WEAK CURVATURE CONDITIONS AND FUNCTIONAL INEQUALITIES

WEAK CURVATURE CONDITIONS AND FUNCTIONAL INEQUALITIES JOHN LOTT AND CEDRIC VILLANI Abstract. We give sufficient conditions for a measured length space (X, d, ?) to admit local and global Poincare inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X, d, ?), defined in terms of transport of measures. We show that DM, together with a doubling condition on ?, implies a scale-invariant local Poincare inequality. We show that if (X, d, ?) 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 by K > 0. Finally we derive a sharp global Poincare inequality. There has been recent work on giving a good notion for a compact measured length space (X, d, ?) to have a “lower Ricci curvature bound”. In our previous work [10] we gave a notion of (X, d, ?) having nonnegative N -Ricci curvature, where N ? [1,∞) is an effective dimension. The definition was in terms of the optimal transport of measures on X. A notion was also given of (X, d, ?) having ∞-Ricci curvature bounded below by K ? R; a closely related definition in this case was given independently by Sturm [13].

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