A NEW MEAN VALUE PROPERTY FOR HARMONIC FUNCTIONS RELATIVE TO THE DUNKL-LAPLACIAN OPERATOR AND APPLICATIONS

作者：Gallardo Leonard^*; Rejeb Chaabane^*

来源：Transactions of the American Mathematical Society, 2016, 368(5): 3727-3753.

DOI：10.1090/tran/6671

• Dunkl-Laplacian operator
• Dunkl transform
• Dunkl harmonic functions
• generalized volume mean value operator
• strong maximum principle
• Harnack's inequality
• Bocher's theorem

For a root system in 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, ..., D-d and the Dunkl-Laplacian Delta(k) = D-1(2)+...+D-d(2). This paper studies the properties of the functions u defined on an open W-invariant set Omega subset of 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 Bocher's theorem for D-harmonic functions.

• 出版日期2016-5