Sharp estimates for the Ornstein-Uhlenbeck operator

Giancarlo Mauceri[1]; Stefano Meda[2]; Peter Sjögren[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 (2004)

  • Volume: 3, Issue: 3, page 447-480
  • ISSN: 0391-173X

Abstract

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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 < ∞ . Then we use this estimate to prove that if b is in [ 0 , ∞ ) and M is a bounded holomorphic function in the sector { z ∈ 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.

How to cite

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Mauceri, Giancarlo, Meda, Stefano, and Sjögren, Peter. "Sharp estimates for the Ornstein-Uhlenbeck operator." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 3.3 (2004): 447-480. <http://eudml.org/doc/84537>.

@article{Mauceri2004,
abstract = {Let $ \{\mathcal \{L\}\}$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $\gamma $ on $\mathbb \{R\}^d.$ We prove a sharp estimate of the operator norm of the imaginary powers of $ \{\mathcal \{L\}\}$ on $L^p(\gamma ),$$1&lt;p&lt;\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 $\lbrace z\in \mathbb \{C\}: ~mod \;\arg (z-b) &lt; \arcsin |2/p-1|\rbrace $ and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator $M( \{\mathcal \{L\}\})$ is bounded on $L^p(\gamma ).$ This improves earlier results of the authors with J. García-Cuerva and J.L. Torrea.},
affiliation = {Dipartimento di Matematica       Università di Genova via Dodecaneso 35 16146 Genova, Italy; Dipartimento di Matematica e Applicazioni Università di Milano-Bicocca via Bicocca degli Arcimboldi 8 20126 Milano, Italy; Department of Mathematics Göteborg University SE-412 96 Göteborg, Sweden},
author = {Mauceri, Giancarlo, Meda, Stefano, Sjögren, Peter},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
language = {eng},
number = {3},
pages = {447-480},
publisher = {Scuola Normale Superiore, Pisa},
title = {Sharp estimates for the Ornstein-Uhlenbeck operator},
url = {http://eudml.org/doc/84537},
volume = {3},
year = {2004},
}

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
PB - Scuola Normale Superiore, Pisa
VL - 3
IS - 3
SP - 447
EP - 480
AB - Let $ {\mathcal {L}}$ be the Ornstein-Uhlenbeck operator which is self-adjoint with respect to the Gauss measure $\gamma $ on $\mathbb {R}^d.$ We prove a sharp estimate of the operator norm of the imaginary powers of $ {\mathcal {L}}$ on $L^p(\gamma ),$$1&lt;p&lt;\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 $\lbrace z\in \mathbb {C}: ~mod \;\arg (z-b) &lt; \arcsin |2/p-1|\rbrace $ and satisfies a Hörmander-like condition of (nonintegral) order greater than one on the boundary, then the operator $M( {\mathcal {L}})$ is bounded on $L^p(\gamma ).$ This improves earlier results of the authors with J. García-Cuerva and J.L. Torrea.
LA - eng
UR - http://eudml.org/doc/84537
ER -

References

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  7. [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. Zbl0995.47010MR1844213
  8. [8] W. Hebisch – G. Mauceri – S. Meda, Holomorphy of spectral multipliers of the Ornstein-Uhlenbeck operator, J. Funct. Anal. 210 (2004), 101-124. Zbl1069.47017MR2052115
  9. [9] L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 104 (1960), 93-140. Zbl0093.11402MR121655
  10. [10] L. Hörmander, “The Analysis of Linear Partial Differential Operators”, Vol. 1 Springer Verlag, Berlin, 1983. Zbl0521.35002
  11. [11] S. Meda, A general multiplier theorem, Proc. Amer. Math. Soc. 110 (1990), 639-647. Zbl0760.42007MR1028046
  12. [12] E. Nelson, The free Markov field, J. Funct. Anal. 12 (1973), 211-227. Zbl0273.60079MR343816
  13. [13] E. M. Stein, “Topics in Harmonic Analysis Related to the Littlewood-Paley Theory”, Annals of Math. Studies, No. 63, Princeton N. J., 1970. Zbl0193.10502MR252961

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