The union problem on complex manifolds
type: Item
instance of: scholarly article
title: The union problem on complex manifolds
zbMATH Open document ID: 1010.32010
DOI: 10.1155/S0161171202007603
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2002-08-12T00:00:00Z
full work available at URL: https://eudml.org/doc/49939
full work available at URL: S0161171202007603.html
review text: Let \(X\) be an \(n\)-dimensional complex manifold with a \(C^\infty\) Hermitian metric. Let \(\Omega_1\Subset \Omega_2\Subset\cdots\) be an increasing sequence of Stein open sets in \(X\) such that their union \(\Omega= \bigcup^\infty_{j=1} \Omega_j\) is relatively compact in \(X\). The following statement is the main result of the paper. Theorem 1. The union \(\Omega\) is Stein if and only if given an \(f\in L^2_{(p,q)}(\Omega)\) \(\overline\partial\)-closed, there is a \(u\in L^2_{(p,q-1)}(\Omega)\) with \(\overline\partial u= f\) and such that \[ \|u\|_{L^2(p,q-1)(\Omega)}\leq K\|f\|_{L^2(p,q)(\Omega)}, \] where \(K\) depends only on \(\Omega\), but not on \(f\).
Mathematics Subject Classification ID: 32E10
Mathematics Subject Classification ID: 32C35
Mathematics Subject Classification ID: 35N15
zbMATH DE Number: 1779145
zbMATH Keywords: \(L^2\)-estimates
zbMATH Keywords: union of Stein open sets
author: Patrick W. Darko
reviewed by: S. Ivashkovich
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1779145