Riesz Transform on the Dual of the Laguerre Hypergroup

In this paper, we define the Riesz transform on the dual of the Laguerre hypergroup associated with Plancherel measure and we give some properties for this transform. Also we investigate $L^p$-boundedness of this operator and under this definition, the higher order Riesz transform is given. Moreover we establish that the Riesz transform can be extended as a principal value singular integral operator and it is a multiplier operator under the Fourier transform.