Bounded holomorphic functional calculus for nonsymmetric Ornstein-Uhlenbeck operators

Abstract

We study bounded holomorphic functional calculus for nonsymmetric infinite dimensional Ornstein-Uhlenbeck operators $\mathcal{L}$. We prove that if $-\mathcal{L}$ generates an analytic semigroup on $L^2({\gamma }_{\infty })$, then $\mathcal{L}$ has bounded holomorphic functional calculus on $L^r({\gamma }_{\infty })$, $1<r<\infty$, in any sector of angle $\theta >{\theta }_r^{\ast }$, where ${\gamma }_{\infty }$ is the associated invariant measure and ${\theta }_r^{\ast }$ the sectoriality angle of $\mathcal{L}$ on $L^r({\gamma }_{\infty })$. The angle ${\theta }_r^{\ast }$ is optimal. In particular our result applies to any nondegenerate finite dimensional Ornstein-Uhlenbeck operator, with dimension-free estimates.

Cited by in corpus (1)

• Dimension-free $L^p$ estimates for vectors of Riesz transforms associated with orthogonal expansions