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Comptes Rendus Mathématique
Volume 353, n° 12
pages 1081-1085 (décembre 2015)
Doi : 10.1016/j.crma.2015.07.016
Received : 21 December 2014 ;  accepted : 24 July 2015
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
 

Paata Ivanisvili a , Nikolay N. Osipov b, c , Dmitriy M. Stolyarov b, d , Vasily I. Vasyunin b , Pavel B. Zatitskiy b, d
a Department of Mathematics, Michigan State University, East Lansing, MI 48823, USA 
b St. Petersburg Department of Steklov Mathematical Institute RAS, Fontanka 27, St. Petersburg, Russia 
c Norwegian University of Science and Technology (NTNU), IME Faculty, Dep. of Math. Sci., Alfred Getz' vei 1, Trondheim, Norway 
d Chebyshev Laboratory, St. Petersburg State University, 14th Line, 29b, St. Petersburg, 199178 Russia 

Abstract

We unify several Bellman function problems treated in [[1], [2], [4], [5], [6], [9], [10], [11], [12], [14], [15], [16], [18], [19], [20], [21], [22], [23], [24], [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  ). 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.

The full text of this article is available in PDF format.
Résumé

Nous unifions plusieurs problèmes concernant la fonction de Bellman traités dans [[1], [2], [4], [5], [6], [9], [10], [11], [12], [14], [15], [16], [18], [19], [20], [21], [22], [23], [24], [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  ). 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.

The full text of this article is available in PDF format.


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