Optimal angle of the holomorphic functional calculus for the classical Ornstein-Uhlenbeck operator on $L^p$

Abstract

We give a simple proof of the fact that the classical Ornstein-Uhlenbeck operator $L$ is R-sectorial of angle $arcsin|1-2/p|$ on $L^p({\mathbb{R}}^n,\exp (-|x{|}^2/2)dx)$ (for $1<p<\infty$). Applying the abstract holomorphic functional calculus theory of Kalton and Weis, this immediately gives a new proof of the fact that $L$ has a bounded $H^{\infty }$ functional calculus with this optimal angle.