Time-frequency analysis of pseudodifferential operators (Q1601789)

We apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn-Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product \(\sigma \sharp \tau \) of two symbols \(\sigma\) and \(\tau\), and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of \(\sigma \sharp \tau \) from the time-frequency content of \(\sigma\) and \(\tau\). As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol \(\sigma \sharp \tau \).