Extremal disks and composition operators on convex domains in \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: Extremal disks and composition operators on convex domains in \(\mathbb{C}^ n\)
zbMATH Open document ID: 0824.32006
author: Peter R. Mercer
published in: Rocky Mountain Journal of Mathematics
publication date: 1995-11-15T00:00:00Z
review text: Let \(\Omega\) be a bounded domain in \(\mathbb{C}^ n\), strongly convex with smooth boundary. Let \(k_ \Omega\) be the Kobayashi distance on \(\Omega\); let \(\varphi \in H(\Delta, \Omega)\) be an associated extremal disk \((k_ \Omega (\varphi (\lambda), \varphi (\mu)) = k_ \Delta (\lambda, \mu)\), \(\Delta\) is the unit disk in \(\mathbb{C})\) and let \(p = p_ \varphi \in H (\Omega, \Omega)\) be an associated retraction. The author proves the following proposition generalizing a theorem of Rudin: \[ | f \circ \varphi^{-1} \circ p |_{H^ (\Omega)} \asymp | f |_{H^ p_{n-2}} (\Delta),\;\forall f \in H^ p_{n-2} (\Delta). \]
Mathematics Subject Classification ID: 32H02
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 757785
zbMATH Keywords: strongly convex domain
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Kobayashi distance
reviewed by: Alexander Russakovskij
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072353
OpenAlex ID: W2015770476
cites work: Q3814016
cites work: Horospheres and iterates of holomorpic maps
cites work: Common fixed points of commuting holomorphic maps
cites work: The Lindelöf principle and the angular derivative in strongly convex domains
cites work: On Boundedness of Composition Operators on H 2 (B 2 )
cites work: Unbounded Composition Operators on H 2 (B 2 )
cites work: Extension dans des classes de Hardy de fonctions holomorphes et estimations de type ''Mesures de Carleson'' pour l'équation \(\bar\partial\)
cites work: Q5608188
cites work: Q3924489
cites work: La métrique de Kobayashi et la représentation des domaines sur la boule
cites work: Q3708310
cites work: Holomorphic invariants, normal forms, and the moduli space of convex domains
cites work: Iterates of holomorphic self-maps of the unit ball in \(C^ n\)
cites work: SPECTRA OF COMPACT COMPOSITION OPERATORS ON HP(RN )
cites work: ON THE ANALYTIC CONTINUATION OF HOLOMORPHIC MAPPINGS
cites work: Q3958740
cites work: Boundary Behavior of Holomorphic Functions of Several Complex Variables. (MN-11)
cites work: Q3949259
cites work: Géodésiques complexes et points fixes d'applications holomorphes
DOI: 10.1216/RMJM/1181072353
description: scientific article; zbMATH DE number 757785