Comptes Rendus
no. 11-12
Mathematical Analysis/Partial Differential Equations
On Bellman function for extremal problems in BMO
[Sur la fonction de Bellman pour des problèmes extrémaux sur lʼespace BMO]

Présenté par : Gilles Pisier

Paata Ivanishvili1, 2 ; Nikolay N. Osipov1, 3 ; Dmitriy M. Stolyarov1, 3 ; Vasily I. Vasyunin3 ; Pavel B. Zatitskiy1, 3
1 Chebyshev Laboratory (SPbU), 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia
2 Saint Petersburg State University, Universitetsky prospekt 28, Peterhof, St. Petersburg, Russia
3 St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 561-564.

Dans cette Note, nous décrivons nos résultats sur la construction de la fonction de Bellman qui résout un problème extrémal pour une grande classe de formes linéaires intégrales sur BMO.

In this Note we describe our results on construction of the Bellman function solving an extremal problem for a large class of integral functionals on BMO.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.06.011

Paata Ivanishvili 1, 2 ; Nikolay N. Osipov 1, 3 ; Dmitriy M. Stolyarov 1, 3 ; Vasily I. Vasyunin 3 ; Pavel B. Zatitskiy 1, 3

1 Chebyshev Laboratory (SPbU), 14th Line 29B, Vasilyevsky Island, St. Petersburg, Russia
2 Saint Petersburg State University, Universitetsky prospekt 28, Peterhof, St. Petersburg, Russia
3 St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia 
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     title = {On {Bellman} function for extremal problems in {BMO}},
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Paata Ivanishvili; Nikolay N. Osipov; Dmitriy M. Stolyarov; Vasily I. Vasyunin; Pavel B. Zatitskiy. On Bellman function for extremal problems in BMO. Comptes Rendus. Mathématique, Volume 350 (2012) no. 11-12, pp. 561-564. doi : 10.1016/j.crma.2012.06.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.06.011/

[1] P. Ivanishvili; N. Osipov; D. Stolyarov; V. Vasyunin; P. Zatitskiy Bellman function for extremal problems in BMO, 2012 (preprint) | arXiv

[2] L. Slavin; V. Vasyunin Sharp Lp estimates on BMO, 2011 (preprint) | arXiv

[3] L. Slavin; V. Vasyunin Sharp results in the integral-form John–Nirenberg inequality, Trans. Amer. Math. Soc., Volume 363 (2011) no. 8, pp. 4135-4169 (preprint, 2007) | arXiv

[4] V. Vasyunin The sharp constant in the John–Nirenberg inequality, 2003 http://www.pdmi.ras.ru/preprint/2003/index.html (preprint POMI, No. 20)

[5] V.I. Vasyunin Mutual estimates of Lp-norms, and the Bellman function, Zapiski POMI, Volume 355 (2008) no. 5, pp. 81-138 (in Russian); English translation in: J. Math. Sci., 156, 2009, pp. 766-798

[6] V. Vasyunin Sharp constants in the classical weak form of the John–Nirenberg inequality, 2011 http://www.pdmi.ras.ru/preprint/2011/eng-2011.html (preprint POMI, No. 10)

[7] Vasily Vasyunin; Alexander Volberg Monge–Ampère equation and Bellman optimization of Carleson embedding theorems, Amer. Math. Soc. Transl. Ser. 2, Volume 226 (2009), pp. 195-238

[8] Vasily Vasyunin; Alexander Volberg Sharp constants in the classical weak form of the John–Nirenberg inequality, 2012 (preprint) | arXiv

  • P. Zatitskii; D. Stolyarov On Locally Concave Functions on Simplest Nonconvex Domains, Journal of Mathematical Sciences, Volume 282 (2024) no. 4, p. 482 | DOI:10.1007/s10958-024-07194-x
  • Dmitriy Stolyarov; Vasily Vasyunin; Pavel Zatitskiy; Ilya Zlotnikov Sharp moment estimates for martingales with uniformly bounded square functions, Mathematische Zeitschrift, Volume 302 (2022) no. 1, p. 181 | DOI:10.1007/s00209-022-03064-x
  • Dmitriy Stolyarov; Pavel Zatitskiy Sharp transference principle for BMO and A, Journal of Functional Analysis, Volume 281 (2021) no. 6, p. 109085 | DOI:10.1016/j.jfa.2021.109085
  • H. Hedenmalm; D.M. Stolyarov; V.I. Vasyunin; P.B. Zatitskiy Sharpening Hölder's inequality, Journal of Functional Analysis, Volume 275 (2018) no. 5, p. 1280 | DOI:10.1016/j.jfa.2018.05.003
  • D.M. Stolyarov; P.B. Zatitskiy Theory of locally concave functions and its applications to sharp estimates of integral functionals, Advances in Mathematics, Volume 291 (2016), p. 228 | DOI:10.1016/j.aim.2015.11.048
  • Leonid Slavin; Vasily Vasyunin Inequalities for BMO on α-Trees, International Mathematics Research Notices, Volume 2016 (2016) no. 13, p. 4078 | DOI:10.1093/imrn/rnv258
  • Adam Osȩkowski Sharp Estimates for Lipschitz Class, The Journal of Geometric Analysis, Volume 26 (2016) no. 2, p. 1346 | DOI:10.1007/s12220-015-9593-7
  • Paata Ivanisvili Inequality for Burkholder’s martingale transform, Analysis PDE, Volume 8 (2015) no. 4, p. 765 | DOI:10.2140/apde.2015.8.765
  • Adam Oscȩkowski Sharp inequalities for BMO functions, Chinese Annals of Mathematics, Series B, Volume 36 (2015) no. 2, p. 225 | DOI:10.1007/s11401-015-0887-7
  • V. I. Vasyunin An Example of Constructing a Bellman Function for Extremal Problems in BMO, Journal of Mathematical Sciences, Volume 209 (2015) no. 5, p. 683 | DOI:10.1007/s10958-015-2521-3
  • Adam Osȩkowski Survey Article: Bellman function method and sharp inequalities for martingales, Rocky Mountain Journal of Mathematics, Volume 43 (2013) no. 6 | DOI:10.1216/rmj-2013-43-6-1759

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