Appendix A
In this appendix, we give a complete proof of Proposition 3.3.
Proof of Proposition 3.3
The estimate of \((-\Delta )^sw_\mu \) base on the estimate of \((-\Delta )^sh_\mu \), where \(h_\mu =|x|^{-\mu }\ (x\ne 0)\). So we first estimate \((-\Delta )^sh_\mu \).
For any given \(x\in \mathbb {R}^N\) such that \(|x|>1\), by changes of variable we have
$$\begin{aligned} \frac{1}{2}(-\Delta )^sh_\mu =&\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_r(x)}\frac{|x|^{-\mu }-|y|^{-\mu }}{|x-y|^{N+2s}}\textrm{d}y\quad (\textrm{set}\ y=|x|y')\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{|y'|^{\mu }-1}{|y'|^{\mu }|y'-\vec {e}_1|^{N+2s}}\textrm{d}y'\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\Bigg (\int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad +\int _{(\mathbb {R}^N{\setminus } B_1(0)){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg ) \end{aligned}$$
(A.1)
where \(\vec {e}_1:=\frac{x}{|x|}\) is a unit vector.
For the coordinate transformation \(y=\frac{y'}{|y'|^2}\) (a inversion of a sphere), we have
$$\begin{aligned} |y||y'|=1,\ dy=|y'|^{-2N}dy'\ (\mathrm{see \ [6, Section\ 2.2.2]}),\ |y-\vec {e}_1||y'|=|y'-\vec {e}_1|, \end{aligned}$$
where we have used that
$$\begin{aligned} |y-\vec {e}_1|^2|y'|^2=(|y|^2-2y\cdot \vec {e}_1+1)|y'|^2=1-2y'\cdot \vec {e}_1+|y'|^2=|y'-\vec {e}_1|^2. \end{aligned}$$
It follows that
$$\begin{aligned} \int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y'|^{\mu }}{|y'|^{N-2s}|y'-\vec {e}_1|^{N+2s}}\textrm{d}y' \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } B_{r}(\vec {e}_1))}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\&+\int _{B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\&-\int _{(B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y, \end{aligned}$$
(A.2)
where
$$\begin{aligned}(B_{r}(\vec {e}_1))^*:=\Bigg \{\frac{y}{|y|^2}\mid y\in B_r(\vec {e}_1)\Bigg \}. \end{aligned}$$
Since \(B_{r/2}(\vec {e}_1)\subset (B_{r}(\vec {e}_1))^*\subset B_{3r/2}(\vec {e}_1)\ \textrm{as}\ r\rightarrow 0,\) we have that
$$\begin{aligned}\frac{|1-|y|^{\mu }|}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\le O\Bigg (\frac{1}{r^{N+2s-1}}\Bigg ),\quad y\in \big (B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*\big )\cup \big (B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)\big ).\end{aligned}$$
On the other hand, we can verify that
$$\begin{aligned}|B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*|+|B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)|=O(r^{N+1})\ \textrm{as}\ r\rightarrow 0,\end{aligned}$$
it follows that
$$\begin{aligned}&\int _{B_{r}(\vec {e}_1){\setminus } (B_{r}(\vec {e}_1))^*}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y=O(r^{2-2s})\rightarrow 0\ \textrm{as}\ r\rightarrow 0, \\&\int _{(B_{r}(\vec {e}_1))^*{\setminus } B_{r}(\vec {e}_1)}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y=O(r^{2-2s})\rightarrow 0\ \textrm{as}\ r\rightarrow 0. \end{aligned}$$
Substituting the estimates above into (A.2), we obtain that
$$\begin{aligned}&\int _{B_1(0){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y|^{\mu }|y-\vec {e}_1|^{N+2s}}\textrm{d}y \nonumber \\ =&\int _{(\mathbb {R}^N\backslash B_1(0)){\setminus } B_{r}(\vec {e}_1))}\frac{1-|y|^{\mu }}{|y|^{N-2s}|y-\vec {e}_1|^{N+2s}}\textrm{d}y+O(r^{2-2s})\ \textrm{as}\ r\rightarrow 0. \end{aligned}$$
(A.3)
Putting (A.3) into (A.1) yields
$$\begin{aligned} \frac{1}{2}(-\Delta )^sh_\mu =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_1(0)){\setminus } B_{r}(\vec {e}_1)}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )\textrm{d}y\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\int _{\mathbb {R}^N{\setminus } B_1(0)}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )\textrm{d}y :=A_\mu \frac{1}{|x|^{\mu +2s}}, \end{aligned}$$
(A.4)
where \(\vec {e}_1\) is not a singular point for the last integral in (A.4) since
$$\begin{aligned}\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}\Bigg ( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}}\Bigg )=O\Bigg (\frac{1}{|x-\vec {e}_1|^{N+2s-2}}\Bigg )\ \ \textrm{as}\ y\rightarrow \vec {e}_1.\end{aligned}$$
Noting the asymptotic behavior of \(\frac{|y|^{\mu }-1}{|y-\vec {e}_1|^{N+2s}}( \frac{1}{|y|^{\mu }}-\frac{1}{|y|^{N-2s}})\) as \(|y|\rightarrow \infty \), it is easy to check that \(A_\mu \) satisfies
$$\begin{aligned} \begin{aligned} \left\{ \begin{array}{ll} A_\mu \in (0,+\infty ),&{}\quad \textrm{if }\ 0<\mu<N-2s,\\ A_\mu =0, &{} \quad \textrm{if }\ \mu =N-2s;\\ A_\mu \in (-\infty ,0),&{} \quad \textrm{if }\ N-2s<\mu <N\\ A_\mu =-\infty ,&{}\quad \textrm{if}\ \mu \ge N. \end{array} \right. \end{aligned} \end{aligned}$$
(A.5)
Now we are ready to estimate \((-\Delta )^sw_{\mu }\) according to different cases stated in (A.5).
Case 1. \(\mu \in (0,N){\setminus }\{N-2s\}\).
By changing variable as in (A.1), we have
$$\begin{aligned} \begin{aligned}&\frac{1}{2}\left| (-\Delta )^sw_\mu -(-\Delta )^sh_\mu \right| \\ \quad \le&\frac{1}{|x|^{\mu +2s}}\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-1+|y|^{-\mu }}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\\:=&\frac{1}{|x|^{\mu +2s}}\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |. \end{aligned} \end{aligned}$$
(A.6)
For any \(M>2\) and \(\rho '\in (r,1/2)\), letting \(|x|>2M\), we have
$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\ \le&\int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y+\Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&+\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y +\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y. \end{aligned}$$
(A.7)
Clearly, \(|L(x,y)|\le 4\) for \(|y-\vec {e}_1|>M\), then
$$\begin{aligned} \int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le 4\int _{|y-\vec {e}_1|>M}\frac{1}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le \frac{C}{M^{2s}}. \end{aligned}$$
(A.8)
For \(|y-\vec {e}_1|<\rho '\), we have \(1/2 \le |y|\le 3/2\), then by Taylor expansion, we have
$$\begin{aligned} |y|^{-\mu }-|\vec {e}_1|=-\mu \vec {e}_1\cdot (y-\vec {e}_1)+O(|y-\vec {e}_1|^2), \end{aligned}$$
(A.9)
$$\begin{aligned}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-(|x|^{-2}+|\vec {e}_1|^2)^{-\frac{\mu }{2}}= -\mu (|x|^{-2}+1)^{-\frac{\mu }{2}-1}\vec {e}_1\cdot (y-\vec {e}_1)+O(|y-\vec {e}_1|^2). \end{aligned}$$
By symmetry,
$$\begin{aligned} \int _{r<|y-\vec {e}_1|<\rho '}\frac{\vec {e}_1\cdot (y-\vec {e}_1)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y=0. \end{aligned}$$
(A.10)
Therefore,
$$\begin{aligned} \Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg | \le C\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{1}{|y-\vec {e}_1|^{N+2s-2}}\textrm{d}y\le C(\rho ')^{2-2s}. \end{aligned}$$
(A.11)
For \(y\in \{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}\), we have
$$\begin{aligned}{} & {} |(|x|^{-2}+1)^{-\mu }-1|\le C|x|^{-2},\\{} & {} \big |(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-|y|^{-\mu }\big |\le \frac{C}{(\rho ')^{\mu +1}}|x|^{-2}, \end{aligned}$$
and thereby,
$$\begin{aligned} \int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y&\le C|x|^{-2}\Bigg (1+\frac{C}{(\rho ')^{\mu +1}}\Bigg )\int _{\{\rho '\le |y-\vec {e}_1|\}}\frac{1}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\le C|x|^{-2}\Bigg (1+\frac{C}{(\rho ')^{\mu +1}}\Bigg )\frac{1}{(\rho ')^{2s}}. \end{aligned}$$
(A.12)
For \(y\in \{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}\), then \(|y-\vec {e}_1|\ge 1-|y|\ge 1/2\) and
$$\begin{aligned}|L(x,y)|=1-(|x|^{-2}+1)^{-\frac{\mu }{2}}+|y|^{-\mu }-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\le 1+|y|^{-\mu },\end{aligned}$$
and consequently,
$$\begin{aligned} \int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|<\rho '\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le&\frac{1}{(1/2)^{N+2s}}\int _{\{|y|<\rho '\}}(1+|y|^{-\mu })\textrm{d}y\nonumber \\ =&C(\rho ')^{N}+C(\rho ')^{N-\mu }. \end{aligned}$$
(A.13)
As a result, we conclude from (A.17)–(A.8) and (A.11)–(A.13) that
$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+(\rho ')^{N}+(\rho ')^{N-\mu }+|x|^{-2}\big (1+\frac{1}{(\rho ')^{\mu +1}}\big )\frac{1}{(\rho ')^{2s}}\Bigg ) \end{aligned}$$
for a constant \(C>0\) independent of \(M>2\) and \(\rho '\in (0,1/2)\). Letting \(M\rightarrow +\infty \) and \(\rho '\rightarrow 0_+\), we have
$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \mathop {\lim }\limits _{\begin{array}{c} M\rightarrow +\infty \\ {\rho '\rightarrow 0_+} \end{array}} \lim _{|x|\rightarrow \infty }C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+(\rho ')^{N}+(\rho ')^{N-\mu } +|x|^{-2}\big (1+\frac{1}{(\rho ')^{\mu +1}}\big )\frac{1}{(\rho ')^{2s}}\Bigg )\nonumber \\&\quad =0. \end{aligned}$$
(A.14)
Then there exists \(R_\mu >0\) such that
$$\begin{aligned} \begin{aligned} \Bigg |\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\le \frac{1}{2}|A_\mu |,\ \ |x|>R_\mu . \end{aligned} \end{aligned}$$
(A.15)
Putting (A.1)–(A.6) and (A.15) together, we infer that
$$\begin{aligned}&0<\frac{A_\mu }{2}\frac{1}{|x|^{\mu +2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le \frac{3A_\mu }{2}\frac{1}{|x|^{\mu +2s}},\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu \in (0,N-2s);\\&\frac{3A_\mu }{2}\frac{1}{|x|^{\mu +2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le \frac{A_\mu }{2}\frac{1}{|x|^{\mu +2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu \in (N-2s,N). \end{aligned}$$
Case 2. \(\mu \ge N\).
Also by changing variable as in (A.1), there holds
$$\begin{aligned} \frac{1}{2}(-\Delta )^sw_\mu =&\frac{1}{|x|^{\mu +2s}}\lim _{r\rightarrow 0}\int _{\mathbb {R}^N{\setminus } B_{r}(\vec {e}_1)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\ =&\frac{1}{|x|^{\mu +2s}}\Bigg (\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\ +\int _{B_{1/2}(0)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y -\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg ). \end{aligned}$$
(A.16)
Same as (A.6), we denote \(L(x,y):=(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}-1+|y|^{-\mu }\). For any \(M>2\) and \(\rho '\in (r,1/2)\), letting \(|x|>2M\), be the same arguments as A.8, (A.11) and (A.12), we have
$$\begin{aligned}&\Bigg |\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \int _{|y-\vec {e}_1|>M}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y+\Bigg |\lim _{r\rightarrow 0}\int _{r<|y-\vec {e}_1|<\rho '}\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\qquad +\int _{\{\rho '\le |y-\vec {e}_1|\le M\}\cap \{|y|\ge \frac{1}{2}\}}\frac{|L(x,y)|}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \le \frac{C}{M^{2s}}+C(\rho ')^{2-2s}+C|x|^{-2}\frac{1}{(\rho ')^{2s}}, \end{aligned}$$
(A.17)
which implies that
$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\Bigg |\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{L(x,y)}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\Bigg |\nonumber \\&\quad \le \mathop {\lim }\limits _{\begin{array}{c} M\rightarrow +\infty \\ {\rho '\rightarrow 0_+} \end{array}}\lim _{|x|\rightarrow \infty }C\Bigg (\frac{1}{M^{2s}}+(\rho ')^{2-2s}+|x|^{-2}\frac{1}{(\rho ')^{2s}}\Bigg )=0. \end{aligned}$$
(A.18)
In view of (A.9) and (A.10), the following integral converges to a constant independent of x as \(r\rightarrow 0\),
$$\begin{aligned}\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1)}\frac{1-|y|^{-\mu }}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y:=C^{*}\in {\mathbb {R}},\end{aligned}$$
and thereby from (A.18),
$$\begin{aligned}&\lim _{|x|\rightarrow \infty }\lim _{r\rightarrow 0}\int _{(\mathbb {R}^N{\setminus } B_{1/2}(0))\backslash B_{r}(\vec {e}_1) }\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}-(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y=C^*. \end{aligned}$$
(A.19)
Obviously,
$$\begin{aligned} \int _{B_{1/2}(0)}\frac{(|x|^{-2}+1)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\le 2^{2s}\omega _N(|x|^{-2}+1)^{-\frac{\mu }{2}}\le 2^{2s}\omega _N. \end{aligned}$$
(A.20)
Letting \(|x|>4\), we have
$$\begin{aligned}&\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \le 2^{N+2s}\int _{B_{1/2}(0)}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\textrm{d}y =2^{N+2s}|x|^{\mu -N}\int _{B_{|x|/2}(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}\textrm{d}y\nonumber \\&\quad \le 2^{N+2s}|x|^{\mu -N}\Bigg (\int _{B_1(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}+\omega _N\int ^{|x|/2}_1\frac{1}{r^{\mu -N+1}}\textrm{d}r\Bigg )\nonumber \\&\quad \le \left\{ \begin{array}{ll} 2^{N+2s}(\omega _N\ln |x|+C),&{} \quad \mu =N, \\ 2^{N+2s}(\frac{\omega _N}{\mu -N}+C)|x|^{\mu -N},&{} \quad \mu >N, \end{array} \right. \end{aligned}$$
(A.21)
where \(\omega _N:=\int _{\partial B_1(0)}dS\). On the other hand,
$$\begin{aligned}&\int _{B_{1/2}(0)}\frac{(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}}{|y-\vec {e}_1|^{N+2s}}\textrm{d}y\nonumber \\&\quad \ge (1/2)^{N+2s}\int _{B_{1/2}(0)}(|x|^{-2}+|y|^2)^{-\frac{\mu }{2}}\textrm{d}y=(1/2)^{N+2s}|x|^{\mu -N}\int _{B_{|x|/2}(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}\textrm{d}y\nonumber \\&\quad \ge (1/2)^{N+2s}|x|^{\mu -N}\Bigg (\int _{B_1(0)}\frac{1}{(1+|y|^2)^{\frac{\mu }{2}}}+\frac{\omega _N}{2^\mu }\int ^{|x|/2}_1\frac{1}{r^{\mu -N+1}}\textrm{d}r\Bigg )\nonumber \\&\quad \ge \left\{ \begin{array}{ll} (1/2)^{N+2s}\left( \frac{\omega _N}{2^N}\ln |x|-C\right) ,&{} \quad \mu =N, \\ \frac{N\omega _N}{2^{2N+2s}}|x|^{\mu -N},&{} \quad \mu >N. \end{array} \right. \end{aligned}$$
(A.22)
Summing up the estimates (A.16)–(A.22) above, there exists \(R_\mu >0\) and \({\tilde{C}}_1,{\tilde{C}}_2, {\tilde{C}}_3, {\tilde{C}}_4>0\) such that
$$\begin{aligned} \begin{aligned}&-\frac{{\tilde{C}}_2\ln |x|}{|x|^{N+2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le -\frac{{\tilde{C}}_1\ln |x|}{|x|^{N+2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu =N;\\&-\frac{{\tilde{C}}_4}{|x|^{N+2s}}\le \frac{1}{2}(-\Delta )^sw_\mu \le -\frac{{\tilde{C}}_3}{|x|^{N+2s}}<0,\ \textrm{if }\ |x|>R_\mu \ \textrm{and}\ \mu >N, \end{aligned} \end{aligned}$$
where
$$\begin{aligned}{\tilde{C}}_1=\frac{\omega _N}{2^{2N+2s+1}},\, {\tilde{C}}_2=2^{N+2s+1}\omega _N,\, {\tilde{C}}_3=\frac{\omega _N}{2^{2N+2s+1}},\,{\tilde{C}}_4=\frac{2^{N+2s+1}}{\mu -N}\omega _N.\end{aligned}$$
Case 3. \(\mu =N-2s\).
In this case, \(w_\mu =(1+|x|^2)^{-\frac{N-2s}{2}}\) is the fundamental solution of the critical fractional equation
$$\begin{aligned}(-\Delta )^su=C_{N-2s}u^{2_s^*-1}\end{aligned}$$
for some constant \(C_{N-2s}>0\) (see [6]).
As a consequence, the proof is completed. \(\square \)