  In this paper, we investigate solutions of the hyperbolic Poisson equation
$\Delta_{h}u(x)=\psi(x)$, where $\psi\in L^{\infty}(\mathbb{B}^{n},
\mathbb{R}^n)$ and \[ \Delta_{h}u(x)= (1-|x|^2)^2\Delta
u(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x)
\] is the hyperbolic Laplace operator in the $n$-dimensional space
$\mathbb{R}^n$ for $n\ge 2$. We show that if $n\geq 3$ and $u\in
C^{2}(\mathbb{B}^{n},\mathbb{R}^n) \cap
C(\overline{\mathbb{B}^{n}},\mathbb{R}^n )$ is a solution to the hyperbolic
Poisson equation, then it has the representation $u=P_{h}[\phi]-G_{ h}[\psi]$
provided that $u\mid_{\mathbb{S}^{n-1}}=\phi$ and
$\int_{\mathbb{B}^{n}}(1-|x|^{2})^{n-1} |\psi(x)|\,d\tau(x)<\infty$. Here
$P_{h}$ and $G_{h}$ denote Poisson and Green integrals with respect to
$\Delta_{h}$, respectively. Furthermore, we prove that functions of the form
$u=P_{h}[\phi]-G_{h}[\psi]$ are Lipschitz continuous.

Analysis of PDEs