Pseudodifferential operators on locally compact abelian groups and Sjöstrand's symbol class

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.