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Let u be a bounded harmonic function on a noncompact rank one symmetric space M = G/'K «s N ~A, N ~AK being a fixed Iwasawa decomposition of G. We prove that if for an a0 e A there exists a limit lim u(na0) m Cq, as n E N ~ goes to infinity, then for any a G A, lim u(na) = c0. For M = SU(n, \)/S(U(n) x i/(l)) = B", the unit ball in C 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 dB", surfaces. 0. Introduction. Let M be a noncompact symmetric space of rank one. M can be expressed as a homogeneous space G/K where G is a semisimple group of isometries of M and K is a maximal compact subgroup of G. Let g, f denote the Lie algebras of G and K, B the Killing form of g, and p the orthogonal complement of f in g relative to B. If it: G -» G/K denotes the canonical projection, its differential at the identity, irm, identifies the subspace p of g with T0(M), the tangent space of M at the origin o = tr(e), and the invariant metric g on M can be chosen so that g0 corresponds to the restriction of B to p X p under the above identification. We denote by A the corresponding (G- in variant) Laplace-Beltrami operator on M. A function u G C°°(A/) is called harmonic if Aw = 0. Let u be a maximal (one-dimen- sional) abelian subspace of p, a and possibly 2a in a*, the corresponding system of positive restricted roots relative to the fixed choice of a "positive part" a+ in a. Let g_a and g_2a denote the root spaces corresponding to —a and — 2a. Then n~=g_0©g_2aisa nilpotent subalgebra of g and one has the Iwasawa decom- position G = N~AK, with N~ = exp n~, A = exp a. The above decomposition shows that every pEM can be uniquely written as p = na- o (n G N ~, a G A). We regard the nilpotent group N ~ as a boundary for the symmetric space M in the following sense. The bounded harmonic functions m on M have boundary values on N~, i.e. lim.og^^ u(na ■ o) =m(/j) exists a.e. (relative to the Haar measure on A/-) and

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