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

Abstract

We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from $\mathrm{BMO}({\mathbb{R}}^n)$ into $\mathrm{BLO}({\mathbb{R}}^n).$ As a consequence, we show 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 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.

32 pages, 4 figures