[Équations de Monge–Ampère et fonctions de Bellman : l'opérateur maximal dyadique]
Nous construisons explicitement la fonction de Bellman pour l'opérateur maximal dyadique sur 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 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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@article{CRMATH_2008__346_9-10_585_0,
author = {Slavin, Leonid and Stokolos, Alexander and Vasyunin, Vasily},
title = {Monge{\textendash}Amp\`ere equations and {Bellman} functions: {The} dyadic maximal operator},
journal = {Comptes Rendus. Math\'ematique},
pages = {585--588},
publisher = {Elsevier},
volume = {346},
number = {9-10},
year = {2008},
doi = {10.1016/j.crma.2008.03.003},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2008.03.003/}
}
TY - JOUR AU - Slavin, Leonid AU - Stokolos, Alexander AU - Vasyunin, Vasily TI - Monge–Ampère equations and Bellman functions: The dyadic maximal operator JO - Comptes Rendus. Mathématique PY - 2008 SP - 585 EP - 588 VL - 346 IS - 9-10 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2008.03.003/ DO - 10.1016/j.crma.2008.03.003 LA - en ID - CRMATH_2008__346_9-10_585_0 ER -
%0 Journal Article %A Slavin, Leonid %A Stokolos, Alexander %A Vasyunin, Vasily %T Monge–Ampère equations and Bellman functions: The dyadic maximal operator %J Comptes Rendus. Mathématique %D 2008 %P 585-588 %V 346 %N 9-10 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2008.03.003/ %R 10.1016/j.crma.2008.03.003 %G en %F CRMATH_2008__346_9-10_585_0
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. http://www.numdam.org/articles/10.1016/j.crma.2008.03.003/
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[2] 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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