Self-improving behaviour of inner functions as multipliers
type: Item
instance of: scholarly article
title: Self-improving behaviour of inner functions as multipliers
zbMATH Open document ID: 1116.46018
published in: Journal of Functional Analysis
publication date: 2007-01-09T00:00:00Z
review text: A bounded analytic function \(\theta\) on the unit disk \(\mathbb D=\{z\in\mathbb{C}:| z| <1\}\) is said to be inner if \(\lim_{r\to1^{-}}| \theta(r\zeta)| =1\) for \(m\)-almost all \(\zeta\in\mathbb T\), where \(\mathbb T=\partial\mathbb{D}\) is the unit circle and \(m\) is the normalized arclength measure on \(\mathbb T\). Let \(X\) and \(Y\) be two classes of analytic functions on \(\mathbb{D}\) with \(X\subset Y\). Then \(\theta\) is called \((X,Y)\)-improving if every function \(f\in X\) satisfying \(f\theta\in Y\) must actually satisfy \(f\theta\in X\), that is, the implication \[ (f\in X,f\theta\in Y)\Longrightarrow f\theta\in X \] holds. Let \(\mathcal{A}\) stand for the disk algebra, i.e., the set of analytic functions on \(\mathbb D\) that admit a continuous extension to \(\mathbb D\cup\mathbb T\) and let the Lipschitz space consisting of those \(f\) which satisfy \[ | f(z_1)-f(z_2)| \leq C| z_1-z_2| ^{\alpha},\qquad 0<\alpha\leq 1,\quad \text{\(z_1\), \(z_2\in\mathbb D\)}, \] be denoted by \(A^{\alpha}\). Further, BMOA (respectively, VMOA) is the analytic subspace of \(\text{BMO}(\mathbb T)\) \(\bigl(\text{respectively, VMO(}\mathbb T\))
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5083456
zbMATH Keywords: inner functions
zbMATH Keywords: Lipschitz spaces
zbMATH Keywords: Bloch space
author: Konstantin M. Dyakonov
reviewed by: Rauno Aulaskari
MaRDI profile type: Publication
full work available at URL: j.jfa.2006.03.020
OpenAlex ID: W2061432055
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DOI: 10.1016/J.JFA.2006.03.020
description: scientific article; zbMATH DE number 5083456