Compact pseudodifferential and Fourier integral operators via localization

TL;DR

This paper introduces a framework for localized operators with matrix coefficients concentrated on the diagonal, unifying and extending results for pseudodifferential and Fourier integral operators, and analyzing their boundedness and compactness on modulation spaces.

Contribution

It develops a general formalism for localized operators, extending existing theorems and providing new results for three-parameter pseudodifferential operators.

Findings

Localized operators are bounded between modulation spaces.

Compactness can be deduced from weak compactness conditions.

Unified approach extends classical results for pseudodifferential and Fourier integral operators.

Abstract

We present a general framework of localized operators, i.e., operators whose matrix coefficients with respect to the Gabor frame are concentrated on the diagonal. We show that localized operators are bounded between modulation spaces, and we deduce their compactness from an easily verifiable weak compactness condition. We apply this abstract formalism to unify and extend existing theorems for pseudodifferential and Fourier integral operators, and to obtain new results for three-parameter pseudodifferential operators.