Characterisation of the Weyl-H\"ormander classes by time-frequency shifts

TL;DR

This paper characterizes Weyl-H"ormander symbol classes using time-frequency shifts and introduces a geometric short-time Fourier transform to analyze these classes through new modulation spaces.

Contribution

It introduces a geometric short-time Fourier transform and characterizes Weyl-H"ormander classes as intersections of new modulation spaces.

Findings

Weyl-H"ormander classes characterized by time-frequency shifts

Introduction of a geometric short-time Fourier transform

Weyl-H"ormander classes expressed as intersections of modulation spaces

Abstract

We characterise the Weyl-H\"ormander symbol classes $S(M,g)$ via the growth of the action of the corresponding $\Psi$DOs on time-frequency shifts of a single test function. For this purpose, we introduce a geometric short-time Fourier transform which is well-suited for the analysis of $S(M,g)$. We define new modulation spaces and achieve the characterisation of the Weyl-H\"ormander classes by showing that they are intersections of such modulation spaces suitable for the time-frequency characterisation.