Optimal angle of the holomorphic functional calculus for the Ornstein–Uhlenbeck operator

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 Lp(Rd,μ) for 1<p<∞, where μ is the standard Gaussian measure with density dμ=(2π)−d2exp(−|x|2∕2)dx. 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∞ functional calculus with this optimal angle.