Sharp integral bounds for Wigner distributions

TL;DR

This paper completely characterizes the modulation spaces where the cross-Wigner distribution is continuous, providing new bounds and boundedness results relevant to quantum mechanics and signal analysis.

Contribution

It offers a full characterization of the modulation spaces for the cross-Wigner distribution's continuity, including weighted spaces, advancing theoretical understanding.

Findings

Full range of modulation spaces for continuity of W(f,g)

New bounds for short-time Fourier transform and ambiguity function

Boundedness results for pseudodifferential operators

Abstract

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.