\(q\)-plurisubharmonicity and \(q\)-pseudoconvexity in \(\mathbb C^n\)
type: Item
instance of: scholarly article
title: \(q\)-plurisubharmonicity and \(q\)-pseudoconvexity in \(\mathbb C^n\)
zbMATH Open document ID: 1116.32025
DOI: 10.5565/PUBLMAT_50206_05
author: Nguyen Quang Dieu
published in: Publicacions Matemàtiques
publication date: 2007-04-05T00:00:00Z
full work available at URL: https://eudml.org/doc/41594
review text: Let \(\varOmega\subset\mathbb C^n\) be open. We say that an upper semicontinuous function \(u:\varOmega\longrightarrow[-\infty,+\infty)\) is \(q\)-plurisubharmonic (\(q\)-psh, \(u\in\mathcal{PSH}_q(\varOmega)\), \(0\leq q\leq n-1\)) if for every complex linear \((q+1)\)-dimensional subspace \(L\), for every closed ball \(B\subset L\cap\varOmega\), and for every smooth plurisuperharmonic function \(g\) defined on an open neighborhood \(U\subset L\cap\varOmega\) of \(B\), if \(u\leq g\) on \(\partial B\), then \(u\leq g\) on \(B\). We say that \(u\) is \textit{strictly \(q\)-psh} on \(\varOmega\) if for every point \(z_0\in\varOmega\) there exist a neighborhood \(U\) and \(\varepsilon>0\) such that the function \(U\ni z\longmapsto u(z)-\varepsilon\|z\|^2\) is \(q\)-psh. A domain \(\varOmega\subset\mathbb C^n\) is \(q\)-pseudoconvex if there exists a neighborhood \(U\) of \(\partial\varOmega\) such that the function \(U\cap\varOmega\ni z\longmapsto -\log\operatorname{dist}(z,\partial\varOmega)\) is \(q\)-psh. A bounded domain \(\varOmega\subset\mathbb C^n\) is \(q\)-hyperconvex if it has a negative continuous \(q\)-psh exhaustion function. A bounded domain \(\varOmega\subset\mathbb C^n\) is \(B_q\)-regular if for every \(\varphi\in\mathcal C(\partial\varOmega, \mathbb R)\) there exists a \(u\in\mathcal C(\overline\varOmega)\cap\mathcal{PSH}_q(\varOmega) \cap(-\mathcal{PSH}_{n-q-1}(\varOmega))\) with \(u=\varphi\) on \(\partial\varOmega\). A compact set \(K\subset\mathbb C^n\) is \(B_q\)-regular if \(\mathcal C(K)\) coincides with the closure in \(\mathcal C(K)\) of the space of all functions that are continuous on \(K\) and \(q\)-psh in a neighborhood of \(K\). The main results of the paper are the following two theorems. (1) Let \(\varOmega\subset\mathbb C^n\) be a bounded domain. (i) If \(\varOmega\) is \(q\)-convex and \(\partial\varOmega\) is \(B_r\)-regular (\(0\leq r\leq n-q\)), then \(\varOmega\) is \(B_{q+r}\)-regular. (ii) Assume that \(\varOmega\) is \(B_q\)-…
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 32F10
zbMATH DE Number: 5140582
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 5140582