A tangential convergence for bounded harmonic functions on a rank one symmetric space

Abstract:

Let $u$ be a bounded harmonic function on a noncompact rank one symmetric space $M = G/K \approx {N^ - }A,{N^ - }AK$ being a fixed Iwasawa decomposition of $G$. We prove that if for an ${a_0} \in A$ there exists a limit $u(n{a_0}) \equiv {c_0}$, as $n \in {N^ - }$ goes to infinity, then for any $a \in A$, $u(na) = {c_0}$. For $M = SU(n,1)/S(U(n) \times U(1)) = {B^n}$, the unit ball in ${{\mathbf {C}}^n}$ with the Bergman metric, this is a result of Hulanicki and Ricci, and in this case it reads (via the Cayley transformation) as a theorem on convergence of a bounded harmonic function to a boundary value at a fixed boundary point, along appropriate, tangent to $\partial {B^n}$, surfaces.