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This short note is motivated by an attempt to understand the distinctionbetween the Laplace operator and the hyperbolic Laplacian on the unit ball of$\mathbb{R}^n$, regarding the Lipschitz continuity of the solutions to thecorresponding 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 alwaysimplies the Lipschitz continuity of the solutions if $\vartheta > 0$, but doesnot when $\vartheta \leq 0$.

Lipschitz continuity of the solutions to the Dirichlet problems for the invariant laplacians