observe that $\tilde v(P)=v(P)$. Accordingly, by the Boundary Harnack Inequality (see Theorem 1 on page 44 of [@MR1438304]), we have that, for all ${\varepsilon}>0$, $$\label{5302345}
\left[ \frac1C,C\right]\ni \Lambda:=\lim_{x_n\searrow 0} \frac{v({\varepsilon},x_n)}{
\tilde v({\varepsilon},x_n)}=
\lim_{\tau\searrow 0} \frac{P_2^\sigma \;v({\varepsilon},\tau)}{
v(P)\;\tau^\sigma},$$ for some $C{\geqslant}1$.

Now, for all ${\varepsilon}>0$ and $\zeta\in S^{n-2}$ we define $$\label{eca2} w(x):=v(x'\cdot\zeta+{\varepsilon},x_n).$$ Also, if $r:=\frac{{\varepsilon}}{2}$ and $x\in B_r\cap\{x_n>0\}$, it follows that $x'\cdot\zeta+{\varepsilon}+\frac{x_n}{\alpha}{\geqslant}-r+{\varepsilon}>0
$, and thus $(x'\cdot\zeta+{\varepsilon},x_n)\in
{\mathcal{C}}_\alpha$. Consequently, by , we have that $$\label{eca4}
{\mbox{$(-\Delta)^\sigma w=0$ in~$B_r\cap\{x_n>0\}$.}}$$ Moreover, using  and , we see that $$\label{eca5}
{\mbox{if~$x_n<0$
then~$w(x)=0$. }}$$ [F]{}rom  and  (see e.g. Section 1.1 in [@MR3694738], or [@MR3293447; @MR3276603]), it also follows that the function ${\rm dist}_{\{x_n<0\}}^{-\sigma} w$ belongs to $C^\infty(B_r\cap\overline{ \{x_n>0\} })$, hence we can define $$\label{2389IKA}
\psi(x'):=\lim_{x_n\searrow0} \frac{w(x',x_n)}{{\rm dist}_{\{x_n<0\}}^\sigma(x',x_n)}
=
\lim_{x_n\searrow0} \frac{w(x',x_n)}{x_n^\sigma}={\mathcal{T}}w(x').$$ This, and  give that $
w\in{\mathcal{H}}_k$. As a consequence, recalling , we have that $$\label{PL} \sum_{{\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}}\atop{|\gamma|{\leqslant}k}}
\omega_\gamma\,
D^\gamma\psi(0)=0.$$ In view of  and , we also observe that $$\label{8343546}
\psi(x')=
\lim_{x_n\searrow0} \frac{v(x'\cdot\zeta+{\varepsilon},x_n)}{x_n^\sigma}.$$ Accordingly, recalling , $$\label{8343546BIS}
\psi(0)=
\lim_{x_n\searrow0} \frac{v({\varepsilon},x_n)}{x_n^\sigma}=\frac{\Lambda\,v(P)}{P_2^\sigma}=:\tilde\Lambda\ne0.$$ Also, by , we can write that $$\label{7u12jd2}
\psi(x')=\psi_0(x'\cdot\zeta),$$ for some $\psi_0:{\ensuremath{\mathbb{R}}}\to[0,+\infty)$. Hence, recalling the homogeneity in , for all $t>0$, $$\begin{aligned}
&&\psi_0\big(t (x'\cdot\zeta+{\varepsilon})-{\varepsilon}\big)=
\psi\big(t (x'+{\varepsilon}\zeta)-{\varepsilon}\zeta\big)=
\lim_{x_n\searrow0} \frac{v\big((t (x'+{\varepsilon}\zeta)-{\varepsilon}\zeta)\cdot\zeta+{\varepsilon}, x_n\big)}{x_n^\sigma}\\
&&\qquad=
\lim_{x_n\searrow0} \frac{v(
t x'\cdot\zeta+t{\varepsilon}, x_n)}{x_n^\sigma}=
\lim_{y_n\searrow0} 
\frac{v\big(
t 