Potential Analysis

, Volume 48, Issue 3, pp 337–360 | Cite as

On the Green Function and Poisson Integrals of the Dunkl Laplacian



We prove the existence and study properties of the Green function of the unit ball for the Dunkl Laplacian △ k in \(\mathbb {R}^{d}\). As applications we derive the Poisson-Jensen formula for △ k -subharmonic functions and Hardy-Stein identities for the Poisson integrals of △ k . We also obtain sharp estimates of the Newton potential kernel, Green function and Poisson kernel in the rank one case in \(\mathbb {R}^{d}\). These estimates contrast sharply with the well-known results in the potential theory of the classical Laplacian.


Dunkl Laplacian Green function Newton kernel Poisson kernel Hardy-Stein identity 

Mathematics Subject Classification (2010)

Primary 31B05 31B25 60J50 Secondary 42B30 51F15 


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© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.LAREMAUniversité d’AngersAngers Cedex 1France
  2. 2.Institut für MathematikUniversität PaderbornPaderbornGermany

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