Sufficient conditions for C^1,α parametrization and rectifiability

2017/09/18 by Silvia Ghinassi, Ghinassi, Silvia
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Metric Geometry (math.MG) #math.CA #math.MG

paper · pdf · doi:10.48550/arxiv.1709.06015

36 pages, 2 figures. To appear in Annales Academiae Scientiarum Fennicae Mathematica

openalex publication_date 2017/09/18 · arxiv created 2019/12/24 · arxiv updated 2019/12/25 · openalex created_date 2024/04/11 · openalex updated_date 2026/07/28

Abstract

We say a measure is C^1,α d-rectifiable if there is a countable union of C^1,α d-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in ℝⁿ to be C^1,α d-rectifiable, with α∈ (0,1]. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the C^1,α constants depend on such a function. Along the way we also give sufficient conditions for C^1,α parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and Toro's Reifenberg parametrizations of sets with holes in the Hölder and Lipschitz categories.

Related

• Characterization of rectifiable measures in terms of α-numbers
• Characterization of n-rectifiability in terms of Jones' square function: Part II
• A square function involving the center of mass and rectifiability
• Identifying 1-rectifiable measures in Carnot groups
• Characterization of n-rectifiability in terms of Jones' square function: Part I