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

TL;DR

This paper investigates the boundedness properties of maximal operators and their commutators in Morrey and Zygmund-Morrey spaces, revealing new boundedness results and characterizations for specific classes of functions.

Contribution

It establishes new boundedness results for maximal operators and their commutators in Morrey-type spaces, including conditions involving BMO functions and radially decreasing functions.

Findings

Maximal operator not bounded on Zygmund-Morrey space for general functions.

Boundedness of $M^2$ from Morrey to weak Morrey space.

Characterization of functions for boundedness of maximal commutator.

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 ${\mathcal {W \! M}}_{L(\log L),\lambda}$ 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.