标题： dyadic rectangles 的$n$线性嵌入定理 显示英文标题

标题： The $n$-linear embedding theorem for dyadic rectangles

摘要： 设 $\sg_i$, $i=1,\ldots,n$表示在 $\R^d$上的反向加倍权，令 $\cdr(\R^d)$表示所有在 $\R^d$上的二进矩形（通常二进区间的笛卡尔积）的集合，令 $K:\,\cdr(\R^d)\to[0,\8)$是一个映射。 在本文中，我们给出了关于二进矩形的 $n$-线性嵌入定理。 也就是说，我们证明了对于二进制矩形 \[ \sum_{R\in\cdr(\R^d)} K(R)\prod_{i=1}^n\lt|\int_{R}f_i\,{\rm d}\sg_i\rt| \le C \prod_{i=1}^n \|f_i\|_{L^{p_i}(\sg_i)} \]，$n$-线性嵌入不等式可以在范围 $1<p_i<\8$ 和 $\sum_{i=1}^n\frac{1}{p_i}>1$内通过简单的测试条件 \[ K(R)\prod_{i=1}^n\sg_i(R) \le C \prod_{i=1}^n\sg_i(R)^{\frac{1}{p_i}} \quad R\in\cdr(\R^d), \]来表征。 作为该定理的一个推论，对于反向倍增权函数，我们验证了多线性强正二进制算子和强分数积分算子的加权范数不等式成立的必要且充分条件。 显示英文摘要

摘要： Let $\sg_i$, $i=1,\ldots,n$, denote reverse doubling weights on $\R^d$, let $\cdr(\R^d)$ denote the set of all dyadic rectangles on $\R^d$ (Cartesian products of usual dyadic intervals) and let $K:\,\cdr(\R^d)\to[0,\8)$ be a~map. In this paper we give the $n$-linear embedding theorem for dyadic rectangles. That is, we prove the $n$-linear embedding inequality for dyadic rectangles \[ \sum_{R\in\cdr(\R^d)} K(R)\prod_{i=1}^n\lt|\int_{R}f_i\,{\rm d}\sg_i\rt| \le C \prod_{i=1}^n \|f_i\|_{L^{p_i}(\sg_i)} \] can be characterized by simple testing condition \[ K(R)\prod_{i=1}^n\sg_i(R) \le C \prod_{i=1}^n\sg_i(R)^{\frac{1}{p_i}} \quad R\in\cdr(\R^d), \] in the range $1<p_i<\8$ and $\sum_{i=1}^n\frac{1}{p_i}>1$. As a~corollary to this theorem, for reverse doubling weights, we verify a~necessary and sufficient condition for which the weighted norm inequality for the multilinear strong positive dyadic operator and for strong fractional integral operator to hold.