Spectral multipliers of Laplace transform type for the Laguerre operator

We shall work with the Laguerre measure on R^d_+ and the associated Laplacian L, by means of which the Laguerre semigroup is defined. Our main result is a multiplier theorem, saying that a function of L which is of Laplace transform type defines an operator of weak type (1,1) for the Laguerre measure. Our starting point is the well-known relationship between the Laguerre and Ornstein-Uhlenbeck semigroups.