In this paper we consider a complete connected noncompact Riemannian manifoldM with bounded geometry and spectral gap. We realize the dual space Y^h(M) ofthe Hardy-type space X^h(M), introduced in a previous paper of the authors, asthe class of all locally square integrable functions satisfying suitableBMO-like conditions, where the role of the constants is played by the space ofglobal k-quasi-harmonic functions. Furthermore we prove that Y^h(M) is also thedual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As aconsequence, if Z is a Banach space and T is a Z-valued linear operator definedon X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if andonly if it is uniformly bounded on X^k-atoms. To obtain these results we provethe global solvability of the generalized Poisson equation L^ku=f with f inL^2_loc(M) and we study some properties of generalized Bergman spaces ofharmonic functions on geodesic balls.
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