2011/10/19 by Jasun Gong, Gong, Jasun
Mathematics · #28A15 #30L05 #46E35 #53C23 #58C20 #FOS: Mathematics #Metric Geometry (math.MG) #math.MG #msc:28A15 #msc:30L05 #msc:46E35 #msc:53C23 #msc:58C20
paper · pdf · doi:10.48550/arxiv.1110.4279
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
arxiv created 2012/08/14 · arxiv updated 2012/08/15
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 Poincaré 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.)