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

TL;DR

This paper unifies various Bellman function problems by defining a class of functions with small mean oscillation, encompassing spaces like BMO, Muckenhoupt, and Gehring classes, and explores their Bellman function problems.

Contribution

It introduces a unified framework for Bellman function problems across multiple function spaces with small mean oscillation.

Findings

Unified description of BMO, Muckenhoupt, and Gehring classes

Solution approach for Bellman function problems in these classes

Insights into integral functionals on small mean oscillation classes

Abstract

We unify several Bellman function problems 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 $\mathbb{R}^2$). We show how the unit ball in the $\mathrm{BMO}$ 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.