Quantitative differentiability on uniformly rectifiable sets

TL;DR

This paper establishes $L^p$ quantitative differentiability estimates for functions on uniformly rectifiable sets, extending classical results to more complex geometric contexts with implications for Sobolev function extensions.

Contribution

It proves a Dorronsoro-type theorem for uniformly rectifiable sets, linking the gradient norm to a new square function measuring affine deviation and flatness.

Findings

$L^p$ estimates for Sobolev functions on rectifiable sets

A new square function characterizing affine deviation and flatness

Extension and trace results for Sobolev functions

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. A corollary dealing with extensions and traces of Sobolev functions may be found in a companion article.