On Lipschitz continuity of solutions of hyperbolic Poisson's equation

In this paper, we investigate solutions of the hyperbolic Poisson equation $Δ_{h}u(x)=ψ(x)$, where $ψ\in L^{\infty}(\mathbb{B}^{n}, \mathbb{R}^n)$ and $Δ_{h}u(x)= (1-|x|^2)^2Δu(x)+2(n-2)(1-|x|^2)\sum_{i=1}^{n} x_{i} \frac{\partial u}{\partial x_{i}}(x)$ 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 $u=P_{h}[ϕ]-G_{ h}[ψ]$ provided that $u\mid_{\mathbb{S}^{n-1}}=ϕ$ and $\int_{\mathbb{B}^{n}}(1-|x|^{2})^{n-1} |ψ(x)|\,dτ(x)<\infty$. Here $P_{h}$ and $G_{h}$ denote Poisson and Green integrals with respect to $Δ_{h}$, respectively. Furthermore, we prove that functions of the form $u=P_{h}[ϕ]-G_{h}[ψ]$ are Lipschitz continuous.

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