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 flatnessquantifying coefficients. The first condition involves the socalled $\alpha$ 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2019
 arXiv:
 arXiv:1904.11004
 Bibcode:
 2019arXiv190411004D
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Analysis of PDEs;
 28A75;
 28A78
 EPrint:
 54 pages, added the proof of Lemma 4.5, minor improvements