Weak curvature conditions and functional inequalities

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 scaleinvariant 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 2N. 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 byK >0. Finally we derive a sharp global Poincaré inequality. © 2006 Elsevier Inc. All rights reserved.