Structure of measures in Lipschitz differentiability spaces

TL;DR

This paper establishes the equivalence of two approaches to generalizing Rademacher's theorem in metric measure spaces, introducing a novel differentiability concept based on Lipschitz curves and characterizing Lipschitz differentiability spaces.

Contribution

It introduces a new approach to differentiability in metric measure spaces using Lipschitz curves and proves the equivalence of this with existing methods, providing new characterizations.

Findings

Equivalence of two generalizations of Rademacher's theorem.

Introduction of a differentiability approach based on Lipschitz curves.

New descriptions of Lipschitz differentiability spaces.

Abstract

We prove the equivalence of two seemingly very different ways of generalising Rademacher's theorem to metric measure spaces. One such generalisation is based upon the notion of forming partial derivatives along a very rich structure of Lipschitz curves in a way analogous to the differentiability theory of Euclidean spaces. This approach to differentiability in this generality appears here for the first time and by examining this structure further, we naturally arrive to several descriptions of Lipschitz differentiability spaces.