  In this paper we establish new $L^1$-type estimates for the classical Riesz
potentials of order $\alpha \in (0, N)$: \[
  \|I_\alpha u\|_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C
\|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] This sharpens the result of Stein
and Weiss on the mapping properties of Riesz potentials on the real Hardy space
$\mathcal{H}^1(\mathbb{R}^N)$ and provides a new family of $L^1$-Sobolev
inequalities for the Riesz fractional gradient.
