Maximal operators and BMO for Banach lattices

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = infinity. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMOX. This turns out to depend strongly on the convexity of the Banach lattice X. We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.