A family of Hardy type spaces on nondoubling manifolds

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