Maximal functions of plurisubharmonic functions
type: Item
instance of: scholarly article
title: Maximal functions of plurisubharmonic functions
zbMATH Open document ID: 0770.31007
author: Hong Oh Kim
author: Yeon Yong Park
published in: Tsukuba Journal of Mathematics
publication date: 1993-02-22T00:00:00Z
review text: Let \(B\) denote the unit ball in \(\mathbb{C}^ n\) \((n\geq 1)\) with boundary \(S\). For a function \(u:B\to\mathbb{C}\), the radial maximal function \({\mathcal M}u\) on \(S\) is defined by \[ {\mathcal M}u(\eta)=\sup\{| u(r\eta)|:0\leq r<1\}. \] For \(\alpha>1\), \(\eta\in S\), let \(D_ \alpha(\eta)=\{z:| 1-\langle z,\eta\rangle|<{\alpha\over 2}(1- | z|^ 2)\}\). The admissible maximal function \({\mathcal M}_ \alpha u\) is defined by \[ {\mathcal M}_ \alpha u(\eta)=\sup\{| u(z)|:z\in D_ \alpha(\eta)\}. \] In the paper the authors prove the following: Theorem I. For \(0<p<\infty\), there is a positive constant \(C\) depending only on \(n,p,\alpha\) such that if \(u\geq 0\) is plurisubharmonic in \(B\), then \[ \int_ S{\mathcal M}_ \alpha u(\eta)^ pd\sigma(\eta)\leq C\int_ S{\mathcal M}u(\eta)^ pd\sigma(\eta), \] where \(\sigma\) is a normalized Lebesgue measure on \(S\).
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 109326
zbMATH Keywords: radial maximal function
zbMATH Keywords: admissible maximal function
reviewed by: Manfred Stoll
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496161827
OpenAlex ID: W1576099236
DOI: 10.21099/TKBJM/1496161827
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description: scientific article; zbMATH DE number 109326