Lipschitz decompositions of domains with bilaterally flat boundaries

TL;DR

This paper extends Lipschitz decomposition techniques from planar domains to higher-dimensional domains with flat or Reifenberg flat boundaries, providing new geometric decompositions with controlled boundary measures.

Contribution

It introduces higher-dimensional Lipschitz decomposition results for domains with Reifenberg flat boundaries, generalizing Jones' planar theorem to more complex geometries.

Findings

Decomposition of higher-dimensional domains with flat boundaries into Lipschitz graph domains.

Bounded overlap decompositions for Reifenberg flat or uniformly rectifiable boundaries.

Controlled total boundary surface area in the decompositions.

Abstract

We study classes of domains in $\mathbb{R}^{d+1},\ d \geq 2$ with sufficiently flat boundaries that admit a decomposition or covering of bounded overlap by Lipschitz graph domains with controlled total surface area. This study is motivated by the following result proved by Peter Jones as a piece of his proof of the Analyst's Traveling Salesman Theorem in the complex plane: Any simply connected domain $\Omega\subseteq\mathbb{C}$ with finite boundary length $\mathcal{H}^1(\partial\Omega)$ can be decomposed into Lipschitz graph domains with total boundary length bounded above by $M\mathcal{H}^1(\partial\Omega)$ for some $M$ independent of $\Omega$. In this paper, we prove an analogous Lipschitz decomposition result in higher dimensions for domains with Reifenberg flat boundaries satisfying a uniform beta-squared sum bound. We use similar techniques to show that domains with general Reifenberg flat or uniformly rectifiable boundaries admit similar Lipschitz decompositions while allowing the constituent domains to have bounded overlaps rather than be disjoint.