Quantitative property A, Poincare inequalities, L^p-compression and L^p-distortion for metric measure spaces

Details Quantitative property A, Poincare inequalities, L^p-compression and L^p-distortion for metric measure spaces Quantitative property A, Poincare inequalities, L^p-compression and L^p-distortion for metric measure spaces

2007-02-13

arxiv.org/abs/math/0702384v2

Mathematics

Metric Geometry

26 pages

Scientific paper

We introduce a quantitative version of Property A in order to estimate the L^p-compressions of a metric measure space X. We obtain various estimates for spaces with sub-exponential volume growth. This quantitative property A also appears to be useful to yield upper bounds on the L^p-distortion of finite metric spaces. Namely, we obtain new optimal results for finite subsets of homogeneous Riemannian manifolds. We also introduce a general form of Poincare inequalities that provide constraints on compressions, and lower bounds on distortion. These inequalities are used to prove the optimality of some of our results.

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