The maximal operator from BMO to BLO on α-trees

The explicit upper Bellman function is found for the natural dyadic maximal operator acting from BMO(Rn) into BLO(Rn). As a consequence, it is shown that the BMO→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 α-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasiperiodic structure depending on α, but also allows a majorant independent of α, 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.