Harmonic Bergman spaces, the Poisson equation and the dual of Hardy-type spaces on certain noncompact manifolds

Abstract. In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Yk (M) of the Hardy-type space Xk (M), introduced in a previous paper of the authors, as the class of all locally square integrable functions satisfying suitable BMO-like conditions, where the role of the constants is played by the space of global k-quasi-harmonic functions. Furthermore we prove that Yk (M) is also the dual of the space Xk fin(M) of finite linear combination of Xk -atoms. As a consequence, if Z is a Banach space and T is a Z-valued linear operator defined on Xk fin(M), then T extends to a bounded operator from Xk (M) to Z if and only if it is uniformly bounded on Xk -atoms. To obtain these results we prove the global solvability of the generalized Poisson equation L ku = f with f 2 L2 loc(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.