On the extension of separately hyperharmonic functions and \(H^ p\)- functions
type: Item
instance of: scholarly article
title: On the extension of separately hyperharmonic functions and \(H^ p\)- functions
zbMATH Open document ID: 0585.32008
author: Juhani Riihentaus
published in: Michigan Mathematical Journal
publication date: 1984-01-01T00:00:00Z
review text: [Shortened version of the original review.] Given the definition of an n-hyperharmonic function and an n-small set in \({\mathbb{C}}^ n\) the author proves the following theorems: Theorem 1: Let G be an open set in \({\mathbb{C}}^ n\). Let \(E\subset G\) be closed in G and n-small. Let \(u: G\setminus E\to (-\infty,\infty]\) be n- hyperharmonic. If u is locally bounded below in G, then u has a unique n- hyperharmonic extension \(u^*: G\to (-\infty,\infty].\) Theorem 2. Let G be an open set in \({\mathbb{C}}^ n\). Let \(E\subset G\) be closed in G and n-small. Let \(f: G\setminus E\to {\mathbb{C}}\) be a holomorphic function such that for some \(p>0\), \(| f|^ p\) has an n-hyperharmonic majorant u in \(G\setminus E\). If u is superharmonic, then f has a unique holomorphic extension \(f^*: G\to {\mathbb{C}}.\) Theorem 3. Let \(E\subset U^ n\) be closed in \(U^ n\) and n-small. Let \(f: U^ n\setminus E\to {\mathbb{C}}\) be a holomorphic function such that for some \(p>0\), \(| f|^ p\) has an n-harmonic majorant u in \(U^ n\setminus E\). Then f has a unique holomorphic extension \(f^*: U^ n\to {\mathbb{C}}\) such that \(| f^*|^ p\) has an n-harmonic majorant in \(U^ n.\) Theorem 3 includes the results of \textit{P. Järvi} [Proc. Am. Math. Soc. 86, 596-598 (1982; Zbl 0532.32004)], and the author in J. Lond. Math. Soc., II. Ser. 27, 281-288 (1983; Zbl 0479.30007)]. Several further theorems and corollaries are proved.
Mathematics Subject Classification ID: 32D15
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A07
zbMATH DE Number: 3937523
zbMATH Keywords: separately hyperharmonic function
zbMATH Keywords: polar sets
zbMATH Keywords: extension of \(H^ p\)- functions
zbMATH Keywords: extension of n-hyperharmonic
zbMATH Keywords: functions
zbMATH Keywords: analytic continuation
zbMATH Keywords: n-small set in \({\mathbb{C}}^ n\)
zbMATH Keywords: n-harmonic majorant
reviewed by: Vikramaditya Singh
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029002968
OpenAlex ID: W2088952779
DOI: 10.1307/MMJ/1029002968
description: scientific article; zbMATH DE number 3937523