On a mean value property associated with the dunkl laplacian operator and applications

Pizzetti Series and Polyharmonicity Associated with the Dunkl Laplacian

“…Mejjaoli and K. Trimèche have defined in [9] the mean value M k u of a function u ∈ C ∞ (R d ) over the sphere S(x, r) of center x and radius r by…”

“…Also, it is shown in [9] that for u ∈ C 2 (R d ) and r > 0 the mean value M k u (r) = M k u (0, r) satisfies the following equation :…”

“…Using these results and the Taylor formula for the Bessel operator, it was established in [9] the following theorem.…”

On mean-value properties for the Dunkl polyharmonic functions

“…Let us point out that formula (4.7) was established in [12,Proposition 4.16]. By the first part of Lemma 4.2 we get an analogue of the Beckenbach-Reade theorem ( [1]) for the Dunkl harmonic functions.…”

“…Afterwards the whole theory related to ∆ κ was elaborated including analogues of Fourier analysis, special functions connected with root systems, algebraic approaches and an application to the solution of quantum Calogero-Sutherland models (see [5] for an excellent survey). In particular, Mejjaoli and Triméche proved in [12] that the operator ∆ κ is hypoelliptic on R n and that smooth Dunkl harmonic functions on R n can be characterized by the Dunkl spherical mean value property. Furthermore, they derived a Pizzetti type formula for smooth functions on R n .…”