Some estimates for the commutators of multilinear maximal function on Morrey-type space
type: Item
instance of: scholarly article
title: Some estimates for the commutators of multilinear maximal function on Morrey-type space
zbMATH Open document ID: 1493.42025
author: Pu Zhang
author: Hongliang Li
author: Xiao Yu
published in: Open Mathematics
publication date: 2021-11-29T00:00:00Z
review text: From the abstract: ``We study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.'' The two main results of the paper are Theorem 1.1 and 1.3. They read as follows: Theorem 1.1 Let \(b_{i}\) be a real-valued, locally integrable function in \(\mathbb{R}^{n}\) with \(i=1, \ldots, m\). Then, the following three assertions are equivalent. (i) The commutator \([\vec{b}, \mathcal{M}](\vec{f})(x)\) is bounded from \(L^{p_{1}, \lambda}\left(\mathbb{R}^{n}\right) \times \cdots \times L^{p_{m}, \lambda}\left(\mathbb{R}^{n}\right)\) to \(L^{p, \lambda}\left(\mathbb{R}^{n}\right)\) for \(1<p_{i}<\infty\) and \(1 / p=\sum_{i=1}^{m} 1 / p_{i}\) with \(p>1\) and \(0 \leq \lambda<n\). (ii) \(b_{i}\) is in \(B M O\) and \(b_{i}^{-}\)belongs to \(L^{\infty}\). (iii) For \(p \in(1, \infty)\), there is \[ \sup _{Q} \frac{1}{|Q|} \int_{Q}\left|b_{i}(x)-M_{Q}\left(b_{i}\right)(x)\right|^{p} \mathrm{~d} x<\infty \] where \(M_{Q}\left(b_{i}\right)(x)=\sup _{Q_{0} \ni x, Q_{0} \subset Q} \frac{1}{\left|Q_{0}\right|} \int_{Q_{0}}\left|b_{i}(t)\right| \mathrm{d} t\). and Theorem 1.3 Let \(b_{i} \in \operatorname{BMO}\left(\mathbb{R}^{n}\right)\) and \(\omega\) satisfy \( \omega(a, 2 t) \leq C_{0} \omega(a, t) \) with \(1<C_{0} \leq 2^{y}\) for any \(0<\gamma<n\). Then, for any \(\beta>0\) and cube \(B=B(x, t)\) with center \(x \in \mathbb{R}^{n}\) and radius \(t>0\), there is \[ \frac{1}{\omega(B)^{m}}\left|\left\{z \in B(x, t): \mathcal{M}_{\vec{b}}(\vec{f})(z)>\beta^{m}\right\}\right|^{m} \leq C\|\vec{b}\|_{\mathrm{BMO}} \prod_{i=1}^{m}\left\|\Phi\left(\frac{\left|f_{i}\right|}{\beta}\right)\right\|_{L_{L \log L}^{1, \omega}\left(\mathbb{R}^{n}\right)}, \] where \(\Phi(x)=t\left(1+\log ^{+} t\right),\|\vec{b}\|_{\mathrm{BMO}}=\sup _{1 \leq i \leq m}\left\|b_{i}\right\|_{\math…
reviewed by: Humberto Rafeiro
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 7435663
zbMATH Keywords: Morrey-type space
zbMATH Keywords: multilinear maximal function
zbMATH Keywords: commutator
zbMATH Keywords: Orlicz function
MaRDI profile type: Publication
full work available at URL: math-2021-0031
OpenAlex ID: W3181718962
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DOI: 10.1515/MATH-2021-0031
description: scientific article; zbMATH DE number 7435663