Refined Kato type inequalities and new vanishing theorems on complete Kähler and quaternionic Kähler manifolds

Given a complete Riemannian manifold satisfying a weighted Poincaré 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 Kähler and quaternionic Kähler manifolds respectively by utilizing the Böchner technique and several refined Kato type inequalities. Moreover, we also prove the vanishing property of pluriharmonic functions with finite $L^p$ energy on complete Kähler manifolds satisfying a Sobolev type inequality.