Corona Decompositions for Parabolic Uniformly Rectifiable Sets

TL;DR

This paper establishes that parabolic uniformly rectifiable sets can be decomposed into simpler structures called corona decompositions using Lip(1,1/2) graphs, linking geometric regularity to analytic decompositions.

Contribution

It proves the existence of bilateral corona decompositions for parabolic uniformly rectifiable sets with respect to Lip(1,1/2) graphs, extending previous work and characterizing these sets.

Findings

Parabolic uniformly rectifiable sets admit corona decompositions with Lip(1,1/2) graphs.

Equivalence of parabolic uniform rectifiability and existence of corona decompositions.

Characterization of these sets as big pieces squared of Lip(1,1/2) graphs.

Abstract

We prove that parabolic uniformly rectifiable sets admit (bilateral) corona decompositions with respect to regular Lip(1,1/2) graphs. Together with our previous work, this allows us to conclude that if $\Sigma\subset\mathbb{R}^{n+1}$ is parabolic Ahlfors-David regular, then the following statements are equivalent. (1) $\Sigma$ is parabolic uniformly rectifiable. (2) $\Sigma$ admits a corona decomposition with respect to regular Lip(1,1/2) graphs. (3) $\Sigma$ admits a bilateral corona decomposition with respect to regular Lip(1,1/2) graphs. (4) $\Sigma$ is big pieces squared of regular Lip(1,1/2) graphs.