Weak curvature conditions and functional inequalities
In the paper the authors give sufficient conditions for a measured length space \((X, d,\nu)\) to admit local and global Poincare inequalities. lt is a continuation of the paper of authors entitled Ricci curvature for metric-measure spaces via optimal transport which will appear in Ann. Math.. Following on the work of Cheeger and Colding they introduce a certain condition DM on a measured length space. Roughly speaking the condition DM says that there is a dynamical democratic transference plan so that a given point is not hit too often by the geodesics. Then they show that this condition together with a doubling condition on \(\nu\), implies a scale invariant local Poincaré inequality. They show that if \((X, d, \nu)\) 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. Based on it they prove a Sobolev inequality for measured length spaces with \(N\)-Ricci curvature bounded below by \(K > 0\). Finally the authors derive a sharp global Poincaré inequality.
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