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

TL;DR

This paper proves new $L^1$-type estimates for Riesz potentials, sharpening classical results and establishing novel $L^1$-Sobolev inequalities related to fractional gradients.

Contribution

It introduces sharper $L^1$-type bounds for Riesz potentials, extending Stein and Weiss's work and providing new inequalities involving fractional gradients.

Findings

Established new $L^1$-type estimates for Riesz potentials.

Sharpened classical results on Riesz potential mapping properties.

Provided new $L^1$-Sobolev inequalities for fractional gradients.

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.