Let $\sigma_i$, $i=1,\ldots,n$, denote positive Borel measures on $\mathbb{ R}^d$, let $\mathcal{ D}$ denote the usual collection of dyadic cubes in $\mathbb{ R}^d$ and let $K:\,\mathcal{ D}\to[0,\infty)$ be a~map. In this paper we give a~characterization of the $n$ linear embedding theorem. That is, we give a~characterization of the inequality $$ \sum_{ Q\in\mathcal{ D}} K(Q)\prod_{ i=1}^n\left|\int_{ Q}f_i\,d\sigma_i\right| \le C \prod_{ i=1}^n \|f_i\|_{ L^{ p_i}(d\sigma_i)} $$ in terms of multilinear Sawyer's checking condition and discrete multinonlinear Wolff's potential, when $1<p_i<\infty$.
Categories: math.CA