spaces, [@GIUMA], (\[aec\]) is valid for $\operatorname{Re}z$ larger than the critical index $z_0(n,p)$ which is smaller than $Z_0(n,p)$. This is a consequence of limitations in our analysis and the lack of an explicit formula for the inverse Abel transform in the general case of a symmetric space. On the other hand, it is interesting that the index $Z_0(n,p)$ only depends on the Euclidean dimension of $X$.

Note also that the multiplier $s_{R}^{z}(\lambda )$ does not extend holomorphically to any tube domain over $\mathfrak{a}^{\ast }$. So, by [@CLERC Theorem 1], the Riesz means operators are not bounded on $L^{p}(X)$ if $p\neq 2$ and consequently the norm summability problem on $L^{p}(X)$, $%
p\neq 2$, is ill posed.

To prove Theorem \[th0\], we split the Riesz means operator in two convolution operators: $S_{R}^{z}=S_{R}^{z,0}+S_{R}^{z,\infty }$. The *local part* $S_{R}^{z,0}$ has a compactly supported kernel around the origin, while the kernel of the *part at infinity* $%
S_{R}^{z,\infty }$ is supported away from the origin. To treat the local part, we follow the approach of [@ALEXOLO; @PAPAGEO]. More precisely, we express the kernel of $S_{R}^{z,0}$ via the heat kernel $p_{t}$ of $X$, and we make use of its estimates, which combined with the fact that 