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

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 ${\mathrm{Op}}_A(a)$ is in the Orlicz Schatten class ${\mathcal{I}}_{\Phi }(L^2({\mathbf{R}}^d))$. Especially we prove Young type results for such classes. We apply the results on Toeplitz operators and prove Orlicz Schatten properties of such operators.

47 pages. In the second version we have generalized some results and especially included a multi-linear version of the dilated convolution result