Poincaré-type inequalities and finding good parameterizations
Poincaré-type inequalities and finding good parameterizations
复制标题
庞加莱型不等式和寻找良好的参数化
DOI：
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发表时间：
2016
影响因子：
0.8
通讯作者：
Jessica Merhej
中科院分区：
文献类型：
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作者：
Jessica Merhej
几何测度论中一个非常重要的问题是集合的几何特征如何转化为关于它的分析信息。 Reifenberg (Bull Am Math Soc 66:312–313, 1960)证明，如果一个集合在每个点和每个尺度上都可以被平面很好地逼近，那么这个集合就是一个平面的双霍尔德图像。今天众所周知，这些近似平面上的卡尔森型条件保证了该集合的双利普希茨参数化。在本文中，我们考虑一个 n -Ahlfors 正则可整流集 $$M 子集 mathbb {R}^{n+d}$$ M ⊂ R n + d ，它满足涉及 Lipschitz 函数及其切向导数的庞加莱型不等式。然后，我们证明 M 的切平面振荡的卡尔森型条件保证 M 包含在 n 平面的双利普希茨图像中。我们还探讨了这里考虑的庞加莱型不等式，并表明它实际上等价于一般度量测度空间上考虑的其他庞加莱型不等式。
A very important question in geometric measure theory is how geometric features of a set translate into analytic information about it. Reifenberg (Bull Am Math Soc 66:312–313, 1960 ) proved that if a set is well approximated by planes at every point and at every scale, then the set is a bi-Hölder image of a plane. It is known today that Carleson-type conditions on these approximating planes guarantee a bi-Lipschitz parameterization of the set. In this paper, we consider an n -Ahlfors regular rectifiable set $$M subset mathbb {R}^{n+d}$$ M ⊂ R n + d that satisfies a Poincaré-type inequality involving Lipschitz functions and their tangential derivatives. Then, we show that a Carleson-type condition on the oscillations of the tangent planes of M guarantees that M is contained in a bi-Lipschitz image of an n -plane. We also explore the Poincaré-type inequality considered here and show that it is in fact equivalent to other Poincaré-type inequalities considered on general metric measure spaces.