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Quantitative differentiability on uniformly rectifiable sets

Jonas Azzam, Mihalis Mourgoglou, Michele Villa

TL;DR

The paper develops a quantitative differentiability theory for Sobolev functions on uniformly rectifiable sets by proving a Dorronsoro-type estimate in the L^p setting. The authors introduce a novel square function G^q built from affine-deviation and flatness coefficients γ_f^q and Ω_f^q, and show that ||∇f||_{L^p(E)} is comparable to ||G^q f||_{L^p(E)} under UR and Poincaré hypotheses, extending classical Euclidean results to non-smooth sets. The core method combines good-λ inequalities, square-function estimates, stopping-time arguments, and a Lipschitz-graph corona decomposition to control multi-scale geometry and affine approximations via Tolsa’s α-numbers, Jones’ β-numbers, and the introduced γ-coefficients. A key result is the two-part achievement: (1) an L^p gradient estimate for Hajłasz-Sobolev functions on UR sets, and (2) a corresponding tangential-gradient bound, with corollaries on extensions and traces in domains with UR boundaries. The framework provides a robust bridge between geometric measure theory and quantitative differentiation, enabling extensions and trace results in domains with UR boundaries and offering tools for further analysis of elliptic problems on non-smooth sets.

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.

Quantitative differentiability on uniformly rectifiable sets

TL;DR

The paper develops a quantitative differentiability theory for Sobolev functions on uniformly rectifiable sets by proving a Dorronsoro-type estimate in the L^p setting. The authors introduce a novel square function G^q built from affine-deviation and flatness coefficients γ_f^q and Ω_f^q, and show that ||∇f||_{L^p(E)} is comparable to ||G^q f||_{L^p(E)} under UR and Poincaré hypotheses, extending classical Euclidean results to non-smooth sets. The core method combines good-λ inequalities, square-function estimates, stopping-time arguments, and a Lipschitz-graph corona decomposition to control multi-scale geometry and affine approximations via Tolsa’s α-numbers, Jones’ β-numbers, and the introduced γ-coefficients. A key result is the two-part achievement: (1) an L^p gradient estimate for Hajłasz-Sobolev functions on UR sets, and (2) a corresponding tangential-gradient bound, with corollaries on extensions and traces in domains with UR boundaries. The framework provides a robust bridge between geometric measure theory and quantitative differentiation, enabling extensions and trace results in domains with UR boundaries and offering tools for further analysis of elliptic problems on non-smooth sets.

Abstract

We prove 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 norm of the gradient of a Sobolev function is comparable to the norm of a new square function measuring both the affine deviation of and how flat the subset is. A corollary dealing with extensions and traces of Sobolev functions may be found in a companion article.
Paper Structure (46 sections, 61 theorems, 369 equations)

This paper contains 46 sections, 61 theorems, 369 equations.

Table of Contents

  1. Introduction and main results
  2. Remarks on Theorems \ref{['t:main-A']} and \ref{['t:main-B']}
  3. Uniformly rectifiable sets
  4. Remarks on the literature
  5. Overview of the proof
  6. Structure of the paper
  7. Notation
  8. Preliminaries: Sobolev spaces and Poincaré inequalities
  9. Preliminaries: dyadic lattices
  10. Preliminaries: uniform rectifiability; the $\alpha$, $\beta$, $\gamma$ and $\Omega$ coefficients
  11. Ahlfors regularity; uniform rectifiability
  12. The geometric coefficients $\alpha$ and $\beta$
  13. Tolsa's $\alpha$ numbers
  14. Jones' $\beta$ numbers
  15. Relation between $\beta$, $\alpha$ and angle between planes
  16. ...and 31 more sections

Key Result

Theorem 1.1

Fix $1 \leq d \in \mathbb{N}$ and $1<p<\infty$. Let $f \in L^p({\mathbb{R}}^d)$ be a real valued function. Then $f \in W^{1,p}({\mathbb{R}}^d)$ if and only if $\|\mathcal{G}_q f\|_{L^p}$ is finite, where $q$ is in the following range: In all these cases, we have where the implicit constant depend on $d,p,q$.

Theorems & Definitions (119)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Remark 1.3: The square function $\mathcal{G}^q$ and the new coefficients $\gamma_f^q$ and $\widetilde{\gamma}_f$
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • Definition 3.4
  • ...and 109 more