Fischer Decomposition and Cauchy Kernel for Dunkl–Dirac operators

Abstract

In this paper we present a Fischer decomposition for Dirac operator and an explicit construction of a Cauchy kernel for Dunkl-monogenic functions.

On the Dunkl Version of Monogenic Polynomials

“…Besides, since the Cauchy kernel C h is Dunkl-monogenic on R n \{0}, P(T) and D h commute and by proposition 3.1 the Kelvin transform I h preserves Dunkl-monogenic functions, then I h ( x |x| 2γ+n P (T )) is a Dunkl-monogenic polynomial. The rest of the assertions is now immediate since (see [2] and [12]) P m = M m ⊕ x P m−1 and by (4.3) we have…”

“…The discussion he had with him during the CIMPA's school Analytical and Probabilistic Aspects of Dunkl Theory (Monastir, 13-25 April 2009), is the real source of this work. He also would like to thank Paula Cerejeiras for sending him the reference [2].…”

Fischer Decomposition in Generalized Fractional Clifford Analysis

“…In order to achieve our goal, we want to study the commutator and the anti-commutator between x α (see (1)) and…”

“…In classical continuous Clifford analysis one obtains a refinement yielding an orthogonal decomposition with respect to the so-called Fischer inner product of homogeneous polynomials in terms of spaces of monogenic polynomials, i.e., null solutions of the Dirac operator (see [3]). Generalizations of the Fischer decomposition in other frameworks can be found, for example, in [1,5,6,9,12,16,17,19] and are mainly based on the establishment of raising and lowering operators acting on a ground state.…”

A Cauchy-Pompeiu formula in super Dunkl-Clifford analysis

“…In 2006, Cerejeiras, Kähler and Ren defined the Dunkl-Dirac operator (see [2]) and constructed the Stokes formula in Clifford analysis by Dunkl transforms (see [15]). The theory of Dunkl-Clifford analysis is further developed in [1], [10], [11], [14], [4] and [17]. In 2013, Fei investigated the fundamental solutions to the Dunkl-Dirac equation, and also obtained the Cauchy integral formula with a Dunkl-Cauchy kernel (see [9]).…”