Carleson's ε2 conjecture in higher dimensions
In this paper we prove a higher dimensional analogue of Carleson's ε2 conjecture. Given two arbitrary disjoint Borel sets Ω+,Ω-⊂Rn+1, and x∈Rn+1, r>0, we denote (Formula presented.) where the infimum is taken over all open affine half-spaces H+ such that x∈∂H+ and we define H-=Rn+1∖H+‾. Our...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2025 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:325490 |
| Online Access: | https://ddd.uab.cat/record/325490 https://dx.doi.org/urn:doi:10.1007/s00222-025-01337-w |
| Access Level: | Open access |
| Summary: | In this paper we prove a higher dimensional analogue of Carleson's ε2 conjecture. Given two arbitrary disjoint Borel sets Ω+,Ω-⊂Rn+1, and x∈Rn+1, r>0, we denote (Formula presented.) where the infimum is taken over all open affine half-spaces H+ such that x∈∂H+ and we define H-=Rn+1∖H+‾. Our first main result asserts that the set of points x∈Rn+1 where (Formula presented.) is n-rectifiable. For our second main result we assume that Ω+, Ω- are open and that Ω+∪Ω- satisfies the capacity density condition. For each x∈∂Ω+∪∂Ω- and r>0, we denote by α±(x,r) the characteristic constant of the (spherical) open sets Ω±∩∂B(x,r). We show that, up to a set of Hn measure zero, x is a tangent point for both ∂Ω+ and ∂Ω- if and only if (Formula presented.) The first result is new even in the plane and the second one improves and extends to higher dimensions the ε2 conjecture of Carleson. |
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