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arXiv · 2008.11544

Coronizations and big pieces in metric spaces

Simon Bortz·John Hoffman·Steve Hofmann·José Luis Luna Garcia·Kaj Nyström
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Abstract

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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Simon Bortz, John Hoffman, Steve Hofmann, José Luis Luna Garcia, Kaj Nyström. 2020-09-09. Coronizations and big pieces in metric spaces. https://arxiv.org/abs/2008.11544

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