Abilov's inequalities in the laguerre hypergroup

Abstract

Let  (K, ¤) be the Laguerre hypergroup 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 L^{2}_{\alpha}(K), by using a generalized translation operator to prove these estimates in certain classes of functions characterized by a generalized continuity modulus.