Characterization of n-rectifiability in terms of Jones' square function: Part II

2015/01/07 by Jonas Azzam, Xavier Tolsa, Azzam, Jonas +1
Mathematics · #28A75 #28A78 #42B20 #Advanced Topology and Set Theory #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #math.AP #math.CA #msc:28A75 #msc:28A78 #msc:42B20

paper · pdf · doi:10.48550/arxiv.1501.01572

A corollary regarding analytic capacity and a few new references have been added

openalex publication_date 2015/01/07 · arxiv created 2015/01/19 · arxiv updated 2015/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a Radon measure μ in \mathbb R^d which is absolutely continuous with respect to the n-dimensional Hausdorff measure Hⁿ is n-rectifiable if the so called Jones' square function is finite μ-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all n-rectifiable measures which are absolutely continuous with respect to Hⁿ. Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth.

Related

• Characterization of n-rectifiability in terms of Jones' square\n function: Part I
• Characterization of rectifiable measures in terms of α-numbers
• Rectifiable measures, square functions involving densities, and the Cauchy transform
• Characterizations of countably n-rectifiable Radon measures by higher-dimensional Menger curvatures
• A square function involving the center of mass and rectifiability