标题： Poincaré不等式的分类和RNP可微空间的局部几何 显示英文标题

标题： Classifying Poincaré Inequalities and the local geometry of RNP-Differentiability Spaces

摘要： 我们把完整的RNP可微空间表征为那些在加倍度量测度空间中可测的，满足某些局部$(1, p)$-Poincaré不等式的空间。 这给出了在Cheeger意义下具有强形式可微结构的空间的完整表征，并提供了他对该定理的部分逆定理。 证明基于一种新的“加厚”构造，该构造可用于将子集扩展为具有Poincaré不等式空间。 我们还引入了一种新的定量连通性概念，该概念表征了满足局部Poincaré不等式空间。 这种表征有独立的兴趣，并且在可微空间之外有若干应用。 我们解决了Tapio Rajala关于满足弱Ricci界的一类$MCP(K, n)$-空间是否存在Poincaré不等式的问题。 我们证明通过Muckenhoupt权对测地度量测度空间进行变形会保持具有Poincaré不等式的性质。 最后，新条件使我们能够证明许多弱、Orlicz和非齐次Poincaré不等式“自我改进”为某些$q \in [1,\infty)$的古典$(1, q)$-Poincaré不等式，这与Keith和Zhong关于Poincaré不等式自我改进的定理有关。 显示英文摘要

摘要： We characterize complete RNP-differentiability spaces as those spaces which are rectifiable in terms of doubling metric measure spaces satisfying some local $(1, p)$-Poincar\'e 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 Poincar\'e inequalities. We also introduce a new notion of quantitative connectivity which characterizes spaces satisfying local Poincar\'e 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 Poincar\'e 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 Poincar\'e inequality. Finally, the new condition allows us to show that many classes of weak, Orlicz and non-homogeneous Poincar\'e inequalities `self-improve' to classical $(1, q)$-Poincar\'e inequalities for some $q \in [1,\infty)$, which is related to Keith's and Zhong's theorem on self-improvement of Poincar\'e inequalities.