Sufficient condition for rectifiability involving Wasserstein distance $W_2$

Abstract

A Radon measure $\mu$ is $n$-rectifiable if it is absolutely continuous with respect to ${\mathcal{H}}^n$ and $\mu$-almost all of $\text{supp}\mu$ can be covered by Lipschitz images of ${\mathbb{R}}^n$. In this paper we give two sufficient conditions for rectifiability, both in terms of square functions of flatness-quantifying coefficients. The first condition involves the so-called $\alpha$ and ${\beta }_2$ numbers. The second one involves ${\alpha }_2$ numbers -- coefficients quantifying flatness via Wasserstein distance $W_2$. Both conditions are necessary for rectifiability, too -- the first one was shown to be necessary by Tolsa, while the necessity of the ${\alpha }_2$ condition is established in our recent paper. Thus, we get two new characterizations of rectifiability.

54 pages, added the proof of Lemma 4.5, minor improvements

References in corpus (2)

• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Absolute continuity and $\alpha$-numbers on the real line

Cited by in corpus (3)

• Necessary condition for rectifiability involving Wasserstein distance $W_2$
• Radon measures and Lipschitz graphs
• An $\alpha$-number characterization of $L^p$ spaces on uniformly rectifiable sets