%0 Journal Article
%T An $L^1$-type estimate for Riesz potentials
%A Armin Schikorra
%A Daniel Spector
%A Jean Van Schaftingen
%J Mathematics
%D 2014
%I arXiv
%X 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.
%U http://arxiv.org/abs/1411.2318v3