Weak curvature conditions and functional inequalities |
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Authors: | John Lott Cédric Villani |
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Institution: | a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109, USA b UMPA (UMR CNRS 5669), ENS Lyon, 46 allée d'Italie, 69364 Lyon Cedex 07, France |
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Abstract: | We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N2. The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality. |
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Keywords: | Poincaré inequality Ricci curvature Metric-measure spaces Sobolev inequality |
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