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

We determine precisely when the Bergman projection $P_{\beta }$ is bound-ed from Lebesgue spaces $L_{\alpha }^p$ to weighted Bergman spaces ${\mathcal{B}}_{\alpha }^p$ 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 ${\mathcal{B}}_{\alpha }^2$ and its partial derivatives. We also consider the projection from $L^{\mathrm{\infty }}$ to the Bloch space $\mathcal{B}$ of $\mathcal{H}$-harmonic functions.