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arXiv · 2306.13017

Quantitative differentiability on uniformly rectifiable sets

Jonas Azzam·Mihalis Mourgoglou·Michele Villa
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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.

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Jonas Azzam, Mihalis Mourgoglou, Michele Villa. 2023-06-22. Quantitative differentiability on uniformly rectifiable sets. https://arxiv.org/abs/2306.13017

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