An L1L^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
An $L^1$-type estimate for Riesz potentials cover
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Abstract

In this paper we establish new L1L^1-type estimates for the classical Riesz potentials of order α(0,N)\alpha \in (0, N):

IαuLN/(Nα)(RN)CRuL1(RN;RN).\|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 H1(RN)\mathcal{H}^1(\mathbb{R}^N) and provides a new family of L1L^1-Sobolev inequalities for the Riesz fractional gradient.

Cite this article

Armin Schikorra, Daniel Spector, Jean Van Schaftingen, An L1L^1-type estimate for Riesz potentials. Rev. Mat. Iberoam. 33 (2017), no. 1, pp. 291–303

DOI 10.4171/RMI/937