$\mathcal{H}$-Harmonic Bergman-Besov Spaces on the Real Hyperbolic Ball

Using characterizations in terms of various differential operators, including partial, normal, and tangential derivatives, we extend the family of Bergman spaces of $\mathcal{H}$-harmonic functions on the real hyperbolic ball from the range $\alpha >-1$ to all $\alpha \in \mathbb{R}$. We then generalize several properties of Bergman spaces; projection, duality, atomic decomposition, and inclusion relations, to this extended family.