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Quantitative Carleson's conjecture for Ahlfors regular domains

Emily Casey, Xavier Tolsa, Michele Villa

TL;DR

This work proves a quantitative version of Carleson's epsilon^2 conjecture in arbitrary dimensions by introducing the geometric coefficients $a$ and ${\varepsilon}_n$ that capture boundary flatness through eigenvalue data and spherical domains. The authors establish an equivalence: ADR boundaries with two-sided corkscrew structure are UR if and only if a Carleson-type integral bound holds for either $a$ or ${\varepsilon}_n$, connecting geometric regularity to harmonic-analytic control. A central strategy is a corona decomposition into Lipschitz (CAD) subdomains, allowing transfer of CAD estimates and the Alt–Caffarelli–Friedman framework to the full domains. The approach also provides a direct bound of ${\varepsilon}_n$ in terms of centered $\beta$-numbers, linking the new coefficients to the classical strong geometric lemma. Overall, the results advance the understanding of higher-codimensional uniform rectifiability and harmonic measure through quantitative geometric coefficients and multiscale decompositions.

Abstract

In this article, we prove a quantitative version of Carleson's $\varepsilon^2$ conjecture in higher dimension: we characterise those Ahlfors-David regular domains in $\mathbb{R}^{n+1}$ for which the Carleson's coefficients satisfy the so-called strong geometric lemma.

Quantitative Carleson's conjecture for Ahlfors regular domains

TL;DR

This work proves a quantitative version of Carleson's epsilon^2 conjecture in arbitrary dimensions by introducing the geometric coefficients and that capture boundary flatness through eigenvalue data and spherical domains. The authors establish an equivalence: ADR boundaries with two-sided corkscrew structure are UR if and only if a Carleson-type integral bound holds for either or , connecting geometric regularity to harmonic-analytic control. A central strategy is a corona decomposition into Lipschitz (CAD) subdomains, allowing transfer of CAD estimates and the Alt–Caffarelli–Friedman framework to the full domains. The approach also provides a direct bound of in terms of centered -numbers, linking the new coefficients to the classical strong geometric lemma. Overall, the results advance the understanding of higher-codimensional uniform rectifiability and harmonic measure through quantitative geometric coefficients and multiscale decompositions.

Abstract

In this article, we prove a quantitative version of Carleson's conjecture in higher dimension: we characterise those Ahlfors-David regular domains in for which the Carleson's coefficients satisfy the so-called strong geometric lemma.
Paper Structure (23 sections, 17 theorems, 149 equations, 3 figures)

This paper contains 23 sections, 17 theorems, 149 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Definition of $a$: spherical domains and their characteristic constants
  3. Definition of ${\varepsilon}_n$: a more explicitly geometric coefficient
  4. An open question
  5. Structure of the article
  6. Preliminaries
  7. Basic notation
  8. Tangent points
  9. Square functions
  10. Ahlfors-David regularity, UR, Carleson measures
  11. Types of domain
  12. Dyadic lattices
  13. The other geometric coefficient: $\beta$
  14. The ACF monotonicity formula
  15. From square function estimates to corkscrews
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\Omega \subset \mathbb{R}^{n+1}$ is an open set, and suppose that $\partial \Omega$ is $n$-ADR. Then $\Omega$ is a two-sided corkscrew open set, and thus UR, if and only if there exists a constant $C\geq 1$ such that for every ball $B$ centered on $\partial \Omega$.

Figures (3)

  • Figure 1: The region $(\partial B(x,r)\cap H^+)\setminus \Omega_1$ is denoted in red.
  • Figure 2: The region $H^+(t)\setminus \Omega^+$ is contained between the equator and the latitude line passing through "bad" point $z_i\in \partial \Omega$. The region on the equator between any two of the partial great circles is an $(n-1)$-ball of radius $\approx \frac{t}{N}$.
  • Figure 3: The arc $A^+(\theta_i)$ and the subarc $C^+(\theta_i)$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2: FTV24
  • Theorem 1.3
  • Remark 2.1
  • Definition 2.2: Harnack chain condition
  • Definition 2.3: NTA domain
  • Definition 2.4: CAD
  • Theorem 2.5
  • Proposition 3.1
  • proof : Proof of Proposition
  • ...and 24 more