In this paper it is shown that the Hardy-Littlewood maximal operator M is not bounded on Zygmund-Morrey space M L(log L),λ , but M is still bounded on M L(log L),λ for radially decreasing functions. The boundedness of the iterated maximal operator M 2 from M L(log L),λ to weak Zygmund-Morrey space WM L(log L),λ is proved. The class of functions for which the maximal commutator C b is bounded from M L(log L),λ to WM L(log L),λ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator M with function b ∈ BMO(R n ) such that b − ∈ L ∞ (R n ) is bounded from M L(log L),λ to WM L(log L),λ . New pointwise characterizations of M α M by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.
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