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

Armin Schikorra; Daniel Spector; Jean Van Schaftingen

doi:10.4171/rmi/937

JournalsrmiVol. 33, No. 1pp. 291–303

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

Armin Schikorra
Universität Freiburg, Germany
Daniel Spector
National Chiao Tung University, Hsinchu, Taiwan
Jean Van Schaftingen
Université Catholique de Louvain, Belgium

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Abstract

In this paper we establish new $L^{1}$ -type estimates for the classical Riesz potentials of order $α \in (0, N)$ :

∥ I_{α} u ∥_{L^{N / (N - α)} (R^{N})} \leq C ∥ R u ∥_{L^{1} (R^{N}; R^{N})} .

This sharpens the result of Stein and Weiss on the mapping properties of Riesz potentials on the real Hardy space $H^{1} (R^{N})$ and provides a new family of $L^{1}$ -Sobolev inequalities for the Riesz fractional gradient.

Cite this article

Armin Schikorra, Daniel Spector, Jean Van Schaftingen, An $L^{1}$ -type estimate for Riesz potentials. Rev. Mat. Iberoam. 33 (2017), no. 1, pp. 291–303

DOI 10.4171/RMI/937