Quantitative property A, Poincaré inequalities, Lp-compression and Lp-distortion for metric measure spaces

We introduce a quantitative version of Property A in order to estimate the Lp-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 Lp-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 Poincaré 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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