Boundary behavior of the pluricomplex Green function
type: Item
instance of: scholarly article
title: Boundary behavior of the pluricomplex Green function
zbMATH Open document ID: 1021.32015
DOI: 10.1007/BF02384773
author: Dan Coman
published in: Arkiv för Matematik
publication date: 2003-06-16T00:00:00Z
review text: The paper studies the behaviour of the pluricomplex Green function \(g_\Omega(z,w)\) of a bounded domain \(\Omega\subset{\mathbf{C}}^n\) when the pole \(w\) tends to \(w_0\in\partial\Omega\). The main result is as follows. Let \(\Omega\) have a plurisubharmonic peak function \(\rho\) at \(w_0\) (i.e., \(\rho\in PSH(\Omega)\cap C(\overline\Omega)\), \(\rho(w_0)=0\) and \(\rho(z)<0\) for all \(z\in \overline\Omega\setminus\{w_0\}\)) that is Hölder continuous at \(w_0\) and such that \[ \max\{{\log|\rho(z)|\over\log|z-w_0|}: z\in\overline\Omega, \;r\leq|z-w_0|\leq{1\over 2}\} = O(\log \log {1\over r}) \] as \(r\to 0\). Then for any compact set \(K\subset \overline\Omega\setminus\{w_0\}\), \[ \lim_{w\to w_0} g_\Omega(z,w)=0 \] uniformly in \(z\in K\). In particular, this is true when \(\Omega\) is pseudoconvex with smooth boundary and \(w_0\) is a point of finite type.
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 31C10
zbMATH DE Number: 1928883
MaRDI profile type: Publication
cites work: Q4730054
cites work: A lower bound on the Kobayashi metric near a point of finite type in \(\mathbb{C} ^ n\)
cites work: Mesures de Monge-Ampère et mesures pluriharmoniques. (Monge-Ampère measures and pluriharmonic measures)
cites work: Extremal plurisubharmonic functions and invariant pseudodistances
full work available at URL: https://doi.org/10.1007/bf02384773
OpenAlex ID: W2059550418
description: scientific article; zbMATH DE number 1928883