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

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

TY  - JOUR
AU  - Mauceri, Giancarlo
AU  - Meda, Stefano
AU  - Sjögren, Peter
TI  - Sharp estimates for the Ornstein-Uhlenbeck operator
JO  - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY  - 2004
SP  - 447
EP  - 480
VL  - 3
IS  - 3
PB  - Scuola Normale Superiore, Pisa
UR  - https://www.numdam.org/item/ASNSP_2004_5_3_3_447_0/
LA  - en
ID  - ASNSP_2004_5_3_3_447_0
ER  -

%0 Journal Article
%A Mauceri, Giancarlo
%A Meda, Stefano
%A Sjögren, Peter
%T Sharp estimates for the Ornstein-Uhlenbeck operator
%J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
%D 2004
%P 447-480
%V 3
%N 3
%I Scuola Normale Superiore, Pisa
%U https://www.numdam.org/item/ASNSP_2004_5_3_3_447_0/
%G en
%F 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, Série 5, Tome 3 (2004) no. 3, pp. 447-480. https://www.numdam.org/item/ASNSP_2004_5_3_3_447_0/

Bibliographie
Cité par

[1] M. Cowling, Harmonic analysis on semigroups, Ann. of Math. 117 (1983), 267-283. | MR | Zbl

[2] M. Cowling - I. Doust - A. Mcintosh - A. Yagi, Banach space operators with a bounded $H^{\infty}$ functional calculus, J. Aust. Math. Soc. 60 (1996), 51-89. | MR | Zbl

[3] M. Cowling - S. Meda, Harmonic analysis and ultracontractivity, Trans. Amer. Math. Soc. 340 (1993), 733-752. | MR | Zbl

[4] E. B. Davies, “Heat Kernels and Spectral Theory”, Cambridge Tract. in Math. 92, Cambridge University Press, Cambridge, 1989. | MR | Zbl

[5] J. B. Epperson, The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel, J. Funct. Anal. 87 (1989), 1-30. | MR | Zbl

[6] J. Garcia-Cuerva - G. Mauceri - P. Sjögren - J.L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999), 281-305. | MR | Zbl

[7] J. García-Cuerva - G. Mauceri - S. Meda - P. Sjögren - J. L. Torrea, Functional Calculus for the Ornstein-Uhlenbeck Operator, J. Funct. Anal. 183 (2001), 413-450. | MR | Zbl

[8] W. Hebisch - G. Mauceri - S. Meda, Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator, J. Funct. Anal. 210 (2004), 101-124. | MR | Zbl

[9] L. Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93-140. | MR | Zbl

[10] L. Hörmander, “The Analysis of Linear Partial Differential Operators”, Vol. 1 Springer Verlag, Berlin, 1983.

[11] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110 (1990), 639-647. | MR | Zbl

[12] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973), 211-227. | MR | Zbl

[13] E. M. Stein, “Topics in Harmonic Analysis Related to the Littlewood-Paley Theory”, Annals of Math. Studies, No. 63, Princeton N. J., 1970. | MR | Zbl