Weighted Modulation Spaces Research Articles

Multilinear Fourier integral operators on modulation spaces

Abstract In this article, we study properties of multilinear Fourier integral operators on weighted modulation spaces. In particular, using the theory of Gabor frames, we study boundedness of multilinear Fourier integral operators on products of weighted modulation spaces. Further, we investigate the periodic multilinear Fourier integral operator. Finally, we study continuity of bilinear pseudo-differential operators on modulation spaces for certain symbol classes, namely 𝐒𝐆 {\mathbf{SG}} -class.

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.

Characterization of modulation spaces by symplectic representations and applications to Schrödinger equations

In the last twenty years modulation spaces, introduced by H. G. Feichtinger in 1983, have been successfully addressed to the study of signal analysis, PDE's, pseudodifferential operators, quantum mechanics, by hundreds of contributions. In 2011 M. de Gosson showed that the time-frequency representation Short-time Fourier Transform (STFT), which is the tool to define modulation spaces, can be replaced by the Wigner distribution. This idea was further generalized to τ-Wigner representations in [11].In this paper time-frequency representations are viewed as images of symplectic matrices via metaplectic operators. This new perspective highlights that the protagonists of time-frequency analysis are metaplectic operators and symplectic matrices A∈Sp(2d,R). We find conditions on A for which the related symplectic time-frequency representation WA can replace the STFT and give equivalent norms for weighted modulation spaces. In particular, we study the case of covariant matrices A, i.e., their corresponding WA are members of the Cohen class.Finally, we show that symplectic time-frequency representations WA can be efficiently employed in the study of Schrödinger equations. In fact, modulation spaces and WA representations are the frame for a new definition of wave front set, providing a sharp result for propagation of micro-singularities in the case of the quadratic Hamiltonians. This new approach may have further applications in quantum mechanics and PDE's.

Qualitative uncertainty principles for the windowed Opdam–Cherednik transform on weighted modulation spaces

The aim of this paper is to establish a few qualitative uncertainty principles for the windowed Opdam–Cherednik transform on weighted modulation spaces associated with this transform. In particular, we obtain Cowling–Price's, Hardy's and Morgan's uncertainty principles for this transform on weighted modulation spaces. The proofs of the results are based on versions of the Phragmén–Lindlöf type result for several complex variables on weighted modulation spaces and the properties of the Gaussian kernel associated with the Jacobi–Cherednik operator.

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.

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators A_a^{varphi _1,varphi _2} with symbols a in the wide weighted modulation space M^infty _{w}({mathbb {R}^{2d}}), and windows varphi _1, varphi _2 in the Gelfand–Shilov space mathcal {S}^{left( 1right) }(mathbb {R}^d). If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of A_a^{varphi _1,varphi _2} have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space mathcal {S}^{(gamma )} (mathbb {R^{d}}) , where the parameter gamma ge 1 is related to the growth of the considered weight. An important role is played by tau -pseudodifferential operators Op_{tau } (sigma ). In that direction we show convenient continuity properties of Op_{tau } (sigma )when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of Op_{tau } (sigma ) when the symbol sigma 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.

Decay and smoothness for eigenfunctions of localization operators

We study decay and smoothness properties for eigenfunctions of compact localization operators Aaφ1,φ2. Operators Aaφ1,φ2 with symbols a in the wide modulation space Mp,∞ (containing the Lebesgue space Lp), p<∞, and windows φ1,φ2 in the Schwartz class S are known to be compact. We show that their L2-eigenfuctions with non-zero eigenvalues are indeed highly compressed onto a few Gabor atoms. Similarly, for symbols a in the weighted modulation spaces Mvs⊗1∞(R2d), s>0 (subspaces of Mp,∞(R2d), p>2d/s) the L2-eigenfunctions of Aaφ1,φ2 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.

Compactness of Fourier integral operators on weighted modulation spaces

Using the matrix representation of Fourier integral operators with respect to a Gabor frame, we study their compactness on weighted modulation spaces. As a consequence, we recover and improve some compactness results for pseudodifferential operators.

Highly oscillatory unimodular Fourier multipliers on modulation spaces

We study the continuity on the modulation spaces $$M^{p,q}$$ of Fourier multipliers with symbols of the type $$e^{i\mu (\xi )}$$ , for some real-valued function $$\mu (\xi )$$ . A number of results are known, assuming that the derivatives of order $$\ge 2$$ of the phase $$\mu (\xi )$$ are bounded or, more generally, that the second derivatives belong to the Sjostrand class $$M^{\infty ,1}$$ . Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space $$W(\mathcal {F}L^1,L^\infty )$$ ; in particular they could have stronger oscillations at infinity such as $$\cos |\xi |^2$$ . Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.

Sharp Integral Bounds for Wigner Distributions

The cross-Wigner distribution $W(f,g)$ of two functions or temperate distributions $f,g$ is a fundamental tool in quantum mechanics and in signal analysis. Usually, in applications in time-frequency analysis $f$ and $g$ belong to some modulation space and it is important to know which modulation spaces $W(f,g)$ belongs to. Although several particular sufficient conditions have been appeared in this connection, the general problem remains open. In the present paper we solve completely this issue by providing the full range of modulation spaces in which the continuity of the cross-Wigner distribution $W(f,g)$ holds, as a function of $f,g$. The case of weighted modulation spaces is also considered. The consequences of our results are manifold: new bounds for the short-time Fourier transform and the ambiguity function, boundedness results for pseudodifferential (in particular, localization) operators and properties of the Cohen class.

Functions operating on modulation spaces and nonlinear dispersive equations

The aim of this paper is two fold. We show that if a complex function F on C operates in the modulation spaces M p , 1 ( R n ) by composition, then F is real analytic on R 2 ≈ C . This answers negatively, the open question posed in Ruzhansky et al. (2012) [21] , regarding the general power type nonlinearity of the form | u | α u . We also characterise the functions that operate in the modulation space M 1 , 1 ( R n ) . The local well-posedness of the NLS, NLW and NLKG equations for the ‘real entire’ nonlinearities are also studied in some weighted modulation spaces M s p , q ( R n ) .

Fractional integral operator on modulation and Wiener amalgam spaces

The purpose of this paper is to investigate the mapping properties of the fractional integral operators on weighted modulation spaces. Based on this result, we also study the boundedness of the bilinear fractional integral operators on product Wiener amalgam spaces. Our results show that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces.

Sharpness of some properties of weighted modulation spaces

In this paper, some properties on weighted modulation and Wiener amalgam spaces are characterized by the corresponding properties on weighted Lebesgue spaces. As applications, sharp conditions for product inequalities, convolution inequalities and embedding on weighted modulation and Wiener amalgam spaces are obtained. These applications improve and extend many known results.

Global Wave-Front Properties for Fourier Integral Operators and Hyperbolic Problems

We illustrate the composition properties for an extended family of SG Fourier integral operators. We prove continuity results for operators in this class with respect to $L^2$ and weighted modulation spaces, and discuss continuity on $\mathscr{S}$, $\mathscr{S}^\prime$ and on weighted Sobolev spaces. We study mapping properties of global wave-front sets under the action of these Fourier integral operators. We extend classical results to more general situations. For example, there are no requirements of homogeneity for the phase functions. Finally, we apply our results to the study of of the propagation of singularities, in the context of modulation spaces, for the solutions to the Cauchy problems for the corresponding linear hyperbolic operators.

Spectral Properties for Matrix Algebras

We consider Banach algebras of infinite matrices defined in terms of a weight measuring the off-diagonal decay of the matrix entries. If a given matrix \(A\) is invertible as an operator on \(\ell ^2\) we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in \({\mathcal B} (\ell ^2),\) the Banach algebra of bounded operators on \(\ell ^2.\) To this end we consider a condition on sequences of weights which extends the notion of GRS-condition. Finally we focus on the behavior of inverses of pseudodifferential operators whose Weyl symbols belong to weighted modulation spaces and the weights lack the GRS condition.

Hypersingular integral operators on modulation spaces for 0 &lt; p &lt; 1

The purpose of this paper is to investigate the mapping properties of the hypersingular integral operators on general weighted modulation spaces M s p , q ( R n ) for 0<p<1. Our results show that modulation spaces are good substitutions for Lebesgue spaces.MSC:42B20, 42B25.

Spectral Properties for Pseudodifferential Operators via Weighted Modulation Spaces

In this paper we deal with this question: considering spectral representation of a positive trace-class integral operator, if its orthonormal eigenvectors are in modulation space $M^p_m$? This question actually provide a new framework for studying the connection between operator theory and modulation spaces. Here we use some Schatten class properties of seudodifferential operators to give a positive answer to this question. Also we investigate convergence conditions for eigenvectors of such operators.

STRONGLY SINGULAR CONVOLUTION OPERATORS ON MODULATION SPACES

The purpose of this paper is to investigate the mapping properties of the strongly singular convolution operators on general weighted modulation spaces [Formula: see text] for 0 &lt; p ≤ ∞, 0 &lt; q ≤ ∞ and s ∈ ℝ. Our results show that modulation spaces are good substitutions for Lebesgue spaces.

Molecular decomposition of the modulation spaces

The aim of this paper is to develop a theory of decomposition in the weighted modulation spaces $M_{p,q}^{s,W}$ with $0 < p,q \le \infty$, $s \in \mathbb{R}$ and $W \in A_{\infty}$, where $W$ belongs to the class of $A_{\infty}$ defined by Muckenhoupt. For this purpose we shall define molecules for the modulation spaces. As an application we give a simple proof of the boundedness of the pseudo-differential operators with symbols in $M_{\infty,\min(1,p,q)}^{0}$. We shall deal with dual spaces as well.

Schatten--von Neumann properties for Fourier integral operators with non-smooth symbols II

We establish continuity and Schatten-von Neumann properties for Fourier integral operators with amplitudes in weighted modulation spaces, when acting on modulation spaces themselves. The phase func ...