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Modulation spaces have received considerable interest recently as it is the natural function spaces to consider low regularity Cauchy data for several nonlinear evolution equations. We establish global well-posedness for 3D Klein-Gordon-Hartree equation u t t − Δ u + u + (| ⋅ | − γ ⁎ | u | 2) u = 0 with initial data in modulation spaces M 1 p , p ′ × M p , p ′ for p ∈ (2 , 54 27 − 2 γ) , 2 < γ < 3. We implement Bourgain's high-low frequency decomposition method to establish global well-posedness, which was earlier used for classical Klein-Gordon equation. This is the first result on low regularity for Klein-Gordon-Hartree equation with large initial data in modulation spaces (which do not coincide with Sobolev spaces). [ABSTRACT FROM AUTHOR]

Hausdorff–Young’s inequality establishes the boundedness of the Fourier transform from L p to L q spaces for 1 ≤ p ≤ 2 and q = p ′ , where p ′ denotes the Lebesgue-conjugate exponent of p. This paper extends this classical result by characterizing the L p - L q boundedness of metaplectic operators, which play a significant role in harmonic analysis. We demonstrate that metaplectic operators are bounded on Lebesgue spaces if and only if their symplectic projection is either free or lower block triangular. As a byproduct, we identify metaplectic operators that serve as homeomorphisms of L p spaces. To achieve this, we leverage a parametrization of the symplectic group by Dopico and Johnson. We use our findings to provide boundedness results within L p spaces for pseudodifferential operators with symbols in Lebesgue spaces, and quantized by means of metaplectic operators. These quantizations consists of shift-invertible metaplectic Wigner distributions, which are essential to measure local phase-space concentration of signals. Using the factorization by Dopico and Johnson, we infer a decomposition law for metaplectic operators on L 2 (R 2 d) in terms of shift-invertible metaplectic operators, establish the density of shift-invertible symplectic matrices in Sp (2 d , R) , and prove that the lack of shift-invertibility prevents metaplectic Wigner distributions to define the so-called modulation spaces M p (R d) . [ABSTRACT FROM AUTHOR]

We review at first the role of localization operators as a meeting point of three different areas of research, namely: signal analysis, quantization and pseudo-differential operators. We extend then the correspondence between symbol and operator which characterizes localization operators to a more general situation, introducing the class of bilocalization operators. We show that this enlargement yields a quantization rule that is closed under composition. Some boundedness results are deduced both for localization and bilocalization operators. In particular for bilocalization operators we prove that square integrable symbols yield bounded operators on L 2 and that the class of bilocalization operators with integrable symbols is a subalgebra of bounded operators on every fixed modulation space. [ABSTRACT FROM AUTHOR]

Inductive and projective type sequence spaces of sub- and superexponential growth, and the corresponding inductive and projective limits of modulation spaces are considered as a framework for almost diagonalization of pseudo-differential operators. Moreover, recent results of the first author and B. Prangoski related to the almost diagonalization of pseudo-differential operators in the context of H?ormander metrics are reviewed. [ABSTRACT FROM AUTHOR]

This paper discusses spectral synthesis for those modulation spaces Msp,q (Rn) which form Banach algebras under pointwise multiplication. An important argument will be the "ideal theory for Segal algebras" by H. Reiter [15]. This paper is a continuation of our paper [5] where the case q = 1 is treated. As a by-product we obtain a variant of Wiener--Lévy theorem for Msp,q(Rn) Fourier--Wermer algebras FLsq(Rn). [ABSTRACT FROM AUTHOR]

This study devotes to Heisenberg's uncertainty principles for Fourier integral operators of types I and II with function-variable symbols, i.e., the symbol σ ∈ S 1 ⊗ v s ∞ (R N) of the type I Fourier integral operator is only w -dependent and the symbol τ ∈ S 1 ⊗ v s ∞ (R N) of the type II Fourier integral operator is only y -dependent. These two special Fourier integral operators are abbreviated as the FIO-I-FV and FIO-II-FV, respectively. We disclose an equivalence relation between the FIO-I-FV and the classical metaplectic transform, as well as the FIO-II-FV and the metaplectic transform, based on which we employ various versions of Heisenberg's uncertainty principles for the metaplectic transform, ranging from the general metaplectic transform of real-valued functions to some specific (e.g., the orthogonal, the orthonormal, the minimum eigenvalue commutative, the maximum eigenvalue commutative) metaplectic transforms of complex-valued functions, in the establishment of the corresponding results for the FIO-I-FV and FIO-II-FV. [ABSTRACT FROM AUTHOR]

Inspired by the recent article Skrettingland (J. Fourier Anal. Appl. 28(2), 1–34 (2022)), this paper is devoted to the study of a suitable class of windows in the framework of bounded linear operators on L 2 (R d) . We establish a natural and complete characterization for the window class such that the corresponding STFT leads to equivalent norms on modulation spaces. The positive bounded linear operators are also characterized by its Cohen’s class distributions such that the corresponding quantities form equivalent norms on modulation spaces. As a generalization, we introduce a family of operator classes corresponding to the operator-valued modulation spaces. Some applications of our main theorems to the localization operators are also concerned. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned about the local well-posedness of Vlasov-Poisson-Fokker-Planck (VPFP) in modulation-Lebesgue space for large initial data. To accomplish this goal, we establish convolution inequalities associated with the fundamental solution to Fokker-Planck equation. The product formula and off-diagonal estimates involving the electronic term are established as well. [ABSTRACT FROM AUTHOR]

We introduce and study continuity properties of the Gabor product ♮ g and relate it to the well-known product formula for the short-time Fourier transform (STFT). We derive a phase space representation of the cubic nonlinear Schrödinger equation in terms of the Gabor product, and discuss how the Gabor product can be used in the study of nonlinear dynamics of mixed quantum states. [ABSTRACT FROM AUTHOR]

We present a different symplectic point of view in the definition of weighted modulation spaces M m p , q (R d) and weighted Wiener amalgam spaces W (F L m 1 p , L m 2 q) (R d). All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ -Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions μ (A) (f ⊗ g ¯) , where μ (A) is the metaplectic operator and A is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13] , the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications. [ABSTRACT FROM AUTHOR]

Motivated by the recent paper of Boggiatto–Garello (J Pseudo-Differ Oper Appl 11:93–117, 2020) where a Gabor operator is regarded as pseudodifferential operator with symbol p (x , ω) periodic on both the variables, we study the continuity and invertibility, on general time frequency invariant spaces, of pseudodifferential operators with completely periodic symbol and general τ quantization. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol ς and two windows functions g 1 and g 2 . We first present some basic properties of the windowed Opdam–Cherednik transform. Then, we use modulation spaces associated with the Opdam–Cherednik transform as appropriate classes for symbols and windows and study the boundedness and compactness of the localization operators associated with the windowed Opdam–Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten–von Neumann class. [ABSTRACT FROM AUTHOR]

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other “geometry-discretizing”) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured “uniformly over the class”, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that the density condition we obtain is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore find that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error. [ABSTRACT FROM AUTHOR]

We obtain Gabor frame characterizations of modulation spaces defined via a broad class of translation-modulation invariant Banach spaces of distributions. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterized by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalizes and unifies several fundamental results for the classical modulation spaces M w p , q and the amalgam spaces W (L p , L w q). Due to the absence of solidity assumptions on the Banach spaces defining these generalized modulation spaces, the methods used for the spaces M w p , q (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution. [ABSTRACT FROM AUTHOR]

We deduce trace properties for modulation spaces (including certain Wiener-amalgam spaces) of Gelfand–Shilov distributions.We use these results to show that Ψ dos with amplitudes in suitable modulation spaces, agree with normal type Ψ dos whose symbols belong to (other) modulation spaces. In particular we extend earlier trace results for modulation spaces, to include quasi-Banach modulation spaces. We also apply our results to extend earlier results on Schatten-von Neumann and nuclear properties for Ψ dos with amplitudes in modulation spaces. [ABSTRACT FROM AUTHOR]

We produce a finite time blow-up solution for nonlinear fractional heat equation ( ∂ t u + (- Δ) β / 2 u = u k ) in modulation and Fourier amalgam spaces on the torus T d and the Euclidean space R d. This complements several known local and small data global well-posedness results in modulation spaces on R d. Our method of proof rely on the formal solution of the equation. This method should be further applied to other non-linear evolution equations. [ABSTRACT FROM AUTHOR]

We generalize the results for Banach algebras of pseudodifferential operators obtained by Gröchenig and Rzeszotnik (Ann Inst Fourier 58:2279–2314, 2008) to quasi-algebras of Fourier integral operators. Namely, we introduce quasi-Banach algebras of symbol classes for Fourier integral operators that we call generalized metaplectic operators, including pseudodifferential operators. This terminology stems from the pioneering work on Wiener algebras of Fourier integral operators (Cordero et al. in J Math Pures Appl 99:219–233, 2013), which we generalize to our framework. This theory finds applications in the study of evolution equations such as the Cauchy problem for the Schrödinger equation with bounded perturbations, cf. (Cordero, Giacchi and Rodino in Wigner analysis of operators. Part II: Schrödinger equations, ). [ABSTRACT FROM AUTHOR]

We prove new local and global well-posedness results for the cubic one-dimensional nonlinear Schrödinger equation in modulation spaces. Local results are obtained via multilinear interpolation. Global results are proven using conserved quantities based on the complete integrability of the equation, persistence of regularity, and by separating off the time evolution of finitely many Picard iterates. [ABSTRACT FROM AUTHOR]

Let A β 1 (T) denote the set of all Lebesgue integrable functions F on the torus T such that ∑ m ∈ Z | F ^ (m) | (1 + | m |) β < ∞ , where { F ^ (m) } m ∈ Z denote the Fourier coefficients of F. We consider necessary and sufficient conditions for all functions F ∈ A β 1 (T) to operate on all real-valued functions in the modulation spaces M s p , q (R) . [ABSTRACT FROM AUTHOR]

In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space ℝ d . In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space (ℝ d) is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods. [ABSTRACT FROM AUTHOR]

In this paper, we establish the asymptotic estimates for the norms of the matrix dilation operators on modulation spaces. As an application, we study the boundedness on modulation spaces of Hausdorff operators. The definition of Hausdorff operators is also revisited for fitting our study. [ABSTRACT FROM AUTHOR]

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation Short-time Fourier Transform (STFT) is replaced by the A - Wigner distribution defined by W A (f) = μ (A) (f ⊗ f ¯) , where A is a 4 d × 4 d symplectic matrix and μ (A) is an associate metaplectic operator. Basic examples are given by the so-called τ -Wigner distributions. Such representations provide a new characterization for modulation spaces when τ ∈ (0 , 1). Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjöstrand class (in particular, in the Hörmander class S 0 , 0 0). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global Hörmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrödinger equations will be given. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for Hartree type equation i u t + u x x = [ K ⁎ | u | 2 ] u , with Cauchy data in modulation spaces M p , q (R). We establish global well-posedness results in M p , p ′ (R) when K (x) = λ | x | γ , (λ ∈ R , 0 < γ < 1) with no smallness condition on initial data, where p ′ is the Hölder conjugate of p. Our proof uses a splitting method inspired by the work of Vargas-Vega, Hyakuna-Tsutsumi, Grünrock and Chaichenets et al. to the modulation space setting and exploits polynomial growth of the Schrödinger propagator on modulation spaces. [ABSTRACT FROM AUTHOR]

We give a new class of equivalent norms for modulation spaces by replacing the window of the short-time Fourier transform by a Hilbert–Schmidt operator. The main result is applied to Cohen’s class of time-frequency distributions, Weyl operators and localization operators. In particular, any positive Cohen’s class distribution with Schwartz kernel can be used to give an equivalent norm for modulation spaces. We also obtain a description of modulation spaces as time-frequency Wiener amalgam spaces. The Hilbert–Schmidt operator must satisfy a nuclearity condition for these results to hold, and we investigate this condition in detail. [ABSTRACT FROM AUTHOR]

We define and investigate α -modulation spaces M p , q s , α (G) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean α -modulation spaces M p , q s , α (R n) that act as intermediate spaces between the modulation spaces ( α = 0 ) in time-frequency analysis and the Besov spaces ( α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases α = 0 , 1 where the spaces M p , q s (G) and B p , q s (G) have non-standard translation and dilation symmetries. Moreover, we show that the spaces M p , q s , α (G) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings Q (G) underlying the α = 0 case M p , q s (G) allows for the existence of geometric embeddings F : M p , q s (R k) ⟶ M p , q s (G) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper. [ABSTRACT FROM AUTHOR]

We identify the generalised modulation spaces associated with tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated with tensor products of various L p spaces. [ABSTRACT FROM AUTHOR]

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V. Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into (L2(R), ||⋅||2), nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space S'(Rd) (d ≥ 1) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes (Qs(Rd), ||⋅||Qs), showing that they are special cases of Katsnelson's setting (only) for s ≥ 0. [ABSTRACT FROM AUTHOR]

Sparsity properties for phase-space representations of several types of operators have been extensively studied in recent articles, including pseudodifferential, Fourier integral and metaplectic operators, with applications to the analysis of Schrödinger-type evolution equations. It has been proved that such operators are approximately diagonalized by Gabor wave packets. While the latter are expected to undergo some spreading phenomenon, there is no record of this issue in the aforementioned results. In this note we prove refined estimates for the Gabor matrix of metaplectic operators, also of generalized type, where sparsity, spreading and dispersive properties are all noticeable. We provide applications to the propagation of singularities for the Schrödinger equation. [ABSTRACT FROM AUTHOR]

In the present paper we propose a new algebraic structure on some famous Banach spaces called modulation space. Our suggestion goes through an analytical properties of a Banach space which would be equipped with a weight function and then a convolution. Furthure we consider the operator theory scheme of these spaces and discus more valuable and applicable properties which will be arise in quantization formulations. The subject here would consider as some elementary properties for any modulation space. [ABSTRACT FROM AUTHOR]

Let H be a Hilbert space of distributions on R d which contains at least one non-zero element of the Feichtinger algebra S 0 and is continuously embedded in D ′ . If H is translation and modulation invariant, also in the sense of its norm, then we prove that H = L 2 , with the same norm apart from a multiplicative constant. [ABSTRACT FROM AUTHOR]

We prove global wellposedness of the Klein-Gordon equation with power nonlinearity |u|α−1u, where α ∈ [1, d/d−2], in dimension d ≥ 3 with initial data in Mp, p'1(Rd) × Mp,p'(Rd) for p sufficiently close to 2. The proof is an application of the high-low method described by Bourgain in [ Global solutions of nonlinear Schrödinger equations , American Mathematical Society, Providence, RI, 1999] where the Klein-Gordon equation is studied in one dimension with cubic nonlinearity for initial data in Sobolev spaces. [ABSTRACT FROM AUTHOR]

Modulation spaces M p , q s were introduced by Feichtinger [11] in 1983. Bényi and Oh [2] defined a modified version to Feichtinger's modulation spaces for which the symmetry scalings are emphasized for its possible applications in PDE. By carefully investigating the scaling properties of modulation spaces and their connections with Bényi and Oh's modulation spaces, we introduce the scaling limit versions of modulation spaces, which contains both Feichtinger's and Bényi and Oh's modulation spaces. As their applications, we will give a local well-posedness and a (small data) global well-posedness results for nonlinear Schrödinger equation in some scaling limit of modulation spaces, which generalize the well posedness results of [3] and [23] , and certain super-critical initial data in H s or in L p are involved in these spaces. [ABSTRACT FROM AUTHOR]