De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the \(\bar \partial\)-Neumann problem
type: Item
instance of: scholarly article
title: De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the \(\bar \partial\)-Neumann problem
zbMATH Open document ID: 0792.32013
DOI: 10.1007/BF02921391
author: Harold P. Boas
author: Emil J. Straube
published in: The Journal of Geometric Analysis
publication date: 1994-07-28T00:00:00Z
review text: Main result of the paper is the following theorem: Let \(\Omega\) be a smooth pseudoconvex domain in \(\mathbb{C}^ n\). Suppose there is a smooth real submanifold \(M\) (with or without boundary of \(b\Omega\) that contains all the points of infinite type of \(b \Omega\) and whose real tangent space at each point is contained in the null space of the Levi form at the point (under the usual identification of \(\mathbb{R}^ n\) with \(\mathbb{C}^ n)\). If the \(H^ 1(M)\) de Rham cohomology class \([\alpha |_ M]\) is zero, then the \(\overline \partial\)-Neumann operators \(N_ q\) and the Bergman projections \(P_ q\) are continuous on the Sobolev space \(W^ s_{(0,q)} (\Omega)\) when \(0 \leq q \leq n\) and \(s \geq 0\). Here \(N_ q\) denotes the inverse of the complex Laplacian \(\overline \partial^*\overline \partial+\overline {\partial \partial}^*\) on \((0,q)\)-forms, and \(P_ q\) is the Bergman orthogonal projection from the space of square-integrable \((0,q)\)-forms onto the subspace of \(\overline \partial\)-closed \((0,q)\)-forms.
reviewed by: Józef Siciak
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 58A12
Mathematics Subject Classification ID: 46E35
zbMATH DE Number: 425631
zbMATH Keywords: \(\overline \partial\)-Neumann problem
zbMATH Keywords: de Rham cohomology
zbMATH Keywords: Sobolev space
Wikidata QID: Q57376097
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921391
OpenAlex ID: W2323641579
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description: scientific article; zbMATH DE number 425631