Sharp estimates for the Ornstein-Uhlenbeck operator

Mauceri, Giancarlo¹  ; Meda, Stefano² ; Sjögren, Peter³

¹ Dipartimento di Matematica       Università di Genova via Dodecaneso 35 16146 Genova, Italy
² Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca via Bicocca degli Arcimboldi 8 20126 Milano, Italy
³ Department of Mathematics Göteborg University SE-412 96 Göteborg, Sweden

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 3 (2004) no. 3, pp. 447-480

Résumé

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.

Détail

Classification : 47A60, 47D03, 60G15