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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2019
- DOI:
- 10.48550/arXiv.1904.11004
- arXiv:
- arXiv:1904.11004
- Bibcode:
- 2019arXiv190411004D
- Keywords:
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- Mathematics - Classical Analysis and ODEs;
- Mathematics - Analysis of PDEs;
- 28A75;
- 28A78
- E-Print:
- 54 pages, added the proof of Lemma 4.5, minor improvements