Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and traceclass properties of pseudodifferential operators. As natural symbol classes, we use the modulation spaces onR2d, which quantify the notion of the time-frequency content of a function or distribution. We show that if a symbol σ lies in the modulation spaceM∞,1 (R2d), then the corresponding pseudodifferential operator is bounded onL2(Rd) and, more generally, on the modulation spacesMp,p (Rd) for 1≤p≤∞. If σ lies in the modulation spaceM2,2s(R2d)=Ls/2(R2d)∩Hs(R2d), i.e., the intersection of a weightedL2-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Kohn-Nirenberg correspondences. Using recent embedding theorems of Lipschitz and Fourier spaces into modulation spaces, we show that these results improve on the classical Calderòn-Vaillancourt boundedness theorem and on Daubechies' trace-class results.