)e^{-1/r^{2}(-\Delta )}\delta _{0}\left( x\right) =h_{j,r}(\sqrt{-\Delta }%
)p_{1/r^{2}}(x).  \label{kj,r}$$Consequently, (\[sum\]) and (\[kj,r\]) imply that $$\kappa _{R}^{z}=\sum\limits_{j\in \mathbb{N}}\kappa _{j,r}.  \label{sumk}$$So, to estimate the kernel $\kappa _{R}^{z}$, it suffices to estimate the kernels $\kappa _{j,r}$, which by (\[kj,r\]) are expressed in terms of the heat kernel $p_{t}$ of $X$ and the functions $h_{j,r}$. For that, we shall first recall from [@ALEXOLO p.214] some properties of the functions $h_{j,r}$ we shall use in the sequel.

There is a $c>0$ such that $$|\operatorname{supp}h_{j,r}|\leq cr2^{-j},  \label{alexo6}$$[@ALEXOLO p.214]. Note that the functions $\chi _{j}$, as well as $h_{j,r}$ are radial and thus invariant by the Weyl group [@AN p.612].

Note also that for every $k\in \mathbb{N}$, there is a $c_{k}>0$, such that $$\Vert \chi _{j,r}^{(k)}\Vert _{\infty }\leq c_{k}r^{-k}2^{kj},\quad \Vert
h_{j,r}^{(k)}\Vert _{\infty }\leq c_{k}r^{-k}2^{-(\operatorname{Re}z-k)j}.
\label{alexo7}$$As it is mentioned in [@ALEXOLO p.214], the estimates (\[alexo6\]) and (\[alexo7\]) imply that for every $k\in \mathbb{N}$, there is a $c_{k}>0$ such that $$\int_{|t|\geq s}|\hat{h}_{j,r}(t)|dt\leq c_{k}s^{-k}r^{-k}2^{(k-\operatorname{Re}%
z)j},\;s>0,  \label{hhat}$$where $\hat{h}_{j,r}$ is the euclidean Fourier transform of $h_{j,r}$.

\[L1ball\] Let $\kappa _{R}^{z}$ be the kernel of the Riesz mean operator $S_{R}^{z}$. Then, there is $c>0$, independent of $R$, such that for $%
\operatorname{Re}z>n/2$, $$\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))}\leq c.$$

For the proof we shall consider different 