A new weighted Bellman function

Pattakos, Nikolaos; Volberg, Alexander

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

Pattakos, Nikolaos ¹ ; Volberg, Alexander ¹

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

Comptes Rendus. Mathématique, Tome 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 :

Pattakos, Nikolaos ¹ ; Volberg, Alexander ¹

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

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     title = {A new weighted {Bellman} function},
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Pattakos, Nikolaos; Volberg, Alexander. A new weighted Bellman function. Comptes Rendus. Mathématique, Tome 349 (2011) no. 21-22, pp. 1151-1154. doi : 10.1016/j.crma.2011.10.007. http://www.numdam.org/articles/10.1016/j.crma.2011.10.007/

Bibliographie
Cité par

[1] Burkholder, D. Boundary value problems and sharp estimates for the martingale transforms, Ann. Probab., Volume 12 (1984), pp. 647-702

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

[3] Dragicevic, O.; Volberg, A. 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] Meyers, N.G. 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] Nazarov, F.; Volberg, A. 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] Petermichl, S.; Volberg, A. 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] Stein, E.; Weiss, G. Interpolation of operators with change of measures, Trans. Amer. Math. Soc., Volume 87 (1958) no. 1, pp. 159-172

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