Wiener Amalgam Spaces Research Articles

Factorizations for quasi-Banach time–frequency spaces and Schatten classes

We deduce factorization properties for Wiener amalgam spaces WLp,q, an extended family of modulation spaces M(ω,ℬ), and for Schatten symbols spw in pseudo-differential calculus under e.g. convolutions, twisted convolutions and symbolic products. Here M(ω,ℬ) can be any quasi-Banach Orlicz modulation space. For example we show that WL1,r∗WLp,q=WLp,q and WL1,r#spw=spw when r∈(0,1], r≤p,q<∞. In particular we improve Rudin’s identity L1∗L1=L1.

Excursus on modulation spaces via metaplectic operators and related time-frequency representations

We provide a comprehensive overview of the theoretical framework surrounding modulation spaces and their characterizations, particularly focusing on the role of metaplectic operators and time-frequency representations. We highlight the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. In particular, this work provides new characterizations via the submanifold of shift-invertible symplectic matrices. Similar results hold for the Wiener amalgam spaces.

Weighted Hardy–Orlicz-amalgam spaces of martingales

Basing on the Wiener amalgam spaces, the purpose of this paper is to introduce the weighted Hardy–Orlicz-amalgam spaces of martingales and to establish the atomic decomposition theorem. As an application, we obtain a duality theorem, namely, the dual of martingale weighted Hardy–Orlicz-amalgam spaces are generalized martingale weighted amalgam Campanato spaces.

Metaplectic Gabor frames and symplectic analysis of time-frequency spaces

We introduce new frames, called metaplectic Gabor frames, as natural generalizations of Gabor frames in the framework of metaplectic Wigner distributions, cf. [7,8,5,17,27,28]. Namely, we develop the theory of metaplectic atoms in a full-general setting and prove an inversion formula for metaplectic Wigner distributions on Rd. Its discretization provides metaplectic Gabor frames.Next, we deepen the understanding of the so-called shift-invertible metaplectic Wigner distributions, showing that they can be represented, up to chirps, as rescaled short-time Fourier transforms. As an application, we derive a new characterization of modulation and Wiener amalgam spaces. Thus, these metaplectic distributions (and related frames) provide meaningful definitions of local frequencies and can be used to measure effectively the local frequency content of signals.

Strichartz estimates for the Dirac flow in Wiener amalgam spaces

AbstractIn this paper, we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non‐relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.

Symplectic analysis of time-frequency spaces

We present a different symplectic point of view in the definition of weighted modulation spaces Mmp,q(Rd) and weighted Wiener amalgam spaces W(FLm1p,Lm2q)(Rd). 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.

Mild solutions and spacetime integral bounds for Stokes and Navier–Stokes flows in Wiener amalgam spaces

Mild solutions and spacetime integral bounds for Stokes and Navier–Stokes flows in Wiener amalgam spaces

Wiener amalgam spaces with respect to orlicz spaces on the affine group

Wiener amalgam spaces with respect to orlicz spaces on the affine group

Wiener amalgam spaces of quasianalytic ultradistributions

We define Wiener amalgam spaces of (quasi)analytic ultradistributions whose local components belong to a general class of translation and modulation invariant Banach spaces of ultradistributions and their global components are either weighted Lp or weighted C0 spaces. We provide a discrete characterisation via so called uniformly concentrated partitions of unity. Finally, we study the complex interpolation method and we identify the strong duals for most of these Wiener amalgam spaces.

Trace mappings on quasi-Banach modulation spaces and applications to pseudo-differential operators of amplitude type

We deduce trace properties for modulation spaces (including certain Wiener-amalgam spaces) of Gelfand–Shilov distributions.We use these results to show that [Formula: see text]dos with amplitudes in suitable modulation spaces, agree with normal type [Formula: see text]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 [Formula: see text]dos with amplitudes in modulation spaces.

Boundedness of bilinear pseudo-differential operators of S0,0-type in Wiener amalgam spaces and in Lebesgue spaces

We extend and improve the known results about the boundedness of the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class BS0,0m(Rn). We consider wider classes of symbols and improve estimates for the corresponding operators. A key idea is to consider the operators in Wiener amalgam spaces.

Equivalent Norms for Modulation Spaces from Positive Cohen\u2019s Class Distributions

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.

Mixed-Norm Amalgam Spaces and Their Predual

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

We consider the Benjamin–Bona–Mahony (BBM) equation of the form ut+ux+uux−uxxt=0,(x,t)∈M×R where M=T or R. We establish norm inflation (NI) with infinite loss of regularity at general initial data in Fourier amalgam and Wiener amalgam spaces with negative regularity. This strengthens several known NI results at zero initial data in Hs(T) established by Bona–Dai (2017) and the ill-posedness result established by Bona–Tzvetkov (2008) and Panthee (2011) in Hs(R). Our result is sharp with respect to the local well-posedness result of Banquet–Villamizar–Roa (2021) in modulation spaces Ms2,1(R) for s≥0.

Boundedness of multilinear pseudo-differential operators with symbols in the Hörmander class S0,0

The multilinear pseudo-differential operators with symbols in the multilinear Hörmander class S0,0 are considered. A complete identification of the cases where those operators define bounded operators between local Hardy spaces is given. Some results for the boundedness between Wiener amalgam spaces are also given. These are extensions and improvements of the results known in the bilinear case.

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of L_{p,{1 / omega }}-norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.

Wiener–Luxemburg amalgam spaces

In this paper we introduce the concept of Wiener–Luxemburg amalgam spaces which are a modification of the more classical Wiener amalgam spaces intended to address some of the shortcomings the latter face in the context of rearrangement-invariant Banach function spaces.We introduce the Wiener–Luxemburg amalgam spaces and study their properties, including (but not limited to) their normability, embeddings between them and their associate spaces. We also study amalgams of quasi-Banach function spaces and introduce a necessary generalisation of the concept of associate spaces. We then apply this general theory to resolve the question whether the Hardy–Littlewood–Pólya principle holds for all r.i. quasi-Banach function spaces. Finally, we illustrate the asserted shortcomings of Wiener amalgam spaces by providing counterexamples to certain properties of Banach function spaces as well as rearrangement

Phase space analysis of the Hermite semigroup and applications to nonlinear global well-posedness

We study the Hermite operator H=−Δ+|x|2 in Rd and its fractional powers Hβ, β>0 in phase space. Namely, we represent functions f via the so-called short-time Fourier, alias Fourier-Wigner or Bargmann transform Vgf (g being a fixed window function), and we measure their regularity and decay by means of mixed Lebesgue norms in phase space of Vgf, that is in terms of membership to modulation spaces Mp,q, 0<p,q≤∞. We prove the complete range of fixed-time estimates for the semigroup e−tHβ when acting on Mp,q, for every 0<p,q≤∞, exhibiting the optimal global-in-time decay as well as phase-space smoothing.As an application, we establish global well-posedness for the nonlinear heat equation for Hβ with power-type nonlinearity (focusing or defocusing), with small initial data in modulation spaces or in Wiener amalgam spaces. We show that such a global solution exhibits the same optimal decay e−ct as the solution of the corresponding linear equation, where c=dβ is the bottom of the spectrum of Hβ. Global existence is in sharp contrast to what happens for the nonlinear focusing heat equation without potential, where blow-up in finite time always occurs for (even small) constant initial data (constant functions belong to M∞,1).

Characterization of Boundedness on Weighted Modulation Spaces of τ-Wigner Distributions

Abstract This paper is devoted to give several characterizations on a more general level for the boundedness of $\tau $-Wigner distributions acting from weighted modulation spaces to weighted modulation and Wiener amalgam spaces. As applications, sharp exponents are obtained for the boundedness of $\tau $-Wigner distributions on modulation spaces with power weights. We also recapture the main theorems of Wigner distribution obtained by Cordero and Nicola [10] and Cordero [6]. As consequences, the characterizations of the boundedness on weighted modulation spaces of several types of pseudodifferential operators are established. In particular, we give the sharp exponents for the boundedness of pseudodifferential operators with symbols in Sjöstrand’s class and the corresponding Wiener amalgam spaces.

Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations

In this paper we obtain some new Strichartz estimates for the wave propagator eit−Δ in the context of Wiener amalgam spaces. While it is well understood for the Schrödinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schrödinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schrödinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.