Lipschitz decompositions of domains with bilaterally flat boundaries

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 $Ω\subseteq\mathbb{C}$ with finite boundary length $\mathcal{H}^1(\partialΩ)$ can be decomposed into Lipschitz graph domains with total boundary length bounded above by $M\mathcal{H}^1(\partialΩ)$ for some $M$ independent of $Ω$. 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.