Characterizing spaces satisfying Poincare Inequalities and applications to differentiability

We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local (1, p)-Poincare inequalities. This gives a full characterization of spaces admitting a strong form of a differentiability structure in the sense of Cheeger, and provides a partial converse to his theorem. The proof is based on a new thickening construction, which can be used to enlarge subsets into spaces admitting Poincare inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincare inequalities. This characterization is of independent interest, and has several applications separate from differentiability spaces. We resolve a question of Tapio Rajala on the existence of Poincare 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 Poincare inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincare inequalities self-improve to classical (1, q)-Poincare inequalities for some q[1,), which is related to Keith's and Zhong's theorem on self-improvement of Poincare inequalities.