Sobolev extensions over Cantor-cuspidal graphs

For a continuous function $f:\mathbb{R}\to\mathbb{R}$, define the corresponding graph by setting $$\Gamma_f := {(x1, f(x1)) : x_1\in\mathbb{R}} .$$ In this paper, we study the Sobolev extension property for the upper and lower domains over the graph $\Gamma_{\psi^\alpha_c}$ for $\psi^\alpha_c(x_1):=d(x_1, \mathcal C)^\alpha$, where $\mathcal C$ is the classical ternary Cantor set in the unit interval and $\alpha\in(0, 1)$.