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

• 论文 • 上一篇    下一篇

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

潘亚丽1,2, 薛庆营3   

  1. 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).

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

Yali PAN1,2, Qingying XUE3   

  1. 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