Boundary regularity in the Dirichlet problem for the invariant Laplacians $\Delta _\gamma $ on the unit real ball

Abstract

We study the boundary regularity in the Dirichlet problem of the differential operators \begin{equation*} \Delta _{\gamma }= (1-|x|^2)\bigg \{ \frac {1-|x|^2}4 \sum _j \frac {\partial ^2} {\partial x_j^2} + \gamma \sum _j x_j \frac {\partial }{\partial x_j} + \gamma \Big (\frac n2 -1 -\gamma \Big )\bigg \}. \end{equation*} Our main result is: if $\gamma >-1/2$ is neither an integer nor a half-integer not less than $n/2-1$, one cannot expect global smooth solutions of $\Delta _\gamma u=0$; if $u\in C^{\infty }(\overline {B}_n)$ satisfies $\Delta _\gamma u=0$, then $u$ must be either a polynomial of degree at most $2\gamma +2-n$ or a polyharmonic function of degree $\gamma +1$.