that the kernel $G_{t}(x,y)$, $x,y\in X$, of the wave operator $\cos t%
\sqrt{-\Delta }$, propagates with finite speed [@TAYLOR], that is $$\operatorname{supp}(G_{t})\subset \{(x,y):\;d(x,y)\leq |t|\}.  \label{finitespeed}$$Since $h_{j,r}$ is even, then by the Fourier inversion formula $$\begin{aligned}
\kappa _{j,r}(x)& =[h_{j,r}(\sqrt{-\Delta })p_{r^{-2}}(\cdot )](x)  \notag
\label{k_j,r} \\
& =(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )]( x)dt.  \notag\end{aligned}$$So, if $x\in A_{p}$, then $$\begin{aligned}
\kappa _{j,r}( x)& =(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}%
_{j,r}(t)[\cos t\sqrt{-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|\leq
2^{p-1}\}}]( x)dt  \notag \\
& +(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|>2^{p-1}\}}]( x)dt.  \notag \\
& =(2\pi )^{-1/2}\int_{|t|\geq 2^{p-1}}\hat{h}_{j,r}(t)[\cos t\sqrt{-\Delta }%
p_{r^{-2}}(\cdot )\mathbf{1}_{_{\{| y|\leq 2^{p-1}\}}}]( x)dt  \notag \\
& +(2\pi )^{-1/2}\int_{-\infty }^{+\infty }\hat{h}_{j,r}(t)[\cos t\sqrt{%
-\Delta }p_{r^{-2}}(\cdot )\mathbf{1}_{\{| y|>2^{p-1}\}}]( x)dt,\end{aligned}$$where in the last equality we have used the finite propagation speed of the wave operator: if $|y|\leq 2^{p-1}$ and $|x|\geq 2^p$, then (\[finitespeed\]) implies that $|t|\geq 2^{p-1}$.

Using (\[k\_j,r\]), the fact that $\Vert \cos t\sqrt{-\Delta }\Vert
_{2\rightarrow 2}\leq 1$, and the inequality $\Vert \widehat{h}_{j,r}\Vert
_{1}\leq \Vert h_{j,r}\Vert _{\infty }$, applying Cauchy-Schwarz we get that $$\begin{aligned}
\Vert \kappa _{j,r}\Vert _{L^{1}(A_{p})}& \leq c|A_{p}|^{1/2}\int_{|t|\geq
2^{p-1}}|\hat{h}_{j,r}(t)|\Vert p_{r^{-2}}\Vert _{2}dt  \label{L1norm} \\
& +c|A_{p}|^{1/2}\Vert h_{j,r}\Vert _{\infty }\Vert p_{r^{-2}}\mathbf{1}%
_{\{|x|>2^{p-1}\}}\Vert _{2}:=I_{1}+I_{2}.  \notag\end{aligned}$$

From (\[alexo7\]), (\[grigoryan\]) and the fact that $2^{i-1}<r\leq
2^{i} $, it follows that 