Dunkl Translation and Uncentered Maximal Operator on the Real Line

On the real line, the Dunkl operators are differential-difference operators introduced in 1989 by Dunkl [1] and are denoted by Λα, where α is a real parameter > −1/2. These operators are associated with the reflection group Z2 on R. The Dunkl kernel Eα is used to define the Dunkl transform α which was introduced by Dunkl in [2]. Rösler in [3] shows that the Dunkl kernels verify a product formula. This allows us to define the Dunkl translation τx, x ∈R. As a result, we have the Dunkl convolution. The Hardy-Littlewood maximal function was first introduced by Hardy and Littlewood in 1930 for functions defined on the circle (see [4]). Later it was extended to various Lie groups, symmetric spaces, some weighted measure spaces (see [5–10]), and different hypergroups (see [11–14]). In this paper, we establish an estimate of the Dunkl translation of the characteristic function τx(χ[−ε,ε])(y), x, y ∈ R, x = 0, based on the inversion formula which extends some results of [11] to the Dunkl operator on R, and we prove the weak type (1,1) of the uncentered maximal operatorM defined for each integrable function f on (R,dμα) by