Pseudodifferential operators on $L^p$, Wiener amalgam and modulation spaces

TL;DR

This paper characterizes the boundedness of pseudodifferential operators with symbols in modulation and Wiener amalgam spaces on various function spaces, providing a complete range of conditions for their continuity.

Contribution

It offers a comprehensive characterization of when pseudodifferential operators with symbols in modulation and Wiener amalgam spaces are bounded on different function spaces.

Findings

Full range of (p,q,r) triples for boundedness on L^r spaces.

Complete characterization for operators on Wiener amalgam and modulation spaces.

Analysis of operators with symbols in W(ℱL^1,L^∞).

Abstract

We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces $M^{p,q}$, acting on a given Lebesgue space $L^r$. Namely, we find the full range of triples $(p,q,r)$, for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space $W(L^r,L^s)$ and even on modulation spaces $M^{r,s}$. Finally the action of pseudodifferential operators with symbols in $W(\Fur L^1,L^\infty)$ is also investigated.