been extensively studied, see for example [@ANJ; @ANO]. Sharp estimates of the heat kernel have been obtained by Davies and Mandouvalos in [@DAVMAN] for the case of real hyperbolic space, while Anker and Ji [@ANJ] and later Anker and Ostellari [@ANO], generalized the results of [@DAVMAN] to all symmetric spaces of noncompact type.

Denote by $\Sigma _{0}^{+}$ the set of positive indivisible roots $\alpha $ of $(\mathfrak{g}, \mathfrak{a})$ and by $m_{\alpha }$ the dimension of the root space $\mathfrak{g}^{\alpha }$. In [@ANO Main Theorem] it is proved the following sharp estimate: $$\begin{aligned}
p_{t}(\exp {H})& \leq ct^{-n/2}\left( \underset{\alpha \in \Sigma _{0}^{+}}{    \prod }(1+\langle \alpha ,H\rangle )(1+t+\langle \alpha ,H\rangle )^{\frac{     m_{\alpha }+m_{2\alpha }}{2}-1}\right) \times \notag \\
& \times e^{-\left\Vert \rho \right\Vert ^{2}t-\langle \rho ,H\rangle
    -|H|^{2}/4t},\quad t>0, \;H\in \overline{\mathfrak{a}   _{+}},  \label{heat}\end{aligned}$$where $n=$dim$X$.

From (\[heat\]), we deduce the following crude estimate $$p_{t}(\exp {H})\leq ct^{-n/2}e^{-|H| ^{2}/4t}, \quad
t>0, \;H\in \overline{\mathfrak{a}_{+}},  \label{ostellari}$$which is sufficient for our purposes.

As it is shown in [@GRIG Lemma 3.2], the estimate (\[ostellari\]) of the heat kernel implies that $$\int_{d(x,x_{0})>a}p_{t}^{2}(x)dx\leq ct^{-n/2}e^{-a^{2}/Dt},
\label{grigoryan}$$for some constant $D>0$ sufficiently large.

Proof of Theorem 1
==================

Let $\kappa _{R}^{z}$ be the kernel of the Riesz means operator. We start with a decomposition of $\kappa _{R}^{z}$: 