On Lipschitz continuity and smoothness up to the boundary of solutions of hyperbolic Poisson's equation

We solve the Dirichlet problem ${u\mid }_{{\mathbb{B}}^n}=\phi ,$ for hyperbolic Poisson’s equation ${\mathrm{\Delta }}_{h}u=\mu$ where $\phi \in L_1(\mathrm{\partial }{\mathbb{B}}^n)$ and $\mu$ is a measure that satisfies a growth condition. Next we present a short proof for Lipschitz continuity of solutions of certain hyperbolic Poisson’s equations, previously established at \cite{ChenRas}. In addition, we investigate some alternative assumptions on hyperbolic Laplacian, which are connected with Riesz’s potential. Also, local Hölder continuity is proved for solution of certain hyperbolic Poisson’s equations. We show that, if $u$ is hyperbolic harmonic in the upper half-space, then $\frac{\mathrm{\partial }u}{\mathrm{\partial }y}(x_0,y)\to 0,y\to 0^{+}$, when boundary function $f$ of the functions $u$ is differentiable at the boundary point $x_0$. As a corollary, we show $C^1(\overline{{\mathbb{H}}^n})$ smoothness of a hyperbolic harmonic function, which is reproduced from the $C_c^1({\mathbb{R}}^{n-1})$ boundary values.