Bellman functions on simple non-convex domains in the plane

TL;DR

This paper generalizes Bellman function techniques to non-convex planar domains, enabling sharp estimates for classes like BMO, A_p, and Gehring, through an algorithm for constructing minimal locally concave functions.

Contribution

It introduces an algorithm for constructing Bellman functions on non-convex domains, extending previous work on convex parabolic strips.

Findings

Derived sharp estimates for BMO, A_p, and Gehring classes.

Extended Bellman function methodology to non-convex domains.

Provided a constructive approach for minimal locally concave functions.

Abstract

The present paper provides a generalization of the previous authors' work on Bellman functions for integral functionals on $\mathrm{BMO}$. Those Bellman functions are the minimal locally concave functions on parabolic strips in the plane. Now we describe the algorithm for constructing minimal locally concave functions on a planar domain that is a difference of two unbounded convex domains. This leads to many sharp estimates for functions in the classes like $\mathrm{BMO}$, $A_p$, or the Gehring classes.