Products on Schatten-von Neumann classes and modulation spaces

TL;DR

This paper investigates the properties of modulation spaces and Schatten-von Neumann classes related to pseudo-differential operators, establishing continuity and convolution results with implications for twisted convolutions on Lebesgue spaces.

Contribution

It introduces new continuity and convolution results for modulation and Schatten-von Neumann spaces, extending their applications to twisted convolutions and operator mappings.

Findings

Proved Hölder-Young and Young type inequalities for these spaces.

Established continuity properties under twisted convolution and Weyl product.

Showed that certain Lebesgue spaces form twisted convolution algebras.

Abstract

We consider modulation space and spaces of Schatten-von Neumann symbols where corresponding pseudo-differential operators map one Hilbert space to another. We prove H\"older-Young and Young type results for such spaces under dilated convolutions and multiplications. We also prove continuity properties for such spaces under the twisted convolution, and the Weyl product. These results lead to continuity properties for twisted convolutions on Lebesgue spaces, e.{}g. $L^p_{(\omega)}$ is a twisted convolution algebra when $1\le p\le 2$ and appropriate weight $\omega$.