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

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 $\vartheta \leq 0$.