Vector-valued Hardy spaces in non-smooth domains
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 \(E\subset \partial D\), and they show that the Hardy space \(H^1_X(D)\) can be described using the atomic decomposition with respect to these atoms.
scientific article; zbMATH DE number 5141590
scientific article