The natural maximal operator on BMO (Q2723487)

The author introduces the \textit{natural maximal operator} \(M^{\#}f(x)= \sup_{Q\ni x}{1\over|Q|} \int_Qf\), a generalization of the Hardy-Littlewood maximal operator \(M\), and shows that it is a bounded operator from functions of bounded mean oscillation (BMO) into functions of bounded lower oscillation (BLO).NEWLINENEWLINENEWLINEThe boundedness of the operator \(M\) from BMO into BLO is then obtained as an immediate consequence. So, a different proof from that given by \textit{F. Chiarenza} and \textit{M. Frasca} [Rend. Mat. Appl., VII. Ser. 7, No. 3/4, 273-279 (1987; Zbl 0717.42023)] is established.NEWLINENEWLINENEWLINEIn the proof of this result it is used the fact that \(M^{\#}\) commutes pointwise (in some sense) with the logarithm on \(A^\infty\). Moreover, this commutation lemma allows the author to prove a precise equivalence between the boundedness of \(M^{\#}\) on BMO and that \(M(A^\infty)\subset A^1\). In this way, the author clarifies the relationship shown by \textit{D. Cruz-Uribe} and \textit{C. J. Neugebauer} [Trans. Am. Math. Soc. 347, No. 8, 2941-2960 (1963; Zbl 0851.42016)] of the behaviour of \(M\) on \(A^\infty\) to the fact that \(M:\text{BMO}\to \text{BLO}\).