Measurable Differentiable Structures on Doubling 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.)
Cite
@article{arxiv.1110.4279,
title = {Measurable Differentiable Structures on Doubling Metric Spaces},
author = {Jasun Gong},
journal= {arXiv preprint arXiv:1110.4279},
year = {2012}
}
Comments
Edits include: (A) a corrected statement of the main theorem, in terms of a countable decomposition of the space and linearly independent sets of derivations, not bases; (B) an improved proof of Lemma 1.10, via generalised Jacobians instead of piecewise linear extensions; (C) the case of finitely generated Lipschitz algebras requires that the measure satisfy the Lebesgue density theorem