[Caractérisation des espaces métriques mesurés rectifiables]
Nous caractérisons les espaces métriques mesurés -rectifiables comme étant les espaces qui admettent une décomposition borélienne dénombrable telle que chaque morceau admet une -densité finie et strictement positive, et vérifie l'une des conditions suivantes : c'est un espace de Lipschitz-différentiabilité de dimension ; il admet représentations d'Alberti indépendantes ; il satisfait à la condition de David pour une carte de dimension . L'outil essentiel est une construction de grille itérative qui nous permet de montrer que l'image par une application de carte d'une boule ayant une grande densité de courbes des représentations d'Alberti contient une grande proportion d'une boule de grand rayon, et donc vérifie la condition de David. Cela nous permet d'appliquer des versions modifiées de résultats connus concernant les « morceaux bilipschitz » [8, 12, 10, 21] sur les cartes.
We characterize -rectifiable metric measure spaces as those spaces that admit a countable Borel decomposition so that each piece has positive and finite -densities and one of the following: is an -dimensional Lipschitz differentiability space; has -independent Alberti representations; satisfies David's condition for an -dimensional chart. The key tool is an iterative grid construction which allows us to show that the image of a ball with a high density of curves from the Alberti representations under a chart map contains a large portion of a uniformly large ball and hence satisfies David's condition. This allows us to apply modified versions of previously known “biLipschitz pieces” results [8, 12, 10, 21] on the charts.
@article{ASENS_2017__50_1_1_0,
author = {Bate, David and Li, Sean},
title = {Characterizations of rectifiable metric measure spaces},
journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
pages = {1--37},
publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
volume = {Ser. 4, 50},
number = {1},
year = {2017},
doi = {10.24033/asens.2314},
mrnumber = {3621425},
zbl = {1369.28002},
language = {en},
url = {http://www.numdam.org/articles/10.24033/asens.2314/}
}
TY - JOUR AU - Bate, David AU - Li, Sean TI - Characterizations of rectifiable metric measure spaces JO - Annales scientifiques de l'École Normale Supérieure PY - 2017 SP - 1 EP - 37 VL - 50 IS - 1 PB - Société Mathématique de France. Tous droits réservés UR - http://www.numdam.org/articles/10.24033/asens.2314/ DO - 10.24033/asens.2314 LA - en ID - ASENS_2017__50_1_1_0 ER -
%0 Journal Article %A Bate, David %A Li, Sean %T Characterizations of rectifiable metric measure spaces %J Annales scientifiques de l'École Normale Supérieure %D 2017 %P 1-37 %V 50 %N 1 %I Société Mathématique de France. Tous droits réservés %U http://www.numdam.org/articles/10.24033/asens.2314/ %R 10.24033/asens.2314 %G en %F ASENS_2017__50_1_1_0
Bate, David; Li, Sean. Characterizations of rectifiable metric measure spaces. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 50 (2017) no. 1, pp. 1-37. doi : 10.24033/asens.2314. http://www.numdam.org/articles/10.24033/asens.2314/
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