Functional calculus for the Laguerre operator

In this paper we study the boundedness of spectral multipliers associated to the multidimensional Laguerre operator Lalpha. It is well known that, for special values of alpha, the analysis of the Laguerre operator can be interpreted as the analysis of the Ornstein-Uhlenbeck operator acting on "polyradial" functions. Exploiting this relation, we prove that if M is a bounded holomorphic function in the sector S = {z is an element of C : \arg z\ < arcsin \2/p - 1\}, satisfying suitable Hormander type conditions on the boundary, then the spectral operator M(L-alpha) is bounded on L-p with respect to the Laguerre measure. We also prove that holomorphy in the sector S is a necessary condition for multipliers whose norm is invariant under dilations.