Time-Frequency Analysis of Pseudodifferential Operators

 In this paper 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 of two symbols σ and τ, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of from the time-frequency content of σ and τ. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol .