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...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2025 |
| País: | España |
| Institución: | Universitat Autònoma de Barcelona |
| Repositorio: | Dipòsit Digital de Documents de la UAB |
| Idioma: | inglés |
| OAI Identifier: | oai:ddd.uab.cat:325490 |
| Acceso en línea: | https://ddd.uab.cat/record/325490 https://dx.doi.org/urn:doi:10.1007/s00222-025-01337-w |
| Access Level: | acceso abierto |
| Sumario: | 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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