Sharp estimates for the Ornstein-Uhlenbeck operator

Mauceri, Giancarlo; Meda, Stefano; Sjögren, Peter

Mauceri, Giancarlo ¹ ; Meda, Stefano ² ; Sjögren, Peter ³

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 $γ$ 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 Zbl

Classification : 47A60, 47D03, 60G15

Affiliations des auteurs :

Mauceri, Giancarlo ¹ ; Meda, Stefano ² ; Sjögren, Peter ³

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     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},
     pages = {447--480},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 3},
     number = {3},
     year = {2004},
     mrnumber = {2099246},
     zbl = {1116.47036},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2004_5_3_3_447_0/}
}

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

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