An implicit function theorem for Lipschitz mappings into metric spaces

Abstract

We prove a version of the implicit function theorem for Lipschitz mappings $f:{\mathbb{R}}^{n+m}\supset A\to X$ 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 $\pi :X\to {\mathbb{R}}^n$ such that $\pi \circ f\circ G^{-1}$, 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.