p-harmonic ℓ-forms on Riemannian manifolds with a weighted Poincaré inequality

Given a Riemannian manifold with a weighted Poincaré inequality, in this paper, we will show some vanishing type theorems for p-harmonic ℓ-forms on such a manifold. We also prove a vanishing result on submanifolds in Euclidean space with flat normal bundle. Our results can be considered as generalizations of the work of Lam, Li-Wang, Lin, and Vieira (see Lam (2008), Li and Wang (2001), Lin (2015), Vieira (2016)). Moreover, we also prove a vanishing and splitting type theorem for p-harmonic functions on manifolds with Spin (9) holonomy provided a (p,p,λ)-Sobolev type inequality which can be considered as a general Poincaré inequality holds true.