On big pieces approximations of parabolic hypersurfaces

Simon Bortz¹ ; John Hoffman² ; Steve Hofmann² ; Jose Luis Luna-Garcia² ; Kaj Nyström³

¹ University of Alabama, Department of Mathematics
² University of Missouri, Department of Mathematics
³ Uppsala University, Department of Mathematics

Annales Fennici Mathematici, Tome 47 (2022) no. 1, pp. 533-571.

Résumé

Let $\Sigma$ be a closed subset of $\mathbb{R}^{n+1}$ which is parabolic Ahlfors-David regular and assume that $\Sigma$ satisfies a 2-sided corkscrew condition. Assume, in addition, that $\Sigma$ is either time-forwards Ahlfors-David regular, time-backwards Ahlfors-David regular, or parabolic uniform rectifiable. We then first prove that $\Sigma$ satisfies a weak synchronized two cube condition. Based on this we are able to revisit the argument of Nyström and Strömqvist (2009) and prove that $\Sigma$ contain suniform big pieces of Lip(1,1/2) graphs. When $\Sigma$ is parabolic uniformly rectifiable the construction can be refined and in this case we prove that $\Sigma$ contains uniform big pieces of regular parabolic Lip(1,1/2) graphs. Similar results hold if $\Omega\subset\mathbb{R}^{n+1}$ is a connected component of $\mathbb{R}^{n+1}\setminus\Sigma$ and in this context we also give a parabolic counterpart of the main result of Azzam et al. (2017) by proving that if $\Omega$ is a one-sided parabolic chord arc domain, and if $\Sigma$ is parabolic uniformly rectifiable, then $\Omega$ is in fact a parabolic chord arc domain. Our results give a flexible parabolic version of the classical (elliptic) result of David and Jerison (1990) concerning the existence of uniform big pieces of Lipschitz graphs for sets satisfying a two disc condition.

Détail

DOI : 10.54330/afm.115417

Keywords: Parabolic Lipschitz graph, parabolic uniform rectifiability, big pieces, parabolic measure, caloric measure