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.