Quantitative control on the Carleson $\epsilon$-function determines regularity

Abstract

Carleson's ${\epsilon }^2$-conjecture states that for Jordan domains in ${\mathbb{R}}^2$, points on the boundary where tangents exist can be characterized in terms of the behavior of the $\epsilon$-function. This conjecture, which was fully resolved by Jaye, Tolsa, and Villa in 2021, established that qualitative control on the rate of decay of the Carleson $\epsilon$-function implies the existence of tangents, up to a set of measure zero. We prove that quantitative control on the rate of decay of this function gives quantitative information on the regularity of the boundary.

Updated references, typos corrected, proof of Lemma 2 edited for clarity; 25 pages, 16 figures