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 $\alpha \in (0,N)$: \[ \|I_αu\|_{L^{N/(N-α)}(\mathbb{R}^N)} \leq C \|Ru\|_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] 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.

13 pages, improves previous version with full spectrum of result and with elementary proof, references display correctly in this version

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