Carleson’s $\varepsilon ^{2}$ conjecture in higher dimensions

Fleschler, Ian; Tolsa, Xavier; Villa, Michele

doi:10.1007/s00222-025-01337-w

Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$

ε^{2}

conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$

Ω^{+}, Ω^{-} \subset R^{n + 1}

, and$$x\in \mathbb{R}^{n+1}$$

x \in R^{n + 1}

,$$r>0$$

r > 0

, we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$

ε_{n} (x, r) : = \frac{1}{r^{n}} inf_{H^{+}} H^{n} (((\partial B (x, r) \cap H^{+}) ∖ Ω^{+}) \cup ((\partial B (x, r) \cap H^{-}) ∖ Ω^{-})),

where the infimum is taken over all open affine half-spaces$$H^{+}$$

H^{+}

such that$$x \in \partial H^{+}$$

x \in \partial H^{+}

and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$

H^{-} = R^{n + 1} ∖ \overline{H^{+}}

. Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$

x \in R^{n + 1}

where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$

\int_{0}^{1} ε_{n} {(x, r)}^{2} \frac{d r}{r} < \infty

is$$n$$

n

-rectifiable. For our second main result we assume that$$\Omega ^{+}$$

Ω^{+}

,$$\Omega ^{-}$$

Ω^{-}

are open and that$$\Omega ^{+}\cup \Omega ^{-}$$

Ω^{+} \cup Ω^{-}

satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$

x \in \partial Ω^{+} \cup \partial Ω^{-}

and$$r>0$$

r > 0

, we denote by$$\alpha ^{\pm }(x,r)$$

α^{\pm} (x, r)

the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$

Ω^{\pm} \cap \partial B (x, r)

. We show that, up to a set of$$\mathcal{H}^{n}$$

H^{n}

measure zero,$$x$$

x

is a tangent point for both$$\partial \Omega ^{+}$$

\partial Ω^{+}

and$$\partial \Omega ^{-}$$

\partial Ω^{-}

if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$

\int_{0}^{1} min (1, α^{+} (x, r) + α^{-} (x, r) - 2) \frac{d r}{r} < \infty .

The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$

ε^{2}

conjecture of Carleson.

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