A smooth pseudoconvex domain in \(\mathbb C^ 2\) for which \(L^ \infty\)- estimates for \(\overline\partial\) do not hold
type: Item
instance of: scholarly article
title: A smooth pseudoconvex domain in \(\mathbb C^ 2\) for which \(L^ \infty\)- estimates for \(\overline\partial\) do not hold
zbMATH Open document ID: 0827.32018
DOI: 10.1007/BF02559484
author: Bo Berndtsson
published in: Arkiv för Matematik
publication date: 1995-12-11T00:00:00Z
review text: It has been known for some time that on a general pseudoconvex domain (bounded and with smooth boundary) one cannot solve \(\overline \partial\) with supnorm estimates [\textit{N. Sibony}, Invent. Math. 62, 235--242 (1980; Zbl 0429.32026)]. Likewise, counterexamples to \({\mathcal L}^p\)-estimates are known [\textit{J. E. Fornaess} and \textit{N. Sibony}, Proc. Symp. Pure Math. 52, Part 3, 129--163 (1991; Zbl 0827.32019)]. These counterexamples are for \(n\geq 3\) and \(p > 2\). In this paper, the author gives an example which is a smooth, bounded, pseudoconvex Hartogs domain in \(\mathbb C^2\), on which supnorm estimates fail for \(\overline \partial\) and for \(\overline \partial_b\). Since in \(\mathbb C^2\), \(\overline \partial_b\) is essentially symmetric (with the usual identifications), this implies failure of \({\mathcal L}^1\)- estimates for \(\overline \partial_b\) as well. The failure of supnorm estimates for \(\overline \partial\) on the Hartogs domain is reduced to failure of certain weighted supnorm estimates for \(\overline \partial\) on the (one-dimensional) base of the Hartogs domain. There is a more recent survey article by the author [Aspects Math. E 26, 27--42 (1994; Zbl 0823.32007)] that contains further results (also concerning Hölder estimates). For example, there is a Hartogs domain in \(\mathbb C^2\), bounded, pseudoconvex, and with smooth boundary, so that \(\overline \partial_b\) does not have closed range in \({\mathcal L}^p\) for any \(p\neq 2\).
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 35N15
Mathematics Subject Classification ID: 32V05
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 556410
zbMATH Keywords: \(\overline \partial_ b\) operator
zbMATH Keywords: closed range
zbMATH Keywords: \(\overline \partial\)
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: Hartogs domain
reviewed by: Emil J. Straube
MaRDI profile type: Publication
cites work: Weighted estimates for \(\overline\partial\) in domains in \(\mathbb{C}\)
cites work: Integral solution operators for the Cauchy-Riemann equations on pseudoconvex domains
cites work: Hölder estimates on domains of complex dimension two and on three- dimensional CR manifolds
cites work: \(L^ 2\) estimates and existence theorems for the \(\partial\)-operator
cites work: Estimates for the Bergman and Szegö kernels in \({\mathbb{C}}^ 2\)
cites work: Integral Representation Formulas and $L^p$-Estimates for the $\bar{\partial}$-Equation.
cites work: Integral kernels and Hölder estimates for \({\bar \partial}\) on pseudoconvex domains of finite type in \({\mathbb{C}}^ 2\)
cites work: Prolongement des fonctions holomorphes bornees et métrique de Caratheodory
description: scientific article; zbMATH DE number 556410