Nets in Euclidean Space and Differentiability of Lipschitz Functions on Metric Measure Spaces

Abstract

We show a discussion of the following questions: for a given real –valued function (i) ( f ∈ ∞ Rn L ) there can not be bi-Lipschitz homeomorphism determinant, det Dφ f = φR : n →n R such that the Jacobian , (ii) a Lipschitz or quasiconformal vector field with div (iii) for a given separated net except for n= 1 y⊂ n R a bi-Lipschitz map u =f φy : , →n Z or if the Lipschitz condition is relaxed to a Hölder condition. We show also an extent to certain metric measure spaces, a generalization of the theorem of Rademacher which asserts that : a real-valued Lipschitz function on Rn is differentiable almost everywhere with respect to Lebesgue measure and the blow ups of a real-valued Lipschitz function converge to a unique linear function.