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 2 N . 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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