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

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] into one setting. 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.Since Slavin [12] and Vasyunin [18] proved the sharp form of the John-Nirenberg inequality (see [15]), there have been many papers where similar principles are used to prove sharp estimates of this kind. However, there is no theory or even a unifying approach; moreover, the class of problems to which the method can be applied has not been described yet. There is a portion of heuristics in the folklore that is each time applied to a new problem in a very similar manner. The first attempt to build a theory (at least for BMO) was made in [16], then the theory was developed in the paper [4] (see the short report [5] also). We would also like to draw the reader's attention to the forthcoming paper [6], which can be considered as a description of the theory for the BMO space in a sufficient generality. Problems of this kind were considered not only in BMO, but in Muckenhoupt classes, Gehring classes, etc (see [1,2,11,13,19,20]). In this short note, we define a class of functions and an extremal problem on it that includes all the problems discussed above. We believe that the unification we offer gives a strong basis for a theory that will distinguish a certain class of problems to which the method is applicable in a direct way. In Section 1 we state the problem and discuss related questions. Section 2 contains a detailed explanation how our classes of functions include the unit ball in BMO as well as the "unit balls" in Muckenhoupt classes and Gehring classes. Finally, in Section 3 we give hints to the solution of the problem (as the reader may expect looking at previous papers, it is rather lengthy and technical, so we omit a description of the solution, but concentrate on an analogy with the case of BMO considered in [4,6,22]).

Bellman function for extremal problems in $\mathrm{BMO}$ II: evolution

“…This conjecture is verified for the case f (t) = e t in the paper [31]. In the light of [12], Conjecture 6.2.2 opens the road towards a theory for dyadic Bellman functions similar to the one described in the present paper (however, the conjecture says nothing about how to find Ω).…”

“…Theorem 2.2.11 is not a miracle now. A relatively short and transparent proof of it (omitting the construction of the Bellman function) was given in [34] (in the general geometric setting of [12]). The idea is that there is a third function (other than B ε and the minimal locally concave B) built as a solution of a certain optimization problem.…”

“…The trick was used in [14] as well. The Bellman function that plays the major role in [27,32] is defined on the exponential strip and falls under the scope of the general setting in [12].…”

“…Indeed, see the papers [1,23,37], where similar questions are studied on Muckenhoupt classes and Gehring (so called reverse Hölder) classes. All these particular cases were unified into a single extremal problem in [12]. The…”

Sharpening Hölder's inequality

“…The closest one is the Bellman function in [13]. See [11] for a more geometric point of view and [9], [10], and [12] for a study of a related problem. We also refer the reader to [15], [16], [19], and [20] for history and basics of the Bellman function theory.…”