Sobolev extensions over Cantor-cuspidal graphs

Abstract

For a continuous function $f:\mathbb{R}\to \mathbb{R}$, define the corresponding graph by setting \[Γ_f := {(x1, f(x1)) : x_1\in\mathbb{R}} .\] In this paper, we study the Sobolev extension property for the upper and lower domains over the graph ${\Gamma }_{{\psi }_c^{\alpha }}$ 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 $\alpha \in (0,1)$.