Convergence of poisson integrals for bounded symmetric domains.

1. The purpose of this note is to describe the extension of the almost everywhere convergence of Poisson integrals to the case of the bounded symmetric domains of Cartan. It is useful to realize such a bounded symmetric domain D as a generalized upper half-plane (i.e., a Siegel domain of type II), and this is done as follows. Let V1 and V2 be two finite-dimensional vector spaces over C. Assume that a real form Re V1 is distinguished in V1, and a proper convex cone Q is given in Re V1. We assume that we are also given a Hermitian symmetric bilinear mapping, D: V2 X V2 -¢ V1 with the property that 4?(w,w) E Q. The domain D is then D ={(z,w): ZEE V1, WE V2, Imz(w,w) CQ},