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

We study the boundary regularity in the Dirichlet problem of the differential operators

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 $\gamma+1$.