On the behavior near a torus of functions holomorphic in the ball
type: Item
instance of: scholarly article
title: On the behavior near a torus of functions holomorphic in the ball
zbMATH Open document ID: 0553.32004
author: Patrick Ahern
published in: Pacific Journal of Mathematics
publication date: 1983-01-01T00:00:00Z
review text: Let B be the open unit ball in \({\mathbb{C}}^ n\). Let \(d\mu (r,\theta)=(1- r)^{\alpha}drd\theta\), \(\alpha >-1\), in polar coordinates. Denote by \(A^ P_{\alpha}\) the space of all holomorphic function in \(L^ P(d\mu)\). The main results of the paper read as follows. Let \(\pi:{\mathbb{C}}^ n\to {\mathbb{C}}\) be given by \(\pi (z_ 1,...,z_ n)=n^{n/2}\prod^{n}_{j=1}z_ j.\) Denote by \(H^ 2(B)\) the Hardy space over B and by BMO(B) the space of all functions of bounded mean oscillation in B. (a) If \(g\in A^ P_{(n-3)/2}\), then \(g\circ \pi \in H^ p(B)\); (b) if g is a Bloch function in the open unit disc U then \(g\circ \pi \in BMO(B)\). - As a corollary of the above (b) it is shown that there is a holomorphic function in BMO(B) which has a finite radial limit at no point of \(T_ n=\{z\in {\mathbb{C}}^ n,\) \(| z_ j| =n^{-1/2},\) \(j=1,...,n\}.\)
Mathematics Subject Classification ID: 32A35
Mathematics Subject Classification ID: 32A07
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 32A30
zbMATH DE Number: 3881034
zbMATH Keywords: open unit ball
zbMATH Keywords: Hardy space
zbMATH Keywords: BMO
zbMATH Keywords: Bloch function
reviewed by: Jan Janas
MaRDI profile type: Publication
full work available at URL: pjm.1983.107.267
OpenAlex ID: W2038504401
DOI: 10.2140/PJM.1983.107.267
description: scientific article; zbMATH DE number 3881034