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 $\mathcal{L}$ is the positive Laplace-Beltrami operator on $M$, then the Riesz-Hardy space $H_{\mathcal{R}}^1(M)$ is the isomorphic image of the Goldberg type space ${\mathfrak{h}}^1(M)$ via the map ${\mathcal{L}}^{1\mathrm{/}2}(\mathcal{I}+\mathcal{L}{)}^{-1\mathrm{/}2}$, a fact that is false in ${\mathbb{R}}^n$. Specifically, $H_{\mathcal{R}}^1(M)$ agrees with the Hardy type space ${\mathfrak{X}}^{1\mathrm{/}2}(M)$ 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 $H_{\mathcal{R}}^1(S)$, the heat Hardy space $H_{\mathcal{H}}^1(S)$ and the Poisson-Hardy space $H_{\mathcal{P}}^1(S)$ are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree.