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In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if L is the positive Laplace–Beltrami operator on M , then the Riesz–Hardy space H R 1 (M) is the isomorphic image of the Goldberg type space h 1 (M) via the map L 1 / 2 (I + L) − 1 / 2 , a fact that is false in R n. Specifically, H R 1 (M) agrees with the Hardy type space X 1 / 2 (M) recently introduced by the first three authors; as a consequence, we prove that H R 1 (M) does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that H R 1 (S) , the heat Hardy space H H 1 (S) and the Poisson–Hardy space H P 1 (S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree. [ABSTRACT FROM AUTHOR]

We introduce a decreasing one-parameter family X γ (M) , γ > 0 , of Banach subspaces of the Hardy–Goldberg space h 1 (M) on certain nondoubling Riemannian manifolds with bounded geometry, and we investigate their properties. In particular, we prove that X 1 / 2 (M) agrees with the space of all functions in h 1 (M) whose Riesz transform is in L 1 (M) , and we obtain the surprising result that this space does not admit an atomic decomposition. [ABSTRACT FROM AUTHOR]