possibly to be taken conveniently small, and we suppose that $$\label{mu1}
f(x')=\frac{(x')^\mu}{\mu!}\qquad{\mbox{for some }}\mu\in{\ensuremath{\mathbb{N}}}^{n-1}.$$ Then, we apply Lemma \[fan\], finding a suitable function $u_\star
\in{\mathcal{H}}_k$ with $\phi_\star:={\mathcal{T}}u_\star$ that satisfies $$\label{mu2} D^\gamma \phi_\star(0)=\begin{cases}
1 & {\mbox{ if }} \gamma=\mu,\\
0 & {\mbox{ if $|\gamma|{\leqslant}|\mu|+k$ and~$ \gamma\ne\mu$.}}
\end{cases}$$ We define $$\begin{aligned}
&&
u_{\varepsilon}(x):= {\varepsilon}^{-\sigma-|\mu|} \,u_\star({\varepsilon}x)\\
{\mbox{and }}&& f_{\varepsilon}(x'):={\varepsilon}^{-|\mu|} \,\phi_\star({\varepsilon}x').\end{aligned}$$ Then, if $x\in B_1\cap\{x_n>0\}$, we have that $(-\Delta)^\sigma u_{\varepsilon}(x)=
{\varepsilon}^{\sigma-|\mu|} \,(-\Delta)^\sigma u_\star({\varepsilon}x)=0$ as long as ${\varepsilon}$ is sufficiently small. In addition, we see that $u_{\varepsilon}<0$ in $\{x_n<0\}$, and that $$\lim_{x_n\searrow0} \frac{u_{\varepsilon}(x',x_n)}{x_n^\sigma}=
\lim_{x_n\searrow0} \frac{{\varepsilon}^{-\sigma-|\mu|} \,u_\star({\varepsilon}x',{\varepsilon}x_n)}{x_n^\sigma}=
\lim_{\tau\searrow0} \frac{{\varepsilon}^{-|\mu|} \,u_\star({\varepsilon}x',\tau)}{\tau^\sigma}={\varepsilon}^{-|\mu|} \,\phi_\star({\varepsilon}x')=
f_{\varepsilon}(x').$$ These observations prove  and .

Furthermore, $$D^\gamma f_{\varepsilon}(x')={\varepsilon}^{|\gamma|-|\mu|} \,D^\gamma \phi_\star({\varepsilon}x').$$ Consequently, if we set $g_{\varepsilon}(x'):=f_{\varepsilon}(x')-f(x')$, we deduce from  and  that $$D^\gamma g_{\varepsilon}(0)=0,$$ for all $\gamma\in{\ensuremath{\mathbb{N}}}^{n-1}$ such that $|\gamma|{\leqslant}|\mu|+k$.

This observation and a Taylor expansion give that, for all $x'\in B_1'$ and all $\zeta
\in{\ensuremath{\mathbb{N}}}^{n-1}$ such that $|\zeta|{\leqslant}k$, $$\begin{aligned}
&&
| D^\zeta f_{\varepsilon}(x') - D^\zeta f(x')|=
|D^\zeta g_{\varepsilon}(x')|{\leqslant}C_k\,\sup_{y'\in B_1'}
\sum_{{\alpha\in{\ensuremath{\mathbb{N}}}^n}\atop{|\alpha| =|\mu|+k+1}}
|D^\alpha g_{\varepsilon}(y')|\\&&\qquad=
C_k\,\sup_{y'\in B_1'}
\sum_{{\alpha\in{\ensuremath{\mathbb{N}}}^n}\atop{|\alpha| =|\mu|+k+1}}
|D^\alpha f_{\varepsilon}(y')|=
C_k\,\sup_{y'\in B_1'}
\sum_{{\alpha\in{\ensuremath{\mathbb{N}}}^n}\atop{|\alpha| =|\mu|+k+1}}
{\varepsilon}^{|\alpha|-|\mu|} \,
|D^\alpha \phi_\star({\varepsilon}x')|\\&&\qquad
= C_k\,{\varepsilon}^{k+1} \,\sup_{y'\in B_1'}
\sum_{{\alpha\in{\ensuremath{\mathbb{N}}}^n}\atop{|\alpha| =|\mu|+k+1}}
|D^\alpha \phi_\star({\varepsilon}x')|{\leqslant}C_k'\,{\varepsilon},\end{aligned}$$ for some $C_k$, $C_k'>0$. This establishes , up to renaming ${\varepsilon}$.

Proof of Theorem \[QC\] {#15kdttd3}
=======================

In this section, 