Sharp estimates of integral functionals on classes of functions with small mean oscillation

Paata Ivanisvili; Nikolay N. Osipov; Dmitriy M. Stolyarov; Vasily I. Vasyunin; Pavel B. Zatitskiy

doi:10.1016/j.crma.2015.07.016

Mathematical analysis/Harmonic analysis

Sharp estimates of integral functionals on classes of functions with small mean oscillation
[Estimations précises de certaines fonctionnelles intégrales sur des classes de fonctions avec une petite oscillation moyenne]

Présenté par : Gilles Pisier

Paata Ivanisvili ¹ ; Nikolay N. Osipov ^2,³ ; Dmitriy M. Stolyarov ^2,⁴ ; Vasily I. Vasyunin ² ; Pavel B. Zatitskiy ^2,⁴

¹ Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA
² St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia
³ Norwegian University of Science and Technology (NTNU), IME Faculty, Dep. of Math. Sci., Alfred Getz' vei 1, Trondheim, Norway
⁴ Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178 Russia

Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1081-1085.

Résumé
Abstract

Nous unifions plusieurs problèmes concernant la fonction de Bellman traités dans [1,2,4–6,9–12,14–16,18–25]. Dans ce but, nous introduisons une classe de fonctions dont l'oscillation moyenne est petite dans un certain sens (cette classe depend de deux sous-ensembles convexes de $R^{2}$ ). Nous démontrons que la boule unité de l'espace BMO, ou de la classe de Muckenhoupt, ou de la classe de Gehring, peut être décrite de cette façon. Finalement, nous considérons un problème de fonction de Bellman sur chacune de ces classes et discutons sa résolution ainsi que des questions voisines.

We unify several Bellman function problems treated in [1,2,4–6,9–12,14–16,18–25]. For that purpose, we define a class of functions that have, in a sense, small mean oscillation (this class depends on two convex sets in $R^{2}$ ). We show how the unit ball in the BMO space, or a Muckenhoupt class, or a Gehring class can be described in such a fashion. Finally, we consider a Bellman function problem on these classes, discuss its solution and related questions.

Reçu le : 2014-12-21
Accepté le : 2015-07-24
Publié le : 2015-10-29

DOI : 10.1016/j.crma.2015.07.016

Affiliations des auteurs :

Paata Ivanisvili ¹ ; Nikolay N. Osipov ^{2, 3} ; Dmitriy M. Stolyarov ^{2, 4} ; Vasily I. Vasyunin ² ; Pavel B. Zatitskiy ^{2, 4}

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     author = {Paata Ivanisvili and Nikolay N. Osipov and Dmitriy M. Stolyarov and Vasily I. Vasyunin and Pavel B. Zatitskiy},
     title = {Sharp estimates of integral functionals on classes of functions with small mean oscillation},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1081--1085},
     publisher = {Elsevier},
     volume = {353},
     number = {12},
     year = {2015},
     doi = {10.1016/j.crma.2015.07.016},
     language = {en},
}

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Paata Ivanisvili; Nikolay N. Osipov; Dmitriy M. Stolyarov; Vasily I. Vasyunin; Pavel B. Zatitskiy. Sharp estimates of integral functionals on classes of functions with small mean oscillation. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1081-1085. doi : 10.1016/j.crma.2015.07.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.07.016/

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