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: ℝ ⁿ × ℝ ^m → ℝ ⁿ (with n-dim. image) can be precomposed with a bi-Lipschitz map ḡ ℝ ⁿ × ℝ ^m → ℝ ⁿ × ℝ ^m such that f ο ḡ will satisfy, when we restrict to a large portion of the domain E ⊂ → ℝ ⁿ × ℝ ^m, that f ο ḡ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map ḡ distorts 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 C ¹ 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 → ℝ ^D gives rise to a large (in an appropriate sense) subset E ⊂ [0,1] ⁿ such that f{pipe} _E is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of 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 ℝ ^D having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in ℝ ^D. This was previously known only for D ≥ 2n + 1 by a result of David and Semmes.