Twisted convolution, pseudo-differential operators and Fourier modulation spaces

TL;DR

This paper investigates the continuity properties of twisted convolution on Fourier modulation and Lebesgue spaces, establishing algebraic structures under certain conditions, which enhances understanding of pseudo-differential operators.

Contribution

It provides new continuity results for twisted convolution on Fourier modulation spaces and Lebesgue spaces, including algebraic properties for weighted L^p spaces.

Findings

Twisted convolution is continuous on weighted Fourier modulation spaces.

Weighted L^p spaces with 1 ≤ p ≤ 2 form algebras under twisted convolution.

Results extend the understanding of pseudo-differential operators in these function spaces.

Abstract

We discuss continuity of the twisted convolution on (weighted) Fourier modulation spaces. We use these results to establish continuity results for the twisted convolution on Lebesgue spaces. For example we prove that if $\omega$ is an appropriate weight and $1\le p\le 2$, then $L^p_{(\omega)}$ is an algebra under the twisted convolution.