Harmonic Bergman spaces with small exponents in the unit ball
The paper deals with weighted harmonic Bergman spaces \(h^p_\alpha\) on the open unit ball \(\mathbb{B}\) in \(\mathbb{R}^n\) where \(n\geq 2\). Let \(\alpha >-1\) and put \[ \|f\|_{p,\alpha} =\Bigl(\int_\mathbb{B} \bigl|f(x) \bigr|^p \bigl(1-|x|^2 \bigr)^\alpha dx \Bigr)^{1/p} \] and \(h^p_\alpha =\{f\) harmonic on \(\mathbb{B}:\|f\|_{p,\alpha} <\infty\}\). The harmonic Bloch space \({\mathcal B}\) consisis of all harmonic functions \(f\) on \(\mathbb{B}\) such that \[ \|f\|_{\mathcal B}=\sup_{x\in\mathbb{B}} \bigl(1-|x |^2\bigr) \bigl|\nabla f(x)\bigr|<\infty. \] For \(1<p <\infty\) it is well-known that \((h^p_\alpha)^* =h^q_\alpha\), where \(p^{-1}+q^{-1}=1\), while \((h^1_\alpha)^* =\mathbb{B}\). Moreover, there are descriptions of the dual spaces for holomorphic Bergman spaces if \(0<p<1\). In the present article, the harmonic case, i.e., \((h^p_\alpha)^*\) for \(0<p<1\), is characterized. Let \(0<p<1\) and \(-1<\alpha <\infty\). Then we have \((h^p_\alpha)^* ={\mathcal B}\) under the pairing \[ \langle f, g\rangle =\lim_{r\to 1^-} \int_\mathbb{B} f(rx)\overline {g(x)}\bigl(1-|x|^2 \bigr)^{-n+(n+ \alpha)/p} dx \] where \(f\in h^p_\alpha\) and \(g\in{\mathcal B}\). The proof is based on estimates of weighted Bergman kernels and the boundedness of weighted Bergman projections.
scientific article; zbMATH DE number 1780853