Characterization of boundedness on weighted modulation spaces of $\tau$-Wigner distributions

TL;DR

This paper provides comprehensive characterizations of the boundedness of $ au$-Wigner distributions on weighted modulation spaces, including sharp exponents and applications to pseudodifferential operators with Sj"{o}strand's class symbols.

Contribution

It introduces general characterizations for boundedness of $ au$-Wigner distributions and applies these to obtain sharp boundedness exponents for various modulation and Wiener amalgam spaces.

Findings

Sharp boundedness exponents for $ au$-Wigner distributions on weighted modulation spaces.

Recovers main theorems of Wigner distribution from previous works.

Characterizes boundedness of pseudodifferential operators with Sj"{o}strand's class symbols.

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 in \cite{CorderoNicola2018IMRNI,Cordero2020a}. 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 Sj\"{o}strand's class and the corresponding Wiener amalgam spaces.