Quantitative differentiability on uniformly rectifiable sets and applications to Sobolev trace theorems

Abstract: We prove $L^p$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-type theorem holds in this context: the $L^p$ norm of the gradient of a Sobolev function $f: E \to \mathbb{R}$ is comparable to the $L^p$ norm of a new square function measuring both the affine deviation of $f$ and how flat the subset $E$ is. As a corollary, given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Haj\l{}asz-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.