A remark on the Weyl quantisation of Paley-Wiener functions
Helge J. Samuelsen
TL;DR
The paper tackles the problem of characterizing when the Weyl quantisation of Paley–Wiener type symbols (tempered distributions with compact Fourier support) lies in Schatten classes. It presents a short proof showing that $L_\tau \in {\mathcal{S}}^p$ if and only if $\tau \in L^p({\mathbb{R}}^{2d})$, and that $L_\tau$ is compact precisely when $\tau \in C_0({\mathbb{R}}^{2d})$, using a quantum version of Wiener's division lemma together with Werner–Young's operator-convolution inequality. The argument relies on the convolution identity $L_\tau \star L_\Phi = \tau * \Phi$ and a boundedness estimate that reduces Schatten-class membership to $L^p$-control of the symbol, complemented by a density argument extending the result to limits of smooth, compactly supported Fourier data. This clarifies the exact $L^p$–Schatten correspondence in the Paley–Wiener setting within quantum harmonic analysis and extends prior work on Fourier restrictions and decoupling in this framework.
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.