Pseudodifferential operators on locally compact abelian groups and Sjotrand’s symbol class, Journal für die reine und angewandte Mathematik

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we intro-duce 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 analy-sis rather than “hard analysis ” and are proved in their natural context and generality.