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On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

Daniel Spector National Chiao Tung University Rahul Garg Technion

Analysis of PDEs mathscidoc:1703.03004

Indiana Univ. Math. J., 64, (6), 1697–1719., 2015
In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new “almost” Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson’s equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.
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@inproceedings{daniel2015on,
  title={On the role of Riesz potentials in Poisson's equation and Sobolev embeddings},
  author={Daniel Spector, and Rahul Garg},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170309063854135348626},
  booktitle={ Indiana Univ. Math. J.},
  volume={64},
  number={6},
  pages={1697–1719.},
  year={2015},
}
Daniel Spector, and Rahul Garg. On the role of Riesz potentials in Poisson's equation and Sobolev embeddings. 2015. Vol. 64. In Indiana Univ. Math. J.. pp.1697–1719.. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170309063854135348626.
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