A proof of Carleson's $\varepsilon^2$-conjecture

TL;DR

This paper proves Carleson's $ ext{ extsterling} ext{ extsterling}$-conjecture, establishing a key characterization of tangent points on Jordan curves via a Carleson square function, advancing geometric measure theory.

Contribution

It provides the first proof of Carleson's $ ext{ extsterling} ext{ extsterling}$-conjecture, linking tangent points to Carleson square functions in Jordan curves.

Findings

Characterization of tangent points via Carleson $ ext{ extsterling} ext{ extsterling}$-square function

Proof of Carleson's $ ext{ extsterling} ext{ extsterling}$-conjecture

Finiteness of the square function characterizes tangent points

Abstract

In this paper we provide a proof of the Carleson $\varepsilon^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon^2$-square function.