# Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps

@article{Azzam2011HardSQ,
title={Hard Sard: Quantitative Implicit Function and Extension Theorems for Lipschitz Maps},
author={Jonas Azzam and Raanan Schul},
journal={Geometric and Functional Analysis},
year={2011},
volume={22},
pages={1062-1123}
}
• Published 21 May 2011
• Mathematics
• Geometric and Functional Analysis
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…
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