follows that to prove the $L^p$-boundedness of the operator $S_{\ast}^{z,0}$, it suffices to prove the $L^p$-boundedness of the operator $\sup\limits_{R\geq\|\rho\|^2}|M^0(-R^{-1}\Delta)|$, and similarly for the $L^1\rightarrow L^{1,w}$ boundedness.

From (\[Mellin\]) and (\[mknorm\]), we have that $$\begin{aligned}
    \|\sup_{R\geq \|\rho\|^2} |M^0(-R^{-1}\Delta)|\|_p &\leq \int_{-\infty}^{+\infty}|\mathcal{M}(\gamma)|\|(-\Delta)^{i\gamma,0}\|_{p\rightarrow p}|\|f\|_pd\gamma\\
    &\leq c|\|f\|_p \int_{-\infty}^{+\infty}(1+|\gamma|)^{-(\operatorname{Re}z+1)})(1+|\gamma|)^{[n/2]+1}d\gamma \\
    &\leq c|\|f\|_p \int_{-\infty}^{+\infty}(1+|\gamma|)^{-(\operatorname{Re}z-[n/2])}d\gamma\leq c\|f\|_p,
    \end{aligned}$$ whenever $\operatorname{Re}z\geq n-\frac{1}{2}$. This completes the proof of Proposition \[proplocal\].

*Proof of Theorem 2.* The proof of Theorem 2 follows from Stein’s complex interpolation, between the $L^p$ result for $p$ close to $1$ and the $L^2$ result (Propositions 12, 13 and 14).

*Proof of Theorem 3.* As it is already mentioned in the Introduction, from Theorem [Th1]{} and Propositions 13 and 14, we deduce the almost everywhere convergence of Riesz means: if $1\leq
p\leq 2$ and $\operatorname{Re}z>\left( n-\frac{1}{2}\right) \left( \frac{2}{p}%
-1\right) $, then $$\lim\limits_{R\rightarrow +\infty }S_{R}^{z}(f)(x)=f(x),\;\text{a.e., for }%
f\in L^{p}(X).$$

[99]{} G. Alexopoulos, N. Lohoué, Riesz means on Lie groups and Riemannian manifolds of nonnegative curvature, *Bull. Soc. Math. France*, **122** (1994), no. 2, 209–223. J.-Ph. Anker, $L^{p}$ Fourier multipliers on Riemannian symmetric spaces of noncompact type, *Ann. of Math.*, **132** (1990), 597–628. J.-Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces. *Duke Math. J.*, 