Inner functions as improving multipliers
type: Item
instance of: scholarly article
title: Inner functions as improving multipliers
zbMATH Open document ID: 1160.30027
author: José Ángel Peláez
published in: Journal of Functional Analysis
publication date: 2008-11-12T00:00:00Z
review text: Let \textbf{D} be the open unit disk in the complex plane, \textbf{T} be the unit circle, and \(H(\)\textbf{D}\()\) be the space of holomorphic functions in \textbf{D}. A function \(\theta \in H(\)\textbf{D}\()\) is said to be inner if \(\theta\) is bounded and has radial limit \(\theta (\xi )\) of modulus \(1\) for almost every \(\xi \in\) \textbf{T}. The singular set \(\sigma (\theta )\) of \(\theta\) is the smallest closed set \(E \subset\) \textbf{T} such that \(\theta\) is analytic across \textbf{T}\(\setminus E\). Since a nontrivial inner function \(\theta\) is highly oscillatory near \(\sigma (\theta )\) one would expect that multiplying by \(\theta\) a function \(f \in H(\)\textbf{D}\()\) that is smooth in some sense on \textbf{T} will destroy smoothness. However, in some sense \(f \theta\) inherits the nice properties of \(f\) and even it can have an added smoothness. Inspired by this phenomenon, \textit{K.~Dyakonov} [J.~Funct.~Anal. 240, 429--444 (2006; Zbl 1116.46018)] introduced the following notion: Let \(X\) and \(Y\) be two classes of analytic functions in \textbf{D}, with \(X \subset Y\). An inner function \(\theta\) is said to be \((X,Y)\)-improving, if every function \(f \in X\) satisfying \(f \theta \in Y\) must actually satisfy \(f \theta \in X\). For \(0 < p < \infty\), the space \(Q_p\) consists of those functions \(f \in H(\)\textbf{D}\()\) such that \[ \sup_{|a| < 1} \int_{|z|<1} |f'(z)|^2 \left( \log \left|{1-\overline{a}z \over a-z}\right| \right)^p \, dA(z) < \infty , \] where \(dA(z)\) is the Lebesgue area measure. It is known that \(Q_p = B\) (the Bloch space) for all \(p > 1\), and \(Q_1 = BMOA\) (the space of those \(H^1\)-functions whose boundary values have bounded mean oscillation on \textbf{T}). For \(0 < \alpha \leq 1\), \(\Lambda^\alpha\) denotes the Lipschitz space, consisting of those functions \(f\) analytic in \textbf{D} and continuous on its closure such that \(|f(z)-f(w)| \leq C|z-w|\) \((z,w \in\) \textbf{D}\()\) for some …
Mathematics Subject Classification ID: 30H05
Mathematics Subject Classification ID: 46E15
zbMATH DE Number: 5365111
zbMATH Keywords: inner functions
zbMATH Keywords: BMOA
zbMATH Keywords: Bloch space
zbMATH Keywords: \(Q_p\)-spaces
zbMATH Keywords: Lipschitz spaces
reviewed by: L. Bernal-González
MaRDI profile type: Publication
full work available at URL: j.jfa.2008.06.010
OpenAlex ID: W1973078412
cites work: Q3050704
cites work: Q4308185
cites work: Some subclasses of BMOA and their characterization in terms of Carleson measures
cites work: On the \(H^ p\) classes of derivatives of functions orthogonal to invariant subspaces
cites work: Q5618011
cites work: Q4458484
cites work: Division and Multiplication by Inner Functions and Embedding Theorems for Star-Invariant Subspaces
cites work: Q5287259
cites work: Q4764254
cites work: Holomorphic functions and quasiconformal mappings with smooth moduli
cites work: Self-improving behaviour of inner functions as multipliers
cites work: Q4497415
cites work: Q3125886
cites work: Q3922907
cites work: Analytic functions of bounded mean oscillation
cites work: INTEGRABILITY OF THE DERIVATIVE OF A BLASCHKE PRODUCT
cites work: On inner functions with derivative in Bergman spaces
cites work: Q4490700
cites work: Derivatives of Blaschke products
cites work: Q3881402
cites work: On the zeros of functions with finite Dirichlet integral and some related function spaces
cites work: Blaschke products satisfying the Carleson-Newman conditions and ideals of the algebra \(H^{\infty}\)
cites work: Q4282702
cites work: Holomorphic \(Q\) classes
cites work: Q5489391
DOI: 10.1016/J.JFA.2008.06.010
description: scientific article; zbMATH DE number 5365111