Mixed Modulation Spaces and Their Application to Pseudodifferential Operators

TL;DR

This paper characterizes when integral and pseudodifferential operators belong to Schatten classes using frame techniques and mixed modulation spaces, providing sharper criteria than previous results.

Contribution

It introduces a novel characterization of Schatten class properties for operators via mixed modulation spaces, improving and sharpening existing criteria.

Findings

Operators with kernels in certain mixed modulation spaces are Schatten p-class.

The results provide sharp conditions, with larger spaces leading to non-Schatten class operators.

The approach uses frame expansions to connect kernel properties to operator classes.

Abstract

This paper uses frame techniques to characterize the Schatten class properties of integral operators. The main result shows that if the coefficients of certain frame expansions of the kernel of an integral operator are in (\ell^{2,p}), then the operator is Schatten p-class. As a corollary, we conclude that if the kernel or Kohn-Nirenberg symbol of a pseudodifferential operator lies in a particular mixed modulation space, then the operator is Schatten p-class. Our corollary improves existing Schatten class results for pseudodifferential operators and the corollary is sharp in the sense that larger mixed modulation spaces yield operators that are not Schatten class.