Weak-type estimates in Morrey spaces for maximal commutator and commutator of maximal function

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 ${\mathcal{W}\text{\hspace{0.17em}⁣}\mathcal{M}}_{L(\log L),\lambda }$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b \in BMO({{\mathbb R}}^n)$suchthat$b^- \in L{\infty}({{\mathbb R}}^n)$ is bounded from ${\mathcal{M}}_{L(\log L),\lambda }$ to ${\mathcal {W ! M}}{L(\log L),\lambda}$\mathrm{.}Newpointwisecharacterizationsof$M{\alpha} M$ by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.