The Quasi CurvatureDimension Condition with applications to subRiemannian manifolds
Abstract
We obtain the best known quantitative estimates for the $L^p$Poincaré and logSobolev inequalities on domains in various ideal subRiemannian manifolds, including ideal Carnot groups and in particular ideal generalized Htype Carnot groups and the Heisenberg groups, the Grushin plane, and various Sasakian and $3$Sasakian manifolds. Moreover, this constitutes the first time that a quantitative estimate independent of the dimension is established on these spaces. For instance, the LiYau / ZhongYang spectralgap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of $4$. We achieve this by introducing a quasiconvex relaxation of the LottSturmVillani $\mathsf{CD}(K,N)$ condition we call the Quasi CurvatureDimension condition $\mathsf{QCD}(Q,K,N)$. Our motivation stems from a recent interpolation inequality along Wasserstein geodesics in the ideal subRiemannian setting due to Barilari and Rizzi. We show that on an ideal subRiemannian manifold of dimension $n$, the Measure Contraction Property $\mathsf{MCP}(K,N)$ implies $\mathsf{QCD}(Q,K,N)$ with $Q = 2^{Nn} \geq 1$, thereby verifying the latter property on the aforementioned spaces. By extending the localization paradigm to completely general interpolation inequalities, we reduce the study of various analytic and geometric inequalities on $\mathsf{QCD}$ spaces to the onedimensional case. Consequently, we deduce that while ideal (strictly) subRiemannian manifolds do not satisfy any type of $\mathsf{CD}$ condition, they satisfy numerous functional inequalities with \emph{exactly the same} quantitative dependence (up to a factor of $Q$) as their $\mathsf{CD}$ counterparts.
 Publication:

arXiv eprints
 Pub Date:
 August 2019
 arXiv:
 arXiv:1908.01513
 Bibcode:
 2019arXiv190801513M
 Keywords:

 Mathematics  Functional Analysis;
 Mathematics  Differential Geometry;
 Mathematics  Metric Geometry;
 Mathematics  Spectral Theory
 EPrint:
 36 pages