Fragment-wise differentiable structures

Abstract

We give a definition of a fragment-wise chart and a fragment-wise differential of a Lipschitz function in general complete metric measure spaces, and show that they exist whenever the space has finite Hausdorff dimension – without any further topological or analytic assumptions. The differential has a natural definition which extends the definition of a differential in Lipschitz differentiability spaces. We also show a cornerstone in the theory of tangent modules: density of directions of curve fragments in the associated tangent space. The key to our approach is the combination of modulus and upper gradient-based methods that draws analogies between the Sobolev and Lipschitz theories. These concepts bridge approaches based on abstract modules, PI-spaces, decomposability bundles and differentiability spaces. In this unified perspective, we give a sharp approximation result, develop duality between ∞-modulus and plans and show the existence of a minimal ∗-upper gradient.