Refined Kato type inequalities and new vanishing theorems on complete K\"ahler and quaternionic K\"ahler manifolds

Abstract

Given a complete Riemannian manifold satisfying a weighted Poincar\'{e} inequality and having a bounded below Ricci curvature, various vanishing theorems for harmonic functions and harmonic 1-forms have been published. We generalized these results to $L^p$-integrable pluriharmonic functions and harmonic 1-forms on complete K\"{a}hler and quaternionic K\"{a}hler manifolds respectively by utilizing the B\"ochner technique and several refined Kato type inequalities. Moreover, we also prove the vanishing property of pluriharmonic functions with finite $L^p$ energy on complete K\"{a}hler manifolds satisfying a Sobolev type inequality.