preprint

Inserted: 4 sep 2024
Last Updated: 4 sep 2024

Year: 2023

Abstract:

In this paper, we prove a higher-dimensional analogue of Carleson's $\varepsilon^2$ conjecture. Given two arbitrary disjoint Borel sets $\Omega^+, \Omega^- \subset \mathbb{R}^{n+1}$, and $x \in \mathbb{R}^{n+1}$, $r > 0$, we denote

\[ \varepsilon_n(x,r) := \frac{1}{r^n}\, \inf_{H^+} \mathcal{H}^n \left( ((\partial B(x,r) \cap H^+) \setminus \Omega^+) \cup ((\partial B(x,r) \cap H^-) \setminus \Omega^-) \right), \]

where the infimum is taken over all open affine half-spaces $H^+$ such that $x \in \partial H^+$, and we define $H^- = \mathbb{R}^{n+1} \setminus \overline{H^+}$. Our first main result asserts that the set of points $x \in \mathbb{R}^{n+1}$ where

\[ \int_0^1 \varepsilon_n(x,r)^2 \, \frac{dr}{r} < \infty \]

is $n$-rectifiable. For our second main result, we assume that $\Omega^+$ and $\Omega^-$ are open and that $\Omega^+ \cup \Omega^-$ satisfies the capacity density condition. For each $x \in \partial \Omega^+ \cup \partial \Omega^-$ and $r > 0$, we denote by $\alpha^\pm(x,r)$ the characteristic constant of the (spherical) open sets $\Omega^\pm \cap \partial B(x,r)$. We show that, up to a set of $\mathcal{H}^n$ measure zero, $x$ is a tangent point for both $\partial \Omega^+$ and $\partial \Omega^-$ if and only if

\[ \int_0^1 \min(1, \alpha^+(x,r) + \alpha^-(x,r) - 2) \frac{dr}{r} < \infty. \]

The first result is new even in the plane, and the second one improves and extends the $\varepsilon^2$ conjecture of Carleson to higher dimensions.