Journal für die reine und angewandte Mathematik
Managing Editor: Weissauer, Rainer
Ed. by Colding, Tobias / Cuntz, Joachim / Huybrechts, Daniel / Hwang, Jun-Muk
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Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class
Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2007, Issue 613, Pages 121–146, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/CRELLE.2007.094, February 2008
- Published Online:
We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjöstrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since “hard analysis” techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjöstrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than “hard analysis” and are proved in their natural context and generality.
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