Characterizing Spaces Satisfying Poincaré Inequalities and Applications to Differentiability

We show that a proper metric measure space is a RNP-differentiability space if and only if it is rectifiable in terms of doubling metric measure spaces with some Poincaré inequality. This result characterizes metric measure spaces that can be covered by spaces admitting Poincaré inequalities, as well as metric measure spaces that admit a measurable differentiable structure which permits differentiation of Lipschitz functions with certain Banach space targets. The proof is based on a new “thickening” construction, which can be used to enlarge subsets into spaces admitting Poincaré inequalities. We also introduce a new characterization in terms of a quantitative connectivity condition of spaces admitting some local (1, p)-Poincaré inequality. This characterization result has several applications of independent interest. We resolve a question of Tapio Rajala on the existence of Poincaré inequalities for the class of MCP (K,n)-spaces which satisfy a weak Ricci-bound. We show that deforming a geodesic metric measure space by Muckenhoupt weights preserves the property of possessing a Poincaré inequality. Finally, the new condition allows us to strengthen the conclusion of the celebrated theorem by Keith and Zhong to show that many classes of weak Poincaré inequalities self-improve to true Poincaré inequalities.