  This short note is motivated by an attempt to understand the distinction
between the Laplace operator and the hyperbolic Laplacian on the unit ball of
$\mathbb{R}^n$, regarding the Lipschitz continuity of the solutions to the
corresponding Dirichlet problems. We investigate the Dirichlet problem
\begin{equation*} \left\{\begin{array}{ll} \Delta_{\vartheta} u = 0, & \text{
in }\, \mathbb{B}^n,\\ u=\phi, & \text{ on }\, \mathbb{S}^{n-1},
\end{array}\right. \end{equation*} where \[ \Delta_{\vartheta} := (1-|x|^2)
\bigg\{ \frac {1-|x|^2} {4} \Delta + \vartheta \sum_{j=1}^n x_{j} \frac
{\partial } {\partial x_j} + \vartheta \left( \frac {n}{2}-1- \vartheta \right)
I\bigg\}. \] We show that the Lipschitz continuity of boundary data always
implies the Lipschitz continuity of the solutions if $\vartheta > 0$, but does
not when $\vartheta \leq 0$.
