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We introduce a generalization of the Hardy-Littlewood maximal operator, the natural maximal operator M , in some sense the maximal operator which most naturally commutes pointwise with the logarithm on A ∞ . This commutation reveals the behavior of M : A ∞ → A 1 to directly correspond to that of M : BMO → BLO; the boundedness of M : BMO → BLO is an immediate consequence.doi:10.1090/s0002-9939-01-05896-8 fatcat:iuvrty2vbvhjzdt7qcpcg3zhna