Acta Mathematica Scientia ( IF 1.2 ) Pub Date : 2021-01-29 , DOI: 10.1007/s10473-021-0219-9 Yali Pan , Qingying Xue
Let 0 < β < 1 and Ω be a proper open and non-empty subset of Rn. In this paper, the object of our investigation is the multilinear local maximal operator \({{\cal M}_\beta }\), defined by
$${{\cal M}_\beta }\left( {\overrightarrow f } \right)\left( x \right) = \mathop {\sup }\limits_{\matrix{{Q \ni x} \cr {Q \in {{\cal F}_\beta }} \cr } } \prod\limits_{i = 1}^m {{1 \over {\left| Q \right|}}\int_Q {\left| {{f_i}\left( {{y_i}} \right)} \right|} {\rm{d}}{y_i},} $$where \({\mathcal{F}_\beta } = \left\{ {Q\left( {x,l} \right):x \in \Omega ,l < \beta d\left( {x,{\Omega ^c}} \right)} \right\},Q = Q\left( {x,l} \right)\) 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 \({{\cal M}_\beta }\) are obtained. A formulation of the Carleson embedding theorem in the multilinear setting is proved.
中文翻译：
多线性局部极大算子的两个权重刻画
令0 < β <1和Ω是R n的适当的开放和非空子集。在本文中，我们调查的对象是多分支当地最大运营商\（{{\ CAL中号} _ \测试} \） ，通过定义
$$ {{\ cal M} _ \ beta} \ left（{\ overrightarrow f} \ right）\ left（x \ right）= \ mathop {\ sup} \ limits _ {\ matrix {{Q \ ni x} \ cr {Q \ in {{\ cal F} _ \ beta}} \ cr}} \ prod \ limits_ {i = 1} ^ m {{1 \ over {\ left | Q \ right |}} \ int_Q {\ left | {{f_i} \ left（{{y_i}} \ right）} \ right |} {\ rm {d}} {y_i}，} $$其中\（{\ mathcal {F} _ \ beta} = \ left \ {{Q \ left（{x，l} \ right）：x \ in \ Omega，l <\ beta d \ left（{x，{ \ Omega ^ c}} \ right）} \ right \}，Q = Q \ left（{x，l} \ right）\）表示为边与轴平行的立方体，x和l表示其中心半边长 获得了多线性局部极大算子\（{{\ cal M} _ \ beta} \）的二重特征。证明了在多线性环境中Carleson嵌入定理的一个公式。
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