Rate of growth of \(p\)th means of invariant potentials in the unit ball of \(\mathbb{C}{}^ n\). II
type: Item
instance of: scholarly article
title: Rate of growth of \(p\)th means of invariant potentials in the unit ball of \(\mathbb{C}{}^ n\). II
zbMATH Open document ID: 0757.31005
DOI: 10.1016/0022-247X(92)90047-H
published in: Journal of Mathematical Analysis and Applications
publication date: 1992-09-27T00:00:00Z
review text: Earlier results [Part I, ibid. 143, 480-499 (1989; Zbl 0697.31008)] on the rate of growth of the means \(M_ p\) of Moebius invariant Green potentials on the unit ball \(B\) in \(\mathbb{C}^ n\) are extended. A typical result is the following: Let \(\beta\in\mathbb{R}\), \(n\leq q\leq p<\infty\) and \(f\geq 0\) such that \(\int_ B(1-| w|^ 2)^ \beta\int^ q(w)d\lambda(w)<\infty\) \((\lambda\) invariant measure on \(B\)). Let \((\beta-n)/q<\alpha<n(1-1/p)+\beta/q\) and let \(M_ p(G_ f^ \alpha,r)\) denote the \(p\)-th mean of \(G_ f^ \alpha=\int(1-| w|^ 2)^ \alpha f(w)G(\cdot,w)d\lambda(w)\) on \(\{z\in\mathbb{C}^ n:| z|=r\}\) \((G\) Moebius invariant Green function on \(B\)). Then \[ \lim_{r\to 1}(1-r^ 2)^{\beta/q-n/p-\alpha}M_ p(G^ \alpha_ f,r)=0. \]
Mathematics Subject Classification ID: 31C15
Mathematics Subject Classification ID: 32A22
zbMATH DE Number: 60454
zbMATH Keywords: Moebius invariant Green potentials
author: Manfred Stoll
reviewed by: Wolfhard Hansen
MaRDI profile type: Publication
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full work available at URL: 0022-247x(92)90047-h
OpenAlex ID: W4247556352
description: scientific article; zbMATH DE number 60454