Calculus of Variations
Monge–Ampère equations and Bellman functions: The dyadic maximal operator
[Équations de Monge–Ampère et fonctions de Bellman : l'opérateur maximal dyadique]

Présenté par : Jean-Pierre Kahane

Slavin, Leonid1 ; Stokolos, Alexander2 ; Vasyunin, Vasily3
1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
2 Department of Mathematics, DePaul University, Chicago, IL 60614, USA
3 St. Petersburg Department of the V. A. Steklov Mathematical Institute, Russian Academy of Sciences, Russia
Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 585-588.

Nous construisons explicitement la fonction de Bellman pour l'opérateur maximal dyadique sur Lp comme solution d'une équation aux dérivées partielles de Bellman de type Monge–Ampère. La fonction a été introduite par A. Melas (2005) sous un angle différent, mais ici nous privilégions notre approche à partir d'une équation aux dérivées partielles. Claire et reproductible, cette approche peut servir de principe unificateur dans les investigations passées et actuelles concernant les fonctions de Bellman.

We find explicitly the Bellman function for the dyadic maximal operator on Lp as the solution of a Bellman partial differential equation of Monge–Ampère type. This function has been previously found by A. Melas (2005) in a different way, but it is our partial differential equation-based approach that is of principal interest here. Clear and replicable, it holds promise as a unifying template for past and current Bellman function investigations.

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DOI : 10.1016/j.crma.2008.03.003
Slavin, Leonid 1 ; Stokolos, Alexander 2 ; Vasyunin, Vasily 3

1 Department of Mathematics, University of Missouri, Columbia, MO 65211, USA
2 Department of Mathematics, DePaul University, Chicago, IL 60614, USA
3 St. Petersburg Department of the V. A. Steklov Mathematical Institute, Russian Academy of Sciences, Russia 
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Slavin, Leonid; Stokolos, Alexander; Vasyunin, Vasily. Monge–Ampère equations and Bellman functions: The dyadic maximal operator. Comptes Rendus. Mathématique, Tome 346 (2008) no. 9-10, pp. 585-588. doi : 10.1016/j.crma.2008.03.003. https://www.numdam.org/articles/10.1016/j.crma.2008.03.003/

[1] Melas, A. The Bellman functions of dyadic-like maximal operators and related inequalities, Adv. Math., Volume 192 (2005) no. 2, pp. 310-340

[2] Nazarov, F.; Treil, S. The hunt for Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis, Algebra i Analiz, Volume 8 (1996) no. 5, pp. 32-162 (in Russian). Translation in St. Petersburg Math. J., 8, 5, 1997, pp. 721-824

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