Sobolev Extensions over Cantor-Cuspidal Graphs

For a continuous function f : \({\mathbb{R}}\) $\mathbb{R}$ → \({\mathbb{R}}\) $\mathbb{R}$ we define the corresponding graph by setting Γ_f ≔ {(x₁, f(x₁)) : x₁ ∈ \({\mathbb{R}}\) $\mathbb{R}$ } . We give the optimal Sobolev extension properties for the upper and lower domains corresponding to the graph \({\Gamma }_{{\psi }_{c}^{\alpha }}\) ${\mathrm{\Gamma }}_{{\psi }_c^{\alpha }}$ for \({\psi }_{c}^{\alpha }\) ${\psi }_c^{\alpha }$ (x₁) = d(x₁, \(\mathcal{C}\) $\mathcal{C}$ )α, where \(\mathcal{C}\) $\mathcal{C}$ is the classical ternary Cantor set in the unit interval and α ∈ (0, 1).