Abilov’s inequalities in the Laguerre hypergroup

Let (K, *?) be the Laguerre hypergroup where K = [0, +?) ? R and *? a convolution product on K coming from the product formula satisfied by the Laguerre functions. In this work, we give new estimates for the Laguerre kernel. We obtain new inequalities for the Fourier-Laguerre transform in the space L2? (K), by using a generalized translation operator to prove these estimates in certain classes of functions characterized by a generalized continuity modulus.