Characterizing spaces satisfying Poincaré 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)-Poincaré 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 Poincaré inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincaré 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 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 show that many classes of weak, Orlicz and non-homogeneous Poincaré inequalities “self-improve” to classical (1, q)-Poincaré inequalities for some q∈[1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${q \in [1,\infty)}$$\end{document}, which is related to Keith’s and Zhong’s theorem on self-improvement of Poincaré inequalities.