On Local Poincare via Transportation

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

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 l…

Uniform Poincaré inequalities on measured metric spaces

Consider a proper geodesic metric space $(X,d)$ equipped with a Borel measure $μ.$ We establish a family of uniform Poincaré inequalities on $(X,d,μ)$ if it satisfies a local Poincaré inequality ($P_{loc}$) and a condition on growth of volume. Consequently if $μ$ is doubling and supports $(P_{loc})$ then it satisfies a $(σ,β,σ)$-Poincaré inequality. If $(X,d,μ)$ is a $δ$-hyperbolic space then using the volume comparison theorem in \cite{BCS} we obtain a uniform Poincaré inequality with exponential growth of the Poincaré constant. If $X$ is the universal cover of a compact $CD(K,\infty)$ space …

马尔可夫核的输运不等式及其应用

Transportation inequalities for Markov kernels and their applications

本文研究度量空间上马尔可夫核的输运不等式，证明压缩不等式与反向Poincaré、反向对数Sobolev及Wang Harnack不等式等价，并应用于马尔可夫过程收敛和热核测度拟不变性。

Local Poincaré inequalities from stable curvature conditions on metric spaces

We prove local Poincaré inequalities under various curvature-dimension conditions which are stable under the measured Gromov-Hausdorff convergence. The first class of spaces we consider is that of weak CD(K,N) spaces as defined by Lott and Villani. The second class of spaces we study consists of spaces where we have a flow satisfying an evolution variational inequality for either the Rényi entropy functional $E_N(ρm) = -\int_X ρ^{1-1/N} dm$ or the Shannon entropy functional $E_\infty(ρm) = \int_X ρ\log ρdm$. We also prove that if the Rényi entropy functional is strongly displacement convex in …

驯服空间的加倍、庞加莱与格罗莫夫-豪斯多夫预紧性

Doubling, Poincaré and Gromov-Hausdorff precompactness for tamed spaces

该研究针对满足特定条件的驯服度量测度空间，通过双李普希茨时间变换建立局部体积加倍与L²-庞加莱不等式，进而得到其有向度量测格罗莫夫-豪斯多夫预紧性。

Doubling and Poincaré inequalities for uniformized measures on Gromov hyperbolic spaces

We generalize the recent results of Björn-Björn-Shanmugalingam \cite{BBS20} concerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces \cite{BHK} by showing that these results also hold in the setting of uniformizing Gromov hyperbolic spaces by Busemann functions that we introduced in \cite{Bu20}. In particular uniformly local doubling and uniformly local Poincaré inequalities for the starting measure transform into global doubling and global Poincaré inequalities for the uniformized measure. We then show in the setting of uniform…