Differentiability and Poincar\'e-type inequalities in metric measure spaces

TL;DR

This paper establishes the necessity and sufficiency of Poincaré-type inequalities in metric measure spaces for Lipschitz differentiability, linking geometric curve structures to functional inequalities.

Contribution

It demonstrates the equivalence between Poincaré inequalities and the existence of rich curve structures in metric measure spaces satisfying Rademacher-type theorems.

Findings

Poincaré inequalities are necessary for RNP Lipschitz differentiability spaces.

Rich structures of connecting curves are both necessary and sufficient for these inequalities.

New characterizations of RNP Lipschitz differentiability spaces are provided.

Abstract

We demonstrate the necessity of a Poincar\'e 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 Poincar\'e 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.