Boundedness of Pseudo-Differential Operators on $L^p$, Sobolev, and Modulation Spaces

TL;DR

This paper introduces new modulation spaces over phase space and demonstrates how they can be used to bound the operator norms of pseudo-differential operators across various function spaces, including $L^p$, Sobolev, and modulation spaces.

Contribution

The paper develops novel modulation spaces over phase space and establishes their role in bounding pseudo-differential operators' norms on multiple function spaces.

Findings

New classes of modulation spaces over phase space introduced

Boundedness of pseudo-differential operators established on $L^p$, Sobolev, and modulation spaces

Use of Kohn-Nirenberg correspondence to connect spaces and operators

Abstract

We introduce new classes of modulation spaces over phase space. By means of the Kohn-Nirenberg correspondence, these spaces induce norms on pseudo-differential operators that bound their operator norms on $L^p$-spaces, Sobolev spaces, and modulation spaces.