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.
Categories: math.FA