\end{array}$$ where $D_1=B_{\frac12}(-e_x)\setminus B_{\frac1{|x|}}(-e_x)$ and $D_2=B_{\frac12}(e_x)\setminus B_{\frac1{|x|}}(e_x)$.*

In fact, for $y\in D_1$, we observe that $$-2\le |e_x-y|^{\tau }-2 \le -1\quad {\rm and }\quad \frac12\le |y|\le \frac32,$$ then $$\begin{aligned}
  \int_{D_1}\frac{I_x(y)}{|y|^{N+2\alpha}}dy &\le& c_{19}\int_{B_{\frac12}(0)\setminus B_{\frac1{|x|}}(0)}|y|^{\tau }dy+\int_{D_1}\frac{|e_x-y|^{\tau }-2}{|y|^{N+2\alpha}}dy
 \\&\le& c_{20}\int_{|x|^{-1}}^{\frac12}r^{\tau +N-1}dr
   \\& \le& c_{21} |x|^{-N-\tau }\end{aligned}$$ and $$\begin{aligned}
  \int_{D_1}\frac{I_x(y)}{|y|^{N+2\alpha}}dy &\ge&
  c_{22}\int_{|x|^{-1}}^{\frac12}r^{\tau +N-1}dr-c_{23}
   \\& \ge& c_{24}|x|^{-N-\tau}-c_{23},\end{aligned}$$ where $c_{18},...,c_{24}$ are positive constants. Since $-N-\tau>0$, there exist $R\ge4$ and $c_{25}>0$ such that for $|x|\ge R$, $$\frac1{c_{25}}|x|^{-N-\tau }\le \int_{D_1}\frac{I_x(y)}{|y|^{N+2\alpha}}dy\le c_{25} |x|^{-N-\tau }.$$ By the fact $$\int_{D_1}\frac{I_x(y)}{|y|^{N+2\alpha}}dy=\int_{D_2}\frac{I_x(y)}{|y|^{N+2\alpha}}dy,$$ we obtain (\[15-09-1\]).

*Claim 2. There exists $c_{26}>0$ such that $$\label{15-09-2}
  | \int_{B_{\frac12}(0)}\frac{I_x(y)}{|y|^{N+2\alpha}}dy|\le c_{26}.$$* Indeed, since function $I_x$ is $C^2$ in $\bar B_{\frac12}(0)$ such that $$I_x(0)=0\quad{\rm and}\quad  I_x(y)=I_x(-y),\quad y\in \bar B_{\frac12}(0),$$ then $\nabla I_x(0)=0$ and there exists $c_{27}>0$ such that $$|D^2 I_x(y)|\le c_{27}, \quad y\in B_{\frac12}(0).$$ Then we have $$I_x(y)\le c_{27}|y|^2,\quad y\in B_{\frac12}(0),$$ which implies that $$|\int_{B_{\frac12}(0)}\frac{I_x(y)}{|y|^{N+2\alpha}}dy|\le c_{27}\int_{B_{\frac12}(0)}\frac{|y|^{2}}{|y|^{N+2\alpha}}dy\le c_{26}.$$

*Claim 3. There exists $c_{28}>0$ such that $$\label{15-09-3}
   |\int_{D_3}\frac{I_x(y)}{|y|^{N+2\alpha}}dy|\le c_{28},$$ where $D_3={\mathbb{R}}^N\setminus (B_{\frac12}(0)\cup B_{\frac12}(e_x)\cup B_{\frac12}(-e_x))=D_0\setminus(D_1\cup D_2\cup B_{\frac12}(0))$.*

In fact, for $y\in D_3$, we observe that there exists $c_{29}>0$ such that $|I_x(y)|\le c_{29}$ and $$|\int_{D_3}\frac{I_x(y)}{|y|^{N+2\alpha}}dy|\le \int_{{\mathbb{R}}^N\setminus B_{\frac12}(0)}\frac{c_{29}}{|y|^{N+2\alpha}}dy\le c_{30},$$ where $c_{30}>0$. Since $\lim_{|x|\to\infty}|x|^{-N-\tau }=\infty$ for $\tau<-N$, by (\[15-09-1\])-(\[15-09-3\]), there exist $R\ge 4$ and $c_{31}>1$ such that for $|x|\ge R$, $$\arraycolsep=1pt
\begin{array}{lll}
\frac1{c_{31}}|x|^{-N-2\alpha}\le  -(-\Delta)^\alpha w(x)
\le 