The Weyl transform of a compactly supported distribution

TL;DR

This paper characterizes when the Weyl transform of a compactly supported distribution is traceable or compact, linking these properties to the Fourier transform's integrability and decay at infinity.

Contribution

It establishes precise conditions relating the Weyl transform's properties to the Fourier transform of distributions, providing new insights into their operator-theoretic behavior.

Findings

Weyl transform is p-power traceable iff Fourier transform is p-power integrable.

Weyl transform is compact iff Fourier transform vanishes at infinity.

Provides criteria connecting distribution support, Fourier transform, and operator properties.

Abstract

If $T$ is a compactly supported distribution on $\mathbb{R}^{2n}$, then the Weyl transform of $T$ is $p$-power traceable if and only if the Fourier transform of $T$ is $p$-power integrable, and the Weyl transform of $T$ is a compact operator if and only if the Fourier transform of $T$ vanishes at infinity.