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

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.