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

Abstract: For $\alpha>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_\alpha$ 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 $\mathcal B^p_\alpha$. Using these we determine precisely when a Bergman space $\mathcal B^p_\alpha$ is included in another Bergman space $\mathcal B^q_\beta$.