ on $L^{2}(\mathbb{R}^{d})$. The Fourier transform of a function $f$ on $\mathbb{R}^{d}$ is normalized as

[TABLE]

The involution $g^{*}$ is given by $g^{*}(t)=\overline{g(-t)}$. Given a measurable and positive weight function $m$ on ${\mathbb{R}^{2d}}$ and $1\leq p<\infty$, we denote by $L^{p}_{m}(\mathbb{R}^{d})$ the Banach space of measurable functions $f:\mathbb{R}^{d}\to\mathbb{C}$ satisfying

[TABLE]

up to the equivalence relation $f\sim g$ if and only if $f(x)=g(x)\,$ for a.e. $x$. For $p=\infty$, we have $f\in L^{\infty}_{m}(\mathbb{R}^{d})$ if $\|f\|_{L^{\infty}_{m}}=$ ess sup${}_{x\in\mathbb{R}^{d}}|f(x)|m(x)<\infty$, again up to the equivalence relation above. For $s\in\mathbb{R}$, recall the Sobolev spaces $H^{s}(\mathbb{R}^{d})=\{f\,:\,\hat{f}(\omega)\langle\omega\rangle^{s}\in L^{2}(\mathbb{R}^{d})\},$ where $\langle\omega\rangle^{s}=(1+|\omega|^{2})^{s/2}$.

We use $T^{*}$ for the adjoint of an operator $T$. Observe that $\pi(z)^{\ast}=\pi(-z)$ and the following commutation relations hold

[TABLE]

For $f,g\in L^{2}(\mathbb{R}^{d})$, the cross-Wigner distribution is defined by:

[TABLE]

Given two normed spaces $A$ and $B$, we denote by $A\hookrightarrow B$ the continuous embedding of $A$ into $B$.

Finally we recall the definition of the Schatten classes $S_{p}$, $0<p<\infty$. The singular values $\{s_{k}(L)\}_{k=1}^{\infty}$ of a compact operator $L$ on a Hilbert space are the eigenvalues of the positive self-adjoint operator $(L^{*}L)^{1/2}$. The Schatten class $S_{p}$ consists of all compact operators whose singular values lie in $\ell^{p}$.

2.1. Weight functions

In the sequel $v$ will always be a continuous, positive, submultiplicative weight function on $\mathbb{R}^{d}$ (or on $\mathbb{Z}^{d}$), i.e., $v(z_{1}+z_{2})\leq v(z_{1})v(z_{2})$, for all $z_{1},z_{2}\in\mathbb{R}^{d}$ (or for all $z_{1},z_{2}\in\mathbb{Z}^{d}$). We say that $m\in\mathcal{M}_{v}(\mathbb{R}^{d})$ (or $m\in\mathcal{M}_{v}(\mathbb{Z}^{d})$) if $m$ is a positive, continuous weight function on $\mathbb{R}^{d}$ (or on $\mathbb{Z}^{d}$) $v$-moderate: $m(z_{1}+z_{2})\leq Cv(z_{1})m(z_{2})$ for all $z_{1},z_{2}\in\mathbb{R}^{d}$ (or for all $z_{1},z_{2}\in\mathbb{Z}^{d}$). We will mainly work with polynomial weights of the type

[TABLE]

Observe that, for $s<0$, $v_{s}$ is $v_{|s|}$-moderate.

Given two weight functions $m_{1},m_{2}$ on $\mathbb{R}^{d}$ (or $\mathbb{Z}^{d}$), we write

[TABLE]

2.2. Spaces of sequences

Definition 2.1**.**

For $0<p,q\leq\infty$, $m\in\mathcal{M}_{v}({\mathbb{Z}^{2d}})$, the space $\ell^{p,q}_{m}({\mathbb{Z}^{2d}})$ consists of all sequences $a=(a_{k,n})_{k,n\in\mathbb{Z}^{d}}$ for which the (quasi-)norm

[TABLE]

(with obvious modification for $p=\infty$ or $q=\infty$) is finite.

For $p=q$, $\ell^{p,q}_{m}({\mathbb{Z}^{2d}})=\ell^{p}_{m}({\mathbb{Z}^{2d}})$, the standard spaces of sequences. Namely, in dimension $d$, for $0<p\leq\infty$, $m$ a weight function on $\mathbb{Z}^{d}$, a sequence $a=(a_{k})_{k\in\mathbb{Z}^{d}}$ is in $\ell^{p}_{m}(\mathbb{Z}^{d})$ if

[TABLE]

Here there are some properties we need in the sequel [26, 27]:

• (i)

Inclusion relations: If $0<p_{1}\leq p_{2}\leq\infty$, then $\ell^{p_{1}}_{m}(\mathbb{Z}^{d})\hookrightarrow\ell^{p_{2}}_{m}(\mathbb{Z}^{d})$, for any positive weight function $m$ on $\mathbb{Z}^{d}$.

• (ii)

Young’s convolution inequality: Consider $m\in\mathcal{M}_{v}(\mathbb{Z}^{d})$, $0<p,q,r\leq\infty$ with

[TABLE]

and

[TABLE]

Then for all $a\in\ell^{p}_{m}(\mathbb{Z}^{d})$ and $b\in\ell^{q}_{v}(\mathbb{Z}^{d})$, we have $a\ast b\in\ell^{r}_{m}(\mathbb{Z}^{d})$, with

[TABLE]

where $C$ is independent of $p,q,r$, $a$ and $b$. If $m\equiv v\equiv 1$, then $C=1$.

• (iii)

Hölder’s inequality: For any positive weight function $m$ on $\mathbb{Z}^{d}$, $0<p,q,r\leq\infty$, with $1/p+1/q=1/r$,

[TABLE]

where the symbol $\hookrightarrow$ denotes that the inclusion is a continuous mapping.

2.3. Wiener Amalgam Spaces [16, 17, 18, 25, 27, 34, 39, 40].

Definition 2.2**.**

Consider $p,q\in(0,\infty]$, a weight function $m\in\mathcal{M}_{v}$ and the compact set $Q=[0,1]^{d}$. The Wiener amalgam space $W\left(L^{p},L_{m}^{q}\right)(\mathbb{R}^{d})$ consists of the functions $f\,:\,\mathbb{R}^{d}\rightarrow\mathbb{C}$ such that $f\in L_{{loc}}^{p}(\mathbb{R}^{d})$ and for which the control function:

[TABLE]

The quasi-norm on $W\left(L^{p},L_{m}^{q}\right)$ is given by

[TABLE]

with suitable adjustments for the cases $p,q=\infty$.

This special definition allows us to grasp the sense of the amalgam: we first view $f$ “locally”  through translations $T_{k}\chi_{Q}$ of the sharp cutoff function $\chi_{Q}$, and measure those local pieces in the $L^{p}$-norm, then we measure the global behavior of those local pieces according to the $\ell_{m}^{q}$-norm. The “window”  through which we view $f$ locally need not be a unit $d$-dimensional cube, cf. [17, 27, 34, 40]. In the sequel we shall use the following properties:

• (i)

Inclusion relations: For $0<p_{1}\leq p_{2}\leq\infty$, $0<q_{2}\leq q_{1}\leq\infty$, we have

[TABLE]

• (ii)

Convolution relations (for the quasi-Banach case see [27, Lemma 2.9]): Consider $m_{i}\in\mathcal{M}_{v}$, $0<p_{i},q_{i}\leq\infty$, $i\in\{1,2,3\}$, and $p_{3}\geq 1$. Assume that $L^{p_{1}}\ast L^{p_{2}}\hookrightarrow L^{p_{3}}$ and $\ell^{q_{1}}_{m_{1}}\ast\ell^{q_{2}}_{m_{2}}\hookrightarrow\ell^{q_{3}}_{m_{3}}$, then

[TABLE]

• (iii)

For $m\in\mathcal{M}_{v}$, $0<p\leq\infty$, we have

[TABLE]

Proposition 2.3** (Multiplication relations).**

Consider $m,w\in\mathcal{M}_{v}$, $0<p_{i},q_{i}\leq\infty$, $i=\{1,2,3\}.$ Assume $\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}$ and $\frac{1}{q_{1}}+\frac{1}{q_{2}}=\frac{1}{q_{3}}$, then

[TABLE]

Proof.

The result is well known for $1\leq p_{i},q_{i}\leq\infty$, cf. [16, 34]. Here we show that the same proof works for quasi-Banach spaces. Indeed, since the standard Hölder inequality holds for Lebesgue exponents in $(0,+\infty]$, for $f_{1}\in W(L^{p_{1}},L^{q_{1}}_{m})$, $f_{2}\in W(L^{p_{2}},L^{q_{2}}_{w/m})$ we have

[TABLE]

Defining $a_{k}=\|f_{1}T_{k}\chi_{Q}\|_{p_{1}}$ and $b_{k}=\|f_{2}T_{k}\chi_{Q}\|_{p_{2}}$ and using Hölder’s inequality for sequences $\ell^{q_{1}}\ell^{q_{2}}\hookrightarrow\ell^{q_{3}}$, for $1/q_{1}+1/q_{2}=1/q_{3}$ ($0<q_{i}\leq\infty$, $i=1,2,3$), we obtain

[TABLE]

This completes the proof. ∎

2.4. Modulation Spaces

We use the extension to quasi-Banach spaces introduced first by Y.V. Galperin and S. Samarah in [27].

Definition 2.4**.**

Fix a non-zero window $g\in\mathcal{S}(\mathbb{R}^{d})$, a weight $m\in\mathcal{M}_{v}$ and $0<p,q\leq\infty$. The modulation space $M^{p,q}_{m}(\mathbb{R}^{d})$ consists of all tempered distributions $f\in\mathcal{S}^{\prime}(\mathbb{R}^{d})$ such that the (quasi-)norm

[TABLE]

(obvious changes with $p=\infty$ or $q=\infty)$ is finite.

The most famous modulation spaces are those $M^{p,q}_{m}(\mathbb{R}^{d})$ with $1\leq p,q\leq\infty$, invented by H. Feichtinger in [19]. In that paper he proved they are Banach spaces, whose norm does not depend on the window $g$, in the sense that different window functions in $\mathcal{S}(\mathbb{R}^{d})$ yield equivalent norms. Moreover, the window class $\mathcal{S}(\mathbb{R}^{d})$ can be extended to the modulation space $M^{1,1}_{v}(\mathbb{R}^{d})$ (so-called Feichtinger algebra).

For shortness, we write $M^{p}_{m}(\mathbb{R}^{d})$ in place of $M^{p,p}_{m}(\mathbb{R}^{d})$ and $M^{p,q}(\mathbb{R}^{d})$ if $m\equiv 1$.

The modulation spaces $M^{p,q}_{m}(\mathbb{R}^{d})$, $0<p,q<1$, where introduced almost twenty years later by Y.V. Galperin and S. Samarah in [27] and then studied in [35, 39, 48, 49] (see also references therein). In this framework, it appears that the largest natural class of windows universally admissible for all spaces $M^{p,q}_{m}(\mathbb{R}^{d})$, $0<p,q\leq\infty$ (with $m$ having at most polynomial growth) is the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$. There are thousands of papers involving modulation spaces with indices $1\leq p,q\leq\infty$, whereas very few works deal with the quasi-Banach case $0<p,q\leq 1$. Indeed, many properties related to the latter case are still unexplored.

In this paper their contribution is fundamental, since they are the key tool for understanding the properties of eigenfunctions of localization operators having symbols with some decay at infinity, measured in the $L^{p}$-mean, $0<p<\infty$.

We observe that for any $0<p\leq\infty$, the window function $g$ can be chosen in a bigger space than the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$, see below [48, Proposition 1.2 (1)].

Proposition 2.1**.**

Consider $m\in\mathcal{M}_{v}$, $r\in(0,1]$, $0<p\leq\infty$ such that $r\leq p$. Then the following is true: if $g\in M_{v}^{r}(\mathbb{R}^{d})\setminus\{0\}$, then $f\in M^{p}_{m}(\mathbb{R}^{d})$ if and only if (20) is finite. In particular, $M^{p}_{m}(\mathbb{R}^{d})$ is independent of the choice of $g\in M_{v}^{r}(\mathbb{R}^{d})\setminus\{0\}$. Moreover, it is a quasi-Banach space under the quasi-norm in (20), and different choices of $g$ give rise to equivalent quasi-norms.

In the sequel we shall use inclusion relations for modulation spaces, we refer to [27, Theorem 3.4] and [28, Theorem 12.2.2] for the proof of the following result.

Theorem 2.5**.**

Let $m\in\mathcal{M}_{v}({\mathbb{R}^{2d}})$. If $0<p_{1}\leq p_{2}\leq\infty$ and $0<q_{1}\leq q_{2}\leq\infty$ then $M^{p_{1},q_{1}}_{m}(\mathbb{R}^{d})\hookrightarrow M^{p_{2},q_{2}}_{m}(\mathbb{R}^{d})$.

The duality properties for modulation spaces with indices $p,q<1$ where studied in [36] and completed in [49, Proposition 6.4, page 163]:

Proposition 2.6**.**

Let $s\in\mathbb{R}$ and $0<p,q<\infty$. If $p\geq 1$ we denote $1/p+1/p^{\prime}=1$ (and similarly for $q$); if $0<p<1$ we denote $p^{\prime}=\infty$. Then $(M^{p,q}_{1\otimes v_{s}}(\mathbb{R}^{d}))^{\prime}=M^{p^{\prime},q^{\prime}}_{1\otimes v_{-s}}(\mathbb{R}^{d})$.

Remark 2.7**.**

In our framework it is important to notice the following inclusion relation for $s>0$:

[TABLE]

This follows from the recent contribution [33, Theorem 1.5]. Hence localization operators $A_{a}^{\varphi_{1},\varphi_{2}}$ with symbols $a$ in $M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})$ and windows $\varphi_{1},\varphi_{2}$ in $\mathcal{S}(\mathbb{R}^{d})$ are compact operators belonging to the Schatten class $S_{p}$, with $p>2d/s$.

We will repeatedly use the following result, cf. [27, Theorem 3.3] (see also [28, Theorem 12.2.1] for $p\geq 1$).

Theorem 2.8**.**

Assume that $m\in\mathcal{M}_{v}$. For $0<p<1$ let $g$ be a non-zero window in $M^{r}_{v}(\mathbb{R}^{d})$, $r\leq p$. For $1\leq p\leq\infty$, the function $g$ can be chosen in the larger space $M^{1}_{v}(\mathbb{R}^{d})$. If $f\in M^{p}_{m}(\mathbb{R}^{d})$, $0<p\leq\infty$, then $V_{g}f\in W(L^{\infty},L^{p}_{m})$ and there exists $C>0$, independent of $f$, such that

[TABLE]

We also need to recall the inversion formula for the STFT (see [28, Proposition 11.3.2]): assume $g\in M^{1}_{v}(\mathbb{R}^{d})\setminus\{0\}$, $1\leq p,q\leq\infty$, $f\in M^{p,q}_{m}(\mathbb{R}^{d})$, then

[TABLE]

and the equality holds in $M^{p,q}_{m}(\mathbb{R}^{d})$.

2.5. Gabor Frames

Let $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$ be a lattice of the time-frequency plane. The set of time-frequency shifts $\mathcal{G}(g,\Lambda)=\{\pi(\lambda)g:\ \lambda\in\Lambda\}$, for a non-zero $g\in L^{2}(\mathbb{R}^{d})$, is called a Gabor system. The set $\mathcal{G}(g,\Lambda)$ is a Gabor frame, if there exist constants $A,B>0$ such that (3) holds true.

If $\mathcal{G}(g,\Lambda)$ is a Gabor frame, then it can be shown that the frame operator

[TABLE]

is a topological isomorphism on $L^{2}(\mathbb{R}^{d})$. Moreover, the system $\mathcal{G}(\gamma,\Lambda)$, where the function $\gamma=S^{-1}g\in L^{2}(\mathbb{R}^{d})$ is the canonical dual window of $g$, is a Gabor frame and we have the reproducing formula

[TABLE]

with unconditional convergence in $L^{2}(\mathbb{R}^{d})$. In particular, if $\gamma=g$ and $\|g\|_{2}=1$ then $A=B=1$, the frame operator is the identity $S=I$ and the Gabor frame is called Parseval Gabor frame. Observe that formula (4) holds true. More generally, any window function $\gamma\in L^{2}(\mathbb{R}^{d})$, such that (23) is satisfied, is called alternative dual window for $g$. In general, given two functions $g,\gamma\in L^{2}(\mathbb{R}^{d})$, it is customary to extend the notion of Gabor frame operator $S_{g,\gamma}$, related to $g,\gamma$, as follows

[TABLE]

whenever the previous operator is well-defined. With this notation the reproducing formula (23) can be rephrased as $S_{g,\gamma}=I$ on $L^{2}(\mathbb{R}^{d})$, with $I$ being the identity operator.

Modulation spaces provide a natural setting for time-frequency analysis, thanks to discrete equivalent norms produced by means of Gabor frames. The key result is the following (see [28, Chapter 12] for $1\leq p,q\leq\infty$, and [27, Theorem 3.7] for $0<p,q<1$).

Theorem 2.9**.**

Assume $m\in\mathcal{M}_{v}({\mathbb{R}^{2d}})$, $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$, $g,\gamma\in\mathcal{S}(\mathbb{R}^{d})$ such that $S_{g,\gamma}=I$ on $L^{2}(\mathbb{R}^{d})$. Then

[TABLE]

with unconditional convergence in $M^{p,q}_{m}(\mathbb{R}^{d})$ if $0<p,q<\infty$ and with weak- convergence in $M^{\infty}_{1/v}(\mathbb{R}^{d})$ otherwise. Furthermore, there are constants $0<A\leq B$ such that, for all $f\in M^{p,q}_{m}(\mathbb{R}^{d})$,*

[TABLE]

independently of $p,q$, and $m$. Similar inequalities hold with $g$ replaced by $\gamma$.

In other words,

[TABLE]

3. Main Results

We first study convolution relations for modulations spaces. Let us recall that, for the Banach cases, convolution relations were studied in [7] and [46, 47]. Our approach is general, the techniques use Gabor frames via the equivalence (25), plus Hölder’s and Young’s inequalities for sequences.

Proposition 3.1**.**

Let $\nu(\omega)>0$ be an arbitrary weight function on $\mathbb{R}^{d}$, $0<p,q,r,t,u,\gamma\leq\infty$, with

[TABLE]

and

[TABLE]

whereas

[TABLE]

For $m\in\mathcal{M}_{v}({\mathbb{R}^{2d}})$, $m_{1}(x)=m(x,0)$ and $m_{2}(\omega)=m(0,\omega)$ are the restrictions to $\mathbb{R}^{d}\times\{0\}$ and $\{0\}\times\mathbb{R}^{d}$, and likewise for $v$. Then

[TABLE]

with norm inequality

[TABLE]

Proof.

We use the key idea in [7, Proposition 2.4] to measure the modulation space norm with respect to the Gaussian windows $g_{0}(x)=e^{-\pi{x^{2}}}$ and $g(x)=2^{-d/2}e^{-\pi x^{2}/2}=(g_{0}\ast g_{0})(x)\in\mathcal{S}(\mathbb{R}^{d})$.

A straightforward computation shows $V_{g}f(x,\omega)=e^{-2\pi ix\omega}(f\ast M_{\omega}g^{*})(x)$ (recall that $g^{*}(x)=\overline{g(-x)}$). Thus, using the same argument as in the proof [7, Proposition 2.4], we recall the easily verified identity $M_{\omega}(g_{0}^{*}\ast g_{0}^{*})=M_{\omega}g_{0}^{*}\ast M_{\omega}g_{0}^{*}$ and write the STFT of $f\ast h$ as follows:

[TABLE]

In the sequel, we first use the norm equivalence (25), written in terms of the STFT as $\|f\|_{M_{m}^{p,q}}\asymp\|(V_{g}f(\lambda))_{\lambda\in\Lambda}\|_{\ell^{p,q}_{{m}}(\Lambda)},$ where $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$. Then we majorize $m$ by

[TABLE]

and finally use Young’s convolution inequality for sequences in the $k$-variable and Hölder’s one in the $n$-variable. The indices $p,q,r,\gamma,t,u$ fulfil the equalities in the assumptions. We show in details the case when $r,\gamma,t,u<\infty$:

[TABLE]

The cases when one among the indexes $r,\gamma,t,u$ is equal to $\infty$ are done similarly. This concludes the proof. ∎

3.1. Weyl Operators

Every continuous operator from $\mathcal{S}(\mathbb{R}^{d})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$ can be represented as a pseudodifferential operator in the Weyl form $L_{\sigma}$, with Weyl symbol $\sigma\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$. The operator is formally defined by

[TABLE]

The crucial relation between the action of the Weyl operator $L_{\sigma}$ on time-frequency shifts and the short-time Fourier transform of its symbol is contained in [29, Lemma 3.1].

Lemma 3.1**.**

Consider $g\in\mathcal{S}(\mathbb{R}^{d})$, $\Phi=W(g,g)$. Then, for $\sigma\in\mathcal{S}^{\prime}({\mathbb{R}^{2d}})$,

[TABLE]

where $j(z_{1},z_{2})=(z_{2},-z_{1})$.

We first recall Schatten class results for the Weyl calculus in terms of modulation spaces, initially proved for $1\leq p\leq\infty$ in [6, Theorem 4.5], for $0<p<1$ we refer to [48, Theorem 3.4].

Theorem 3.2**.**

If the Weyl symbol $\sigma\in M^{p,1}({\mathbb{R}^{2d}})$ for some $0<p<\infty$, then the operator $L_{\sigma}$ belongs to the Schatten class $S_{p}$ with

[TABLE]

In particular, $L_{\sigma}$ is a compact operator on $L^{2}(\mathbb{R}^{d})$.

Theorem 3.3**.**

(i) Assume $p,q,\gamma\in(0,\infty]$ such that

[TABLE]

*If $\sigma\in M^{p,\min\{1,\gamma\}}({\mathbb{R}^{2d}})$, then the pseudodifferential operator $L_{\sigma}$, from $\mathcal{S}(\mathbb{R}^{d})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, extends uniquely to a bounded operator from ${M}^{q}(\mathbb{R}^{d})$ to ${M}^{\gamma}(\mathbb{R}^{d})$.

(ii) If $s,r\geq 0$, $t\geq r+s$, and the symbol $\sigma\in M^{\infty,1}_{v_{s}\otimes v_{t}}(\mathbb{R}^{2d})$, then the pseudodifferential operator $L_{\sigma}$, from $\mathcal{S}(\mathbb{R}^{d})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, extends uniquely to a bounded operator from $M^{2}_{v_{r}}(\mathbb{R}^{d})$ into $M^{2}_{v_{r+s}}(\mathbb{R}^{d})$.*

Proof.

$(i)$ Assume $\gamma\geq 1$, then, by (32), $p\geq\gamma\geq 1$ and $q\geq\gamma\geq 1$. The claim was proved by Toft in [46, Theorem 4.3]. The case $\gamma<1$, $p,q\in(0,\infty]$ was again proved by Toft in [48, Theorem 3.1].

$(ii)$ Let $g\in\mathcal{S}(\mathbb{R}^{d})$ with $\|g\|_{L^{2}}=1$. From the inversion formula (22),

[TABLE]

The desired result thus follows if we can prove that the map $M(\sigma)$ defined by

[TABLE]

is continuous from $L^{2}_{v_{r}}({\mathbb{R}^{2d}})$ into $L^{2}_{v_{r+s}}({\mathbb{R}^{2d}})$. Using (31), we see that it is sufficient to prove that the integral operator with integral kernel

[TABLE]

is bounded on $L^{2}({\mathbb{R}^{2d}})$. This follows from Schur’s test (see, e.g., [28, Lemma 6.2.1 (b)]). Indeed, by assumption $\sigma\in M^{\infty,1}_{v_{s}\otimes v_{t}}$, so that

[TABLE]

and similarly

[TABLE]

Hence it is sufficient to prove that for some positive constant $C>0$ we have

[TABLE]

Let us prove the estimate (33). Setting $x=z+w$, $y=w-z$, the inequality (33) can be rephrased as

[TABLE]

For $|x|<2|y|$, observe that $|x+y|<3|y|$ and since $t\geq r+s$ we get the estimate (34). For $|x|\geq 2|y|$, we use $\langle x+y\rangle\asymp\langle x-y\rangle\asymp\langle x\rangle$ and (34) immediately follows. ∎

We remark that

[TABLE]

the Shubin-Sobolev spaces, cf. [6, 41]. In particular, for $s=0$, $M^{2}(\mathbb{R}^{d})=L^{2}(\mathbb{R}^{d})$. Thus, we recover the known result, c.f. [47, Theorem 4.3]:

Corollary 3.4**.**

If $s,r\geq 0$, $t\geq r+s$, and the symbol $\sigma\in M^{\infty,1}_{v_{s}\otimes v_{t}}(\mathbb{R}^{2d})$, then the pseudodifferential operator $L_{\sigma}$, from $\mathcal{S}(\mathbb{R}^{d})$ to $\mathcal{S}^{\prime}(\mathbb{R}^{d})$, extends uniquely to a bounded operator from $Q_{r}(\mathbb{R}^{d})$ into ${Q}_{{r+s}}(\mathbb{R}^{d})$.

An application of the previous theorem concerns the study of eigenfunctions’ properties for Weyl operators. Observe that a Weyl operator having symbol $\sigma\in M^{p,\gamma}({\mathbb{R}^{2d}})$ with $p<\infty$ is a compact operator on $L^{2}(\mathbb{R}^{d})$ by Theorem 3.2.

Proposition 3.5**.**

Consider a Weyl symbol $\sigma\in M^{p,\gamma}$ for some $0<p<\infty$ and every $\gamma>0$. If $\lambda\in\sigma_{P}(L_{\sigma})\setminus\{0\}$, then any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ is in $\cap_{\gamma>0}M^{\gamma}(\mathbb{R}^{d})$.

Proof.

By Theorem 3.3, if the symbol $\sigma$ is in $M^{p,\gamma}({\mathbb{R}^{2d}})$, for every $\gamma>0$, then the Weyl operator acts continuously from $M^{2}(\mathbb{R}^{d})$ into $M^{\gamma_{1}}(\mathbb{R}^{d})$, with $1/p+1/2=1/{\gamma_{1}}$ and, since $p<\infty$, we have $\gamma_{1}<2$. Thus, for $f\in M^{2}(\mathbb{R}^{d})$ eigenfunction with eigenvalue $\lambda\not=0$, we have $f=\frac{1}{\lambda}L_{\sigma}f\in M^{\gamma_{1}}(\mathbb{R}^{d})$. Starting with $f\in M^{\gamma_{1}}(\mathbb{R}^{d})$, we repeat the same argument, obtaining that the eigenfunction $f$ is in the smaller modulation space $M^{\gamma_{2}}(\mathbb{R}^{d})$, with

[TABLE]

(observe $\gamma_{2}<\gamma_{1}$ since $p<\infty$). Continuing this way we construct a decreasing sequence of indices $\gamma_{n}>0$ and such that $\lim_{n\to\infty}\gamma_{n}=0$. This proves the claim. ∎

Recall that, for $s,t>0$, by inclusion relations for modulation spaces we have $M^{\infty,1}_{v_{s}\otimes v_{t}}({\mathbb{R}^{2d}})\subset M^{p,1}({\mathbb{R}^{2d}})$, for any $p>2d/s$, [33, Theorem 1.5 and Lemma 4.10]. Hence the operators below are compact operators by Theorem 3.2.

Proposition 3.6**.**

Consider a Weyl symbol $\sigma\in M^{\infty,1}_{v_{s}\otimes v_{t}}({\mathbb{R}^{2d}})$ for some $s>0$ and every $t>0$. If $\lambda\in\sigma_{P}(L_{\sigma})\setminus\{0\}$, then any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ is in $\mathcal{S}(\mathbb{R}^{d})$.

Proof.

By Theorem 3.3, if the symbol $\sigma$ is in $M^{\infty,1}_{v_{s}\otimes v_{t}}({\mathbb{R}^{2d}})$, for some $s>0$ and every $t>0$, then the Weyl operator acts continuously from $L^{2}(\mathbb{R}^{d})$ into $M^{2}_{v_{s}}(\mathbb{R}^{d})=Q_{s}(\mathbb{R}^{d})$. Starting now with the eigenfunction $f$ in $Q_{s}(\mathbb{R}^{d})$ and repeating the same argument with $t\geq s$ we obtain that the eigenfunction is in $Q_{2s}(\mathbb{R}^{d})$. Proceeding this way we infer that $f\in\cap_{n\in\mathbb{N}_{+}}Q_{ns}(\mathbb{R}^{d})$. The inclusion relations for Shubin-Sobolev spaces and the property (see e.g., [7, 20, 47])

[TABLE]

prove the claim. ∎

3.2. Localization Operators

The study of eigenfunctions for a localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ uses its representation as a Weyl one:

[TABLE]

(cf. [7] and references therein) where $W(\varphi_{2},\varphi_{1})$ is the the cross-Wigner distribution defined in (8). Therefore, the Weyl symbol of $A_{a}^{\varphi_{1},\varphi_{2}}$ is given by

[TABLE]

We will deduce properties for $A_{a}^{\varphi_{1},\varphi_{2}}$ using its Weyl form $L_{a\ast W(\varphi_{2},\varphi_{1})}$, as detailed below. Precisely, we shall focus on properties of eigenfunctions of compact localization operators.

Theorem 3.7**.**

Consider a symbol $a\in M^{p,\infty}({\mathbb{R}^{2d}})$, for some $0<p<\infty$, and non-zero windows $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$. Assume that $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}\not=\emptyset$. If $\lambda\in\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$, any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ satisfies $f\in\bigcap_{\gamma>0}M^{\gamma}(\mathbb{R}^{d})$.

Proof.

Since the windows $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$, the cross-Wigner distribution is in $\mathcal{S}({\mathbb{R}^{2d}})\subset M^{q,\gamma}({\mathbb{R}^{2d}})$, for every $0<q,\gamma\leq\infty$. We next apply the convolution relations for modulation spaces (29). Namely, if $p\geq 1$, choose $q=1$, so that $r=p$, and

[TABLE]

If $0<p<1$, choose $p=q=r$ so that

[TABLE]

In both cases we obtain that $\sigma\in M^{p,\gamma}({\mathbb{R}^{2d}})$, for every $\gamma>0$. Hence the claim immediately follows by Proposition 3.5. ∎

As a consequence, the eigenfunctions are extremely concentrated on the time-frequency space, having very few Gabor coefficients large whereas all the others are negligible.

Consider a Parseval Gabor frame $\mathcal{G}(g,\Lambda)$ for $L^{2}(\mathbb{R}^{d})$, with $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$ and $g\in\mathcal{S}(\mathbb{R}^{d})$. For $N\in\mathbb{N}_{+}$, we defined in (5) the space $\Sigma_{N}$ to be the set of all linear combinations of Gabor atoms consisting of at most $N$ terms.

Given a function $f\in L^{2}(\mathbb{R}^{d})$, the $N$-term approximation error in $L^{2}(\mathbb{R}^{d})$ is recalled in (6). Namely, $\sigma_{N}(f)$ is the error produced when $f$ is approximated optimally by a linear combination of $N$ Gabor atoms.

Assume $f\in M^{p}(\mathbb{R}^{d})$ for some $0<p<2$ (thus, in particular, $f\in L^{2}(\mathbb{R}^{d})$). The series of Gabor coefficients in (4) are absolutely convergent, hence also unconditionally convergent. Thus we can rearrange the Gabor coefficients $|\langle f,\pi(\lambda)g\rangle|$ in a decreasing order. Precisely, set $c_{k,n}=\langle f,\pi(\alpha k,\beta n)g\rangle$, $k,n\in\mathbb{Z}^{d}$, and let $\iota:\mathbb{N}_{+}\to\mathbb{Z}^{d}\times\mathbb{Z}^{d}$ be any bijection satisfying

[TABLE]

The sequence $(\tilde{c}_{m})_{m\in\mathbb{N}_{+}}=(|c_{\iota(m)}|)_{m\in\mathbb{N}_{+}}$ is called the non-increasing rearrangement of $(c_{k,n})_{k,n}$ above. With this notations, denoting by $p_{opt}$ the best approximation of $f$ in $\Sigma_{N}$, the $N$-term approximation error becomes

[TABLE]

By abuse of notation, given $a=(a_{m})_{m}$, $a_{m}\geq 0$ for every $m$, a non-increasing sequence ($a_{1}\geq a_{2}\geq\cdots\geq a_{m}\geq a_{m+1}\geq\cdots$) we write

[TABLE]

The key tool is now the following lemma, see [42] and [15] (we also refer to [28, Lemma 12.4.1]):

Lemma 3.2**.**

Let $a=(a_{m})_{m}$, $a_{m}\geq 0$ for every $m$, be a non-increasing sequence and consider $0<p<2$. Set

[TABLE]

Then there exists a constant $C=C(p)>0$, such that

[TABLE]

Proposition 3.8**.**

Assume $f\in M^{p}(\mathbb{R}^{d})$ for some $0<p<2$. Then, there exists $C=C(p)>0$ such that the $N$-term approximation error satisfies

[TABLE]

where $\gamma>0$ is defined in (37).

Proof.

If $\mathcal{G}(g,\Lambda)$ is a Parseval Gabor frame with $g\in\mathcal{S}(\mathbb{R}^{d})$, and $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$, $\alpha,\beta>0$, then the sequence of Gabor coefficients of $f$, given by $(\langle f,\pi(\alpha k,\beta n)g\rangle)_{k,n\in\mathbb{Z}^{d}}$, are in $\ell^{p}(\Lambda)$ by Theorem 2.9, with

[TABLE]

and the sequence $(|\langle f,\pi(\alpha k,\beta n)g\rangle|)_{k,n}$ can be rearranged in a non-increasing one $(a_{m})$, as explained above. Applying Lemma 3.2 to such a sequence, from the right-hand side inequality in (38) and using the majorization in (36) we infer (39). ∎

Corollary 3.9**.**

Consider a Parseval Gabor frame $\mathcal{G}(g,\Lambda)$, with $g\in\mathcal{S}(\mathbb{R}^{d})$, and $\Lambda=\alpha\mathbb{Z}^{d}\times\beta\mathbb{Z}^{d}$, $\alpha,\beta>0$. Under the assumptions of Theorem 3.7, any $f$ eigenfunction of $A_{a}^{\varphi_{1},\varphi_{2}}$ (with eigenvalue $\lambda\not=0$) is highly compressed onto a few Gabor atoms $\pi(\lambda)g$, in the sense that its $N$-term approximation error satisfies the following property: for every $r>0$ there exists $C=C(r,f)>0$ with

[TABLE]

Proof.

By Theorem 3.7, the eigenfunction fulfils $f\in M^{p}(\mathbb{R}^{d})$, for every $p>0$. Hence the assumptions of Proposition 3.8 are satisfied for every $0<p<2$. This immediately yields the claim. ∎

We next consider the case of compact localization operators with symbols $a\in M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})$, $s>0$. In this case $L^{2}$ eigenfunctions reveal to be Schwartz functions, as shown below.

Theorem 3.10**.**

Consider a symbol $a\in M^{\infty}_{v_{s}\otimes 1}({\mathbb{R}^{2d}})$, for some $s>0$, and non-zero windows $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$. Assume that $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}\not=\emptyset$. If $\lambda\in\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$, any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ belongs to the Schwartz class $\mathcal{S}(\mathbb{R}^{d})$.

Proof.

The assumption $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$ implies $W(\varphi_{2},\varphi_{1})\in\mathcal{S}({\mathbb{R}^{2d}})\subset M^{1,1}_{v_{r}\otimes v_{t}}({\mathbb{R}^{2d}})$, for every $r,t>0$. We next apply the convolution relations for modulation spaces (29), obtaining that $A_{a}^{\varphi_{1},\varphi_{2}}=L_{\sigma}$ with $\sigma\in M^{\infty,1}_{v_{s}\otimes v_{t}}({\mathbb{R}^{2d}})$, for some $s>0$ and every $t>0$. Hence the claim immediately follows by Proposition 3.6. ∎

Remark 3.11**.**

*The nice properties of eigenfunctions for localization operators studied so far seem to depend on the fact that such operators are not only compact but belong to the Schatten class $S_{p}$, $0<p<\infty$ (cf. [8, Theorem 1]). Fernández and Galbis in [22, Theorem 3.15] characterize compact localization operators. Namely, fix $g_{0}\in\mathcal{S}(\mathbb{R}^{d})$ and $a\in M^{\infty}({\mathbb{R}^{2d}})$, then the following conditions are equivalent:

(i) The localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ is compact on $L^{2}(\mathbb{R}^{d})$ for every $\varphi_{1},\varphi_{2}$ in $\mathcal{S}(\mathbb{R}^{d})$;

(ii) For every $R>0$,*

[TABLE]

It seems that for symbols satisfying condition (41) the techniques developed above do not work anymore. It would be very interesting to know whether for compact operators that are not in the Schatten class $S_{p}$, $0<p<\infty$, the $L^{2}$ eigenfunctions do gain any additional smoothness and regularity. This topic will be investigated in a subsequent paper.

4. Symbols in $L^{p}({\mathbb{R}^{2d}})$ spaces

We now consider localization operators with symbols in weighted Lebesgue spaces. Let us recall that any localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ with windows in $\mathcal{S}(\mathbb{R}^{d})$ and symbol $a\in L^{q}({\mathbb{R}^{2d}})$, with $1\leq q<\infty$, is a compact operator, cf. [50, Proposition 13.3]. The case of weighted Lebesgue spaces and, more generally, Potential Sobolev spaces was treated in [6]: let us stress that any localization operator $A_{a}^{\varphi_{1},\varphi_{2}}$ with Schwartz windows and symbol $a$ in $L^{q}_{m}({\mathbb{R}^{2d}})$, with $1\leq q<\infty$ is a compact operator on $L^{2}(\mathbb{R}^{d})$.

Theorem 4.1**.**

Let $m\in\mathcal{M}_{v}$, $m(z)\geq 1$ for every $z\in{\mathbb{R}^{2d}}$, $a\in L^{q}_{m}({\mathbb{R}^{2d}})$, $1\leq q<\infty$, and non-zero windows $\varphi_{1},\varphi_{2}\in\mathcal{S}(\mathbb{R}^{d})$. Assume that $\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}\not=\emptyset$. If $\lambda\in\sigma_{P}(A_{a}^{\varphi_{1},\varphi_{2}})\setminus\{0\}$, any eigenfunction $f\in L^{2}(\mathbb{R}^{d})$ with eigenvalue $\lambda$ satisfies $f\in\bigcap_{p>0}M^{p}_{m}(\mathbb{R}^{d})$.

Proof.

By assumption and using (18), we start with a symbol $a$ in $L^{q}_{m}({\mathbb{R}^{2d}})=W(L^{q},L^{q}_{m})({\mathbb{R}^{2d}})$. Consider the eigenvector $f\in L^{2}(\mathbb{R}^{d})$ and the window $\varphi_{1}\in\mathcal{S}(\mathbb{R}^{d})$. Then by Theorem 2.8 the STFT $V_{\varphi_{1}}f$ is in the Wiener amalgam space $W(L^{\infty},L^{2})({\mathbb{R}^{2d}})$. Proposition 2.3 yields that $aV_{\varphi_{1}}f\in W(L^{q},L^{p_{1}}_{m})({\mathbb{R}^{2d}})$, with

[TABLE]

so that the index $p_{1}$ satisfies $p_{1}<\min\{q,2\}$. Consider now a non-zero window $g\in\mathcal{S}(\mathbb{R}^{d})$. Using (7),

[TABLE]

so that,

[TABLE]

We estimate

[TABLE]

Observing that $W(L^{\infty},L^{1}_{v})({\mathbb{R}^{2d}})\hookrightarrow W(L^{q^{\prime}},L^{1}_{v})({\mathbb{R}^{2d}})$ and applying the convolution relations (17) we infer $|V_{g}(A_{a}^{\varphi_{1},\varphi_{2}}f)|\in W(L^{\infty},L^{p_{1}}_{m})\hookrightarrow L^{p_{1}}_{m}$. This proves that $A_{a}^{\varphi_{1},\varphi_{2}}f\in M^{p_{1}}_{m}(\mathbb{R}^{d})$.

Recalling the assumption $A_{a}^{\varphi_{1},\varphi_{2}}f=\lambda f$, $\lambda\not=0$, we infer $f\in M^{p_{1}}_{m}(\mathbb{R}^{d})$.

We now repeat the previous argument starting with $f\in M^{p_{1}}_{m}(\mathbb{R}^{d})$. By Theorem 2.8 the STFT $V_{\varphi_{1}}f\in W(L^{\infty},L^{p_{1}}_{m})({\mathbb{R}^{2d}})$ and $aV_{\varphi_{1}}f\in W(L^{q},L^{p_{2}}_{m^{2}})\hookrightarrow W(L^{q},L^{p_{2}}_{m})$, (since $m^{2}\geq m$), with

[TABLE]

so that $p_{2}<p_{1}.$ Arguing as above we infer $|V_{g}(A_{a}^{\varphi_{1},\varphi_{2}}f)(w)|\in W(L^{\infty},L^{p_{2}}_{m})\hookrightarrow L^{p_{2}}_{m}$. Thus, the eigenfunction $f$ belongs to the smaller space $M^{p_{2}}_{m}$.

Continuing this way we construct a strictly decreasing sequence of indices $p_{n}>0$ and such that

[TABLE]

By induction and using the same argument as above one immediately obtains that if $f\in M^{p_{n}}_{m}(\mathbb{R}^{d})$ then $f\in M^{p_{n+1}}_{m}(\mathbb{R}^{d})$. This concludes the proof. ∎

Acknowledgments

The last two authors were partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).