On harmonic function spaces (Q2569281)

The author investigates \(a\)-Bloch, Hardy, Bergman, BMO\(_p\) and Dirchlet spaces of harmonic functions on the unit ball \(B\) in \(\mathbb R^n\) (\(n\geq 2\)). For \(g:[0,1]\to\mathbb R\) a measurable function, the Hardy-Littlewood operator \(L_g\) is defined on the space of measurable functions on \(B\) by \[ L_g(f)(x) = \int_0^1 f(tx)g(t)\,dt, \quad x\in B, \] provided the integral exists. The author proves that for \(g\in L^1([0,1])\), \(L_g\) is a bounded operator on the \(a\)-Bloch and BMO\(_p\) spaces of harmonic functions. Sufficient conditions are also given in order that \(L_g\) be a bounded operator on weighted Bergman spaces, Hardy spaces, and Dirichlet spaces. In Theorem 10 of the paper the author proves that if \(u\) is in the Dirichlet space \(\mathcal D_{\alpha}^2\), \(\alpha\in (-1,1]\), then \(u\) is in the harmonic Hardy space \(\mathcal H^p\) for all \(p\in (0,\frac{2n-2}{n+\alpha-2}]\). Remark: The author may not be aware that Theorem 10, along with a converse, has also been proved by the reviewer in [J. Math. Anal. Appl. 274, 788--811 (2002; Zbl 1019.31002)] for eigenfunctions of the Laplacian on bounded domains in \(\mathbb R^n\) with \(C^{1,1}\) boundary.