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 nearby points, similar in nature to Semmes's pencil of curves for the standard Poincaré inequality. Using techniques similar to Cheeger–Kleiner [12], 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 [5].