Carleson's ε2 conjecture in higher dimensions
Fleschler, Ian; Tolsa, Xavier; Villa, Michele (2025-06-03)
Springer
03.06.2025
Fleschler, I., Tolsa, X. & Villa, M. Carleson’s ε2 conjecture in higher dimensions. Invent. math. 241, 207–307 (2025). https://doi.org/10.1007/s00222-025-01337-w
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© The Author(s) 2025. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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https://urn.fi/URN:NBN:fi:oulu-202506114337
https://urn.fi/URN:NBN:fi:oulu-202506114337
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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 \(\Omega ^{+}\), \(\Omega ^{-}\) 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 to higher dimensions the \(\varepsilon^{2}\) conjecture of Carleson.
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 \(\Omega ^{+}\), \(\Omega ^{-}\) 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 to higher dimensions the \(\varepsilon^{2}\) conjecture of Carleson.
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