the wave operator $\cos t\sqrt{-\Delta }$ of $X$ propagates with finite speed, allow us to prove that $S_{R}^{z,0}$ is continuous on $L^{p}(X)$ for all $p\geq 1$. To treat the part at infinity of the operator, we proceed as in [LOMAanna]{}, and obtain estimates of its kernel by using the support preserving property of the *Abel transform*. We combine known ingredients to study Riesz means for noncompact symmetric spaces of arbitrary rank, an area that remained inactive since the seminal work [@GIUMA] in 1991.

This paper is organized as follows. In Section 2 we present the necessary ingredients for our proofs. In Section 3 we deal with the local part and the part at infinity, of the Riesz mean operator and we prove Theorem \[th0\]. In Section 4 we prove Theorem \[Th1\] and we deduce Theorem \[Th2\].

Preliminaries
=============

In this section we recall some basic facts about symmetric spaces. For details see for example [@AN; @FOMAMA; @HEL; @LOMAjga].

Symmetric spaces
----------------

Let $G$ be a semisimple Lie group, connected, noncompact, with finite center and let $K$ be a maximal compact subgroup of $G$. We denote by $X$ the noncompact symmetric space $G/K$. In the sequel we assume that $\operatorname{dim}%
X=n 