On the Green function and Poisson integrals of the Dunkl Laplacian

2016/07/29 by Graczyk, Piotr, Luks, Tomasz, Rösler, Margit
#31B25 #51F15 #60J50 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Primary 31B05 #Secondary 42B30

paper · doi:10.48550/arxiv.1607.08746

Abstract

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

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