An implicit function theorem for Lipschitz mappings into metric spaces
An implicit function theorem for Lipschitz mappings into metric spaces
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Lipschitz 映射到度量空间的隐函数定理
DOI：
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发表时间：
2018
期刊：
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通讯作者：
Scott Zimmerman
中科院分区：
文献类型：
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作者：
P. Hajłasz;Scott Zimmerman
我们证明了任意度量空间中Lipschitz映射$f:mathbb{R}^{n+m}集a o X$的隐函数定理的一个版本。只要Hausdorff内容$mathcal{H}_{infty}^n$被$f$拉回在一个正Lebesgue测度集合上具有正的$n$-密度，那么，在$mathbb{R}}^{n+m}$中存在一个局部微分同构$G$和一个Lipschitz映射$pi: x o mathbb{R}^n$，使得$picirc fcirc G^{-1}$，当限制于正测度$ a $的某个子集时，是$mathbb{R}} {n+m}$在前$n$-坐标上的正交投影。这可以看作是Azzam和Schul的一个类似结果的定性版本。我们证明的主要工具是Hajlasz和Malekzadeh在一篇论文中引入的变量度规变换。
We prove a version of the implicit function theorem for Lipschitz mappings $f:mathbb{R}^{n+m}supset A o X$ into arbitrary metric spaces. As long as the pull-back of the Hausdorff content $mathcal{H}_{infty}^n$ by $f$ has positive upper $n$-density on a set of positive Lebesgue measure, then, there is a local diffeomorphism $G$ in $mathbb{R}^{n+m}$ and a Lipschitz map $pi:X o mathbb{R}^n$ such that $picirc fcirc G^{-1}$, when restricted to a certain subset of $A$ of positive measure, is a the orthogonal projection of $mathbb{R}^{n+m}$ onto the first $n$-coordinates. This may be seen as a qualitative version of a similar result of Azzam and Schul. The main tool in our proof is the metric change of variables introduced in a paper of Hajlasz and Malekzadeh.