Coronizations and big pieces in metric spaces

TL;DR

This paper generalizes the concept of coronizations implying big pieces squared from Euclidean spaces to metric spaces with arbitrary regular sets, broadening the scope of geometric measure theory and its applications.

Contribution

It proves that coronizations with respect to arbitrary d-regular sets imply big pieces squared in general metric spaces, extending classical Euclidean results to a broader setting.

Findings

Coronizations imply big pieces squared in metric spaces.

The results apply to metric spaces with arbitrary regular sets.

Application to parabolic uniform rectifiability.

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.