On Hölder and BMO estimates for \(\overline \partial\) on convex domains in \(\mathbb{C}^ 2\)
type: Item
instance of: scholarly article
title: On Hölder and BMO estimates for \(\overline \partial\) on convex domains in \(\mathbb{C}^ 2\)
zbMATH Open document ID: 0768.32013
DOI: 10.1007/BF02921578
published in: The Journal of Geometric Analysis
publication date: 1993-02-10T00:00:00Z
review text: Let us consider the equation \(\overline\partial u=f\), where \(f=\sum^ n_{j=1} f_ jd\bar z_ j\) is a \(\overline\partial\)-closed (0,1)-form on a bounded pseudoconvex domain \(D\subset\mathbb{C}^ n\). We denote by \(\Lambda_ \alpha(D)\) the classical Lipschitz space and \(\text{BMO}(D)\) the space of functions of bounded mean oscillation on \(D\). The author proves: Let \(D\subset\mathbb{C}^ 2\) be convex with \(C^ \infty\) boundary. Then there is an integral solution operator \(T: C_{0,1}(\overline D)\to C(D)\) for \(\overline\partial\), such that: (i) \(| Tf|_{\Lambda_ \alpha(D)}\leq C_ \alpha| f|_{\Lambda_ \alpha(D)}\) for all \(f\) with \(\overline\partial f=0\) and all \(\alpha>0\); (ii) \(\| Tf\|_{\text{BMO}(D)}\leq C\| f\|_{L^ \infty}\).
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 32A37
zbMATH DE Number: 118437
zbMATH Keywords: Cauchy-Riemann equations
zbMATH Keywords: Hölder regularity
zbMATH Keywords: bounded pseudoconvex domain
zbMATH Keywords: Lipschitz space
zbMATH Keywords: space of functions of bounded mean oscillation
zbMATH Keywords: integral solution operator
author: R. Michael Range
reviewed by: A. Klíč
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02921578
OpenAlex ID: W2327511081
description: scientific article; zbMATH DE number 118437