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

Daniel Spector National Chiao Tung University Armin Schikorra University of Frieberg Jean Van Schaftingen Catholique University of Louvain

Functional Analysis mathscidoc:1703.12003

Rev. Mat. Iberoam., 33, (1), 291-304., 2017
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.
fractional gradient, L1 estimate
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@inproceedings{daniel2017an,
  title={ An $L^1$-type estimate for Riesz potentials},
  author={Daniel Spector, Armin Schikorra, and Jean Van Schaftingen},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170309065031112820629},
  booktitle={Rev. Mat. Iberoam.},
  volume={33},
  number={1},
  pages={291-304.},
  year={2017},
}
Daniel Spector, Armin Schikorra, and Jean Van Schaftingen. An $L^1$-type estimate for Riesz potentials. 2017. Vol. 33. In Rev. Mat. Iberoam.. pp.291-304.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170309065031112820629.
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