Multipliers of Möbius invariant \(Q_{s}\) spaces
type: Item
instance of: scholarly article
title: Multipliers of Möbius invariant \(Q_{s}\) spaces
zbMATH Open document ID: 1170.30016
author: Jordi Pau
author: José Ángel Peláez
published in: Mathematische Zeitschrift
publication date: 2009-03-26T00:00:00Z
review text: The space \(Q_s\), \(0\leq s<\infty\), consists of those analytic functions in the unit disc \(\mathbb D\) which satisfy \[ \sup_{a\in\mathbb D}\int_{\mathbb D}|f^{\prime}(z)|^2(1-|\varphi_a(z)|^2)^s\,dA(z)<\infty,\quad \text{where}\quad \varphi_a(z)=\frac{a-z}{1-\overline{a}z}. \] In the paper under review, the authors give a complete description of \(M(Q_s)\), the algebra of pointwise multipliers of \(Q_s\). They accomplish this by proving Xiao's conjecture: \(f\in M(Q_s)\) if and only if \(f\in H^{\infty}\) and \[ \sup_{a\in\mathbb D}\left( \log\frac{2}{1-|a|}\right)^2 \int_{\mathbb D}|f^{\prime}(z)|^2(1-|\varphi_a(z)|^2)^s\,dA(z)<\infty. \]
reviewed by: George Csordas
Mathematics Subject Classification ID: 30D45
Mathematics Subject Classification ID: 30D50
Mathematics Subject Classification ID: 30H05
zbMATH DE Number: 5533138
zbMATH Keywords: \(Q_s\) spaces
zbMATH Keywords: pointwise multipliers
zbMATH Keywords: Carleson measures
MaRDI profile type: Publication
full work available at URL: s00209-008-0338-1
OpenAlex ID: W2483775702
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DOI: 10.1007/S00209-008-0338-1
description: scientific article; zbMATH DE number 5533138