Date: 11 Mar 2010

Multiplication properties in pseudo-differential calculus with small regularity on the symbols

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 Hölder–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 ≤ p ≤ 2 and ω is an appropriate weight.