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Abstract

Let $\sigma_i$, $i=1,2,3$, denote positive Borel measures on $\mathbb R^n$, let ${\mathcal D}$ denote the usual collection of dyadic cubes in $\mathbb R^n$ and let $K:\,{\mathcal D}\to[0,\infty)$ be a map. We give a characterization of a trilinear embedding theorem, that is, of the inequality $$ \sum_{Q\in{\mathcal D}} K(Q)\prod_{i=1}^3\left|\int_{Q}f_i\,d\sigma_i\right| \le C \prod_{i=1}^3 \|f_i\|_{L^{p_i}(d\sigma_i)} $$ in terms of a discrete Wolff potential and Sawyer's checking condition, when $1< p_1,p_2,p_3 < \infty$ and $1/{p_1}+1/{p_2}+1/{p_3}\ge 1$.