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

We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\bR^n\times \bR^m\to\bR^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\bR^n\times \bR^m\to \bR^n\times \bR^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \bR^n\times \bR^m$, 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 $\bR^{n+m}$ 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 {\bf 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 $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere. On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \bR^D$ 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 {\bf all} of $\bR^n$. The most interesting case is the case $D=n$. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\bR^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\bR^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.