Excursus on modulation spaces via metaplectic operators and related time-frequency representations

Abstract: Modulation spaces were originally introduced by Feichtinger in 1983. Since the 2000s there have been thousands of contributions using them as correct framework; they range from PDEs, pseudodifferential operators, quantum mechanics, signal analysis. This justifies a deep study of such spaces and the related Wiener ones. Recently, metaplectic Wigner distributions, which contain as special examples the $\tau$-Wigner distributions, the ambiguity function and the Short-time Fourier transform, have proved to characterize modulation spaces, under suitable assumptions. We investigate the metaplectic action which is hidden in their construction and guarantees equivalent (quasi-)norms for such spaces. We add a new result on this topic and conclude with an exhaustive vision of these characterizations. Similar results hold for the Wiener amalgam ones.