On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

Let P(z,beta) be the Poisson kernel in the unit disk U, and let P(lambda)f(z) = integral(partial derivative U)P(z,phi)(1/2+lambda)f(phi) d phi be the lambda-Poisson integral of f, where f is an element of L-p(partial derivative U). We let P(lambda)f be the normalization P(lambda)f/P(lambda)1. If lambda > 0, we know that the best (regular) regions where P(lambda)f converges to f for a.a. points on partial derivative U are of nontangential type. If lambda = 0 the situation is different. In a previous paper, we proved a result concerning the convergence of p(0)f toward f in an L-p weakly tangential region. if f is an element of L-p(partial derivative U) and p > 1. In the present paper we will extend the result to symmetric spaces X of rank 1. Let f be an L-p function on the maximal distinguished boundary K/M of X. Then P(0)f(x) will converge to f(kM) as x tends to kM in an L-p weakly tangential region, for a.a. kM is an element of K/M.