A remark on the Weyl quantisation of Paley-Wiener functions

TL;DR

This paper provides a concise proof linking the Schatten class membership of Weyl quantisations of Paley-Wiener functions to the $L^p$-integrability of their symbols, using quantum harmonic analysis tools.

Contribution

It introduces a simplified proof connecting Schatten class properties of Weyl quantisations with symbol integrability, utilizing Werner-Young's inequality and a quantum Wiener's division lemma.

Findings

Weyl quantisation of Paley-Wiener functions is in Schatten $p$-class iff the symbol is $L^p$-integrable.

The proof leverages quantum harmonic analysis techniques.

The result clarifies the relationship between symbol regularity and operator class membership.

Abstract

We present a short proof of the fact that the Weyl quantisation of a tempered distribution with compactly supported Fourier transform is in the Schatten $p$-class if and only if the symbol is $L^p$-integrable. The proof is based on Werner-Young's inequality from quantum harmonic analysis and a quantum version of Wiener's division lemma.