A new weighted Bellman function

Nikolaos Pattakos; Alexander Volberg

doi:10.1016/j.crma.2011.10.007

Mathematical Analysis/Harmonic Analysis

A new weighted Bellman function
[Une nouvelle fonction pondérée de Bellman]

Présenté par : Gilles Pisier

Nikolaos Pattakos ¹ ; Alexander Volberg ¹

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1151-1154.

Résumé
Abstract

Nous construisons une nouvelle fonction de Bellman qui nous permet de donner des estimations précises de la norme des transformées de Riesz dans les espaces pondérés $L^{p} (w d x)$ , quand la caractéristique du poids est proche de 1.

We give sharp in p and w estimates of operator norms of Riesz transforms in the $L^{p} (w d x)$ spaces, when the $A_{p}$ characteristic of the weight is close to 1 (flat case). This is done by proving the existence of a certain Bellman function.

Reçu le : 2011-05-19
Accepté le : 2011-10-10
Publié le : 2011-10-22

DOI : 10.1016/j.crma.2011.10.007

Affiliations des auteurs :

Nikolaos Pattakos ¹ ; Alexander Volberg ¹

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA

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     author = {Nikolaos Pattakos and Alexander Volberg},
     title = {A new weighted {Bellman} function},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1151--1154},
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     volume = {349},
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     year = {2011},
     doi = {10.1016/j.crma.2011.10.007},
     language = {en},
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Nikolaos Pattakos; Alexander Volberg. A new weighted Bellman function. Comptes Rendus. Mathématique, Volume 349 (2011) no. 21-22, pp. 1151-1154. doi : 10.1016/j.crma.2011.10.007. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.10.007/

Bibliographie
Cité par

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[2] J.G. Conlon; T. Spencer A strong central limit theorem for a class of random surfaces | arXiv

[3] O. Dragicevic; A. Volberg Bellman function, Littlewood–Paley estimates and asymptotics for the Ahlfors–Beurling operator in $L^{p} (C)$ , Indiana Univ. Math. J., Volume 54 (2005) no. 4, pp. 971-995

[4] N.G. Meyers An $L^{p}$ estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa (3), Volume 17 (1963), pp. 189-206

[5] F. Nazarov; A. Volberg Heating of the Ahlfors–Beurling operator and estimates of its norms, St. Petersburg Math. J., Volume 15 (2004) no. 4, pp. 563-573 S1061-0022(04)00822-2

[6] N. Pattakos, A. Volberg, Continuity of weighted estimates in $A_{p}$ norm, Proc. Amer. Math. Soc., forthcoming, article ID: PROC11165.

[7] S. Petermichl; A. Volberg Heating of the Ahlfors–Beurling operator: Weakly quasiregular maps on the plane are quasiregular, Duke Math. J., Volume 112 (2002) no. 2, pp. 281-305

[8] E. Stein; G. Weiss Interpolation of operators with change of measures, Trans. Amer. Math. Soc., Volume 87 (1958) no. 1, pp. 159-172

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