Weak Curvature Conditions and Functional Inequalities

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 . 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.