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

TL;DR

This paper advances the Bellman function method for extremal problems in BMO by developing an evolutional approach that removes previous restrictions, enabling a more natural and dynamic analysis of the function's behavior.

Contribution

It introduces an evolutional framework for Bellman functions in BMO, allowing analysis without restrictive assumptions and highlighting the dynamical aspects of the problem.

Findings

Development of an evolutional approach to Bellman functions in BMO

Removal of previous restrictions on generating functions

Enhanced understanding of the dynamical evolution of Bellman functions

Abstract

In the paper "Bellman function for extremal problems in $\mathrm{BMO}$", the authors built the Bellman function for integral functionals on the $\mathrm{BMO}$ space. The present paper provides a development of the subject. We abandon the majority of unwanted restrictions on the function that generates the functional. It is the new evolutional approach that allows us to treat the problem in its natural setting. What is more, these new considerations lighten dynamical aspects of the Bellman function, in particular, evolution of its picture.