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Modulation spaces and pseudodifferential operators
Authors:Karlheinz Gröchenig  Christopher Heil
Institution:(1) Department of Mathematics U-9, University of Connecticut, 06269-3009 Storrs, Connecticut, USA;(2) School of Mathematics, Georgia Institute of Technology, 30332-0160 Atlanta, Georgia, USA
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 sgr lies in the modulation spaceM infin,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 1lepleinfin. If sgr lies in the modulation spaceM 2,2 s (R 2d )=L s /2 (R 2d )capH 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.
Keywords:Primary 35S05  47G30  Secondary 42C15  47B10
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