A stochastic approach to the Shilov boundary
type: Item
instance of: scholarly article
title: A stochastic approach to the Shilov boundary
zbMATH Open document ID: 0634.32012
DOI: 10.1016/0022-1236(87)90032-2
author: Hiroshi Kaneko
author: Setsuo Taniguchi
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
review text: Soit D un domaine borné pseudoconvexe dans \({\mathbb{C}}^ n\), sur lequel existent N fonctions \(p_ i<0\) plurisousharmoniques minorées telles que \(q=-\sum_{i}\ln | p_ i|\) ait pour limite \(+\infty\) en tout point \(\in \partial D\) et que, sur tout D'\(\subset \subset D\), \(dd^ cq\geq C' dd^ c| z|^ 2\) pour un nombre C' dépendant de D'. Etant donné sur D un courant positif fermé \(\theta\) de bidegré \((n-1,n-1),\) on note \(\Gamma\) l'ensemble des points \(\xi\in \partial D\) pour lesquels existent \(\epsilon >0\), w plurisousharmonique bornée sur D et U voisinage de \(\xi\) tels que \(dd^ cw\wedge \theta \geq \epsilon dd^ cq\wedge \theta\) sur \(D\cap U:\Gamma\) est une partie ouverte de \(\partial D\) disjointe à la frontière de Shilov, et l'on indique quelques cas où \(\Gamma\) en est complémentaire dans \(\partial D\). Prenant \(\theta =(dd^ cq)^{n-1}\), on montre le principe du minimum suivant: si u et v plurisousharmonique bornées sur D vérifient \((dd^ cu)^ n\leq (dd^ cv)^ n\) sur D, \(\liminf (u-v)\geq 0\) en tout point \(\in \partial D\setminus \Gamma\), enfin \(dd^ c(u+v)\leq Ce^{-\alpha q}dd^ cq\) sur \(D\cap U\) pour un \(\alpha >0\), un nombre C et un ouvert \(U\supset \Gamma\), alors \(u\geq v\) sur D. Soit d'autre part \(\phi\) holomorphe \(D\to {\mathbb{C}}\), \(\phi\) (D) ouvert de Green, \(\phi\) application propre \(D\to \phi (D):\) presque tout (pour la mesure harmonique de \(\phi\) (D)) point-frontière y de \(\phi\) (D) est valeur asymptotique de \(\phi\) en un point \(\in \partial \setminus \Gamma\), dépendant de y. Les démonstrations utilisent les martingales conformes.
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 60G46
Mathematics Subject Classification ID: 31C10
Mathematics Subject Classification ID: 46J20
zbMATH DE Number: 4032077
zbMATH Keywords: Shilov boundary
zbMATH Keywords: minimum principle
zbMATH Keywords: pseudoconvex domain
zbMATH Keywords: plurisubharmonic functions
zbMATH Keywords: conformal martingales
reviewed by: Rose-Marie Hervé
MaRDI profile type: Publication
full work available at URL: 0022-1236(87)90032-2
OpenAlex ID: W2056353427
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description: scientific article; zbMATH DE number 4032077