Sobolev extensions over Cantor-cuspidal graphs

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