with $g\in L^{p}$ and $h\in L^{q}$. The norm of $f\in
(L^{p}+L^{q})(X)$ is given by $$\Vert f\Vert _{\left( p,q\right) }=\inf \left\{ \Vert f\Vert _{p}+\Vert
g\Vert _{q}:\text{ for all decompositions }f=g+h\right\} .$$For $q\geq 1$, denote by $q^{\prime }$ its conjugate. In the present work we prove the following results.

\[th0\]Let $z\in \mathbb{C}$ with $\operatorname{Re}z\geq n-\frac{1}{2}$ and consider $q>2$. Then, for every $p$ such that $1\leq p\leq q^{\prime }$, and for every $r\in \lbrack qp^{\prime }/(p^{\prime }-q),\infty ]$, $S_{R}^{z}$ is uniformly bounded from $L^{p}(X)$ to $(L^{p}+L^{r})(X)$.

Next we deal with the maximal operator $S_{\ast }^{z}$ associated with Riesz means: $$S_{\ast }^{z}(f)(x)=\sup_{R>\Vert \rho \Vert ^{2}}|S_{R}^{z}(f)(x)|,\;f\in
L^{p}({X}),\;1\leq p\leq 2.$$Set $$Z_0(n,p)=\left( n-\frac{1}{2}\right) \left( \frac{2}{p}-1\right) .$$We have the following result.

\[Th1\] Let $1\leq p\leq 2$ and consider $q>2$. If $\operatorname{Re}z>Z_0(n,p)$, then for every $s\geq pq/(2-p+pq-q)$, there is a constant $c\left( z\right) >0$, such that for every $f\in L^{p}(X)$, $$\Vert S_{\ast }^{z}f\Vert _{\left( p,s\right) }\leq c(z)\Vert f\Vert _{p}.$$

As a corollary of the Theorem \[Th1\], we obtain the almost everywhere convergence of Riesz means.

\[Th2\] Let $1\leq p\leq 2$. If $\operatorname{Re}z>Z_0(n,p)$, then for $f\in
L^{p}(X)$, $$\lim\limits_{R\rightarrow +\infty }S_{R}^{z}f(x)=f(x),\;\text{a.e.. }
\label{aec}$$

Note that in the setting of $\mathbb{R}^{n}$, [@STEIN27], as well as in case of the rank one symmetric 