Quantitative control on the Carleson ε-function determines regularity

2024/10/24 by Casey, Emily
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2410.18422

openalex publication_date 2024/10/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Carleson's ε²-conjecture states that for Jordan domains in ℝ², points on the boundary where tangents exist can be characterized in terms of the behavior of the ε-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 ε-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.

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