Characterisation of the Weyl-Hörmander classes by time-frequency shifts

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