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

Abstract

Let u u be a bounded harmonic function on a noncompact rank one symmetric space M = G / K ≈ N − A , N − A K M = G/K \approx {N^ - }A,{N^ - }AK being a fixed Iwasawa decomposition of G G . We prove that if for an a 0 ∈ A {a_0} \in A there exists a limit u ( n a 0 ) ≡ c 0 u(n{a_0}) \equiv {c_0} , as n ∈ N − n \in {N^ - } goes to infinity, then for any a ∈ A a \in A , u ( n a ) = c 0 u(na) = {c_0} . For M = S U ( n , 1 ) / S ( U ( n ) × U ( 1 ) ) = B n M = SU(n,1)/S(U(n) \times U(1)) = {B^n} , the unit ball in C n {{\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 ∂ B n \partial {B^n} , surfaces.