Spectral multipliers of Laplace transform type for the Laguerre operator

Abstract

We shall work with the Laguerre measure on and the associated Laplacian ℒα, by means of which the Laguerre semigroup is defined. Our main result is a multiplier theorem, saying that a functions of ℒα 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.