Differentiability of Lipschitz Functions on Doubling Metric Measure Spaces

Details Differentiability of Lipschitz Functions on Doubling Metric Measure Spaces Differentiability of Lipschitz Functions on Doubling Metric Measure Spaces

2011-10-19

arxiv.org/abs/1110.4279v1

Mathematics

Metric Geometry

27 pages

Scientific paper

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.

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