An implicit function theorem for Lipschitz mappings into metric spaces

2018/09/18 by Hajłasz, Piotr, Zimmerman, Scott
#28A75 #53C23 #54E40 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1809.06829

Abstract

We prove a version of the implicit function theorem for Lipschitz mappings f:ℝ^n+m⊃ A → X into arbitrary metric spaces. As long as the pull-back of the Hausdorff content H_∞ⁿ by f has positive upper n-density on a set of positive Lebesgue measure, then, there is a local diffeomorphism G in ℝ^n+m and a Lipschitz map π:X→ ℝⁿ such that π∘ f∘ G^-1, when restricted to a certain subset of A of positive measure, is a the orthogonal projection of 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.

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