Measurable Differentiable Structures on Doubling Metric Spaces

TL;DR

This paper establishes a differentiability theorem for Lipschitz functions on doubling metric spaces, linking it to the existence of nontrivial derivations satisfying certain conditions, extending previous results by Cheeger and Keith.

Contribution

It introduces new conditions under which Lipschitz functions are differentiable on doubling metric spaces, generalizing prior differentiability results.

Findings

Proves equivalence between differentiability and nontrivial derivations in doubling spaces.

Establishes new rank bounds for derivations with doubling measures.

Extends Cheeger and Keith's differentiability results to broader classes of metric spaces.

Abstract

On metric spaces equipped with doubling measures, we prove that a differentiability theorem holds for Lipschitz functions if and only if the space supports nontrivial (metric) derivations in the sense of Weaver that satisfy an additional infinitesmal condition. In particular it extends the case of spaces supporting Poincar\'e inequalities, as first proven by Cheeger, as well as the case of spaces satisfying the Lip-lip condition of Keith. The proof relies on generalised "change of variable" arguments that are made possible by the linear algebraic structure of derivations. As a crucial step in the argument, we also prove new rank bounds for derivations with respect to doubling measures. (Note: this is an updated version of an earlier preprint, titled "Differentiability of Lipschitz functions on doubling metric measure spaces." The edits are listed in the comments.)