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Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{\lambda }f\left(z\right)={ʃ}_{\partial }P{(z,\phi )}^{1/2+\lambda }f\left(\phi \right)d\phi$ be the λ -Poisson integral of f, where $f\in L^p\left(\partial \right)$. We let $P_{\lambda }f$ be the normalization $P_{\lambda }f/P_{\lambda }1$. If λ >0, we know that the best (regular) regions where $P_{\lambda }f$ 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 $P_{0}f$ toward f in an $L^p$ weakly tangential region, if $f\in L^p\left(\partial \right)$ and p > 1. In the present paper we will extend the result to symmetric spaces X of...