Bellman function for extremal problems in BMO

Abstract

In this paper we develop the method of finding sharp estimates by using a Bellman function. In such a form the method appears in the proofs of the classical John-Nirenberg inequality and L p estimations of BMO functions. In the present paper we elaborate a method of solving the boundary value problem for the homogeneous Monge-Ampère equation in a parabolic strip for sufficiently smooth boundary conditions. In such a way, we have obtained an algorithm for constructing an exact Bellman function for a large class of integral functionals on the BMO space.

Sharp inequalities for BMO functions

“…Finally, we would like to mention here the recent work of Ivanishvili et al [7], which treats the above BMO estimates from a much wider perspective. More specifically, it provides the detailed description of the machinery which can be used to prove a general estimate in the BMO setting (under some regularity conditions on the boundary value function).…”

“…More specifically, it provides the detailed description of the machinery which can be used to prove a general estimate in the BMO setting (under some regularity conditions on the boundary value function). Consult also [6] for the short discussion on the subject.…”

“…As in the aforementioned papers, our approach will rest on the Bellman function method. We would like to point out here that the desired estimate does not fall into the scope of the (general) bounds covered by [6][7], since the corresponding boundary value function is not sufficiently regular. However, the arguments developed in these papers have turned out to be quite helpful during the search.…”

Distribution of martingales with bounded square functions

“…We refer the reader to the paper [7] for treatment of the general case. The paper [6] considers less general case (the authors make additional assumptions on the structure of f ), however, provides a much shorter presentation. The short report [5] outlines the results.…”

A sharp estimate for Muckenhoupt class $$A_\infty $$ A ∞ and BMO

“…Since then, the technique has been successfully applied in numerous settings: see e.g. [2,5,6,[13][14][15][16][17] and consult the references therein.…”