c_{31} |x|^{-N-2\alpha}.
\end{array}$$ $(ii)$ The case of $\tau=-N$. Similarly to (\[5.5\]) and (\[5.6\]), we have that for $|x|>4$, $$\arraycolsep=1pt
\begin{array}{lll}
 -(-\Delta)^\alpha w_{\tau }(x) \le\frac{|x|^{-N-2\alpha}\log^{\gamma_0}(\varrho_0+|x|)}2\int_{D_0}\frac{II_x(y)}{|y|^{N+2\alpha}}dy+c_{32}|x|^{-N-2\alpha}
\end{array}$$ and $$\arraycolsep=1pt
\begin{array}{lll}
 -(-\Delta)^\alpha w_{\tau }(x) \ge\frac{|x|^{-N-2\alpha}\log^{\gamma_0}(\varrho+|x|)}2\int_{D_0}\frac{II_x(y)}{|y|^{N+2\alpha}}dy-
 c_{32}|x|^{-2N-2\alpha}\log^{\gamma_0}|x|,
\end{array}$$ where $c_{32}>0$ and $$\arraycolsep=1pt
\begin{array}{lll}
II_x(y) =\frac{\log^{\gamma_0}(\varrho_0+|x||e_x+y|)}{\log^{\gamma_0}(\varrho_0+|x|)}|e_x+y|^{-N }+\frac{\log^{\gamma_0}(\varrho_0+|x||e_x-y|)}{\log^{\gamma_0}(\varrho_0+|x|)}
|e_x-y|^{-N}-2
\end{array}.$$ For $y\in D_1$, we have that $\frac{\log^{\gamma_0}(\varrho_0+|x||e_x+y|)}{\log^{\gamma_0}(\varrho_0+|x|)}\le 1$, then $$\begin{aligned}
  \int_{D_1}\frac{II_x(y)}{|y|^{N+2\alpha}}dy &\le& c_{33}\int_{B_{\frac12}(0)\setminus B_{\frac1{|x|}}(0)}|y|^{-N}dy+c_{34}
   \\& \le& c_{35} \log|x|+c_{34}\end{aligned}$$ and $$\begin{aligned}
  \int_{D_1}\frac{II_x(y)}{|y|^{N+2\alpha}}dy &\ge&
  c_{33}\int_{|x|^{-1}}^{\frac12}r^{-1}\frac{\log^{\gamma_0}(\varrho_0+|x|r)}{\log^{\gamma_0}(\varrho_0+|x|)}dr-c_{34}
  \\& =&   c_{33}\int_{1}^{\frac12|x|}s^{-1}\frac{\log^{\gamma_0}(\varrho_0+s)}{\log^{\gamma_0}(\varrho_0+|x|)}ds-c_{34}
   \\& \ge& c_{35}\log|x|-c_{34},\end{aligned}$$ where $c_{33},c_{34},c_{35}>0$. By the fact that $$\int_{D_1}\frac{II_x(y)}{|y|^{N+2\alpha}}dy=\int_{D_2}\frac{II_x(y)}{|y|^{N+2\alpha}}dy,$$ there exists $c_{36}>0$ such that $$\label{7.1}
 \arraycolsep=1pt
 \begin{array}{lll}
  \frac1{c_{36}}\log|x|\le \int_{D_1\cup D_2}\frac{II_x(y)}{|y|^{N+2\alpha}}dy\le c_{36} \log|x|.
 \end{array}$$ Similarly to (\[15-09-1\])-(\[15-09-3\]), there exists $c_{37}>0$ such that $$\label{7.2}
 \arraycolsep=1pt
 \begin{array}{lll}
 |\int_{{\mathbb{R}}^N\setminus (B_{\frac12}(e_x)\cup B_{\frac12}(-e_x))}\frac{II_x(y)}{|y|^{N+2\alpha}}dy|\le c_{37},
 \end{array}$$ which, together with (\[7.1\]), imply (\[5.4\]).

$(iii)$ The case that $\tau\in(-N,-N+2\alpha)$. By Lemma 3.1 and Lemma 3.2 in [@FQ2], we have $$\label{7.3}
  (-\Delta)^\alpha |x|^{\tau }=c(\tau)|x|^{\tau -2\alpha},$$ where $c(\tau)<0$.

Let $\tilde w(x)=w_{\tau }(x)-|x|^{\tau }$ for $x\in {\mathbb{R}}^N\setminus\{0\}$, then $\tilde w=0$ in $B_1^c(0)$. For $|x|>4$, we have that $$\label{5.7}
\arraycolsep=1pt
\begin{array}{lll}
  |(-\Delta)^\alpha\tilde w(x)| \le\int_{B_1(0)}\frac{w_\tau(z)+|z|^{\tau }}{|z-x|^{N+2\alpha}}dz \\[3mm]
  \phantom{------}
   \le(|x|-1)^{-N-2\alpha}  \int_{B_1(0)}(w_\tau(z)+|z|^{\tau })dz,
\end{array}$$ which, together with (\[7.3\]), imply (\[5.05\]). We complete the proof.$\Box$

\[lm 5.2\] Let $\eta:{\mathbb{R}}^N\to[0,1]$ be a $C^2$ function with support in $B_2(0)$ and $\eta=1$ in $B_1(0)$, $\bar w(x)=\eta(x)|x|^{-N+2\alpha}$ for $x\in {\mathbb{R}}^N$. 