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 $${\mathfrak {X}}^{1/2}(M)$$ 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.