Sharp estimates for the Ornstein-Uhlenbeck operator

Let G be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure gamma on R-d. We prove a sharp estimate of the operator norm of the imaginary powers of L on L-P(gamma), 1 < p < infinity. Then we use this estimate to prove that if b is in [0, infinity] and M is a bounded holomorphic function in the sector {z is an element of C : \arg(z - b)\ < arcsin \2/p - 1\} and satisfies a Hormander-like condition of (nonintegral) order greater than one on the boundary, then the operator M(L) is bounded on L-P(gamma). This improves earlier results of the authors with J. Garcia-Cuerva and J.L. Torrea.