Inclusions and noninclusions of Hardy type spaces on certain nondoubling manifolds

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 HR1(M) is the isomorphic image of the Goldberg type space h1(M) via the map L1/2(I+L)−1/2, a fact that is false in Rn. Specifically, HR1(M) agrees with the Hardy type space X1/2(M) recently introduced by the first three authors; as a consequence, we prove that HR1(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 HR1(S), the heat Hardy space HH1(S) and the Poisson–Hardy space HP1(S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.