Weighted Sobolev inequalities under lower Ricci curvature bounds

We obtain sharp weighted Poincaré and Sobolev inequalities over complete, noncompact Riemannian manifolds with polynomial volume growth and a quadratically decaying lower bound on Ricci. This improves and extends earlier work of Tian-Yau and Minerbe. We deduce a sharp existence result for bounded solutions of the Poisson equation on such manifolds, highlighting the well-known distinction between spaces of volume growth ≤ 2 ≤ 2 and > 2 > 2 in terms of their Green’s functions. We also show that if the manifold is nonparabolic and carries a smooth function which behaves like the radius function of a cone, then these solutions almost decay at the rates expected from a cone.