Sharp integral bounds for Wigner distributions

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.

23 pages, some typos corrected

Cited by in corpus (5)

• Linear perturbations of the Wigner distribution and the Cohen's class
• Norm Estimates for $\tau$-Pseudodifferential Operators in Wiener Amalgam and Modulation Spaces
• Boundedness of Pseudodifferential Operators with symbols in Wiener amalgam spaces on Modulation Spaces
• Comparisons between Fourier and STFT multipliers: the smoothing effect of the Short-time Fourier Transform
• Modulation spaces associated to tensor products of amalgam spaces