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June 2018 Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function
Müjdat AǦCAYAZI, Amiran GOGATISHVILI, Rza MUSTAFAYEV
Tokyo J. Math. 41(1): 193-218 (June 2018). DOI: 10.3836/tjm/1502179258

Abstract

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

Citation

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Müjdat AǦCAYAZI. Amiran GOGATISHVILI. Rza MUSTAFAYEV. "Weak-type Estimates in Morrey Spaces for Maximal Commutator and Commutator of Maximal Function." Tokyo J. Math. 41 (1) 193 - 218, June 2018. https://doi.org/10.3836/tjm/1502179258

Information

Published: June 2018
First available in Project Euclid: 18 December 2017

zbMATH: 06966864
MathSciNet: MR3830814
Digital Object Identifier: 10.3836/tjm/1502179258

Subjects:
Primary: 42B25
Secondary: 42B35

Rights: Copyright © 2018 Publication Committee for the Tokyo Journal of Mathematics

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Vol.41 • No. 1 • June 2018
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