Harmonic Bergman Spaces on the Real Hyperbolic Ball: Atomic Decomposition, Interpolation and Inclusion Relations

For $\alpha >-1$ and $0<p<\mathrm{\infty }$, we study weighted Bergman spaces ${\mathcal{B}}_{\alpha }^p$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for ${\mathcal{B}}_{\alpha }^p$. Using these we determine precisely when a Bergman space ${\mathcal{B}}_{\alpha }^p$ is included in another Bergman space ${\mathcal{B}}_{\beta }^q$.