On the Geometry of Rectifiable Sets with Carleson and Poincar\'e-type Conditions
On the Geometry of Rectifiable Sets with Carleson and Poincar\'e-type Conditions
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论具有卡尔森和庞加莱型条件的可整流集的几何
DOI：
10.1512/iumj.2017.66.6161
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发表时间：
2015
期刊：
影响因子：
--
通讯作者：
Jessica Merhej
中科院分区：
文献类型：
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作者：
Jessica Merhej
几何测度论中的一个中心问题是集合的几何性质能否转化为解析性质。1960年，E. R. Reifenberg证明了如果$\mathbb{R}^{n+k}$的一个$n$维子集$M$在每一点和每一尺度上都被$n$-平面很好地逼近，那么$M$是$n$-平面的一个局部双H\“lder像。从那时起，Reifenberg的定理已经在几个方面完善，以确保$M$是一个双Lipschitz图像的$n$-平面。本文证明了$\mathbb{R}^{n+1}$的$n$-Ahlfors正则可求长子集$M$的单位法线振动的Carleson条件满足Poincar\'e型不等式，从而证明了$M$包含在$\mathbb{R}^{n+1}$的$n$维仿射子空间的双Lipschitz像中.我们还证明了这个Poincar\'e型不等式编码了关于$M$的几何信息，即它隐含着$M$是拟凸的。
A central question in geometric measure theory is whether geometric properties of a set translate into analytical ones. In 1960, E. R. Reifenberg proved that if an $n$-dimensional subset $M$ of $\mathbb{R}^{n+k}$ is well approximated by $n$-planes at every point and at every scale, then $M$ is a locally bi-H\"older image of an $n$-plane. Since then, Reifenberg's theorem has been refined in several ways in order to ensure that $M$ is a bi-Lipschitz image of an $n$-plane. In this paper, we show that a Carleson condition on the oscillation of the unit normal of an $n$-Ahlfors regular rectifiable subset $M$ of $\mathbb{R}^{n+1}$ satisfying a Poincar\'e-type inequality is sufficient to prove that $M$ is contained inside a bi-Lipschitz image of an $n$-dimensional affine subspace of $\mathbb{R}^{n+1}$. We also show that this Poincar\'e-type inequality encodes geometrical information about $M$, namely it implies that $M$ is quasiconvex.