Inequalities for BMO on $\alpha$-trees

TL;DR

This paper extends Bellman function techniques to BMO spaces on $ ext{alpha}$-trees, deriving sharp inequalities including the John--Nirenberg inequality and relating $L^1$ and $L^2$ oscillations, with broad applicability.

Contribution

It introduces new technical tools for Bellman functions on $ ext{alpha}$-trees, enabling sharp inequalities and reformulations for continuous BMO.

Findings

Proved sharp integral John--Nirenberg inequality for BMO on $ ext{alpha}$-trees.

Established an inequality relating $L^1$ and $L^2$ oscillations with explicit constants.

Reformulated the John--Nirenberg inequality for continuous BMO in terms of special martingales.

Abstract

We develop technical tools that enable the use of Bellman functions for BMO defined on $\alpha$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $\alpha$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be used for any function class that corresponds to a non-convex Bellman domain.