Convergence for the square root of the Poisson kernel (Q1080986)

Let X be a Riemannian symmetric space and f an integrable function on its boundary \(\partial X\). The 0-Poisson integral \(P_ 0f\) is the function on X obtained by integrating f against the square root of the Poisson kernel. We give Fatou theorems saying that the normalized function \(P_ 0f/P_ 01\) converges almost everywhere to f on \(\partial X\). Many such results are known for \(\lambda\)-Poisson integrals \(P_{\lambda}f\) with \(\lambda\) in the positive Weyl chamber. But the case \(\lambda =0\) is different, since larger regions of convergence can be used. Some of our results are general, some are given for the bidisk or SL(3,\({\mathbb{R}})/SO(3)\). The paper extends previous results by the author for the disk and the bidisk.