Spectral multipliers for the Ornstein-Uhlenbeck semigroup

The setting of this paper is Euclidean space with the Gaussian measure. We let L be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The 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 Gaussian measure. The (distribution) kernel of this operator is determined, in terms of an integral involving the kernel of the Omstein-Uhlenbeck semigroup. This applies in particular to the imaginary powers of L. It is also verified that the weak type constant of these powers increases exponentially with the absolute Value of the exponent.