On Lipschitz continuity of solutions of hyperbolic Poisson's equation

In this paper, we investigate solutions of the hyperbolic Poisson equation ${\mathrm{\Delta }}_{h}u(x)=\psi (x)$, where $\psi\in L^{\infty}($\mathbb{B}$^{n}, $\mathbb{R}$^n)$ and $${\mathrm{\Delta }}_{h}u(x)=(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^2\mathrm{\Delta }u(x)+2(n-2)(1-\mathrm{\mid }x{\mathrm{\mid }}^2)\sum _{i=1}^n{x}_i\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}(x)$$ is the hyperbolic Laplace operator in the $n$-dimensional space ${\mathbb{R}}^n$ for $n\ge 2$. We show that if $n\ge 3$ 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 $u=P_h[\varphi ]-G_h[\psi ]$ provided that $u{\mid }_{{\mathbb{S}}^{n-1}}=\varphi$ and ${\int }_{{\mathbb{B}}^n}(1-\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{n-1}\mathrm{\mid }\psi (x)\mathrm{\mid }d\tau (x)<\mathrm{\infty }$. Here $P_h$ and $G_h$ denote Poisson and Green integrals with respect to ${\mathrm{\Delta }}_h$, respectively. Furthermore, we prove that functions of the form $u=P_h[\varphi ]-G_h[\psi ]$ are Lipschitz continuous.