(\[2.0.9\]), we have $$\label{2.0.10}
{\|u_n\|}_{C^{2\alpha+\beta}(\mathcal{K})}\le c_{17} {\|\nu\|}_{\mathfrak{M}^b(\bar\Omega)},$$ for any compact set $\mathcal{K}$ and some $\beta\in(0,\alpha)$. Then $u_n$ is $C^{2\alpha+\beta}$ locally in $\Omega$. Together with the fact that $u_{n,t}$ is classical solution of (\[2.2.1\]), we derive by Theorem 2.2 in [@CFQ] that $u_n$ is a classical solution of (\[1.1\]). $\Box$

For $\lambda>0$, let us define $$\label{Slambda}
  \tilde S_\lambda=\{x\in \Omega:\mathbb{G}_{\alpha}[\frac{\partial^\alpha |\nu|}{\partial \vec{n}^\alpha}](x)>\lambda\}\quad{\rm and}\quad \tilde m(\lambda)=\int_{S_\lambda} \rho_{\partial\Omega}^\alpha(x)dx.$$

\[lm 00\] For $\nu\in\mathfrak{M}^b_{\partial\Omega}(\bar\Omega)$, then there exist $\lambda_0>1$ and $c_{18}>0$ such that for any $\lambda\ge \lambda_0$, $$\label{annex 0}
\tilde m(\lambda)\le c_{18}\lambda^{-\frac{N+\alpha}{N-\alpha}}.$$

[**Proof.**]{} From Lemma \[lm 2.1\], we see that $$\mathbb{G}_\alpha[\frac{\partial^\alpha|\nu|}{\partial \vec{n}^\alpha}](x)\le \int_{\partial\Omega}\frac{c_5}{|x-y|^{N-\alpha}}d|\nu(y)|,\qquad x\in\Omega.$$ For $\Lambda>0$ and $y\in\partial\Omega$, we denote $$\tilde A_\Lambda(y)=\{x\in\Omega: \frac{c_5}{|x-y|^{N-\alpha}}>\Lambda\}\ \ {\rm
{and}}\quad \tilde m_\Lambda(y)=\int_{\tilde A_\Lambda(y)}\rho_{\partial\Omega}^\alpha(x) dx.$$ For any $(x,y)\in\Omega\times\partial\Omega$, it infers by(\[annex 01\]) that $$\begin{aligned}
\tilde A_\Lambda(y)\subset B_{r_0}(y),\end{aligned}$$ where $r_0=(\frac{c_5}{\Lambda})^{\frac1{N-\alpha}}$.

Since $\Omega$ is $C^2$, there exists $\Lambda_0>1$ such that for $\Lambda>\Lambda_0$ such that $$\rho_{\partial\Omega}(x)\le |x-y|,\quad \forall x\in \tilde A_\Lambda(y)$$ and $$\label{annex 1xhw}
\tilde m_\Lambda(y)\le \int_{ B_{r_0}(y)}|x-y|^\alpha dx\le  c_{19}\Lambda^{-\frac{N+\alpha}{N-\alpha}}.$$

For $y\in\partial\Omega$, we have that $$\begin{aligned}
\int_{\tilde S_\lambda} \frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le
\int_{\tilde A_\Lambda(y)}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx+\Lambda\int_{\tilde S_\lambda}
\rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ By integration by parts, we obtain $$\begin{aligned}
\int_{\tilde A_\Lambda(y)}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx&=&\Lambda \tilde m_\Lambda(y)+ \int_\Lambda^\infty\tilde m_s(y)ds
\\&\le& c_{20} \Lambda^{1-\frac{N+\alpha}{N-\alpha}},\end{aligned}$$ where $c_{20}>0$. Thus, $$\begin{aligned}
\int_{\tilde S_\lambda}\frac{c_5}{|x-y|^{N-\alpha}}\rho_{\partial\Omega}^\alpha(x)dx\le
c_{20}\Lambda^{1-\frac{N+\alpha}{N-\alpha}}+\Lambda \int_{\tilde S_\lambda}
\rho_{\partial\Omega}^\alpha(x)dx.\end{aligned}$$ Since $S_{\tilde \lambda_1}\subset S_{\tilde \lambda_2}$ if 