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

Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{λ}f(z) = ʃ_{∂} P(z,φ)^{1//2+λ} f(φ)dφ$ be the λ -Poisson integral of f, where $f ∈ L^p(∂)$. We let $P_{λ}f$ be the normalization $P_{λ}f//P_{λ}1$. If λ >0, we know that the best (regular) regions where $P_{λ}f$ converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_0f$ toward f in an $L^p$ weakly tangential region, if $f ∈ L^p(∂)$ 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 ∈ K/M.