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Entire $ s$-harmonic functions are affine

Author: Mouhamed Moustapha Fall
Journal: Proc. Amer. Math. Soc. 144 (2016), 2587-2592
MSC (2010): Primary 35R11, 42B37
Published electronically: January 27, 2016
MathSciNet review: 3477075
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Abstract: In this paper, we prove that solutions to the equation $ (-\Delta )^s u=0$ in $ \mathbb{R}^N$, for $ s\in (0,1)$, are affine. This allows us to prove the uniqueness of the Riesz potential $ \vert x\vert^{2s-N}$ in Lebesgue spaces.

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Mouhamed Moustapha Fall
Affiliation: African Institute for Mathematical Sciences in Senegal, KM 2, Route de Joal, B.P. 14 18. Mbour, Sénégal

Keywords: Fractional Laplacian, Liouville theorem, uniqueness, Riesz kernel, entire $\alpha$-harmonic, Cauchy estimates.
Received by editor(s): July 24, 2014
Received by editor(s) in revised form: April 9, 2015, and August 1, 2015
Published electronically: January 27, 2016
Additional Notes: This work was supported by the Alexander von Humboldt foundation and the author would like to thank Tobias Weth and Krzysztof Bogdan for useful discussions. This work was completed while the author was visiting the Goethe-Universität Frankfurt am Main and the Technische Universität Chemnitz. The author is also very grateful to the referee for the detailed comments. The variety of substantial suggestions helped the author to improve the earlier versions of this manuscript
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2016 American Mathematical Society

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