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Support properties of the intertwining and the mean value operators in Dunkl theory


Authors: Léonard Gallardo and Chaabane Rejeb
Journal: Proc. Amer. Math. Soc. 146 (2018), 145-152
MSC (2010): Primary 31B05, 33C52, 47B39; Secondary 43A32, 51F15
DOI: https://doi.org/10.1090/proc/13478
Published electronically: September 27, 2017
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Abstract: In this paper we show that the representing measures of the Dunkl intertwining operator associated to a Coxeter-Weyl group $ W$ in $ \mathbb{R}^d$ and to a multiplicity function $ k\geq 0$, have $ W$-invariant supports under the condition $ k>0$. This property enables us to determine explicitly the supports of the measures representing the volume mean operator, a fundamental tool for the study of harmonic functions relative to the Dunkl-Laplacian operator.


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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: Laboratoire de Mathématiques et Physique Théorique CNRS-UMR 7350, Université de Tours, Campus de Grandmont, 37200 Tours, France – and – Université de Tunis El Manar, Faculté des Sciences de Tunis, Laboratoire d’Analyse Mathématiques et Applications LR11ES11, 2092 El Manar I, Tunis, Tunisia
Email: chaabane.rejeb@gmail.com

DOI: https://doi.org/10.1090/proc/13478
Keywords: Dunkl-Laplacian operator, Dunkl's intertwining operator, generalized volume mean operator and harmonic kernel, R\"osler's measure, Dunkl harmonic functions
Received by editor(s): June 1, 2016
Received by editor(s) in revised form: September 13, 2016
Published electronically: September 27, 2017
Communicated by: Mourad Ismail
Article copyright: © Copyright 2017 American Mathematical Society

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