  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.
