Let X be a symmetric space
and f an integrable function on its boundary ∂X. The 0-Poisson integral
P0f 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 P0f∕P01 converges almost everywhere to f on ∂X. Many such
results are known for λ-Poisson integrals Pλf with λ in the positive Weyl
chamber. But the case λ = 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,R)∕SO(3). The paper extends previous results by the author for the disk and
the bidisk.
