On fractional maximal function and fractional integral on the Laguerre hypergroup

Let K=[0,∞)×R be the Laguerre hypergroup which is the fundamental manifold of the radial function space for the Heisenberg group. In this paper we obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional maximal operator and the fractional integral operator on the Laguerre hypergroup from the spaces Lp(K) to the spaces Lq(K) and from the spaces L1(K) to the weak spaces WLq(K).