An $L^1$-type estimate for Riesz potentials

In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $\alpha \in (0,N)$: $$\mathrm{\parallel }{I}_{\alpha }u{\mathrm{\parallel }}_{L^{N\mathrm{/}(N-\alpha )}({\mathbb{R}}^N)}\le C\mathrm{\parallel }Ru{\mathrm{\parallel }}_{L^1({\mathbb{R}}^N;{\mathbb{R}}^N)}\mathrm{.}$$ 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.