An integral operator on \(H^ p\) and Hardy's inequality.
type: Item
instance of: scholarly article
title: An integral operator on \(H^ p\) and Hardy's inequality.
zbMATH Open document ID: 1061.30025
DOI: 10.1007/BF02788078
author: Alexandru Aleman
author: Joseph A. Cima
published in: Journal d'Analyse Mathématique
publication date: 2002-07-10T00:00:00Z
review text: The authors study the properties of the mapping which to a function \(f\) analytic in the unit disk associates the function given by \[ T_g f(z)= \int_0^z f(\zeta) g'(\zeta) d\zeta , \] where \(g\) is a given analytic function (the symbol of the operator). In the case where \(g\) is the identity, this operator just amounts to taking the primitive of \(f\), and Hardy and Littlewood had already shown that this sends the usual Hardy space \(H^p\) to \(H^{p/(1-p)}\), for \(0<p<1\). Other special cases of the operator \(T_g\) have been studied since, see \textit{A. G. Siskakis}, Proc. Am. Math. Soc. 110, 461--462 (1990; Zbl 0719.47020), \textit{Miao Jie}, Proc. Am. Math. Soc. 116, 1077--1079 (1992; Zbl 0787.47029), and also for case where \(p=2\) and \(g \in BMOA\), \textit{C. Pommerenke}, Comment. Math. Helv. 52, 591--602 (1977; Zbl 0369.30012). The authors prove the following results, which extend the results in [\textit{A. Aleman} and \textit{A. Siskakis}, An integral operator on \(H^p\), Complex Variables, Theory Appl. 28, 149--158 (1995; Zbl 0837.30024)] and yield an improvement of the Hardy-Littlewood theorem: (i) For \(1/q > 1/p\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g \in H^s\), where \(s=1/q-1/p\); (ii) For \(p=q\), \(T_g\) maps \(H^p\) into itself if and only if \(g \in BMOA\); (iii) For \(1/p > 1/q \geq 1/p -1\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g\) is in the analytic Lipschitz class \(\Lambda_{1/p-1/q}\); (iv) For \(1/q < 1/p -1\), \(T_g\) maps \(H^p\) into \(H^q\) if and only if \(g\) is constant (and thus \(T_g=0\)). Furthermore, they characterize the cases where \(T_g\) is a compact operator between the spaces in question (in case (i), \(T_g\) is compact whenever it is bounded, in cases (ii) and (iii), one needs the ''little oh'' versions of the ''big oh'' conditions given on \(g\)). They also derive from their main results mapping properties of \(T_g\) from \(H^p\) to the analytic Lipschitz spaces, namely, if one assumes t…
reviewed by: Pascal J. Thomas
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47G10
zbMATH DE Number: 1765843
zbMATH Keywords: Hardy's inequality
zbMATH Keywords: Cesaro operator
zbMATH Keywords: bounded mean oscillation
zbMATH Keywords: Hardy spaces
zbMATH Keywords: integral operators
MaRDI profile type: Publication
cites work: An Integral Operator on Hp
cites work: Q3954045
cites work: Q5618011
cites work: The Cesaro Operator is Bounded on H p for 0 < p < 1
cites work: Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation
cites work: The Cesaro Operator is Bounded on H 1
full work available at URL: https://doi.org/10.1007/bf02788078
OpenAlex ID: W2349697008
description: scientific article; zbMATH DE number 1765843