TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

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TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

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doi:10.1007/s10473-021-0219-9

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (2): 596-608.doi: 10.1007/s10473-021-0219-9

潘亚丽^1,2, 薛庆营³

1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China;
2. Department of Mathematics, School of Information, Huaibei Normal University, Huaibei 235000, China;
3. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

收稿日期:2020-03-05 修回日期:2020-09-19 出版日期:2021-04-25 发布日期:2021-04-29
通讯作者: Qingying XUE E-mail:qyxue@bnu.edu.cn
作者简介:Yali PAN,E-mail:yalipan@zjnu.edu.cn
基金资助:
The first author was supported partly by the Natural Science Foundation from the Education Department of Anhui Province (KJ2017A847). The second author was supported by NSFC (11671039, 11871101) and NSFC-DFG (11761131002).

Yali PAN^1,2, Qingying XUE³

1. College of Mathematics and Computer Science, Zhejiang Normal University, Jinhua 321004, China;
2. Department of Mathematics, School of Information, Huaibei Normal University, Huaibei 235000, China;
3. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received:2020-03-05 Revised:2020-09-19 Online:2021-04-25 Published:2021-04-29
Contact: Qingying XUE E-mail:qyxue@bnu.edu.cn
About author:Yali PAN,E-mail:yalipan@zjnu.edu.cn
Supported by:
The first author was supported partly by the Natural Science Foundation from the Education Department of Anhui Province (KJ2017A847). The second author was supported by NSFC (11671039, 11871101) and NSFC-DFG (11761131002).

摘要： Let $0<\beta <1$ and $\Omega$ be a proper open and non-empty subset of $\mathbf{R}^n$ . In this paper, the object of our investigation is the multilinear local maximal operator $\mathcal{M}_{\beta}$ , defined by

$\mathcal{M}_{\beta}(\vec{f})(x)= \sup_{\substack{Q \ni x \\ Q\in{\mathcal{F}_{\beta}}}} \prod_{i=1}^m \frac{1}{|Q|} \int_Q |f_i(y_i)|{\rm d}y_i,$ where

$\mathcal{F}_{\beta}=\{Q(x,l):x \in \Omega, l< \beta {\rm d}(x, \Omega^c)\}$ ,

$Q=Q(x,l)$ is denoted as a cube with sides parallel to the axes, and

$x$ and

$l$ denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator

$\mathcal{M}_{\beta}$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.

关键词: Multilinear local maximal operators, $A_{(\vec{p},q)}^{\beta}$ weights, two-weight inequalities

Abstract: Let $0<\beta <1$ and $\Omega$ be a proper open and non-empty subset of $\mathbf{R}^n$ . In this paper, the object of our investigation is the multilinear local maximal operator $\mathcal{M}_{\beta}$ , defined by

$\mathcal{M}_{\beta}(\vec{f})(x)= \sup_{\substack{Q \ni x \\ Q\in{\mathcal{F}_{\beta}}}} \prod_{i=1}^m \frac{1}{|Q|} \int_Q |f_i(y_i)|{\rm d}y_i,$ where

$\mathcal{F}_{\beta}=\{Q(x,l):x \in \Omega, l< \beta {\rm d}(x, \Omega^c)\}$ ,

$Q=Q(x,l)$ is denoted as a cube with sides parallel to the axes, and

$x$ and

$l$ denote its center and half its side length. Two-weight characterizations for the multilinear local maximal operator

$\mathcal{M}_{\beta}$ are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.

Key words: Multilinear local maximal operators, $A_{(\vec{p},q)}^{\beta}$ weights, two-weight inequalities

中图分类号:

42B20

潘亚丽, 薛庆营. TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS[J]. 数学物理学报(英文版), 2021, 41(2): 596-608.

Yali PAN, Qingying XUE. TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS[J]. Acta mathematica scientia,Series B, 2021, 41(2): 596-608.

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