Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem

Dragičević, Oliver; Volberg, Alexander

doi:10.1016/j.crma.2005.03.021

Harmonic Analysis

Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the

p - 1

problem
[Fonction de Bellman pour obtenir des estimations de type Littlewood–Paley et de type assymptotique pour le problème

p - 1

]

Présenté par : Gilles Pisier

Dragičević, Oliver ¹ ; Volberg, Alexander ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Tome 340 (2005) no. 10, pp. 731-734.

Résumé
Abstract

On utilise la technique de la fonction de Bellman pour obtenir les estimations nouvelles et assez générales du type de Littlewood–Paley. Comme la premier consequence de nos estimation du type de Littlewood–Paley on derive les resultats classiques concernants les bornes libre de dimension pour les transformations de Riesz. La deuxième consequence est une amilioration de la borne dans $L^{p} (C)$ de transformation de Ahlfors–Beurling quand $p \to \infty$ .

We utilize the method of Bellman functions to derive new $L^{p}$ -estimates of Littlewood–Paley type involving $p - 1$ . Among the applications to singular integrals we improve the $2 (p - 1)$ bounds for the Ahlfors–Beurling operator on $L^{p} (C)$ when $p \to \infty$ . In addition, dimensionless estimates of Riesz transforms in the classical as well as in the Ornstein–Uhlenbeck setting are attained.

Reçu le : 2005-03-03
Accepté le : 2005-03-22
Publié le : 2005-05-10

DOI : 10.1016/j.crma.2005.03.021

Affiliations des auteurs :

Dragičević, Oliver ¹ ; Volberg, Alexander ²

¹ Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
² Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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Dragičević, Oliver; Volberg, Alexander. Bellman function for the estimates of Littlewood–Paley type and asymptotic estimates in the $ p-1$ problem. Comptes Rendus. Mathématique, Tome 340 (2005) no. 10, pp. 731-734. doi : 10.1016/j.crma.2005.03.021. http://archive.numdam.org/articles/10.1016/j.crma.2005.03.021/

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