Dimension-free $H^{\infty }$-calculus of angle $<\pi /2$ for UMD-valued Ornstein--Uhlenbeck operators

Abstract

Let $1<p<\infty$, let $X$ be a UMD Banach space, let $1<p<\infty$, and let $L_d$ be the generator of the Ornstein--Uhlenbeck semigroup $(P_d(t){)}_{t\ge 0}$ on $L^p({\mathbb{R}}^d,{\gamma }_d;X)$. We prove that the operators $-L_d$ are $R$-sectorial with a common angle strictly smaller than $\pi /2$ and with bounds independent of $d$. Combining this with the Hieber--Prüss transference theorem and the Kalton--Weis angle comparison, we deduce that the operators $-L_d$ admit bounded $H^{\infty }$-calculi of a common angle strictly smaller than $\pi /2$, again with dimension-free bounds. The corresponding Walsh $R$-analyticity estimates are proved first and transferred to the Ornstein--Uhlenbeck setting by a central limit argument.

22 pages, submitted for publication