no. 5
Coronizations and big pieces in metric spaces
[Construction en couronnes et grands morceaux dans des espaces métriques]
Bortz, Simon1 ; Hoffman, John2 ; Hofmann, Steve2 ; Luna-Garcia, Jose Luis3 ; Nyström, Kaj4
1 Department of Mathematics University of Alabama Tuscaloosa, AL, 35487 (USA)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 3L8 (Canada)
4 Department of Mathematics Uppsala University S-751 06 Uppsala (Sweden)
Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 2037-2078.

Nous prouvons que la construction en couronnes relative à des ensembles d-réguliers arbitraires (non nécessairement des graphes) fournit de larges pièces carrées de ces ensembles (approximants). Cela est connu (et dû à David et Semmes dans le cas d’une co-dimension assez grande, et à Azzam et Schul en général) dans le cadre (classique) des espaces Euclidiens avec une mesure de Hausdorff de dimension entière, où les ensembles approximants sont des graphes Lipschitziens. Nos travaux sont une généralisation d’envergure de ces résultats, et nous montrons que le fait que la construction en couronnes fournisse de larges pièces carrées est une propriété générique. En particulier, nos résultats, convenablement interprétés, s’appliquent dans des espaces métriques de dimension strictement positive fixée (éventuellement non entière), munisavec une mesure de Borel régulière et avec des espaces approximants arbitraires. Comme application nouvelle, nous illustrons comment on peut utiliser ces outils généraux dans le cadre de la rectifiabilité parabolique uniforme.

We prove that coronizations with respect to arbitrary d-regular sets (not necessarily graphs) imply big pieces squared of these (approximating) sets. This is known (and due to David and Semmes in the case of sufficiently large co-dimension, and to Azzam and Schul in general) in the (classical) setting of Euclidean spaces with Hausdorff measure of integer dimension, where the approximating sets are Lipschitz graphs. Our result is a far reaching generalization of these results and we prove that coronizations imply big pieces squared is a generic property. In particular, our result applies, when suitably interpreted, in metric spaces having a fixed positive (perhaps non-integer) dimension, equipped with a Borel regular measure and with arbitrary approximating sets. As a novel application we highlight how to utilize this general setting in the context of parabolic uniform rectifiability.

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Révisé le :
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DOI : 10.5802/aif.3518
Classification : 28A75, 30L99, 43A85
Keywords: Carleson measures, corona decompositions, big pieces, geometric lemmas.
Mot clés : Mesures de Carleson, décomposition en couronnes, grands morceaux, lemmes géométriques.
Bortz, Simon 1 ; Hoffman, John 2 ; Hofmann, Steve 2 ; Luna-Garcia, Jose Luis 3 ; Nyström, Kaj 4

1 Department of Mathematics University of Alabama Tuscaloosa, AL, 35487 (USA)
2 Department of Mathematics University of Missouri Columbia, MO 65211 (USA)
3 Department of Mathematics & Statistics McMaster University Hamilton, ON L8S 3L8 (Canada)
4 Department of Mathematics Uppsala University S-751 06 Uppsala (Sweden)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {Coronizations and big pieces in metric spaces},
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     pages = {2037--2078},
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Bortz, Simon; Hoffman, John; Hofmann, Steve; Luna-Garcia, Jose Luis; Nyström, Kaj. Coronizations and big pieces in metric spaces. Annales de l'Institut Fourier, Tome 72 (2022) no. 5, pp. 2037-2078. doi : 10.5802/aif.3518. https://aif.centre-mersenne.org/articles/10.5802/aif.3518/

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