Sharp estimates for the Ornstein-Uhlenbeck operator

Let $ℒ$ 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 $L^p\left(\gamma \right),$ $1<p<\infty .$ Then we use this estimate to prove that if $b$ is in $[0,\infty )$ and $M$ is a bounded holomorphic function in the sector $\{z\in ℂ:modarg(z-b)<arcsin|2/p-1\left|\right\}$ and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator $M\left(ℒ\right)$ is bounded on $L^p\left(\gamma \right).$ This improves earlier results of the authors with J. García-Cuerva and J.L. Torrea.