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Pseudodifferential Operators on Locally Compact Abelian Groups and Sjoestrand's Symbol Class

Karlheinz Grochenig·Thomas Strohmer
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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 Sjoestrand'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. Sjoestrand'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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Karlheinz Grochenig, Thomas Strohmer. 2006-04-12. Pseudodifferential Operators on Locally Compact Abelian Groups and Sjoestrand's Symbol Class. https://arxiv.org/abs/math/0604294

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