Potential Analysis

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

On the Green Function and Poisson Integrals of the Dunkl Laplacian

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Abstract

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.

Keywords

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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Copyright information

© 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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