On the weighted ∂-Neumann problem on unbounded domains

2009/12/04 by Klaus Gansberger, Gansberger, Klaus
Mathematics · #32W05 #46E35 #Algebraic and Geometric Analysis #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.0912.0841

openalex publication_date 2009/12/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Ω be an unbounded, pseudoconvex domain in \Bbb Cⁿ and let φ be a \mathcal C²-weight function plurisubharmonic on Ω. We show both necessary and sufficient conditions for existence and compactness of a weighted ∂-Neumann operator N_φ on the space L²_(0,1)(Ω,e^-φ) in terms of the eigenvalues of the complex Hessian (∂ ²φ/∂ z_j∂ z_k)_j,k of the weight. We also give some applications to the unweighted ∂-Neumann problem on unbounded domains.

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