The Donnelly-Fefferman theorem on \(q\)-pseudoconvex domains
type: Item
instance of: scholarly article
title: The Donnelly-Fefferman theorem on \(q\)-pseudoconvex domains
zbMATH Open document ID: 1214.32015
author: Nguyen Quang Dieu
author: Heungju Ahn
published in: Osaka Journal of Mathematics
publication date: 2009-12-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.ojm/1256564197
review text: The authors extend the notion of a \(q\)-subharmonic function introduced in the \(C^{2}\) class by \textit{L.-H. Ho} [Math. Ann. 290, No.~1, 3--18 (1991; Zbl 0714.32006)] to the class of upper semicontinuous functions, and use these functions to obtain \(L^{2}\)-estimations of Donnelly-Fefferman type for the \(\overline{\partial}\)-equations on certain domains with non-smooth boundaries. More precisely, let \(\varphi\) be an upper semicontinuous function on an open set in \(\mathbb{C}^{n}\); it is called \(q\)-subharmonic in \(U\) if, for every linear complex space \(H\) of dimension \(q\), and for every compact set \(K\subset U\cap H\), it has the following property: if \(h\) is a continuous harmonic function on \(K\) and \(h\leq\varphi\) on \(\partial K\), then \(h\leq\varphi\) on \(K\). Plurisubharmonicity is equivalent to 1-subharmonicity; an open set \(D\) on \(\mathbb{C}^{n}\) is called \(q\)-pseudoconvex if there is a \(q\)-subharmonic exhaustion function for \(D\). The main result is the following: Let \(D\) be a \(q\)-pseudoconvex domain in \(\mathbb{C}^{n}\), let \(\varphi\) be a given \(q\)-subharmonic function in \(D\), and let \(\psi\in C^{2}(D)\) be strictly plurisubharmonic such that \(-e^{-\psi}\) is \(q\)-subharmonic. Further, let \(\varepsilon\in(0,1)\). Then for every \(\overline{\partial}\)-closed \((0,r)\)-form \(g\), with \(q\leq r\leq n\), there is a solution of \(\overline{\partial}u=g\) such that \[ \int_{D}|u|^{2}e^{-\varphi+\varepsilon\psi}dV\leq\frac{4}{\varepsilon(1-\varepsilon)^{2}}\frac{1}{r}\mathop{\sum{'}}\limits_{| K|=r-1}\mathop{\sum}\limits_{j,k}\int_{D}\psi^{j\overline{k}}g_{jK}\overline{g}_{kK}e^{-\varphi+\varepsilon\psi}dV\tag{*} \] whenever the right hand side of \((*)\) is bounded. Here, \((\psi^{j\overline{k}})=(\psi_{\nu\overline{\mu}})^{-1}\). As a corollary, the authors prove the following approximation result: Let \(D\) be as above, \(h\) a continuous \(q\)-subharmonic function in \(D\), and \(K=\left\{ z\in D: h(z)\leq0…
reviewed by: Gheorghe Gussi
Mathematics Subject Classification ID: 32W05
Mathematics Subject Classification ID: 32F10
zbMATH DE Number: 5644228
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: integral estimates
zbMATH Keywords: approximation of forms
MaRDI profile type: Publication
cites work: The extension theorem of Ohsawa-Takegoshi and the theorem of Donnelly-Fefferman
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description: scientific article; zbMATH DE number 5644228