Weighted Poincaré inequality and the Poisson Equation

Munteanu, Ovidiu; Sung, Chiung-Jue; Wang, Jiaping

doi:10.1090/tran/8291

Weighted Poincaré inequality and the Poisson Equation
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by Ovidiu Munteanu, Chiung-Jue Anna Sung and Jiaping Wang PDF

Trans. Amer. Math. Soc. 374 (2021), 2167-2199 Request permission

Abstract:

We develop Green’s function estimates for manifolds satisfying a weighted Poincaré inequality together with a compatible lower bound on the Ricci curvature. This estimate is then applied to establish existence and sharp estimates of solutions 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 Kähler manifolds. Consequently, such Kähler manifolds must be connected at infinity.

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