Translation operator and maximal function for the (k,1)-generalized Fourier transform

In this paper we study a translation operator associated with the n-dimensional (k,1)-generalized Fourier transform, where k is a multiplicity function for the Dunkl operators. In particular, we prove that the translation is a positivity-preserving operator acting on a suitable space of radial functions on Rⁿ. We then use it to define a Hardy-Littlewood type maximal operator, where weak-type (1,1) and strong-type (p,p) estimates for the maximal operator are established with a precise behavior in n and k.