Wasserstein Distance and the Rectifiability of Doubling Measures: Part I

TL;DR

This paper establishes a quantitative link between the rectifiability of doubling measures in Euclidean space and their Wasserstein distance to flat measures, providing a decomposition of the support into rectifiable pieces under certain conditions.

Contribution

It introduces a new measure, based on Wasserstein distance, to analyze rectifiability of doubling measures and decomposes the support into rectifiable parts with additional control under Carleson measure assumptions.

Findings

Decomposition of measure support into rectifiable pieces.

Quantitative relation between Wasserstein distance and rectifiability.

Enhanced control of measure size under Carleson estimates.

Abstract

Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $\mu$ and its distance to flat measures. More precisely, for $x$ in the support $\Sigma$ of $\mu$ and $r > 0$, we introduce a number $\alpha(x,r)\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the minimal distance between the restriction of $\mu$ to $B(x,r)$ and a multiple of the Lebesgue measure on an affine subspace that meets $B(x,r/2)$. We show that the set of points of $\Sigma$ where $\int_0^1 \alpha(x,r) \frac{dr}{r} < \infty$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $\mu$ when we assume that some Carleson measure estimates hold. Soit $\mu$ une mesure doublante dans $\mathbb{R}^n$. On \'etudie des relations quantifi\'ees entre la rectifiabilit\'e de $\mu$ et la distance entre $\mu$ et les mesures plates. Plus pr\'ecis\'ement, on utilise une variante de la $L^1$-distance de Wasserstein pour d\'efinir, pour $x$ dans le support $\Sigma$ de $\mu$ et $r>0$, un nombre $\alpha(x,r)$ qui mesure la distance minimale entre la restriction de $\mu$ \`a $B(x,r)$ et une mesure de Lebesgue sur un sous-espace affine passant par $B(x,r/2)$. On d\'ecompose l'ensemble des points $x\in \Sigma$ tels que $\int_0^1 \alpha(x,r) \frac{dr}{r} < \infty$ en parties rectifiables de dimensions diverses, et on obtient un meilleur contr\^ole de ces parties et de la taille de $\mu$ quand les $\alpha(x,r)$ v\'erifient certaines conditions de Carleson.