Sharp estimates for the Ornstein-Uhlenbeck operator

Mauceri, Giancarlo; Meda, Stefano; Sjögren, Peter

Mauceri, Giancarlo; Meda, Stefano; Sjögren, Peter

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, p. 447-480

Abstract

Let $ℒ$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $γ$ on $ℝ^{d} .$ We prove a sharp estimate of the operator norm of the imaginary powers of $ℒ$ on $L^{p} (γ),$ $1 < p < \infty .$ Then we use this estimate to prove that if $b$ is in $[0, \infty)$ and $M$ is a bounded holomorphic function in the sector ${z \in ℂ : m o d arg (z - b) < arcsin | 2 / p - 1 |}$ and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator $M (ℒ)$ is bounded on $L^{p} (γ) .$ This improves earlier results of the authors with J. García-Cuerva and J.L. Torrea.

MR 2099246 | Zbl 1116.47036

Classification: 47A60, 47D03, 60G15

@article{ASNSP_2004_5_3_3_447_0,
     author = {Mauceri, Giancarlo and Meda, Stefano and Sj\"ogren, Peter},
     title = {Sharp estimates for the Ornstein-Uhlenbeck operator},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {3},
     year = {2004},
     pages = {447-480},
     zbl = {1116.47036},
     mrnumber = {2099246},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_447_0}
}

Mauceri, Giancarlo; Meda, Stefano; Sjögren, Peter. Sharp estimates for the Ornstein-Uhlenbeck operator. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 3 (2004) no. 3, pp. 447-480. http://www.numdam.org/item/ASNSP_2004_5_3_3_447_0/

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