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 $\mathscr{ L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H^1_\mathscr{ R}(M)$ is the isomorphic image of the Goldberg type space $\mathfrak{ h}^1(M)$ via the map $\mathscr{ L}^{ 1/2} (\mathscr{ I} + \mathscr{ L})^{ -1/2}$, a fact that is false in $\mathbb{ R}^n$. Specifically, $H^1_\mathscr{ R}(M)$ agrees with the Hardy type space $\mathfrak{ X}^{ 1/2}(M)$ recently introduced by the the first three authors; as a consequence, we prove that $\mathfrak{ h}^1(M)$ 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 $H^1_\mathscr{ R}(S)$, the heat Hardy space $H^1_\mathscr{ H}(S)$ and the Poisson-Hardy space $H^1_\mathscr{ P}(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.
Categories: math.FA math.CA
