\(C^{k}\)-estimates for the \(\bar\partial\)-equation on convex domains of finite type
type: Item
instance of: scholarly article
title: \(C^{k}\)-estimates for the \(\bar\partial\)-equation on convex domains of finite type
zbMATH Open document ID: 1114.32005
author: William Alexandre
published in: Mathematische Zeitschrift
publication date: 2006-03-21T00:00:00Z
full work available at URL: http://www.numdam.org/item
review text: Let \(\Omega\) be a bounded convex domain in \({\mathbb C}^n\) with smooth boundary of finite type \(m\). Let \(C^0_{0,q}(\bar\Omega)\) be the Banach space of \((0,q)\)-forms with continuous coefficients on \(\bar\Omega\). Here \(1\leq q\leq n-1\). The author constructs a bounded linear operator \[ T_q: C^0_{0,q}(\bar\Omega)\,\rightarrow\, C^{\frac{1}{m}}_{0,q-1}(\bar\Omega) \] such that \(\bar\partial \,T_q(f)=f\) for all \(f\in C^0_{0,q}(\bar\Omega)\) with \(\bar\partial f=0\). The author also obtains estimates in Sobolev spaces. More precisely, there exists a linear operator \[ T_q^\ast:\, C_{0,q}(\bar\Omega)\,\rightarrow\, C_{0,q-1}(\bar\Omega) \] such that for all \(k\in {\mathbb N}\) and all \(\bar\partial\)-closed \(f\in C^k_{0,q}(\bar\Omega)\), one has \(\bar\partial \,T_q^\ast(f)=f\) and \[ \| T_q^\ast(f)\| _{\bar\Omega, k+\frac{1}{m}}\leq c_k\| f\| _{\bar\Omega, k}. \]
Mathematics Subject Classification ID: 32F18
Mathematics Subject Classification ID: 32F32
zbMATH DE Number: 5013682
zbMATH Keywords: Cauchy-Riemann equation
zbMATH Keywords: convex domain
zbMATH Keywords: finite type
zbMATH Keywords: \(C^k\)-estimates
reviewed by: Der-Chen Chang
MaRDI profile type: Publication
OpenAlex ID: W2149450333
cites work: Zero varieties for the Nevanlinna class in convex domains of finite type in \(\mathbb{C}^n\)
cites work: Support functions for convex domains of finite type
cites work: Hölder estimates on convex domains of finite type
cites work: Zero varieties for the Nevanlinna class on all convex domains of finite type
cites work: \(L^p\) estimates on convex domains of finite type
cites work: Lösungsoperatoren für den Cauchy-Riemann-Komplex mit \(\mathcal C^K\)- Abschätzungen
cites work: Q1821916
cites work: Convex domains of finite type
cites work: Estimates on the Bergman kernels of convex domains
cites work: Q3719028
cites work: Extension of C ∞ Functions Defined in a Half Space
DOI: 10.1007/S00209-005-0812-Y
description: scientific article; zbMATH DE number 5013682