The sharp Sobolev inequality on metric measure spaces with lower Ricci curvature bounds

DOI10.1007/s11118-015-9485-2zbMath1333.53050OpenAlexW849247667MaRDI QIDQ905628

Angelo Profeta

Publication date: 27 January 2016

Published in: Potential Analysis (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/s11118-015-9485-2

zbMATH Keywords

Nash inequalitymetric measure spacescurvature-dimension conditionsharp Sobolev inequality

Mathematics Subject Classification ID

Nonsmooth analysis (49J52) Sobolev spaces and other spaces of ``smooth functions, embedding theorems, trace theorems (46E35) Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces (53C23) Methods of global Riemannian geometry, including PDE methods; curvature restrictions (53C21) Inequalities involving derivatives and differential and integral operators (26D10)

Related Items

Diffusion, optimal transport and Ricci curvature ⋮ Kac regularity and domination of quadratic forms ⋮ Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds ⋮ Metric measure spaces and synthetic Ricci bounds: fundamental concepts and recent developments ⋮ Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds ⋮ Some functional inequalities on non-reversible Finsler manifolds ⋮ Extremal of log-Sobolev functionals and Li-Yau estimate on \(\mathrm{RCD}^* (K,N)\) spaces ⋮ Sobolev's inequality under a curvature-dimension condition ⋮ Sobolev inequalities on a weighted Riemannian manifold of positive Bakry-Émery curvature and convex boundary ⋮ The measure preserving isometry groups of metric measure spaces ⋮ On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces ⋮ Weighted Sobolev inequalities in CD(0,N) spaces

Cites Work

• Unnamed Item
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• On the Bakry-Émery condition, the gradient estimates and the local-to-global property of \(\mathrm{RCD}^*(K,N)\) metric measure spaces
• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in \(\text{RCD}(K, \infty)\) metric measure spaces
• Local Poincaré inequalities from stable curvature conditions on metric spaces
• An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature
• On the equivalence of the entropic curvature-dimension condition and Bochner's inequality on metric measure spaces
• Constantes explicites pour les inégalités de Sobolev sur les variétés Riemanniennes compactes
• Localization and tensorization properties of the curvature-dimension condition for metric measure spaces
• Hypercontractivity and the Bakry-Emery criterion for compact Lie groups
• Upper bounds for symmetric Markov transition functions
• Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators
• Diffusion on compact Riemannian manifolds and logarithmic Sobolev inequalities
• Equations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire
• Best constant in Sobolev inequality
• Problèmes isoperimetriques et espaces de Sobolev
• Ricci curvature for metric-measure spaces via optimal transport
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Metric measure spaces with Riemannian Ricci curvature bounded from below
• On the geometry of metric measure spaces. I
• On the geometry of metric measure spaces. II
• Non-branching geodesics and optimal maps in strong \(CD(K,\infty)\)-spaces
• Li-Yau and Harnack type inequalities in \(\text{RCD}^\ast (K,N)\) metric measure spaces
• Heat Flow and Calculus on Metric Measure Spaces with Ricci Curvature Bounded Below—The Compact Case
• Sobolev Inequalities in Familiar and Unfamiliar Settings
• The Yamabe problem
• Sobolev met Poincaré
• Sobolev inequalities in disguise
• Analysis and Geometry of Markov Diffusion Operators
• Optimal Transport