Quantitative estimates for the Green function and an application to the Bergman metric
type: Item
instance of: scholarly article
title: Quantitative estimates for the Green function and an application to the Bergman metric
zbMATH Open document ID: 0960.32022
author: K. Diederich
author: Gregor Herbort
publication date: 2000-07-19T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75454
review text: Let \(D\Subset\mathbb{C}^n\) be a bounded pseudoconvex domain. There is a bounded plurisubharmonic exhaustion function \(\rho: D\to [-1,0)\), such that for some number \(0<\alpha<1\) and constants \(C_1,C_2>0\) the inequality \[ C_2{\delta^\alpha}_D(z)\leq |\rho(z)|\leq C_1{\delta^\alpha}_D(z) \] holds everywhere on \(D\). Here \(\delta_D(\cdot)\) denotes the boundary distance in \(D\). It is proved that for any fixed number \(0<t\ll 1\) there exists a constant \(\widetilde{C}_t\gg 1\), such that for all compact subsets \(K\subset D\) and all \(w\in D\) satisfying \[ \delta_D(w)< \widetilde{C}_t^{-1}\delta_D(K) \] the pluricomplex Green function \(G_D\) of \(D\) can be estimated by \[ \sup_{K}\bigl|G_D(\cdot,w)\bigr|\quad\leq\widetilde{C}_t\left(\left(\frac{\delta_D^{1-2nt}(w)}{\delta_D(K)}\right)^{\alpha /n} \left(\log\frac{2R_D}{\delta_D(w)}\right)^{1/n}+\delta_D^{t\alpha}(w)\bigl|\log\delta_D(w)\bigr|\right) \] where \(R_D\) denotes the diameter of \(D\). The authors show two applications of the last inequality to questions about the boundary behavior of the Bergman differential metric \(B_D\) of \(D\).
Mathematics Subject Classification ID: 32U35
Mathematics Subject Classification ID: 32F45
Mathematics Subject Classification ID: 32T25
Mathematics Subject Classification ID: 32T27
zbMATH DE Number: 1478820
zbMATH Keywords: pluricomplex Green function
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: order of pseudoconvex extendability
zbMATH Keywords: Bergman metric
MaRDI profile type: Publication
OpenAlex ID: W2021121576
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DOI: 10.5802/AIF.1790
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 1478820