Generalized Fredholm properties for invariant pseudodifferential operators

2011/01/24 by Joe J. Perez, Perez, Joe J.
Mathematics · #35S05 #58J40 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1101.4614

openalex publication_date 2011/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We define classes of pseudodifferential operators on G-bundles with compact base and give a generalized L² Fredholm theory for invariant operators in these classes in terms of von Neumann's G-dimension. We combine this formalism with a generalized Paley-Wiener theorem, valid for bundles with unimodular structure groups, to provide solvability criteria for invariant operators. This formalism also gives a basis for a G-index for these operators. We also define and describe a transversal dimension and its corresponding Fredholm theory in terms of anisotropic Sobolev estimates, valid also for similar bundles with nonunimodular structure group.

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