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 ${\displaystyle P_{\lambda }f\left(z\right)={ʃ}_{\partial }P{(z,\phi )}^{1/2+\lambda }f(\phi )d\phi }$ be the λ -Poisson integral of f, where ${\displaystyle f\in L^p(\partial )}$. We let ${\displaystyle P_{\lambda }f}$ be the normalization ${\displaystyle P_{\lambda }f/P_{\lambda }1}$. If λ >0, we know that the best (regular) regions where ${\displaystyle P_{\lambda }f}$ 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 ${\displaystyle P_{0}f}$ toward f in an ${\displaystyle L^p}$ weakly tangential region, if ${\displaystyle f\in L^p(\partial )}$ and p > 1. In the present paper we will extend the result to symmetric spaces X of rank 1. Let f be an ${\displaystyle L^p}$ function on the maximal distinguished boundary K/M of X. Then ${\displaystyle P_{0}f\left(x\right)}$ will converge to f(kM) as x tends to kM in an ${\displaystyle L^p}$ weakly tangential region, for a.a. kM ∈ K/M.