Characterization of $C^{1,\alpha }$-domains in terms of Carleson-type $\epsilon$-coefficients

Let $\mathrm{\Omega }={\mathrm{\Omega }}^{+}\subset {\mathbb{R}}^{n+1}$ be a bounded open set with $n$-Ahlfors regular boundary and let ${\mathrm{\Omega }}^{-}={\mathbb{R}}^{n+1}\setminus \overline{{\mathrm{\Omega }}^{+}}$. For $x\in \mathrm{\partial }\mathrm{\Omega }$, $r>0$, define \begin{equation*} \varepsilon_{n,Ω}(x,r):=\frac{1}{r^n} \inf_{H^+}$\mathcal{H}$^n\left(((\partial B(x,r)\cap H^+)\setminus Ω^+) \cup ((\partial B(x,r)\cap H^-)\setminus Ω^-)\right), \end{equation*} where ${\mathcal{H}}^n$ stands for $n$-dimensional Hausdorff measure and the infimum is taken over all open affine half-spaces $H^{+}$ such that $x\in \mathrm{\partial }{H}^{+}$. Here $H^{-}={\mathbb{R}}^{n+1}\setminus \overline{H^{+}}$. In this work we prove that $\mathrm{\Omega }$ is a $C^{1,\alpha }$ open set, with $0<\alpha <1$, if and only if there exists some constant $C>0$ such that ${\epsilon }_{n,\mathrm{\Omega }}(x,r)\le C{r}^{\alpha }$ for all $x\in \mathrm{\partial }\mathrm{\Omega }$ and all $0<r\le \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{\partial }\mathrm{\Omega })$.