论文标题
Lipschitz分解具有双侧平坦边界的域
Lipschitz decompositions of domains with bilaterally flat boundaries
论文作者
论文摘要
我们在$ \ mathbb {r}^{d+1}中研究域的类别，\ d \ geq 2 $具有足够平坦的边界，这些界限允许由Lipschitz Graph Graph域对具有控制的总表面积的分解或覆盖有限的重叠。这项研究是由彼得·琼斯（Peter Jones）证明了以下结果的动机$ M \ MATHCAL {H}^1（\partialΩ）$ for $ m $独立于$ω$。在本文中，我们证明了类似的Lipschitz分解会导致较高的尺寸，该域具有reifenberg平坦边界，满足均匀的beta平方和结合的均匀尺寸。我们使用类似的技术来表明，具有一般Reifenberg平坦或均匀校正边界的域允许相似的Lipschitz分解，同时允许组成域的重叠界定而不是不相交。
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.
