Riesz means on symmetric spaces

Abstract

Let $X$ be a non-compact symmetric space of dimension $n$. We prove that if $f\in L^p(X)$, $1\le p\le 2$, then the Riesz means $S_R^z\left(f\right)$ converge to $f$ almost everywhere as $R\to \infty$, whenever $\mathrm{Re}z>\left(n-\frac{1}{2}\right)\left(\frac{2}{p}-1\right)$.