Weak curvature conditions and functional inequalities

John Lott; Cedric Villani

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arXiv · math/0506481

Weak curvature conditions and functional inequalities

John Lott·Cedric Villani

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Abstract

We give sufficient conditions for a measured length space (X,d,m) to admit local and global Poincare inequalities. We first introduce a condition DM on (X,d,m), defined in terms of transport of measures. We show that DM, along with a doubling condition on m, implies a scale-invariant local Poincare inequality. We show that if (X,d,m) 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 prove a sharp global inequality.

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BibTeX RIS

John Lott, Cedric Villani. 2006-10-22. Weak curvature conditions and functional inequalities. https://arxiv.org/abs/math/0506481

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