2023/10/18 by Ian Fleschler, Xavier Tolsa, Fleschler, Ian +3
Mathematics · #28A75 28A78 35R35 #Advanced Harmonic Analysis Research #Analysis of PDEs (math.AP) #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Approximation and Integration
paper · pdf · doi:10.48550/arxiv.2310.12316
openalex publication_date 2023/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04
In this paper we prove a higher dimensional analogue of Carleson's ε2 conjecture. Given two arbitrary disjoint open sets Ω+,Ω-⊂ ℝn+1, and x∈ℝn+1, r>0, we denote εn(x,r) := (1)/(rn) infH+ Hn ( ((∂ B(x,r)∩ H+) ∖ Ω+) ∪ ((∂ B(x,r)∩ H-) ∖ Ω-)), 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 any Borel subset of \x∈ℝn+1 : ∫01 εn(x,r)2 (dr)/(r)lt;∞\ 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∫01 min(1,α+(x,r) + α-(x,r) -2) (dr)/(r) lt; ∞. The first result is new even in the plane and the second one improves and extends to higher dimensions the ε2 conjecture of Carleson.