Calculus of Variations and Geometric Measure Theory

A. Schikorra - D. Spector - J. Van Schaftingen

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

created by spector on 13 Nov 2014
modified on 01 Sep 2016


Rev. Mat. Iberoam.

Inserted: 13 nov 2014
Last Updated: 1 sep 2016

Pages: 12
Year: 2014
Doi: arXiv:1411.2318


In this paper we establish new $L^1$-type estimates for the classical Riesz potentials of order $\alpha \in (0, N)$: \[ \
I_\alpha u\
_{L^{N/(N-\alpha)}(\mathbb{R}^N)} \leq C \
_{L^1(\mathbb{R}^N;\mathbb{R}^N)}. \] 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.