Functional Inequalities on Weighted Riemannian Manifolds Subject to Curvature-Dimension Conditions

We establish new sharp inequalities of Poincar\'{e} or log-Sobolev type, on geodesically-convex weighted Riemannian manifolds $(M,\mathfrak{g},\mu)$ whose (generalized) Ricci curvature $Ric_{\mathfrak{g},\mu,N}$ with effective dimension parameter $N\in (-\infty,\infty]$ is bounded from below by a constant $K\in\mathbb{R}$, and whose diameter is bounded above by $D\in (0,\infty]$ (Curvature-Dimension-Diameter conditions $CDD(K,N,D)$). To this end we establish a general method which complements the `localization' theorem which has recently been established by B. Klartag. Klartag's Theorem is based on optimal transport techniques, leading to a disintegration of the manifold measure into marginal measures supported on geodesics of the manifold. This leads to a reduction of the problem of proving a n-dimensional inequality into an optimization problem over a class of measures with 1-dimensional supports. In this work we firstly develop a general approach which leads to a reduction of this optimization problem into a simpler optimization problem, on a subclass of `model measures'. This reduction is based on functional analytic techniques, in particular a classification of extreme points of a specific subset of measures, and showing that the solution to the optimization problem is attained on this set of extreme points. Finally we solve the optimization problems associated with the Poincar\'{e}, p-Poincar\'{e} and the log-Sobolev inequalities subject to specific $CDD(K,N,D)$ conditions. Notably, we prove new sharp Poincar\'{e} inequalities for $N\in (-\infty,0]$. We find that for $N\in (-1,0]$ the characterization of the sharp lower bound on the Poincar\'{e} constant is of different nature; in addition we derive new lower bounds on the log-Sobolev constant under $CDD(K,\infty,D)$ conditions where $K\in\mathbb{R}$ and $D\in (0,\infty]$, which up to numeric constants are best possible.