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

Abstract: 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.