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 $\alpha \in (0,N)$:

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.