cases. Recall that $R\geq \Vert
\rho \Vert ^{2}$*.*

*Case 1:* $\Vert \rho \Vert ^{2}\leq R\leq \Vert \rho \Vert ^{2}+1$*.*

Combining (\[ostellari\]) and the heat semigroup property, we get that $$\begin{aligned}
\Vert p_{t}\Vert _{L^{2}(X)}& =\left( \int_{X}p_{t}(x,y)p_{t}(y,x)dy\right)
^{1/2}  \label{L2heat} \\
& \leq p_{2t}(x,x)^{1/2}\leq ct^{-n/4}.  \notag\end{aligned}$$Thus, using (\[alexo7\]), (\[kj,r\]) and (\[L2heat\]), we have $$\begin{aligned}
\Vert \kappa _{j,r}\Vert _{L^{1}(B(0,1))}& \leq |B(0,1)|^{1/2}\Vert \kappa
_{j,r}\Vert _{L^{2}(X)}  \notag \\
& \leq c\Vert h_{j,r}(\sqrt{-\Delta })\Vert _{L^{2}\rightarrow L^{2}}\Vert
p_{1/r^{2}}\Vert _{L^{2}(X)}  \notag \\
& \leq c\Vert h_{j,r}\Vert _{\infty }(1/r^{2})^{-n/4}  \label{case1} \\
& \leq c2^{-j\operatorname{Re}z},  \notag\end{aligned}$$where $c=c(\Vert \rho \Vert )$. So, $$\Vert \kappa _{R}^{z}\Vert _{L^{1}(B(0,1))}\leq \sum_{j\in \mathbb{N}}\Vert
\kappa _{j,r}\Vert _{L^{1}(B(0,1))}\leq c\sum_{j\in \mathbb{N}}2^{-j\operatorname{Re}%
z}\leq c,$$ since $\operatorname{Re}z>0$.

*Case 2:* $R\geq \Vert \rho \Vert ^{2}+1.$

Recall that $r=\sqrt{R}$. So, the ball $B(0,1/r)$ is contained in the unit ball. Next, let $i\geq 0$ be such that $2^{i-1}<r\leq 2^{i}$ and consider the annulus $A_{p}=\{x\in X:2^{p}\leq |x|\leq 2^{p+1}$, with $p \geq -i$. We write $$B(0,1)\subset B(0,1/r)\bigcup_{p=-i}^{0}A_{p}.$$

Applying (\[alexo7\]), (\[kj,r\]) and (\[L2heat\]), and proceeding as in Case 1, one can show that $$\Vert \kappa _{j,r}\Vert _{L^{1}(B(0,1/r))}\leq c2^{-j\operatorname{Re}z}.
\label{B1/r}$$

So, to finish the proof of the lemma it remains to prove estimates of the kernels $\kappa _{j,r}$ on the annulus $A_{p}$. For that, we shall use the fact 