The Sharp Sobolev Inequality on Metric Measure Spaces with Lower Ricci Curvature Bounds

2015 ◽  
Vol 43 (3) ◽  
pp. 513-529 ◽  
Author(s):  
Angelo Profeta
Keyword(s):  
Ricci Curvature ◽  
Sobolev Inequality ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Measure Spaces

scholarly journals Lagrangian calculus for nonsymmetric diffusion operators

2020 ◽  
Vol 13 (4) ◽  
pp. 361-383
Author(s):  
Christian Ketterer
Keyword(s):  
Ricci Curvature ◽  
Vector Fields ◽  
Probabilistic Approach ◽  
Gradient Estimates ◽  
Metric Measure Spaces ◽  
Curvature Bounds ◽  
Maximal Diameter ◽  
Measure Spaces ◽  
Evolution Variational Inequality ◽  
Dual Semigroup

AbstractWe characterize lower bounds for the Bakry–Emery Ricci tensor of nonsymmetric diffusion operators by convexity of entropy and line integrals on the {L^{2}}-Wasserstein space, and define a curvature-dimension condition for general metric measure spaces together with a square integrable 1-form in the sense of [N. Gigli, Nonsmooth differential geometry—an approach tailored for spaces with Ricci curvature bounded from below, Mem. Amer. Math. Soc. 251 2018, 1196, 1–161]. This extends the Lott–Sturm–Villani approach for lower Ricci curvature bounds of metric measure spaces. In generalized smooth context, consequences are new Bishop–Gromov estimates, pre-compactness under measured Gromov–Hausdorff convergence, and a Bonnet–Myers theorem that generalizes previous results by Kuwada [K. Kuwada, A probabilistic approach to the maximal diameter theorem, Math. Nachr. 286 2013, 4, 374–378]. We show that N-warped products together with lifted vector fields satisfy the curvature-dimension condition. For smooth Riemannian manifolds, we derive an evolution variational inequality and contraction estimates for the dual semigroup of nonsymmetric diffusion operators. Another theorem of Kuwada [K. Kuwada, Duality on gradient estimates and Wasserstein controls, J. Funct. Anal. 258 2010, 11, 3758–3774], [K. Kuwada, Space-time Wasserstein controls and Bakry–Ledoux type gradient estimates, Calc. Var. Partial Differential Equations 54 2015, 1, 127–161] yields Bakry–Emery gradient estimates.

scholarly journals Riemannian Ricci curvature lower bounds in metric measure spaces with $\sigma $-finite measure

2015 ◽  
Vol 367 (7) ◽  
pp. 4661-4701 ◽  
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Andrea Mondino ◽  
Tapio Rajala
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Finite Measure ◽  
Metric Measure Spaces ◽  
Measure Spaces

scholarly journals On interpolation and curvature via Wasserstein geodesics

2017 ◽  
Vol 10 (2) ◽  
pp. 125-167 ◽  
Author(s):  
Martin Kell
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Similar Condition ◽  
Interpolation Inequality ◽  
Metric Measure Spaces ◽  
Main Ingredient ◽  
Curvature Condition ◽  
Finsler Manifolds ◽  
Measure Spaces ◽  
Positively Curved

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

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