Convolutions of Orlicz spaces and Orlicz Schatten classes, with applications to Toeplitz operators

TL;DR

This paper investigates convolution properties of Orlicz Schatten classes associated with pseudo-differential operators, establishing Young-type results and applying them to analyze Toeplitz operators within this framework.

Contribution

It introduces new convolution properties for Orlicz Schatten classes and applies these to study Toeplitz operators with Orlicz Schatten class symbols.

Findings

Established Young-type convolution results for Orlicz Schatten classes.

Proved Orlicz Schatten properties of Toeplitz operators.

Extended the understanding of symbol classes in pseudo-differential operator theory.

Abstract

Let $\Phi$ be a Young function. We study convolution properties for symbol classes $s_{A,\Phi}$, which consist of all $a$ such that the pseudo-differential operator $\operatorname{Op} _A(a)$ is in the Orlicz Schatten class $\mathscr I _\Phi (L^2(\mathbf R^d))$. Especially we prove Young type results for such classes. We apply the results on Toeplitz operators and prove Orlicz Schatten properties of such operators.