Harmonic Bloch Space on the Real Hyperbolic Ball

Abstract

We study the Bloch and the little Bloch spaces of harmonic functions on the real hyperbolic ball. We show that the Bergman projections from $L^{\infty }(\mathbb{B})$ to $\mathcal{B}$, and from $C_0(\mathbb{B})$ to ${\mathcal{B}}_0$ are onto. We verify that the dual space of the hyperbolic harmonic Bergman space ${\mathcal{B}}_{\alpha }^1$ is $\mathcal{B}$ and its predual is ${\mathcal{B}}_0$. Finally, we obtain an atomic decomposition of Bloch functions as a series of Bergman reproducing kernels.

19 pages