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

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