Admissible convergence of Poisson integrals in symmetric spaces


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 $\mathscr{P}_\lambda f$. The method of proof uses maximal functions, and it also gives a new proof of almost everywhere restricted convergence of $Pf$ and $\mathscr{P}_\lambda f$ for $f\in L^1$.


Peter Sjögren