  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.
