Quasi-Banach Schatten-von Neumann properties in Weyl-H\"ormander calculus

TL;DR

This paper investigates Schatten-von Neumann properties of pseudo-differential operators within the Weyl-H"ormander calculus, extending classical results to quasi-Banach spaces and applying findings to SG and Shubin operators.

Contribution

It establishes new Schatten-$p$ properties for pseudo-differential operators with symbols satisfying Wiener-Lebesgue bounds, especially in the quasi-Banach setting for $p o 0$, within the Weyl-H"ormander calculus.

Findings

Schatten-$p$ properties are derived for $p o 0$ in the Weyl-H"ormander calculus.

Results apply to global SG and Shubin pseudo-differential operators.

Structural properties of Wiener-Lebesgue spaces are analyzed for slowly varying metrics.

Abstract

We study structural properties of Wiener-Lebesgue spaces with respect to a slowly varying metrics and certain Lebesgue parameters. For $p\in (0,1]$, we deduce Schatten-$p$ properties for pseudo-differential operators whose symbols, together with their derivatives, obey suitable Wiener-Lebesgue-boundedness conditions. Especially, we perform such investigations for the Weyl-H\"ormander calculus. Finally, we apply our results to global-type SG and Shubin pseudo-differential operators.