Carleson's ε2 conjecture in higher dimensions

• 成果类型: Article
• 署名作者: Fleschler, Ian; Tolsa, Xavier; Villa, Michele
• 署名单位: Princeton University; ICREA; Autonomous University of Barcelona; Centre de Recerca Matematica (CRM); University of Oulu; University of Basque Country
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01337-w
• 发表日期: 2025
• 页码: 207-307
• 关键词: n-rectifiability; terms; sets

摘要

In this paper we prove a higher dimensional analogue of Carleson's epsilon 2 conjecture. Given two arbitrary disjoint Borel sets Omega(+),Omega(-)subset of Rn+1, and x is an element of Rn+1, r>0, we denote epsilon(n)(x,r):=1/r(n) inf(H+)H(n)(((partial derivative B(x,r)boolean AND H+)\Omega(+))boolean OR((partial derivative B(x,r)boolean AND H-)\Omega(-))), where the infimum is taken over all open affine half-spaces H+ such that x is an element of partial derivative H+ and we define H-=Rn+1\H+. Our first main result asserts that the set of points x is an element of Rn+1 where integral(1)(0)epsilon(n)(x,r)(2)dr/r0, we denote by alpha(+/-)(x,r) the characteristic constant of the (spherical) open sets Omega(+/-)boolean AND partial derivative B(x,r). We show that, up to a set of H-n measure zero, x is a tangent point for both partial derivative Omega(+) and partial derivative Omega(-) if and only if integral(1)(0)min(1,alpha(+)(x,r)+alpha(-)(x,r)(-)2)dr/r