Corona Decompositions for Parabolic Uniformly Rectifiable Sets

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

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

Metric Geometry 2023-02-08 v3 Analysis of PDEs

Authors: Simon Bortz , John Hoffman , Steve Hofmann , José Luis Luna Garcia , Kaj Nyström

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.

Keywords

corona theorem

Cite

@article{arxiv.2103.12497,
  title  = {Corona Decompositions for Parabolic Uniformly Rectifiable Sets},
  author = {Simon Bortz and John Hoffman and Steve Hofmann and José Luis Luna Garcia and Kaj Nyström},
  journal= {arXiv preprint arXiv:2103.12497},
  year   = {2023}
}

Comments

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 $\delta$

Corona Decompositions for Parabolic Uniformly Rectifiable Sets

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