Modulation spaces and pseudodifferential operators |
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Authors: | Karlheinz Gröchenig Christopher Heil |
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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 |
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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 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 1p. If lies in the modulation spaceM
2,2
s
(R
2d
)=L
s
/2
(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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Keywords: | Primary 35S05 47G30 Secondary 42C15 47B10 |
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