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

Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measure dx. Given a C-2 positive bounded integrable function M on G, we give a sufficient condition for an L-2 Poincare inequality with respect to the measure M(x) dx to hold on G. We then establish a nonlocal 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.