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A new mean value property for harmonic functions relative to the Dunkl-Laplacian operator and applications


Authors: Léonard Gallardo and Chaabane Rejeb
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
Published electronically: May 22, 2015
MathSciNet review: 3451892
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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.


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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
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
Email: chaabane.rejeb@gmail.com

DOI: https://doi.org/10.1090/tran/6671
Keywords: Dunkl-Laplacian operator, Dunkl transform, Dunkl harmonic functions, generalized volume mean value operator, strong maximum principle, Harnack's inequality, B\^ocher's theorem
Received by editor(s): February 25, 2014
Received by editor(s) in revised form: January 19, 2015
Published electronically: May 22, 2015
Article copyright: © Copyright 2015 American Mathematical Society