Abstract
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 onR 2d, 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 (R 2d), then the corresponding pseudodifferential operator is bounded onL 2(R d) and, more generally, on the modulation spacesM p,p (R d) for 1≤p≤∞. If σ lies in the modulation spaceM s2,2 (R 2d)=L /2 s (R 2d)∩H s(R 2d), i.e., the intersection of a weightedL 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 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.
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Gröchenig, K., Heil, C. Modulation spaces and pseudodifferential operators. Integr equ oper theory 34, 439–457 (1999). https://doi.org/10.1007/BF01272884
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DOI: https://doi.org/10.1007/BF01272884