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Revista Matemática Iberoamericana

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Volume 33, Issue 1, 2017, pp. 291–303
DOI: 10.4171/RMI/937

Published online: 2017-02-22

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

Armin Schikorra[1], Daniel Spector[2] and Jean Van Schaftingen[3]

(1) Universität Freiburg, Germany
(2) National Chiao Tung University, Hsinchu, Taiwan
(3) Université Catholique de Louvain, Belgium

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.

Keywords: Riesz potentials, Riesz transforms, Sobolev inequalities, fractional gradient

Schikorra Armin, Spector Daniel, Van Schaftingen Jean: An $L^1$-type estimate for Riesz potentials. Rev. Mat. Iberoam. 33 (2017), 291-303. doi: 10.4171/RMI/937