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

TL;DR

This paper establishes that a Radon measure absolutely continuous with respect to Hausdorff measure is n-rectifiable if its Jones' square function is finite almost everywhere, providing a key characterization of rectifiable measures.

Contribution

It proves the sufficiency of the finiteness of Jones' square function for n-rectifiability of measures, complementing a previous converse result, thus classifying rectifiable measures absolutely continuous with respect to Hausdorff measure.

Findings

Finite Jones' square function implies n-rectifiability.

Relationship between Jones' square function and Menger curvature.

Complete classification of absolutely continuous n-rectifiable measures.

Abstract

We show that a Radon measure $\mu$ in $\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $\mu$-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 $H^{n}$. 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.