Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators Aaφ1,φ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_a^{\varphi _1,\varphi _2}$$\end{document} with symbols a in the wide weighted modulation space Mw∞(R2d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ M^\infty _{w}({\mathbb {R}^{2d}})$$\end{document}, and windows φ1,φ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \varphi _1, \varphi _2 $$\end{document} in the Gelfand–Shilov space S1(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}^{\left( 1\right) }(\mathbb {R}^d)$$\end{document}. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of Aaφ1,φ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_a^{\varphi _1,\varphi _2}$$\end{document} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space S(γ)(Rd)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal {S}^{(\gamma )} (\mathbb {R^{d}}) $$\end{document}, where the parameter γ≥1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma \ge 1 $$\end{document} is related to the growth of the considered weight. An important role is played by τ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}-pseudodifferential operators Opτ(σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Op_{\tau } (\sigma )$$\end{document}. In that direction we show convenient continuity properties of Opτ(σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Op_{\tau } (\sigma )$$\end{document}when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Opτ(σ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Op_{\tau } (\sigma )$$\end{document} when the symbol σ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma $$\end{document} belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

frequency plane [36]. They are defined as integral transforms determined by a suitable choice of a symbol a and window functions ϕ 1 , ϕ 2 . In this paper we examine the conditions which the symbol and windows of an operator should satisfy to ensure an ultra-rapid decay and regularity of its eigenfunctions. It is known that the eigenfunctions of time-frequency localization operators, also known as Daubechies operators and denoted by A ϕ 1 ,ϕ 2 a , with Gaussian windows ϕ 1 (t) = ϕ 2 (t) = π −d/4 exp(−t 2 /2) and with a radial symbol a ∈ L 1 (R 2d ), are Hermite functions, i.e. they have superexponential decay in phase space [16]. In a different terminology, those eigenfunctions belong to the smallest projective Gelfand-Shilov space S {1/2} (R) (cf. Definition 2.7).
The investigations in [16,36] are motivated by some questions in signal analysis. However, localization operators appear in different mathematical contexts. For example, they were used (under the name anti-Wick operators) in the study of quantization problem in quantum mechanics, cf. [3,37]. In abstract harmonic analysis, localization operators on a locally compact group G and Lebesgue spaces L p (G), 1 ≤ p ≤ ∞, were studied in [50]. We also mention their presence in the form of Toeplitz operators in complex analysis [4]. Here we do not intend to discuss different manifestations of localization operators and refer to e.g. [17] for a survey.
Our results fits in the framework of time-frequency analysis. An important step forward in that direction was made by the seminal paper [9]. Thereafter the subject has been considered by many authors including [1,5,10,13,26,41,44], where among others one can find different continuity, Schatten class and lifting properties of localization operators. The time-frequency analysis approach is based on the use of modulation spaces as appropriate functional analytic framework. Another issue established in [5,9] is the identification of localization operators as Weyl pseudodifferential operators.
The focus of this paper is to consider the properties of eigenfunctions of compact localization operators. Our investigations are inspired by the recent work [2]. Indeed, there it is shown that if the symbol a belongs to the modulation space M ∞ v s ⊗1 (R 2d ), s > 0 (see Definition 2.17) and ϕ 1 , ϕ 2 ∈ S(R d ), then the eigenfunctions of A ϕ 1 ,ϕ 2 a are actually Schwartz functions. Moreover, similar result is proved for the Weyl pseudodifferential operators whose symbol belongs to M ∞,1 v s ⊗v t (R 2d ), for some s > 0 and every t > 0, cf. [2,Proposition 3.6]. Here v s (z) = (1 + |z| 2 ) s/2 , s ∈ R, z ∈ R d .
We extend the framework of the Schwartz space of test functions and its dual space of tempered distributions given in [2] by replacing it with a more subtle family of Gelfand-Shilov spaces and their duals, spaces of ultra-distributions. This is motivated by the fact that Gelfand-Shilov spaces describe smoothness and decay properties which go beyond the scope of the Schwartz space, cf. [7,22,23,34,35,40,44]. To that end, as an important technical tool, we consider a class of weights which contains the weights of subexponential growth, apart from polynomial type weights. As explained in [27], replacing polynomial weights with weights of faster growth at infinity is not a mere routine. Indeed, weights of ultra-rapid growth and decay give rise to ultradistributional framework for modulation spaces. To underline this difference, we sometimes refer to ultra-modulation spaces when modulation spaces are allowed to contain such ultra-distributions.
One of the main tools in our analysis is the (cross-)τ -Wigner distribution W τ ( f , g), f , g ∈ L 2 (R d ), see Definition 2.11. The relation between W τ ( f , g) and the short-time Fourier transform V g ( f ) (another relevant time-frequency representation) serves as a bridge between properties of modulation spaces and τ -pseudodifferential operators. More precisely, we extend the recent result [2, Theorem 3.3] to a more general class of operators and weights (Theorem 3.3). Although this result follows from [47,Theorem 3.1] and [45,Theorem A.2], our proof is independent and based on different arguments.
Our first main result concerns decay properties of the eigenfunctions of τpseudodifferential operators. In fact, by using iterated actions of the operator we conclude that its eigenfunctions belong to the Gelfand-Shilov space S (γ ) (R d ) (Theorem 3.7). As already mentioned, this gives an information about regularity and decay properties of eigenfunctions which can not be captured within the Schwartz class.
Finally, we use Theorem 3.7 and convolution relation for modulation spaces (Proposition 2.24) to show that the eigenfunctions of localization operators A , and if w is of a certain ultra-rapid growth. We use the representation of localization operators as pseudodifferential operators. Evidently, the Weyl form of localization operators suggests to introduce and consider τ -localization operators by using τpseudodifferential operators and the (cross-)τ -Wigner distribution. However, it turns out the such approach does not extend the class of localization operators given by Definition 2.15 (cf. Proposition 2.16).
We end this introduction with a brief report of the content of the paper. In Preliminaries we collect relevant background material. Apart from the review of known results, it contains some new results and proofs. In Sect. 3 we prove our main results: continuity properties of τ -pseudodifferential operators on modulation spaces, estimates for eigenfunctions of τ -pseudodifferential operators and decay and smoothness properties of eigenfunctions of localization operators. Appendix contains the proofs of two auxiliary technical results.

Notation
We denote the Euclidean scalar product on R d by x y := x · y and the Euclidean norm by |x| := √ x · x. We put N 0 := N ∪ {0}. A B means that for given constants A and B there exists a constant c > 0 independent of A and B such that A ≤ cB, and we write A B if both A B and B A.
We define the involution g * of a function g by g * (t) := g(−t). Given a function f on R d its Fourier transform is normalized to be Given two spaces A and B, we denote by A → B the continuous embedding of A into B. S(R d ) denotes the Schwartz class, and its topological dual, the space of tempered distributions, is indicated by S (R d ). By the brackets f , g we mean the extension of the L 2 -inner product f , g := f (t)g(t) dt to any dual pair.
Consider 0 < p < ∞ and a positive and measurable function m on R d , then L p m (R d ) denotes the (quasi-)Banach space of measurable functions f : If the restriction of f to any compact set belongs to L p (R d ), then we write f ∈ L p loc (R d ). For given Hilbert space H and compact operator T on H its singular values {s k (T )} ∞ k=1 are the eigenvalues of (T * T ) 1/2 , which is a positive and self-adjoint operator. The Schatten class S p (H ), with 0 < p < ∞, is the set of all compact operators on H such that their singular values are in p . For consistency, we define S ∞ (H ) := B(H ), the set of all linear and bounded operators on H . We shall deal with H = L 2 (R d ).
By σ P (T ) we denote the point spectrum of the operator T . If T is a compact mapping on L 2 (R d ), then the spectral theory for compact operators yields is the spectrum of the operator. For compact operators on L 2 (R d ), we have 0 ∈ σ (T ) and the point spectrum σ P (T ) {0} (possibly empty) is at most a countable set.

Preliminaries
In this section we collect background material and prove some auxiliary results. More precisely, we provide definitions and basic facts on weights, sequence spaces, Gelfand-Shilov spaces, time-frequency representations, pseudodifferential and localization operators, modulation spaces and Gabor frames. We also give some new results and proofs (Lemma 2.4, Proposition 2.16, Proposition 2.24) which will be used in Sect. 3.

Weight functions
By weight m on R d (or on Z d ) we mean a positive function m > 0 such that m ∈ L ∞ loc (R d ) and 1/m ∈ L ∞ loc (R d ). A weight m is said to be submultiplicative if it is even and Given a weight m on R d and a positive function v Therefore submultiplicative weights are moderate and the previous inequality implies the following estimates: For a submultiplicative weight v there are convenient ways to find a smooth weight v 0 which is equivalent to v in the sense that there is a constant C > 0 such that see e.g. [15,25,44]. Next we introduce some weights which will be used in the sequel. Given k, γ > 0 we define w γ k (x) := e k|x| 1/γ , x ∈ R d . Sometimes we shall use the above expression for k = 0 also, with obvious meaning. If γ > 1 the above functions are called subexponential weights, and when γ = 1 we write w k instead of w 1 k . Note that (sub-)exponential weights w γ k are submultiplicative [this follows from (29)]. When 0 < γ < 1 we obtain weights of super-exponential growth at infinity. We shall work with the following weight classes defined for γ > 0: For 0 < γ 2 < γ 1 we have Moreover, for 0 < γ < 1 we have P E = P E,γ = P 0 E,γ ; see [6, Remark 2.6] and [49]. The next lemma states that if m ∈ P E , then it is w k -moderate fore some k > 0 large enough. This implies P E = P E,1 .
We remark that P E contains the weights of polynomial type, i.e. weights moderate with respect to some polynomial.
In the sequel P * E,γ means P E,γ or P 0 E,γ . The following lemma follows by easy calculations and we leave the proof for the reader (see also [44]). Observe that due to the equality P E,1 = P E,γ = P 0 E,γ , 0 < γ < 1, it is sufficient to consider γ ≥ 1.

Lemma 2.2
Consider γ > 0. Then P * E,γ (R d ) is a group under the pointwise multiplication and with the identity m ≡ 1.
Given a function f defined on R 2d we denote its restrictions to R d ×{0} and {0}×R d as follows: Given two functions g, h defined on R d their tensor product is the function on R 2d defined in the following manner: The families P * E,γ turn out to be closed under restrictions and tensor products in the sense of the following lemma. The proof is omitted, since it follows from definitions and properties of the Euclidean norm.
). Next we exhibit a lemma which will play a key role in the sequel, see Proposition 3.7. The proof is given in the appendix.
Then for every x, ω, y, η ∈ R d the following estimate holds true: We finish this subsection by introducing some polynomial weights which will be used in Theorem 3.4 and Lemma 3.5. Let τ ∈ [0, 1] and u ≥ 0, then we define the weights of polynomial type

Remark 2.5
If v u and m τ u are given by (4) and (5) respectively, then we notice that which will be used in Lemma 3.5. Indeed:

Spaces of sequences
When p = q we recover the standard spaces of sequences In the following proposition we collect some properties that we shall use later on, see [20,21]. Proposition 2.6 (i) Inclusion relations: consider 0 < p 1 ≤ p 2 ≤ ∞ and let m be any positive weight function on Z d . Then (ii) Young's convolution inequality: where the constant C > 0 is independent of p, q, r , a and b. If m ≡ v ≡ 1, then C = 1. (iii) Hölder's inequality: let m be any positive weight function on Z d and 0 < p, q, r ≤ ∞ such that
In the sequel we will also use the following notation: where * stands for (γ ) or {γ }.
The Gelfand-Shilov spaces enjoy beautiful symmetric characterizations which also involve the Fourier transform of their elements. The following result has been reinvented several times, in similar or analogous terms, see [7,28,30,34]. Theorem 2.9 Let γ, τ ≥ 1/2. The following conditions are equivalent: (ii) There exist (resp. for every) constants A, B > 0 such that (iii) There exist (resp. for every) constants A, B > 0 such that (iv) There exist (resp. for every) constants h, k > 0 such that (v) There exist (resp. for every) constants h, B > 0 such that Moreover, we could consider any L p -norm, 1 ≤ p < ∞ instead of L ∞ -norm in Theorem 2.9, cf. [30].
By using Theorem 2.9 it can be shown that the Fourier transform is a topological isomorphism between S Similar considerations hold for partial Fourier transforms with respect to some choice of variables. In particular, if γ = τ and γ ≥ 1/2 then , and similarly for their distribution spaces. Due to this fact, corresponding dual spaces are referred to as tempered ultra-distributions (of Roumieu and Beurling type respectively), see [35].
The combination of global regularity with suitable decay properties at infinity [cf. (8)] which is built in the very definition of S γ τ (R d ) and γ τ (R d ), makes them suitable for the study of different problems in mathematical physics [22,23,34]. We refer to [13,14,40,41] for the study of localization operators in the context of Gelfand-Shilov spaces. See also [44,47,48] for related studies.

Time-frequency representations
In this subsection we recall the definitions and basic properties of the short-time Fourier transform and the (cross-)τ -Wigner distribution.
Given a function f on R d and x, ω ∈ R d , the translation operator T x and the modulation operator M ω are defined as and their composition π(x, ω) := M ω T x is called time-frequency shift. We can now introduce two most commonly used time-frequency representations of a signal f , the so-called short-time Fourier transform (STFT) and the (cross-)Wigner distribution.
with respect to g is the function defined on the phase-space as follows: We refer to Gröchenig [24,Chapter 3] for the properties and different equivalent forms of the STFT.
When τ = 1/2, W 1/2 ( f , g) is simply called the cross-Wigner distribution of f and g and is denoted by W ( f , g) for short. Both STFT and W τ are well defined for then the connection between the STFT and τ -Wigner distribution is described as follows.
where R( f , g) denotes the Rihaczek distribution of f and g.
Proof The proof is straightforward, and we show only (i) for the sake of completeness (see also [15,Proposition 1.3.30]). After the change of variables s = x + τ t in (9) we obtain Notice that when τ = 1/2, we have A 1/2 g(t) = g(−t), and (10) becomes Definitions 2.10 and 2.11 are uniquely extended to f ∈ (S (1) ) (R d ) by duality. We will also use the following fact related to time-frequency representations of the Gelfand-Shilov spaces. 1]. Then the following are true: Proof The proof for the STFT and W 1/2 can be found in several sources, see e.g. [28,38,44]. The case τ ∈ [0, 1], τ = 1/2, can be proved in a similar fashion and is left for the reader as an exercise.

Pseudodifferential and localization operators
Next we introduce τ -quantizations as pseudodifferential operators acting on S (1) (R d ).
We address the reader to the textbooks [15,24] in which the framework is mostly the one of S(R d ) and S (R d ), and we suggest [34,38,40,44,47,48] for the framework of Gelfand-Shilov spaces and their spaces of ultra-distributions. Definition 2.14 Let τ ∈ [0, 1]. Given a symbol σ ∈ S (1) (R 2d ), the τ -quantization of σ is the pseudodifferential operator or, in a weak sense, The correspondence between the symbol σ and the operator Op τ (σ ) given by (11) is known as the Shubin τ -representation [37]. By a change of variables and an interchange of the order of integration, it can be shown that Op τ (σ ), σ ∈ S (1) (R 2d ), and the (cross-)τ -Wigner distribution are related by the following formula: Thus, for τ = 1/2 (the Weyl quantization) we recover the Weyl pseudodifferential operators, and when τ = 0 we obtain the Kohn-Nirenberg operators. Commonly used equivalent notation for the Weyl operators in the literature are Op W (σ ), Op w (σ ), L σ or σ w . The Weyl calculus reveals to be extremely important since every continuous and linear operator from can be written as the Weyl transform of some (Weyl) symbol σ ∈ S (1) (R 2d ). This is due to the Schwartz kernel theorem when extended to the duality between S (1) (R d ) and S (1) (R d ), see [32,40]. Next we introduce localization operators in the form of the STFT multipliers, and discuss their relation to τ -quantizations given above.

Definition 2.15
Let there be given window functions ϕ 1 , ϕ 2 ∈ S (1) (R d ) {0} and a symbol a ∈ S (1) (R 2d ). Then the localization operator is the continuous and linear mapping formally defined by or, in a weak sense, It can be proved that every localization operator A ϕ 1 ,ϕ 2 a can be written in the Weyl form, i.e. identified with the Weyl pseudodifferential operator due to the following formula and . We refer to [5, Lemma 2.4] or [19] for the proof, see also [41].
It turns out that the class of localization operators given by (15) coincides to the one given by Definition 2.15, see [45]. We give an independent proof based on the kernel argument, and point out that the result is essentially a restatement of (1.23) in [45].
Proof By the Schwartz kernel theorem for S (1) where the kernel k of the operator A ϕ 1 ,ϕ 2 a is given by It remains to calculate the kernel of A ϕ 1 ,ϕ 2 a,τ . By the commutation relation T x M ω = e −2πi xω M ω T x , and the covariance property of τ -Wigner transform: we calculate a * W τ (ϕ 2 , ϕ 1 ) and obtain Now by using a suitable interpretation of the oscillatory integrals in the distributional sense, and appropriate change of variables (cf. [41]) we get where k is given by (16). By the uniqueness of the kernel we conclude that a,τ and the proof is finished.

Ultra-modulation spaces
We use the terminology ultra-modulation spaces in order to emphasize that such spaces may contain ultra-distributions, contrary to the most usual situation when members of modulation spaces are tempered distributions. However, ultra-modulation spaces belong to the family of modulation spaces introduced in [18]. We refer to e.g. [46,48] for a general approach to the broad class of modulation spaces.

Definition 2.17 Fix a non-zero window
and m of at most polynomial growth at infinity, were invented by Feichtinger [18] and called modulation spaces. There it was proved that they are Banach spaces and that different window functions in S(R d ) {0} yield equivalent norms. Moreover, the window class can be enlarged to the Feichtinger algebra where v is a submultiplicative weight of at most polynomial growth at infinity such that m is v-moderate.
It turned out that properties analogous to the Banach case hold in the quasi-Banach one as well, see [21]. Moreover, such properties remain valid also in the more general setting of Definition 2.17. We collect them in the following theorem in the same manner as it is done in e.g. [46,47], see also the references given there.

Theorem 2.18
Consider 0 < p, p 1 , p 2 , q, q 1 , q 2 ≤ ∞ and weights m, m 1 , m 2 ∈ P E (R 2d ). Let · M p,q m be given by (17) for a fixed g ∈ S (1) (R d ) {0}. Then: is a quasi-Banach space whenever at least one between p and q is strictly smaller than 1, otherwise it is a Banach space;

then it induces a (quasi-)norm equivalent to
· M p,q m ; (iii) if p 1 ≤ p 2 , q 1 ≤ q 2 and m 2 m 1 , then; and similarly for q .

Remark 2.19
Point (ii) of the previous theorem tell us that the definition of M p,q m (R d ) is independent of the choice of the window. Moreover, it can be shown that the class for window functions can be extended from S (1) where r ≤ p, p , q, q and v ∈ P E (R 2d ) is submultiplicative and such that m is v-moderate [47].

Proposition 2.20
Consider γ ≥ 1 and 1 ≤ p, q ≤ ∞. Then In some situations it is convenient to consider ultra-modulation spaces as subspaces of S {1/2} (R d ) (taking the window g in S {1/2} (R d )), see for example [8,47]. However, for our purposes it is sufficient to consider the weights in P E (R 2d ), and then M p,q m (R d ) is a subspace of S (1) (R d where the equality holds in M p,q The embeddings between modulation spaces are studied by many authors. We recall the recent contribution [29,Theorem 4.11], which is convenient for our purposes and which will be used in Lemma 3.5.

Theorem 2.22
Let 0 < p j , q j ≤ ∞, s j , t j ∈ R for j = 1, 2 and consider the polynomial weights v t j , v s j defined as in (4). Then if the following two conditions hold true: p 1 , p 2 , t 1 , t 2 ) satisfies one of the following conditions: (ii) (q 1 , q 2 , s 1 , s 2 ) satisfies one of the conditions (C 1 ) or (C 2 ) with p j and t j replaced by q j and s j respectively.

Gabor frames
Consider a lattice := αZ d ×βZ d ⊂ R 2d for some α, β > 0. Given g ∈ L 2 (R d ) {0}, the set of time-frequency shifts G(g, ) := {π(λ)g : λ ∈ } is called a Gabor system. The set G(g, ) is a Gabor frame if there exist constants A, B > 0 such that If G(g, ) is a Gabor frame, then the frame operator is a topological isomorphism on L 2 (R d ). Moreover, if we define h := S −1 g ∈ L 2 (R d ), then the system G(h, ) is a Gabor frame and we have the reproducing formulae with unconditional convergence in L 2 (R d ). The window h is called the canonical dual window of g. In particular, if h = g and g L 2 = 1, then A = B = 1, the frame operator is the identity on L 2 (R d ) and the Gabor frame is called Parseval Gabor frame. In particular, from (19) we can recover exactly the L 2 -norm of every vector: Any window h ∈ L 2 (R d ) such that (20) is satisfied is called alternative dual window for g. Given two functions g, h ∈ L 2 (R d ) we are able to extend the notion of Gabor frame operator to the operator S g,h = S g,h in the following way: whenever this is well defined. With this notation the reproducing formulae (20) can be rephrased as S g,h = I = S h,g , where I is the identity on L 2 (R d ). Discrete equivalent norms produced by means of Gabor frames make of ultramodulation spaces a natural framework for time-frequency analysis. We address the reader to Galperin and Samarah [21], Gröchenig [24], and Toft [46,47].
independently of p, q, and m. Equivalently: Similar inequalities hold with g replaced by h.
Now we are able to prove the convolution relations for ultra-modulations spaces which will be used to prove our main results in Sect. 3. For the Banach cases with weight of at most polynomial growth at infinity, convolution relations were studied in e.g [9,42,43]. We modify the technique used in [2] to the Gelfand-Shilov framework presented so far. The essential tool is the equivalence between continuous and discrete norm (21).
Proof First observe that due to Lemmas 2.2 and 2.3 it follows that the ultra-modulation spaces which came into play are well defined. The main tool is the idea contained in [9,Proposition 2.4]. We take the ultramodulation norm with respect to the Gaussian windows g 0 (x) := e −π x 2 ∈ S {1/2} (R d ) and g(x) Since the involution operator g * (x) = g(−x) and the modulation operator M ω commute, by a direct computation we have Thus, by using the associativity and commutativity of the convolution product, we obtain We use the norm equivalence (21) for a suitable = αZ d × βZ d , and then the v-moderateness in order to majorize m: m(αk, βn) m(αk, 0)v(0, βn) = m| 1 (αk)v| 2 (βn).
Eventually Young's convolution inequality for sequences is used in the k-variable and Hölder's one in the n-variable. Indeed both inequalities can be used since p, q, r , γ, t, u fulfill the assumptions of the proposition. We write in details the case when r , γ, t, u < ∞ and leave to the reader the remaining cases, when one among the indices r , γ, t, u is equal to ∞, which can be done analogously.
This concludes the proof.

Main results
From the previous observations it follows that localization operators are in fact a particular class of pseudodifferential operators, cf. (14) and (15). Therefore, in the study of eigenfunctions of localization operators we first observe operators Op τ (σ ) when σ belongs to weighted modulation space M ∞,1 w (R 2d ), where w is of a certain subexponential growth at infinity.
We first prove some continuity properties of Op τ (σ ) on ultra-modulation spaces (Theorem 3.3), and then use this result to prove subexponential decay of eigenfunctions of certain τ -pseudodifferential operators (Theorem 3.7). Finally, we conclude the paper with subexponential decay and regularity properties of eigenfunctions of localization operators in terms of Gelfand-Shilov spaces (Theorem 3.8).
An important relation between the action of an operator Op τ (σ ) on time-frequency shifts and the STFT of its symbol σ is explained in [12]. The setting given there is the one of S(R d ) and S (R d ), but it is easy to see that the claim is still valid when dealing with S (1) (R d ) and S (1) (R d ). Moreover, S (1) (R d ) and its dual can be replaced by S {γ } (R d ) and S {γ } (R d ) as it is done in [11] when τ = 1/2. Thus, the proof of the following lemma is omitted, since it follows by a slight modification of the proof of [12,Lemma 4.1]. (22) where z = (z 1 , z 2 ), w = (w 1 , w 2 ) ∈ R 2d and T τ and J are defined as follows: The following lemma can be viewed as a form of the inversion formula (18). The independent proof is given in the Appendix. (24) in the sense that Next we show how the τ -quantization Op τ (σ ), τ ∈ [0, 1], can be extended between ultra-modulation spaces under suitable assumptions on the weights. We remark that the following theorem is contained in more general [47,Theorem 3.1]. A more elementary proof of the same claim when Lebesgue parameters are greater than or equal to 1 is given in [45,Theorem A.2]. In contrast to [45,47] we use different arguments. Namely, our proof is based on the Schur test in combination with Lemmas 3.1 and 3.2. We note that [2, Theorem 3.3] is a particular case of Theorem 3.3 when restricted to polynomial weights and the duality between S(R d ) and S (R d ).
Due to the normalization chosen g L 2 = ĝ L 2 and we recall the inversion formula (18) which can be seen as a pointwise equality between smooth functions in this case (see [24,Proposition 11.2.4] By Lemma 3.2 we have In the next step we prove that the map M τ (σ ) : G → M τ (σ )G, defined by Using (22), we see that it is equivalent to prove that the integral operator with kernel where T τ ans J are defined in (23), is bounded on L p (R 2d ). We do this using the Schur test (see, e.g., [24, Lemma 6.2.1 (b)]). First we majorize K τ with another integral kernel Q τ using the condition (25) with w = (x, ω) ∈ R 2d and z = (y, η) ∈ R 2d : We now show that Q τ satisfies the Schur conditions. By appropriate change of variables (w ≡ w z (w) Furthermore, by the change of variables w ≡ w w (z) := J (w − z) for every w fixed, we obtain Hence from the Schur test it follows that M τ (σ ) is continuous, and due to (26) we notice that where the right hand-side is continuous and takes elements of S (1) . Therefore Op τ (σ ) is linear, continuous and densely defined. This concludes the proof.
Schatten class properties for various classes of pseudodifferential operators in the framework of time-frequency analysis are studied by many authors, let us mention just [10,24,33,47]. However, for our purposes it is convenient to recall [31,Theorem 1.2] about Schatten class property for pseudodifferential operators Op τ (σ ) with symbols in modulation spaces.
where m τ u is defined as in (5). Then .
Proof The first inclusion is due to the inclusion relations between ultra-modulation spaces since v l ⊗ v j w γ s ⊗ w γ t . The last inclusion follows similarly since m τ u v u ⊗ v u , as it is shown in Remark 2.5.
On account of the following corollary all the operators considered in Theorem 3.7 are compact on L 2 (R d ). Proof The claim follows by Lemma 3.5 with u satisfying (27), after choosing any 0 < p < 2, in addition with Theorem 3.4. Now we prove the decay property of the eigenfunctions of Op τ (σ ) when the symbol belongs to certain weighted modulation spaces. This result improves [2, Proposition 3.6], in the sense that we show how faster decay of the symbol implies stronger regularity and decay properties for the eigenfunctions of the corresponding operator. More precisely, [2, Proposition 3.6] deals with polynomial decay, whereas Theorem 3.7 allows to consider sub-exponential decay as well. for every t ≥ 0.