COMMUTATORS OF THE MAXIMAL FUNCTIONS ON BANACH FUNCTION SPACES

Let M and M-# be Hardy-Littlewood maximal operator and sharp maximal operator, respectively. In this article, we present neces-sary and sufficient conditions for the boundedness properties for commu-tator operators [M, b] and [M-#, b] in a general context of Banach func-tion spaces when b belongs to BMO(R-n) spaces. Some applications of the results on weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces and Musielak-Orlicz spaces are also given.