Vector-valued Hardy spaces in non-smooth domains
type: Item
instance of: scholarly article
title: Vector-valued Hardy spaces in non-smooth domains
zbMATH Open document ID: 1122.46019
author: Salvador Pérez-Esteva
author: Jorge Rivera-Noriega
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-04-10T00:00:00Z
review text: The authors consider Hardy spaces of harmonic functions from an \(n\)-dimensional (\(n>2\)) domain with Lipschitz boundary into a real Banach space \(X\). The main result establishes that the Radon--Nikodym property (RNP) of a Banach space \(X\) can also be described in terms of the existence of boundary values of harmonic functions in Hardy spaces with values in \(X\) defined on such domains and also that there exists the corresponding atomic decomposition for the case \(p=1\) replacing functions by measures. Their results extend some known results for the unit disc (see \textit{O.\,Blasco} and \textit{J.\,García-Cuerva} [Math.\ Nachr.\ 132, 57--65 (1987; Zbl 0632.46032)], \textit{W.\,Hensgen} [Arch.\ Math.\ 57, No.\,1, 88--96 (1991; Zbl 0696.46031)], \textit{A.\,V.\thinspace Bukhvalov} and \textit{A.\,A.\thinspace Danilevich} [Math.\ Notes 31, 104--110 (1982); translation from Mat.\ Zametki 31, 203--214 (1982; Zbl 0496.30029)]) and now their techniques, instead of the Poisson kernel, use the representation arising from the harmonic measure \(\omega\) associated to the domain [see \textit{B.\,E.\thinspace Dahlberg}, Stud.\ Math.\ 66, 13--24 (1979; Zbl 0422.31008)]. The Hardy spaces \(H_X^p(D)\) are defined using harmonic functions with non-tangential maximal function in \( L^p(\omega)\) for \(1\leq p\leq\infty\). It is shown that \(X\) has the RNP if and only if any \(u\in H_X^p(D)\) has non-tangential limits \(\omega\)-a.e. They also show that under the RNP, if \(f\in L^p(\omega)\) for \(2<p<\infty\), there exists a harmonic function \(u\in H^p_X(D)\) with boundary values \(f\) and \(\| u^*\| _{L^p(\omega)}\leq C \| f\| _{L^p(\omega)}\). Moreover, the authors suggest the use of ``atom'' as an \(X\)-valued measure \(\mu\) such that \(\mu(\partial D)=0\), \(\text{supp}\mu\subset \Delta=B\cap\partial D\) for some ball \(B\) centered at a point in \(\partial D\), and \(\| \mu(E)\| \leq \frac{\omega(E)}{\omega(\Delta)}\) for all measurable set \(E\subset \partial D\), …
Mathematics Subject Classification ID: 46E40
Mathematics Subject Classification ID: 46B22
Mathematics Subject Classification ID: 30D55
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 31C05
zbMATH DE Number: 5141590
zbMATH Keywords: vector-valued harmonic functions
zbMATH Keywords: atomic decomposition
zbMATH Keywords: Radon-Nikodym property
reviewed by: Oscar Blasco
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.06.097
OpenAlex ID: W2076271677
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DOI: 10.1016/J.JMAA.2006.06.097
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description: scientific article; zbMATH DE number 5141590