Bounds on pseudodifferential operators and Fourier restriction for Schatten classes

TL;DR

This paper establishes a precise characterization of when pseudodifferential operators with symbols supported in Fourier space belong to Schatten classes, linking phase space Lebesgue spaces to operator classes, and provides a new proof of Fourier restriction results.

Contribution

It proves that pseudodifferential operators with compact Fourier support are in Schatten class $ ext{S}^p$ if and only if their symbols are in $L^p$, offering a new proof of related Fourier restriction results.

Findings

Characterization of Schatten class membership via symbol $L^p$ spaces.

New proof of Fourier restriction estimates for Schatten classes.

Connection between phase space Lebesgue spaces and operator classes.

Abstract

As main result, we show that a pseudodifferential operator in the Weyl calculus, whose symbol has compact Fourier support, lies in the Schatten class $\mathcal S^p$ if and only if its symbol lies in the Lebesgue space $L^p$ on phase space. As an immediate consequence, this gives an alternative and very lucid proof of a recent result by Luef and Samuelsen, who had discovered that for compactly supported measures $\mu,$ classical Fourier restriction estimates with respect to the measure $\mu$ are equivalent to quantum restriction estimates for the Fourier-Wigner transform for Schatten classes.