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

Abstract

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 ${\mathcal{W}\mathcal{M}}_{L(\log L),\lambda }$ is proved. The class of functions for which the maximal commutator $C_b$ is bounded from ${\mathcal{M}}_{L(\log L),\lambda }$ to ${\mathcal{W}\mathcal{M}}_{L(\log L),\lambda }$ are characterized. It is proved that the commutator of the Hardy-Littlewood maximal operator $M$ with function $b\in BMO({\mathbb{R}}^n)$ such that $b^{-}\in L_{\infty }({\mathbb{R}}^n)$ is bounded from ${\mathcal{M}}_{L(\log L),\lambda }$ to ${\mathcal{W}\mathcal{M}}_{L(\log L),\lambda }$. New pointwise characterizations of $M_{\alpha }M$ by means of norm of Hardy-Littlewood maximal function in classical Morrey spaces are given.

21 pages

References in corpus (1)

• A note on boundedness of the Hardy-Littlewood maximal operator on Morrey spaces

Cited by in corpus (2)

• Some notes on commutators of the fractional maximal function on variable Lebesgue spaces
• Characterization of the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces