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

Abstract: 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 $\vartheta \leq 0$.