\(L^ 2\) estimates and existence theorems for the tangential Cauchy-Riemann complex
type: Item
instance of: scholarly article
title: \(L^ 2\) estimates and existence theorems for the tangential Cauchy-Riemann complex
zbMATH Open document ID: 0581.35057
DOI: 10.1007/BF01394783
published in: Inventiones Mathematicae
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/143281
review text: The author proves that if \(\alpha\) is a \({\bar \partial}\)-closed (p,q) form on the boundary \(\partial \Omega\) of a smoothly bounded pseudoconvex domain \(\Omega \subseteq {\mathbb{C}}^ n\), \(0<q<n-2\), then there is a (p,q- 1) form u on \(\partial \Omega\) such that \({\bar \partial}_ bu=\alpha\) and u satisfies Sobolev estimates. The starting point for this work is an observation of Rosay that this result with \(C^{\infty}\) estimates follows from earlier work of Kohn and Kohn/Rossi. The author uses these ideas, together with a two-sided extension theorem for forms \(\alpha\), to obtain the more precise Sobolev estimates. These results and techniques should prove useful in later work.
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 3929567
zbMATH Keywords: tangential Cauchy-Riemann equations
zbMATH Keywords: \({\bar \partial }\)-closed (p,q) form
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: Sobolev estimates
zbMATH Keywords: two-sided extension theorem
author: Mei-Chi Shaw
reviewed by: Steven G. Krantz
MaRDI profile type: Publication
cites work: Q5670927
cites work: Q4111841
cites work: The Neumann Problem for the Cauchy-Riemann Complex. (AM-75)
cites work: Estimates for the \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \partial \limits^ - _b $\end{document} complex and analysis on the heisenberg group
cites work: The Identity of Weak and Strong Extensions of Differential Operators
cites work: Fundamental solutions in complex analysis. I. The Cauchy Riemann operator
cites work: Q4130476
cites work: Q3843833
cites work: \(L^ 2\) estimates and existence theorems for the \(\partial\)-operator
cites work: Q5512342
cites work: Q5552354
cites work: Global Regularity for $\overline \partial$ on Weakly Pseudo-Convex Manifolds
cites work: Non‐coercive boundary value problems
cites work: On the extension of holomorphic functions from the boundary of a complex manifold
cites work: Strongly pseudoconvex CR structures over small balls. I: An a priori estimate
cites work: Equation de Lewy-résolubilité globale de l'équation \(\partial_ bu=f\) sur la frontière de domaines faiblement pseudo-convexes de \(C^ 2\) (ou \(C^ n)\)
cites work: A simplification of Rosay's theorem on global solvability of tangential Cauchy-Riemann equations
cites work: Global Solvability and Regularity for $\overline\partial$ on an Annulus Between Two Weakly Pseudo-Convex Domains
description: scientific article; zbMATH DE number 3929567