Carleson's ${\epsilon }^2$ conjecture in higher dimensions

Abstract

In this paper we prove a higher dimensional analogue of Carleson's ${\epsilon }^2$ conjecture. Given two arbitrary disjoint open sets ${\Omega }^{+},{\Omega }^{-}\subset {\mathbb{R}}^{n+1}$, and $x\in {\mathbb{R}}^{n+1}$, $r>0$, we denote $${\epsilon }_n(x,r):=\frac{1}{r^n}\underset{H^{+}}{\inf }{\mathcal{H}}^n\left(((\partial B(x,r)\cap H^{+})\setminus {\Omega }^{+})\cup ((\partial B(x,r)\cap H^{-})\setminus {\Omega }^{-})\right),$$ 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 any Borel subset of $$\left\{x\in {\mathbb{R}}^{n+1}:{\int }_0^1{\epsilon }_n(x,r{)}^2\frac{dr}{r}<\infty \right\}$$ is $n$-rectifiable. For our second main result we assume that ${\Omega }^{+},{\Omega }^{-}$ 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\begin{equation*} \int_0^{1} \min(1,α^+(x,r) + α^-(x,r) -2) \frac{dr}{r} < \infty. \end{equation*} The first result is new even in the plane and the second one improves and extends to higher dimensions the ${\epsilon }^2$ conjecture of Carleson.

86 pages, 7 figures. V3: first main result is extended to the case of ${\Omega }^{\pm }$ being just Borel subsets. Correction of minor typos