Corona Decompositions for Parabolic Uniformly Rectifiable Sets

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.

We have improved the exposition in section 4. In particular, we have treated the case where `contact set' in the Whitney-type construction is empty. This case seemed to be overlooked in David and Semmes' original construction, but only required a modest fix. We have also corrected the (egregious) typo in what is now equation (4.42), which now has $\epsilon$ rather than $\delta$