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

TL;DR

This paper establishes a characterization of $n$-rectifiability of measures in Euclidean space using Jones' square function and $eta$ coefficients, linking geometric measure theory with harmonic analysis.

Contribution

It provides a new necessary condition for $n$-rectifiability based on Jones' square function and introduces an analogous condition involving Wasserstein distance variants.

Findings

Finite $eta_{rac{p}{n}}^n$ integrals characterize $n$-rectifiability.

Necessary conditions for rectifiability are established using Jones' coefficients.

Analogous conditions involving Wasserstein distances are also proved.

Abstract

In this paper it is shown that if $\mu$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty \beta_{\mu,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $\mu$-a.e. $x\in\mathbb R^d$,}$$ where $$\beta_{\mu,p}^n(x,r) = \inf_L (\frac1{r^n} \int_{\bar B(x,r)} (\frac{\mathrm dist(y,L)}{r})^p\,d\mu(y))^{1/p},$$ with the infimum taken over all the $n$-planes $L\subset \mathbb R^d$. The $\beta_{\mu,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessary condition for $n$-rectifiability in terms of other coefficients involving some variant of the Wasserstein distance $W_1$ is also proved.