On the relationship between derivations and measurable differentiable structures on metric measure spaces

Abstract

We investigate the relationship between measurable differentiable structures on doubling metric measure spaces and derivations. We prove: [1] a decomposition theorem for the module of derivations into free modules; [2] the existence of a measurable differentiable structure assuming that one can control the pointwise upper Lipschitz constant of a function through derivations; [3] an extension of a result of Keith about the choice of chart functions.