Subspaces of $L^{1}(\mathbb{R} ^{d})$

The relationship of the Hardy space $H^{1}(R^{d})$ and the space of integrable functions $L^{1}(R^{d})$ is examined in terms of intermediate spaces of functions that are described as sums of atoms. It is proved that these spaces have dual spaces that lie between the space of functions of bounded mean oscillation, $BMO$, and $L^{\infty }$. Furthermore, the spaces intermediate to $H^{1}$ and $L^{1}$ are shown to be dual to spaces similar to the space of functions of vanishing mean oscillation. The proofs are extensions of classical proofs.