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 \(\int _0^1 \alpha (x,r) {dr \over r} < \infty \) 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.
Résumé en Français
Soit \(\mu \) une mesure doublante dans \({\mathbb {R}}^n\). On étudie des relations quantifiées entre la rectifiabilité de \(\mu \) et la distance entre \(\mu \) et les mesures plates. Plus précisément, on utilise une variante de la \(L^1\)-distance de Wasserstein pour définir, 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 \(\mu \) à \(B(x,r)\) et une mesure de Lebesgue sur un sous-espace affine passant par \(B(x,r/2)\). On décompose l’ensemble des points \(x\in \Sigma \) tels que \(\int _0^1 \alpha (x,r) {dr \over r} < \infty \) en parties rectifiables de dimensions diverses, et on obtient un meilleur contrôle de ces parties et de la taille de \(\mu \) quand les \(\alpha (x,r)\) vérifient certaines conditions de Carleson.
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Acknowledgments
The authors are grateful to Alessio Figalli and Xavier Tolsa for helpful discussions. The first author would like to thank IPAM for its hospitality, part of this manuscript was written while he was in residence there.
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Jonas Azzam was partially supported by NSF RTG Grant 0838212. Guy David acknowledges the generous support of the Institut Universitaire de France, and of the ANR (programme blanc GEOMETRYA, ANR-12-BS01-0014). Tatiana Toro was partially supported by an NSF Grants DMS-0856687 and DMS-1361823, a Grant from the Simons Foundation (# 228118) and the Robert R. and Elaine F. Phelps Professorship in Mathematics.
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Azzam, J., David, G. & Toro, T. Wasserstein distance and the rectifiability of doubling measures: part I. Math. Ann. 364, 151–224 (2016). https://doi.org/10.1007/s00208-015-1206-z
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DOI: https://doi.org/10.1007/s00208-015-1206-z