Multiplier Theorems for the Dunkl Transform

Abstract

For a family of weight functions invariant under a finite reflection group, we prove a transference theorem between the L ^p multiplier of h-harmonic expansions on $\mathbb{S}^{d}$ and that of the Dunkl transform. This theorem is stated together with some related definitions and notations in Section 7.1. The proof of this transference theorem is, however, rather long, so we split it into three parts, which are given in the Sections 7.2, 7.3, and 7.4, respectively.