In this paper it is shown that the Hardy-Littlewood maximal operator $M$ isnot bounded on Zygmund-Morrey space $mathcal{M}_{L(log L),lambda}$, but $M$is still bounded on $mathcal{M}_{L(log L),lambda}$ for radially decreasingfunctions. The boundedness of the iterated maximal operator $M^2$ from$mathcal{M}_{L(log L),lambda}$ to weak Zygmund-Morrey space ${mathcal {W !M}}_{L(log L),lambda}$ is proved. The class of functions for which themaximal commutator $C_b$ is bounded from $mathcal{M}_{L(log L),lambda}$ to${mathcal {W ! M}}_{L(log L),lambda}$ are characterized. It is proved thatthe commutator of the Hardy-Littlewood maximal operator $M$ with function $bin BMO({{mathbb R}}^n)$ such that $b^- in L_{infty}({{mathbb R}}^n)$ isbounded from $mathcal{M}_{L(log L),lambda}$ to ${mathcal {W ! M}}_{L(logL),lambda}$. New pointwise characterizations of $M_{lpha} M$ by means ofnorm of Hardy-Littlewood maximal function in classical Morrey spaces are given.
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