Differentiability and Poincaré-type inequalities in metric measure spaces

Abstract

We demonstrate the necessity of a Poincaré type inequality for those metric measure spaces that satisfy Cheeger's generalization of Rademacher's theorem for all Lipschitz functions taking values in a Banach space with the Radon-Nikodym property. This is done by showing the existence of a rich structure of curve fragments that connect near by points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger-Kleiner, we show that our conditions are also sufficient. We also develop another characterization of "RNP Lipschitz differentiability spaces" by connecting points by curves that form a rich structure of partial derivatives that were first discussed in work by the first author.

v2: Significant reorganization of the paper and simplification of the construction of a non differentiable RNP valued Lipschitz function. New introduction. v3: incorporate referee's comments