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： 我们证明，一个合适的度量测度空间是一个RNP可微空间当且仅当它在具有某些Poincaré不等式的加倍度量测度空间的意义下是可求长的。 这个结果刻画了可以被接受Poincaré不等式的空间覆盖的度量测度空间，以及接受可测可微结构的度量测度空间，该结构允许对具有某些巴拿赫空间目标的Lipschitz函数进行微分。 证明基于一种新的“加厚”构造，该构造可用于将子集扩展为接受Poincaré不等式的空间。 我们还引入了一种新的表征方式，该方式以接受某种局部$(1,p)$-Poincaré不等式的空间的定量连通性条件为基础。 这一表征结果有几个独立感兴趣的用途。 我们解决了Tapio Rajala关于满足弱 Ricci 界的$MCP(K,n)$-空间类中Poincaré不等式存在性的问题。 我们证明，通过Muckenhoupt权重对测地度量测度空间进行变形，可以保持具有Poincaré不等式的性质。 最后，新条件使我们能够加强Keith和Zhong的著名定理的结论，以表明许多类型的弱Poincaré不等式自我改进为真正的Poincaré不等式。 显示英文摘要

摘要： 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 Poincare inequality. This result characterizes metric measure spaces that can be covered by spaces admitting Poincare 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 Poincare inequalities. We also introduce a new characterization in terms of a quantitative connectivity condition of spaces admitting some local $(1,p)$-Poincare inequality. This characterization result has several applications of independent interest. 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 strengthen the conclusion of the celebrated theorem by Keith and Zhong to show that many classes of weak Poincare inequalities self-improve to true Poincare inequalities.