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Characterizing spaces satisfying Poincaré Inequalities and applications to differentiability

  • Sylvester Eriksson-BiqueEmail author
Article

Abstract

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 \({q \in [1,\infty)}\), which is related to Keith’s and Zhong’s theorem on self-improvement of Poincaré inequalities.

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Notes

Acknowledgements

The author is thankful to professor Bruce Kleiner for suggesting the problem on the local geometry of Lipschitz differentiability spaces, for numerous helpful discussions on the topic and several comments that improved the exposition of this paper. Kleiner was instrumental in restructuring the proofs in the third and fifth sections and thus helped greatly simplify the presentation. The author also thanks a number of people who have given useful comments in the process of writing this paper, such as Sirkka-Liisa Eriksson, Jana Björn, Nagesvari Shanmugalingam, Pekka Koskela, Guy C. David and Ranaan Schul. Some results of the paper were heavily influenced by conversations with Tatiana Toro and Jeff Cheeger. An earlier draft of this paper had a more complicated construction used to resolve Theorem 1.15. This construction is here rephrased in terms of a modified hyperbolic filling which is much clearer than the earlier version. This modification was encouraged by Bruce Kleiner, and suggested to the author by Daniel Meyer. We also thank the anonymous referees for numerous comments and corrections. The research was supported by a NSF graduate student fellowship DGE-1342536 and NSF Grant DMS-1405899.

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Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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