Sufficient conditions for $C^{1,\alpha}$ parametrization and rectifiability

TL;DR

This paper establishes quantitative conditions involving a Bishop-Jones type square function that ensure a Radon measure in Euclidean space is $C^{1,eta}$ rectifiable, and provides parametrization criteria for Reifenberg flat sets.

Contribution

It introduces new sufficient, quantitative conditions for $C^{1,eta}$ rectifiability of measures and sets, extending Reifenberg parametrization techniques with explicit dependence on a square function.

Findings

Provides criteria for $C^{1,eta}$ rectifiability based on a Bishop-Jones type square function.

Establishes sufficient conditions for $C^{1,eta}$ parametrizations of Reifenberg flat sets.

Quantifies the dependence of $C^{1,eta}$ constants on the square function.

Abstract

We say a measure is $C^{1,\alpha}$ $d$-rectifiable if there is a countable union of $C^{1,\alpha}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be $C^{1,\alpha}$ $d$-rectifiable, with $\alpha \in (0,1]$. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,\alpha}$ constants depend on such a function. Along the way we also give sufficient conditions for $C^{1,\alpha}$ 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 H\"{o}lder and Lipschitz categories.