Modulation spaces and pseudodifferential operators

We use methods from time-frequency analysis to study boundedness and trace- class properties of pseudodifferential operators. As natural symbol classes , we use the modulation spaces on R-2d, which quantify the notion of the ti me-frequency content of a function or distribution. We show that if a symbo l a lies in the modulation space M-infinity,M-1(R-2d), then the correspondi ng pseudodifferential operator is bounded on L-2(R-d) and, more generally, on the modulation spaces M-p,M-p(R-d) for 1 less than or equal to p less th an or equal to infinity. If sigma lies in the modulation space M-2,2(s)(R-2 d) = L-s(2)(R-2d) boolean AND H-s(R-2d), i.e., the intersection of a weight ed L-2-space and a Sobolev space, then the corresponding operator lies in a specified Schatten class. These results hold for both the Weyl and the Koh n-Nirenberg correspondences. Using recent embedding theorems of Lipschitz a nd Fourier spaces into modulation spaces, we show that these results improv e on the classical Calderon-Vaillancourt boundedness theorem and on Daubech ies' trace-class results.