  In this paper we consider a complete connected noncompact Riemannian manifold
M with bounded geometry and spectral gap. We realize the dual space Y^h(M) of
the Hardy-type space X^h(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 Y^h(M) is also the
dual of the space X^k_fin(M) of finite linear combination of X^k-atoms. As a
consequence, if Z is a Banach space and T is a Z-valued linear operator defined
on X^k_fin(M), then T extends to a bounded operator from X^k(M) to Z if and
only if it is uniformly bounded on X^k-atoms. To obtain these results we prove
the global solvability of the generalized Poisson equation L^ku=f with f in
L^2_loc(M) and we study some properties of generalized Bergman spaces of
harmonic functions on geodesic balls.
