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

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In this paper we consider a complete connected noncompact Riemannian manifold M with bounded geometry and spectral gap. We realize the dual space Yʰ(M) of the Hardy-type space Xʰ(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 Yʰ(M) is also the dual of the space Xᵏfin(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 Xᵏfin(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 Lᵏu=f with f in L²ₗoc(M) and we study some properties of generalized Bergman spaces of harmonic functions on geodesic balls.