A new characterization for Carleson measures and some applications
type: Item
instance of: scholarly article
title: A new characterization for Carleson measures and some applications
zbMATH Open document ID: 1248.47036
published in: Integral Equations and Operator Theory
publication date: 2011-11-30T00:00:00Z
review text: For \(s\in{\mathbb R}\), a non-negative measure \(\mu\) on \({\mathbb D}\) is called \(s\)-Carleson if \(\sup\limits_{I\subset\partial {\mathbb D}}\frac{\mu(S(I))}{|I|^s}<\infty\), and it is said to be compact \(s\)-Carleson if \(\lim\limits_{|I|\to 0}\frac{\mu(S(I))}{|I|^s}\), where \(S(I)\) is the Carleson square based on an arc \(I\subset\partial{\mathbb D}\). Let \(\Gamma(\zeta)=\{z\in{\mathbb D}:|\zeta-z|<2(1-|z|)\}\), where \(\zeta\in\partial{\mathbb D}\). For \(a\in{\mathbb D}\), \(\tau>0\), \(s\in{\mathbb R}\), and \(0<p\leq\infty\), put \({\mathcal A}_\mu(a,\tau,s,p)=\left\|\int_{\Gamma(\cdot)}\frac{(1-|a|^2)^\tau}{|1-\overline{a}z|^{\tau+s}}\frac{d\mu(z)}{1-|z|}\right\|_{L^p}\). Natural questions arise: whether an \(s\)-Carleson measure (resp., a compact \(s\)-Carleson measure) can be characterized by \(\sup\limits_{a\in{\mathbb D}}{\mathcal A}_\mu(a,\tau,s,p)<\infty\) (resp., by \({\mathcal A}_\mu(a,\tau,s,p)\to 0\) as \(|a|\to 1_-\))? The paper is devoted to answer these questions and to provide some applications to boundedness and compactness of Volterra type operators.
reviewed by: Oleksiy Karlovych
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 32A35
zbMATH DE Number: 5982597
zbMATH Keywords: Carleson measures
zbMATH Keywords: Hardy spaces
zbMATH Keywords: Calderón-Zygmund decomposition
author: Z. Wu
MaRDI profile type: Publication
full work available at URL: s00020-011-1892-1
OpenAlex ID: W2082907841
cites work: Q4880972
cites work: An integral operator on \(H^ p\) and Hardy's inequality.
cites work: An Interpolation Problem for Bounded Analytic Functions
cites work: Generalized area operators on Hardy spaces
cites work: Extension of a theorem of Carleson
cites work: Bounded Analytic Functions
cites work: Area operators from \(\mathcal{H}^p\) spaces to \(L^q\) spaces
cites work: Area operator on Bergman spaces
cites work: \(Q\) spaces and Morrey spaces
DOI: 10.1007/S00020-011-1892-1
description: scientific article; zbMATH DE number 5982597