Nonlocal Poincaré 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 $L^2$ Poincaré inequality with respect to the measure $M(x)\, dx$ to hold on $G$. We then establish a nonlocal Poincaré inequality on $G$ with respect to $M(x)\, dx$. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.