Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps
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Hard Sard：Lipschitz 映射的定量隐式函数和可拓定理
DOI：
10.1007/s00039-012-0189-0
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发表时间：
2011
影响因子：
2.2
通讯作者：
Raanan Schul
中科院分区：
文献类型：
--
作者：
Jonas Azzam;Raanan Schul
We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be precomposed with a bi-Lipschitz map $${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ such that $${f \circ \bar{g}}$$ will satisfy, when we restrict to a large portion of the domain $${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ , that $${f \circ \bar{g}}$$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $${\bar{g}}$$ distorts $${\mathbb{R}^{n+m}}$$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C1 map from [0, 1]3 onto [0, 1]2 with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function $${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$$ gives rise to a large (in an appropriate sense) subset $${E \subset [0,1]^{n}}$$ such that $${f|_E}$$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of $${\mathbb{R}^{n}}$$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in $${\mathbb{R}^{D}}$$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $${\mathbb{R}^{D}}$$ . This was previously known only for D ≥ 2n + 1 by a result of David and Semmes.
We prove a global implicit function theorem. In particular we show that any Lipschitz map $${f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}$$ (with n-dim. image) can be precomposed with a bi-Lipschitz map $${\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ such that $${f \circ \bar{g}}$$ will satisfy, when we restrict to a large portion of the domain $${E \subset \mathbb{R}^{n} \times \mathbb{R}^{m}}$$ , that $${f \circ \bar{g}}$$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $${\bar{g}}$$ distorts $${\mathbb{R}^{n+m}}$$ in a controlled manner so that the fibers of f are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set E on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman’s 1979 construction of a C1 map from [0, 1]3 onto [0, 1]2 with rank ≤ 1 everywhere. On route we prove an extension theorem which is of independent interest. We show that for any D ≥ n, any Lipschitz function $${f : [0,1]^{n} \rightarrow \mathbb{R}^{D}}$$ gives rise to a large (in an appropriate sense) subset $${E \subset [0,1]^{n}}$$ such that $${f|_E}$$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of $${\mathbb{R}^{n}}$$ . This extends results of Jones and David, from 1988. As a simple corollary, we show that n-dimensional Ahlfors–David regular spaces lying in $${\mathbb{R}^{D}}$$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $${\mathbb{R}^{D}}$$ . This was previously known only for D ≥ 2n + 1 by a result of David and Semmes.