A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications
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- by Léonard Gallardo and Chaabane Rejeb PDF
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Abstract:
For a root system in $\mathbb {R}^d$ furnished with its Coxeter-Weyl group $W$ and a multiplicity nonnegative function $k$, we consider the associated commuting system of Dunkl operators $D_1,\ldots ,D_d$ and the Dunkl-Laplacian $\Delta _k=D^{2}_{1}+\ldots +D^{2}_{d}$. This paper studies the properties of the functions $u$ defined on an open $W$-invariant set $\Omega \subset \mathbb {R}^d$ and satisfying $\Delta _k u=0$ on $\Omega$ (D-harmonicity). In particular, we introduce and give a complete study of a new mean value operator which characterizes D-harmonicity. As applications we prove a strong maximum principle, a Harnack’s type theorem and a Bôcher’s theorem for D-harmonic functions.References
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Additional Information
- Léonard Gallardo
- Affiliation: Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350, Université de Tours, Campus de Grandmont, 37200 Tours, France
- MR Author ID: 70775
- Email: Leonard.Gallardo@lmpt.univ-tours.fr
- Chaabane Rejeb
- Affiliation: Département de Mathématiques, Faculté des Sciences de Tunis, 1060 Tunis, Tunisie – and – Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350, Université de Tours, Campus de Grandmont, 37200 Tours, France
- MR Author ID: 1095811
- Email: chaabane.rejeb@gmail.com
- Received by editor(s): February 25, 2014
- Received by editor(s) in revised form: January 19, 2015
- Published electronically: May 22, 2015
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 368 (2016), 3727-3753
- MSC (2010): Primary 31B05, 43A32, 42B99, 33C52; Secondary 51F15, 33C80, 47B38
- DOI: https://doi.org/10.1090/tran/6671
- MathSciNet review: 3451892