Inner functions in \(W_{\alpha}\) as improving multipliers
type: Item
instance of: scholarly article
title: Inner functions in \(W_{\alpha}\) as improving multipliers
zbMATH Open document ID: 1451.30109
author: D. Kharzeev
published in: Annals of Functional Analysis
publication date: 2020-07-17T00:00:00Z
review text: Let \(\mathbb{D}\) denote the open unit disk of the complex plane. For \(\alpha >-1\), the Dirichlet space \(\mathcal{D}_\alpha\) consists of all analytic functions \(f\) on the open unit disk for which \[\Vert f\Vert ^2_{\mathcal{D}_\alpha}=|f(0)|^2+\int_{\mathbb{D}} |f^\prime (z)|^2 dA_\alpha (z)<\infty,\] where \(dA_\alpha (z)=\pi^{-1} (\alpha +1)(1-|z|^2)^\alpha dxdy\). Similarly, the space \(\mathcal{W}_\alpha\) is defined as the space of all analytic functions \(f\) on \(\mathbb{D}\) such that \[\Vert f\Vert ^2_{\mathcal{W}_\alpha}=\sup_{\| g\|_{\mathcal{D}_\alpha \le 1}} \left (\int_{\mathbb{D}}|g(z)|^2 |f^\prime (z)|^2 dA_\alpha (z)\right )^{1/2}<\infty.\] These spaces are known to satisfy \(\mathcal{W}_\alpha\subseteq \mathcal{D}_\alpha\). Let \(X\) and \(Y\) be two spaces of analytic functions on the unit disk such that \(X\subseteq Y\). An inner function \(\theta\) (a bounded analytic function on the unit disk whose boundary function has modulus \(1\) almost everywhere on the unit circle) is said to be \((X,Y)\)-improving if for each \(f\in X\) with \(\theta f\in Y\), we have \(\theta f\in X\). The main result of the paper under review states that for \(0<\alpha <1\),\, \(g\in \mathcal{D}_\alpha\), and an inner function \(\theta\), the following are equivalent: 1) \(\theta \in \mathcal{W}_\alpha\). 2) \(\theta\) is a Blaschke product with sequence of zeros \(\{z_j\}\) satisfying \[\sup_{\| g\|_{\mathcal{D}_\alpha \le 1}}\sum_{j=1}^\infty |g(z_j)|^2(1-|z_j|^2)^\alpha <\infty.\] 3) \(\theta\) is \((\mathcal{W}_\alpha, \mathrm{BMOA})\)-improving, where \(\mathrm{BMOA}\) is the space of analytic functions of bounded mean oscillation. 4) \(\theta\) is \((\mathcal{W}_\alpha, \mathcal{B})\)-improving, where \(\mathcal{B}\) is the Bloch space.
reviewed by: A. Abkar
Mathematics Subject Classification ID: 30H30
Mathematics Subject Classification ID: 30H35
Mathematics Subject Classification ID: 30H99
Mathematics Subject Classification ID: 30J05
Mathematics Subject Classification ID: 30J10
zbMATH DE Number: 7222409
zbMATH Keywords: Dirichlet space
zbMATH Keywords: \(\mathcal W_{\alpha}\) space
zbMATH Keywords: inner functions
zbMATH Keywords: Blaschke products
MaRDI profile type: Publication
full work available at URL: s43034-019-00037-w
OpenAlex ID: W3012119704
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DOI: 10.1007/S43034-019-00037-W
description: scientific article; zbMATH DE number 7222409