Weighted Poincare inequality and the Poisson equation

We develop Green's function estimate for manifolds satisfying a weighted Poincare inequality together with a compatible lower bound on the Ricci curvature. The estimate is then applied to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds. As an application, a Liouville property for finite energy holomorphic functions is proven on a class of complete K\"ahler manifolds. Consequently, such K\"ahler manifolds must be connected at infinity.