The $\mathrm{BMO}\rightarrow\mathrm{BLO}$ action of the maximal operator on $\alpha$-trees

The explicit upper Bellman function is found for the natural dyadic maximal operator acting from $\mathrm {BMO}(\mathbb{R}^n)$ into $\mathrm {BLO}(\mathbb{R}^n)$. As a consequence, it is shown that the $\mathrm {BMO}\to \mathrm {BLO}$ 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 consequence of a theorem for the so-called $\alpha$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasiperiodic structure depending on $\alpha$, but also allows a majorant independent of $\alpha$, hence a dimension-free norm constant. Also, the decay of the norm is described with respect to the growth of 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.