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

In this paper we consider a complete connected noncompact Rieman- nian 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 fc-quasi-harmonic functions. Furthermore we prove that Yk(M) is also the dual of the space Xn(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 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&ku = / with / e L2.ioc (M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.