Weighted -Inequalities for the Dunkl Transform

We give, for , weighted -inequalities for the Dunkl transform, using, respectively, the modulus of continuity of radial functions and the Dunkl convolution in the general case. As application, we obtain, in particular, the integrability of this transform on Besov-Lipschitz spaces. 1. Introduction Dunkl theory is a far reaching generalization of Euclidean Fourier analysis. It started twenty years ago with Dunkl’s seminal work [1] and was further developed by several mathematicians (see [2–6]) and later was applied and generalized in different ways by many authors (see [7–11]). The Dunkl operators are commuting differential-difference operators , . These operators, attached to a finite root system and a reflection group acting on , can be considered as perturbations of the usual partial derivatives by reflection parts. These reflection parts are coupled by parameters, which are given in terms of a nonnegative multiplicity function . The Dunkl kernel has been introduced by Dunkl in [12]. For a family of weight functions invariant under a reflection group , we use the Dunkl kernel and the weighted Lebesgue measure to define the Dunkl transform , which enjoys properties similar to those of the classical Fourier transform. If the parameter then , so that becomes the classical Fourier transform and the , , reduce to the corresponding partial derivatives , (see next section, Remark 1). The classical Fourier transform behaves well with the translation operator , which leaves the Lebesgue measure on invariant. However, the measure is no longer invariant under the usual translation. Trimèche has introduced in [6] the Dunkl translation operators , , on the space of infinitely differentiable functions on . At the moment an explicit formula for the Dunkl translation of a function is unknown in general. However, such formula is known when the function is radial (see next section). In particular, the boundedness of is established in this case. As a result one obtains a formula for the Dunkl convolution . An important motivation to study Dunkl operators originates from their relevance for the analysis of quantum many body systems of Calogero-Moser-Sutherland type. These describe algebraically integrable systems in one dimension and have gained considerable interest in mathematical physics (see [13]). Let be a function in , , where denote the space with the weight function associated with the Dunkl operators given by with a fixed positive root system (see next section). The modulus of continuity of first order of a radial function in is defined by where is the unit