  In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is
not 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 decreasing
functions. 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 the
maximal 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 that
the commutator of the Hardy-Littlewood maximal operator $M$ with function $b
\in BMO({{\mathbb R}}^n)$ such that $b^- \in L_{\infty}({{\mathbb R}}^n)$ is
bounded from $\mathcal{M}_{L(\log L),\lambda}$ to ${\mathcal {W \! M}}_{L(\log
L),\lambda}$. New pointwise characterizations of $M_{\alpha} M$ by means of
norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.
