Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

TL;DR

This paper develops a method to extend Sobolev boundary data smoothly into corkscrew domains with uniformly rectifiable boundaries, using advanced harmonic analysis tools and geometric measure theory.

Contribution

It constructs a surjective trace map from domain functions to boundary Sobolev spaces, leveraging the Dorronsoro theorem for UR sets, advancing boundary extension theory.

Findings

Constructed a surjective trace map for Sobolev spaces

Utilized the Dorronsoro theorem for UR sets

Established boundary extension results in corkscrew domains

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.