For $x=k_1(\exp H)k_2\in G$, with $|x|>1$ and $k\in \mathbb{N}$ with $k>\frac{n}{2}-\frac{l}{4}$, we have that $$|\kappa _{R}^{z}(x)|\leq c\varphi _{0}(x)\left( \int\limits_{|H|>|x|-\frac{1%
}{2}}\left( \sum_{|\alpha |\leq 2k}|\partial _{H}^{\alpha }(\mathcal{F}%
^{-1}s_{R}^{z})(H)|\right) ^{2}\right) ^{1/2}.  \label{kernelLOMA}$$

Thus, to estimate the kernel for $|x|>1$, it suffices to obtain estimates for the derivatives of the euclidean inverse Fourier transform of $%
s_{R}^{z}(\lambda )$. Denote by $\mathcal{J}_{\nu }(t)=t^{-\nu }J_{\nu }(t)$, $t>0$, where $J_{\nu }$ is the Bessel function of order $\nu $. Then, it holds $$(\mathcal{F}^{-1}s_{R}^{z})(\exp H)=c(n,z)R^{-z}(R-\Vert \rho \Vert
^{2})^{z+l/2}\mathcal{J}_{z+l/2}\left( \sqrt{R-\Vert \rho \Vert ^{2}}%
|H|\right) ,  \label{Fourier}$$ [@DAVCHANG; @GIUMA], and we shall need the following auxiliary lemma.

For every multi-index $\alpha $, it holds that $$|\partial _{H}^{\alpha }\mathcal{J}_{z+l/2}(\sqrt{R-\Vert \rho \Vert ^{2}}%
|H|)|\leq c(R-\Vert \rho \Vert ^{2})^{\frac{|\alpha |}{2}-(\frac{\operatorname{Re}z}{%
2}+\frac{l+1}{4})}|H|^{-(\operatorname{Re}z+\frac{l+1}{2})}.  \label{Bessel deriv}$$

Using the identity $\mathcal{J}_{\nu }^{\prime }(t)=-t\mathcal{J}_{\nu
+1}(t) $, it is straightforward to get that $$\mathcal{J}_{\nu }^{(a)}(t)=(-1)^{a}t^{a}\mathcal{J}_{\nu
+a}(t)+\sum_{j=1}^{[a/2]}c_{j}^{a}t^{a-2j}\mathcal{J}_{\nu +a-j}(t),\;a\in
\mathbb{N},  \label{nth derivative}$$for some constants $c_{j}^{a}$, where $[a]$ denotes the integer part of $a$. Applying the inequality $$\ |\mathcal{J}_{\mu }(t)|\leq c_{\mu }t^{-(\operatorname{Re}\mu +1/2)},\text{ for
all }t>0,$$[@GIUMA], it follows that $$|\partial _{H}^{\alpha }\mathcal{J}_{\nu }(\sqrt{R-\Vert \rho \Vert ^{2}}%
|H|)|\leq c(R-\Vert \rho \Vert ^{2})^{\frac{|\alpha |}{2}-(\frac{\operatorname{Re}%
\nu }{2}+\frac{1}{4})}|H|^{-(\operatorname{Re}\nu +\frac{1}{2})}$$and (\[Bessel deriv\]) follows by taking $\nu =z+l/2$.

\[infinity\] If $R\geq \|\rho\|^2+1$, then $$\label{kappainf}
|\kappa_R^z(x)|\leq c\varphi_{0 }(x)R^{-\frac{1}{2}(\operatorname{Re}z-n+\frac{1}{2}%
)}|x|^{-\operatorname{Re}z-\frac{1}{2}}, 