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Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measure dx. Given a C positive bounded integrable function M on G, we give a sufficient condition for an L Poincare inequality with respect to the measure M(x)dx to hold on G. We then establish a non-local Poincare inequality on G with respect to M(x)dx. We also give analogous Poincare inequalities on Riemannian manifolds and deal with the case of Hardy inequalities. AMS classification: 26D10, 42B35.

Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds