Estimates of harmonic measures associated with degenerate Laplacian on strictly pseudoconvex domains
type: Item
instance of: scholarly article
title: Estimates of harmonic measures associated with degenerate Laplacian on strictly pseudoconvex domains
zbMATH Open document ID: 0706.31013
author: Hitoshi Arai
published in: Proceedings of the Japan Academy. Series A
publication date: 1990-01-01T00:00:00Z
review text: Let D be a smooth, bounded, strictly pseudoconvex domain in \({\mathbb{C}}^ n\) and suppose that D is given by \(D=\{z\in U:\lambda (z)<0\}\), where U is an open neighbourhood of the closure of D and \(\lambda\) is an infinitely differentiable, strictly plurisubharmonic function on U such that \(d\lambda\neq 0\) on the boundary \(\partial D\) of D. The author announces results on harmonic measures associated with the Laplace- Beltrami operator L of the complete Kähler metric \(-\partial {\bar \partial} \log (-\lambda)\) on D. Theorem 1 states that the L-harmonic measure \(d\omega\) relative to a fixed point z of D and the induced Euclidean measure \(d\sigma\) on \(\partial D\) are mutually absolutely continuous; moreover there is a function \(k_ z\) on \(\partial D\) such that \(k_ z,k_ z^{-1}\in L^{\infty}(d\sigma)\) and \(d\omega =k_ z d\sigma\). Theorem 2, which is obtained from Theorem 1 and some other results, concerns the existence of admissible limits of L-harmonic functions in D. Indications of proofs are given; details are to appear elsewhere.
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 31C05
Mathematics Subject Classification ID: 32T99
zbMATH DE Number: 4159068
zbMATH Keywords: harmonic measures
zbMATH Keywords: Laplace-Beltrami operator
zbMATH Keywords: Kähler metric
zbMATH Keywords: absolutely continuous
MaRDI profile type: Publication
full work available at URL: pjaa.66.13
OpenAlex ID: W2060228895
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DOI: 10.3792/PJAA.66.13
description: scientific article; zbMATH DE number 4159068