we treat the general case of noncompact symmetric spaces of *all* ranks.

To state our results, we need to introduce some notation. Let $G$ be a semi-simple, noncompact, connected Lie group with finite center and let $K$ be a maximal compact subgroup of $G$. We consider the symmetric space of noncompact type $X=G/K$. Denote by $\mathfrak{g}$ and $\mathfrak{k}$ the Lie algebras of $G$ and $K$, respectively. We have the Cartan decomposition $%
\mathfrak{g}=\mathfrak{p}\oplus \mathfrak{k}$. Let $\mathfrak{a}$ be a maximal abelian subspace of $\mathfrak{p}$ and $\mathfrak{a}^{\ast }$ its dual. If $\operatorname{dim}\mathfrak{a}=l$, we say that $X$ has rank $l$.

Denote by $\rho $ the half sum of positive roots, counted with their multiplicities. Fix $R\geq \Vert \rho \Vert ^{2}$ and $z\in \mathbb{C}$ with $\operatorname{Re}z\geq 0$, and consider the bounded function $$s_{R}^{z}(\lambda )=\left( 1-\frac{\Vert \rho \Vert ^{2}+\Vert \lambda \Vert
^{2}}{R}\right) _{+}^{z},\;\lambda \in \mathfrak{a^{\ast }}.  \label{mult}$$

Denote by $\kappa _{R}^{z}$ its inverse spherical Fourier transform in the sense of distributions and consider the so-called Riesz means operator $%
S_{R}^{z}$: $$S_{R}^{z}(f)(x)=\int_{G}\kappa _{R}^{z}(y^{-1}x)f(y)dy=(\kappa _{R}^{z}\ast
f)(x),\quad f\in C_{0}(X).  \label{operatorX}$$

For every pair $p,q$ such that $1\leq p,q\leq \infty $, denote by $%
(L^{p}+L^{q})(X)$ the Banach space of all functions $f$ on $X$ which admit a decomposition $f=g+h$ 