# The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $\alpha$-trees

## TL;DR

This paper derives an explicit Bellman function for the dyadic maximal operator from BMO to BLO spaces, establishing a dimension-free norm of 1 and analyzing the operator's behavior on generalized $ ext{α}$-trees.

## Contribution

It introduces a new Bellman function for the BMO to BLO operator on $ ext{α}$-trees, revealing a quasi-periodic structure and a dimension-free norm constant.

## Key findings

- The BMO to BLO norm of the maximal operator equals 1 in all dimensions.
- The Bellman function exhibits a quasi-periodic structure depending on $ ext{α}$.
- Explicit decay rates of the norm with respect to function averages are provided.

## Abstract

We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural operator equals 1 for all $n,$ and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called $\alpha$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $\alpha,$ but also allows a majorant independent of $\alpha,$ hence the dimension-free norm constant. We also describe the decay of the norm with respect to the difference between the average of a function on a cube and the infimum of its maximal function on that cube. An explicit norm-optimizing sequence is constructed.