Subelliptic estimates for the \({\bar \partial}\)-Neumann problem on pseudoconvex domains
type: Item
instance of: scholarly article
title: Subelliptic estimates for the \({\bar \partial}\)-Neumann problem on pseudoconvex domains
zbMATH Open document ID: 0627.32013
DOI: 10.2307/1971347
published in: Annals of Mathematics. Second Series
publication date: 1987-01-01T00:00:00Z
review text: This article gives a necessary and sufficient condition for the \({\bar \partial}\)-Neumann problem on a bounded pseudoconvex domain \(\Omega\) in \({\mathbb{C}}^ n\) with smooth boundary \(b\Omega =\{r=0\}\) to satisfy local subelliptic estimate at a boundary point \(z_ 0\in b\Omega\). The condition is given by \(D_ q(z_ 0)<\infty\), where q is the degree of the antiholomorphic part of the space of forms under consideration and \(D_ q(z_ 0)=\sup_{V^ q} \{\tau (V^ q,z_ 0)\}.\) Here the supremum is taken over germs \(V^ q\) of q-dimensional subvarieties passing through \(z_ 0\) and lying outside \(\Omega\), \(\tau (V^ q,z_ 0)\) denotes the maximum value of \(ord_ 0(r\circ \gamma^ k_ S)/ord_ 0(\gamma^ k_ S)\) among the one-dimensional components \(V^ q_{S,k}=\{z=\gamma^ k_ S(t):({\mathbb{C}},0)\to ({\mathbb{C}}^ n,z_ 0)\}\), \(k=1,...,P\) of the intersection \(V^ q\cap S\) by an \((n-q+1)-\) dimensional plane S in general position. This result gives a final answer to the problem initiated and studied by J. J. Kohn.
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 35J55
zbMATH DE Number: 4019444
zbMATH Keywords: \({\bar \partial }\)-Neumann problem
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: subelliptic estimate
Wikidata QID: Q56701612
author: David W. Catlin
reviewed by: Akira Kaneko
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/1971347
OpenAlex ID: W2330408022
description: scientific article; zbMATH DE number 4019444