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

2006/04/12 by Karlheinz Gröchenig, Thomas Strohmer, Grochenig, Karlheinz +1
Computer Science · Earth and Planetary Sciences · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Mathematical Analysis and Transform Methods #Underwater Acoustics Research

paper · pdf · doi:10.48550/arxiv.math/0604294

openalex publication_date 2006/04/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Pseudodifferential operators and Banach algebras in mobile communications
• Projective Modules over Higher-Dimensional Non-Commutative Tori

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