Tokyo Journal of Mathematics

Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function


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In this paper it is shown that the Hardy-Littlewood maximal operator $M$ is not bounded on Zygmund-Morrey space $\mathcal{M}_{L(\log L),\lambda}$, $0 < \lambda < n$, 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 $b \in \text{BMO}(\mathbb{R}^n)$ 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.

Article information

Tokyo J. Math., Volume 41, Number 1 (2018), 193-218.

First available in Project Euclid: 18 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis


GOGATISHVILI, Amiran; MUSTAFAYEV, Rza; AǦCAYAZI, Müjdat. Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function. Tokyo J. Math. 41 (2018), no. 1, 193--218.

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