Weighted norm estimates for the maximal operator of the Laguerre functions heat diffusion semigroup

We obtain weighted L-p boundedness, with weights of the type y(delta), delta > -1, for the maximal operator of the heat semigroup associated to the Laguerre functions, {L-k(alpha)}(k), when the parameter alpha is greater than - 1. It is proved that when - 1 < alpha < 0, the maximal operator is of strong type (p, p) if p > 1 and 2 (1 + delta) / (2 + alpha) < p < 2 (1 + delta)/(-alpha), and if alpha >= 0 it is of strong type for 1 < p <= infinity and 2(1 + delta)/(2 + alpha) < p. The behavior at the end points of the intervals where there is strong type is studied in detail and sharp results about the existence or not of strong, weak or restricted types are given.