Quantitative Carleson's conjecture for Ahlfors regular domains

TL;DR

This paper provides a quantitative characterization of Ahlfors-David regular domains in higher dimensions where Carleson's coefficients meet the strong geometric lemma, advancing understanding of geometric measure theory.

Contribution

It proves a quantitative version of Carleson's $ ext{ extbackslash epsilon}^2$ conjecture for higher-dimensional Ahlfors-David regular domains, linking Carleson's coefficients to geometric properties.

Findings

Characterization of domains satisfying the strong geometric lemma

Quantitative version of Carleson's $ ext{ extbackslash epsilon}^2$ conjecture

Extension of results to higher dimensions

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.