Admissible convergence of Poisson integrals in symmetric spaces (Q1103752)

Let f be an \(L^ p\) function, \(p>1\), on the distinguished boundary of a Furstenberg-Satake compactification of a symmetric space. We prove that the Poisson integral Pf of f converges admissibly at almost all components of each boundary of the compactification. This was known previously only for large p. In particular, Pf converges admissibly to f almost everywhere at the distinguished boundary. A similar result is obtained for the normalized \(\lambda\)-Poisson integral \({\mathcal P}_{\lambda}f\). The method of proof uses maximal functions, and it also gives a new proof of almost everywhere restricted convergence of Pf and \({\mathcal P}_{\lambda}f\) for \(f\in L^ 1\).