The Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time

van Neerven, Jan; Portal, Pierre

doi:10.24033/bsmf.2771

The Weyl calculus with respect to the Gaussian measure and restricted

L^{p}

-

L^{q}

boundedness of the Ornstein-Uhlenbeck semigroup in complex time
[Calcul fonctionnel de Weyl pour la mesure gaussienne et estimées

L^{p}

-

L^{q}

restreintes du semigroupe d’Ornstein-Uhlenbeck en temps complexe]

van Neerven, Jan ¹ ; Portal, Pierre ²

¹Delft University of Technology Delft Institute of Applied Mathematics P.O. Box 5031 2600 GA Delft The Netherlands
²Mathematical Sciences Institute, John Dedman Building The Australian National University ACTON ACT 0200 Australia and Laboratoire Paul Painlevé Université Lille 1 F-59655 Villeneuve D’Ascq France

Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 4, pp. 691-712

Abstract (VO)
Résumé

In this paper, we introduce a Weyl functional calculus $a \mapsto a (Q, P)$ for the position and momentum operators $Q$ and $P$ associated with the Ornstein-Uhlenbeck operator $L = - Δ + x \cdot \nabla$ , and give a simple criterion for restricted $L^{p}$ - $L^{q}$ boundedness of operators in this functional calculus. The analysis of this non-commutative functional calculus is simpler than the analysis of the functional calculus of $L$ . It allows us to recover, unify, and extend old and new results concerning the boundedness of $exp (- z L)$ as an operator from $L^{p} (ℝ^{d}, γ_{α})$ to $L^{q} (ℝ^{d}, γ_{β})$ for suitable values of $z \in ℂ$ with $Re z > 0$ , $p, q \in [1, \infty)$ , and $α, β > 0$ . Here, $γ_{τ}$ denotes the centered Gaussian measure on $ℝ^{d}$ with density ${(2 π τ)}^{- d / 2} {exp (- | x |}^{2} / 2 τ)$ .

Ce papier introduit un calcul fonctionnel de Weyl $a \mapsto a (Q, P)$ adapté aux opérateurs de position et d’impulsion $Q$ et $P$ associés à l’opérateur d’Ornstein-Uhlenbeck $L = - Δ + x \cdot \nabla$ , et fournit un critère simple pour prouver des estimées $L^{p}$ - $L^{q}$ restreintes de ce calcul fonctionnel. L’analyse de ce calcul fonctionnel non-commutatif se révèle être plus simple que celle du calcul fonctionnel de $L$ . Ceci nous permet de redémontrer, d’unifier, et d’étendre des résultats anciens et nouveaux sur les propriétés de bornitude de $exp (- z L)$ de $L^{p} (ℝ^{d}, γ_{α})$ dans $L^{q} (ℝ^{d}, γ_{β})$ pour les valeurs appropriées de $z \in ℂ$ (avec $Re z > 0$ , $p, q \in [1, \infty)$ , et $α, β > 0$ ). La notation $γ_{τ}$ est utilisée pour le mesure Gaussienne centrée sur $ℝ^{d}$ de densité ${(2 π τ)}^{- d / 2} {exp (- | x |}^{2} / 2 τ)$ .

Reçu le : 2017-02-13
Révisé le : 2017-08-14
Accepté le : 2017-09-13
Publié le : 2018-01-28

MR Zbl

DOI : 10.24033/bsmf.2771

Classification : 47A60, 47D06, 47G30, 60H07, 81S05
Keywords: Weyl functional calculus, canonical commutation relation, Schur estimate, Ornstein-Uhlenbeck operator, Mehler kernel, restricted $L^p$-$L^q$-boundedness, restricted Sobolev embedding
Mots-clés : Calcul fonctionnel de Weyl, relations de commutation canoniques, estimées de Schur, opérateur d’Ornstein-Uhlenbeck, estimées $L^p$-$L^q$ restreintes, injections de Sobolev restreintes.

Affiliations des auteurs :

van Neerven, Jan ¹ ; Portal, Pierre ²

¹ Delft University of Technology Delft Institute of Applied Mathematics P.O. Box 5031 2600 GA Delft The Netherlands
² Mathematical Sciences Institute, John Dedman Building The Australian National University ACTON ACT 0200 Australia and Laboratoire Paul Painlevé Université Lille 1 F-59655 Villeneuve D’Ascq France

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     title = {The {Weyl} calculus with respect to the {Gaussian} measure and restricted $L^p$-$L^q$ boundedness of the {Ornstein-Uhlenbeck} semigroup in complex time},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {691--712},
     year = {2018},
     publisher = {Soci\'et\'e math\'ematique de France},
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van Neerven, Jan; Portal, Pierre. The Weyl calculus with respect to the Gaussian measure and restricted $L^p$-$L^q$ boundedness of the Ornstein-Uhlenbeck semigroup in complex time. Bulletin de la Société Mathématique de France, Tome 146 (2018) no. 4, pp. 691-712. doi: 10.24033/bsmf.2771

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