The $n$-linear embedding theorem for dyadic rectangles

Abstract

Let $\sg_i$, $i=1,\dots ,n$, denote reverse doubling weights on ${\mathbb{R}}^d$, let $\cdr(\R^d)$ denote the set of all dyadic rectangles on ${\mathbb{R}}^d$ (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.