Schatten class Fourier Integral Operators

TL;DR

This paper investigates the conditions under which Fourier integral operators with smooth phases and localized symbols belong to Schatten classes, revealing sharp bounds based on mixed modulation space localization.

Contribution

It establishes sharp criteria linking symbol localization in mixed modulation spaces to Schatten class membership for Fourier integral operators.

Findings

Operators with well-localized symbols are Schatten p-class for p in [1,2]

Mixed modulation spaces characterize symbol localization and Schatten class inclusion

Sharpness of results shown by larger spaces containing non-Schatten operators

Abstract

Fourier integral operators with sufficiently smooth phase act on the time-frequency content of functions. However time-frequency analysis has only recently been used to analyze these operators. In this paper, we show that if a Fourier integral operator has a smooth phase function and its symbol is well-localized in time and frequency, then the operator is Schatten \( p \)-class for \( p \in [1,2] \), with inclusion of the symbol in mixed modulation spaces serving as the appropriate measure of time-frequency localization. Our main results are sharp in the sense that larger mixed modulation spaces necessarily contain symbols of Fourier integral operators that are not Schatten \( p \)-class.