On the radial cluster sets for holomorphic functions in the unit ball of \({\mathbb{C}}^ n\)
type: Item
instance of: scholarly article
title: On the radial cluster sets for holomorphic functions in the unit ball of \({\mathbb{C}}^ n\)
zbMATH Open document ID: 0662.32005
DOI: 10.1007/BF01215254
author: Andrei Iordan
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174085
review text: Let \(\omega(r)\) be an increasing function defined for \(0\leq r<1\) such that \(\lim_{r\to 1}\omega(r)=\infty\). In a paper of J. P. Kahane and Y. Katznelson and in a paper of M. Ortel and W. Schneider, there is proved that for every measurable function \(\phi\) on the boundary of the unit disc D, there exists a holomorphic function \(f\) in \(D\) such that \(| f(z)| \leq \omega (| z|)\) for every \(z\in D\) and \(\lim_{r\to 1}f(r\zeta)=\phi (\zeta)\) almost everywhere on \(\partial D.\) In this paper we prove that for every measurable function \(\phi\) on the boundary of the unit ball \(B\) of \({\mathbb{C}}^ n\) there exist an increasing sequence \(\{r_ j\}\), \(\lim_{j\to \infty} r_ j=1\) and a holomorphic function \(f\) in \(B\) such that \(| f(z)| \leq \omega (| z|)\) for every \(z\in B\) and \(\lim_{n\to \infty} f_{r_ n}(\zeta)=\phi (\zeta)\) almost everywhere on \(\partial B\). As a consequence we obtain that for every measurable function \(\phi\) on \(\partial B\) there exists a holomorphic function \(f\) in \(B\), \(f\in \cap_{1\leq p<\infty}L^ p(B)\) such that \(\phi\) (\(\zeta)\) belongs to the cluster set of \(f\) almost everywhere on \(\partial B\).
Mathematics Subject Classification ID: 32A40
Mathematics Subject Classification ID: 30E25
zbMATH DE Number: 4081947
zbMATH Keywords: boundary behaviour
zbMATH Keywords: radial cluster sets for holomorphic functions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4081947