$\mathcal H$-Harmonic Bergman Projection on the Hyperbolic Ball

We determine precisely when the Bergman projection $P_\beta$ is bound\-ed from Lebesgue spaces $L^p_\alpha$ to weighted Bergman spaces $\mathcal B^p_\alpha$ of $\mathcal H$-harmonic functions on the hyperbolic ball, and verify a recent conjecture of M. Stoll. We obtain upper estimates for the reproducing kernel of the $\mathcal H$-harmonic Bergman space $\mathcal B^2_\alpha$ and its partial derivatives. We also consider the projection from $L^\infty$ to the Bloch space $\mathcal B$ of $\mathcal H$-harmonic functions.