\quad |x|>1.$$

From (\[Bessel deriv\]), we get that $$\begin{aligned}
I^{2}& :=\int\limits_{|H|>|x|-\frac{1}{2}}\left( \sum_{|\alpha |\leq
2k}\left\vert \partial _{H}^{a}\mathcal{J}_{z+l/2}\left( \sqrt{R-\Vert \rho
\Vert ^{2}}|H|\right) \right\vert \right) ^{2}dH  \notag  \label{estimate} \\
& \leq c\left( \sum_{|\alpha |\leq 2k}(R-\Vert \rho \Vert ^{2})^{a/2}\right)
^{2}\times\notag \\ &\times \int\limits_{|H|>|x|-\frac{1}{2}}\left( (R-\Vert \rho \Vert ^{2})^{-(%
\frac{\operatorname{Re}z}{2}+\frac{l+1}{4})}|H|^{-(\operatorname{Re}z+\frac{l+1}{2})}\right)
^{2}dH  \notag \\
& \leq c(R-\Vert \rho \Vert ^{2})^{-2(\frac{\operatorname{Re}z}{2}+\frac{l+1}{4}%
)+2k}\int\limits_{u>|x|-\frac{1}{2}}u^{-(l+1)-2\operatorname{Re}z}u^{l-1}du  \notag
\\
& \leq c(R-\Vert \rho \Vert ^{2})^{-2(\frac{\operatorname{Re}z}{2}+\frac{l+1}{4}%
)+2k}\left( |x|-\frac{1}{2}\right) ^{-2\operatorname{Re}z-1}.\end{aligned}$$ For $R\geq \Vert \rho \Vert ^{2}+1$, since $k>\frac{n}{2}-\frac{l}{4}$, we have that $$I\leq c(R-\Vert \rho \Vert ^{2})^{-(\frac{\operatorname{Re}z+l-n}{2}+\frac{1}{4}%
)}\left( |x|-\frac{1}{2}\right) ^{-\operatorname{Re}z-\frac{1}{2}}.
\label{estimate1}$$Using (\[estimate1\]) and (\[Fourier\]), from (\[kernelLOMA\]) we obtain that $$\begin{aligned}
|\kappa _{R}^{z}(x)|& \leq c\varphi _{0}(x)R^{-\operatorname{Re}z}(R-\Vert \rho
\Vert ^{2})^{\operatorname{Re}z+\frac{l}{2}}\times \notag \\ &\times (R-\Vert \rho \Vert ^{2})^{-(\frac{\operatorname{%
            Re}z+l-n}{2}+\frac{1}{4})}\left( |x|-\frac{1}{2}\right) ^{-\operatorname{Re}z-\frac{1%
}{2}} \\
& \leq c\varphi _{0}(x)R^{-\frac{1}{2}(\operatorname{Re}z-n+\frac{1}{2})}|x|^{-\operatorname{%
Re}z-\frac{1}{2}},\quad |x|>1.\end{aligned}$$

Using the estimate (\[estimate1\]) and proceeding as above, one can prove the following result.

\[extracase\]If $\Vert \rho \Vert ^{2}\leq R\leq \Vert \rho \Vert
^{2}+1, $ then $$|\kappa _{R}^{z}(x)|\leq c\varphi _{0}(x)|x|^{-\operatorname{Re}z-\frac{1}{2}%
},\;|x|>1.$$

Finally, we shall prove the following result, which, combined with Proposition \[T0X\], finishes the proof of Theorem 1.

\[11\] \[TinfX\] Let $\operatorname{Re}z\geq n-\frac{1}{2}$ and consider $q>2$. Then for every $p$ such that $1\leq p\leq q^{\prime }$, $S_{R}^{z,\infty }$ is continuous from $L^{p}(X)$ to $L^{r}(X)$ for every $r\in \lbrack qp^{\prime
