Mean Lipschitz spaces and bounded mean oscillation
type: Item
instance of: scholarly article
title: Mean Lipschitz spaces and bounded mean oscillation
zbMATH Open document ID: 0878.30028
author: Daniel Girela
published in: Illinois Journal of Mathematics
publication date: 1997-11-20T00:00:00Z
review text: Assume that \(f(z)\) is analytic in the unit disk and has a non-tangential limit \(f(e^{i\theta})\) at a.e. point of the unit circle. The integral modulus of continuity of order \(p\), \(1\leq p < +\infty\), of the boundary function \(f(e^{i\theta})\) is \(\omega (\delta , f) = \sup_{0<|t|\leq \delta} \bigl({1 \over {2\pi}} \int_{-\pi}^{\pi} |f(e^{i(\theta + t)}) - f(e^{i\theta})|^{p}d\theta \bigr)^{1/p}\). The essential supremum is used in the usual fashion for \(\omega_{\infty}(\delta , f)\). The mean Lipschitz space \(\Lambda_{\alpha}^{p}\) consists of those functions \(f(z)\) for which \(\omega_{p}(\delta , f) = O(\delta^{\alpha})\) as \(\delta \to 0\). For a continuous and increasing function \(\omega:[0,\pi]\to [0,\infty)\) with \(\omega(0)=0\), the generalized mean Lipschitz space \(\Lambda(p,\omega)\) consists of those functions \(f(z)\) for which \(\omega_{p}(\delta , f) = O(\omega(\delta))\) as \(\delta \to 0\). For a continuous and increasing function \(\omega:[o,\pi]\to [0,\infty]\) with \(\omega(0)=0\) the generalized mean Lipschitz space \(\Lambda(p,\omega)\) consists of those functions \(f(z)\) for which \(\omega_p(\delta,f)=O(\omega(\delta))\) as \(\delta\to 0\). In Theorem 1 conditions on \(\omega(\delta)\) are given under which the class \(\Lambda(p,\omega)\), \(1<p<\infty\), contains a function \(f(z)\) which is not a Bloch function. Further, in Theorem 2 conditions on \(\omega(\delta)\) are given under which the class \(\Lambda(p,\omega)\), \(1<p<\infty\), contains a function \(f(z)\) which is not a normal function. Since the class \(BMOA\) lies in the class of Bloch functions which in turn lies in the class of normal functions and \(\Lambda(p,\omega) \not\subset BMOA\) for \(\omega\) in Theorem 1, this demonstrates the sharpness of the known inclusion \(\Lambda_{1/p}^{p} \subset BMOA\) due to \textit{P. Bourdon}, \textit{J. Shapiro} and \textit{W. Sledd} [Lond. Math. Soc. Lect. Note Ser. 137, 81-110 (1989; Zbl 0672.42003)].
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 30D45
zbMATH DE Number: 1029012
zbMATH Keywords: Mean Lipschitz space
zbMATH Keywords: BMOA
zbMATH Keywords: Bloch function
zbMATH Keywords: Normal function
reviewed by: Stephen Dragosh
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1029012