Boundary behavior of Poisson integrals on symmetric spaces

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Its distinguished boundary is the space of real points (x1,.. ., x). An unrestricted nontangential domain at (xO,. . ., xo) is a set {(z1, ., zn) I Ixj-xJl 0. A restricted nontangential domain is a subset of the above, satisfying yj 1. With the aid of these one defines the notion of restricted or unrestricted convergence of a function F on D to a boundary function f. Generalizing the classical Fatou theorem Marcinkiewicz and Zygmund have shown [18, Chapter XVII] that if F is the Poisson integral of f then F converges to f almost everywhere, unrestrictedly if f E LP (1 <p ? oo) and restrictedly iff E L1. For a product of discs one can make analogous definitions and one has analogous results. Since halfplanes and discs are equivalent under conformal transformations which extend to the boundary almost everywhere, the results about products of discs are equivalent to the results about products of halfplanes. The results of Marcinkiewicz and Zygmund were generalized in [9] to products of unit balls in complex n-space. In this case the natural generalization of nontangential convergence is no longer nontangential in the geometric sense; in [9] it was given the name admissible convergence. One still has the distinction between restricted and unrestricted admissible convergence. In trying to generalize these notions to arbitrary symmetric spaces of noncompact type several new features appear. First, a space may have more than one boundary (the Furstenberg-Satake boundaries). To each boundary there corresponds a different Poisson integral, and to each one has to define a different notion of restricted (and unrestricted) admissible convergence. Second, since a symmetric space in general has no natural imbedding as a domain in Euclidean space, one has to define restricted and unrestricted admissible convergence in an intrinsic way. Finally, the notion of restricted admissible convergence which seems to be natural depends (for each fixed boundary) on the arbitrary choice of an element H in a certain Weyl chamber. In fact, this arbitrary choice also appears in the case of a