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Abstract
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.
Received: 2006-03-16
Revised: 2006-10-11
Published Online: 2008-02-05
Published in Print: 2007-12-19
© Walter de Gruyter
