Lecture 15 : Grasshopper criteria . Gromov ’ s doubling condition
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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 nonne...
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Abstract. We give sufficient conditions for a measured length space (X, d, ν) to admit local and global Poincaré inequalities. 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 ha...
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