Sufficient condition for compactness of the ∂-Neumann operator using the Levi core

2022/09/02 by John N. Treuer, Treuer, John N.
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2209.01162

openalex publication_date 2022/09/02 · openalex created_date 2022/09/06 · openalex updated_date 2026/07/28

Abstract

On a smooth, bounded pseudoconvex domain Ω in ℂⁿ, to verify that Catlin's Property (P) holds for bΩ, it suffices to check that it holds on the set of D'Angelo infinite type boundary points. In this note, we consider the support of the Levi core, S_\mathfrakC(N), a subset of the infinite type points, and show that Property (P) holds for bΩ if and only if it holds for S_\mathfrakC(N). Consequently, if Property (P) holds on S_\mathfrakC(N), then the ∂-Neumann operator N₁ is compact on Ω.

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