Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries
Jonas Azzam, Mihalis Mourgoglou, Michele Villa
TL;DR
This work develops a sharp trace-extension theory for corkscrew domains with uniformly rectifiable boundaries, establishing a bounded, surjective trace map from a gradient-controlled domain space $\mathcal{V}_p(\Omega)$ to the Hajłasz-Sobolev boundary space $M^{1,p}(\partial\Omega)$. The extension $u$ is constructed via a Whitney-partition of unity using affine approximants, and is shown to satisfy $\|\mathcal{N}(\nabla u)\|_{L^p(\partial\Omega)}$ and $\|\mathcal{S}(d_\Omega\nabla^2 u)\|_{L^p(\partial\Omega)}$ bounds in terms of a Hajłasz upper gradient of the boundary data. Central to the argument are Dorronsoro-type estimates on UR sets (via the coefficients $\Omega$ and $\gamma$) and a careful analysis of non-tangential cones, Whitney averages, and compatible corkscrew choices. The results guarantee both trace well-definedness and surjectivity, and in Lipschitz settings yield non-tangential convergence of $\nabla u$ to the boundary tangential gradient, thereby bridging boundary Sobolev data with interior $L^p$-control in non-smooth geometries.
Abstract
Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Hajlasz-Sobolev space on the boundary from the space of functions on the domain with $L^p$ norm involving the non-tangential maximal function of the gradient and the conical square function of the Hessian. This fundametally uses the Dorronsoro theorem for UR sets proven in a companion paper.