Boundary Behavior of Poisson Integrals on Boundaries of Symmetric Spaces

In this paper we investigate the boundary behavior of L p -Poisson integrals for various boundaries of Riemannian Symmetric Spaces of the noncompact type.In particular, we show that if a function F on a Riemannian symmetric space G/K is solution of some invariant differential system associated to a standard parabolic subgroup P E of G then F is the Poisson integral of an L p -function on the boundary component G/P E if and only if it satisfies a Hardy type condition on a family of K -orbits.