The n Linear Embedding Theorem

Abstract

Let σ _i , i = 1,…, 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

$$\sum\limits_{Q\in\mathcal{D}} K(Q)\prod\limits_{i=1}^{n}\left|{\int}_{Q}f_{i}\,d{\sigma}_{i}\right| \le C\prod\limits_{i=1}^{n} \|f_{i}\|_{L^{p_{i}}(d{\sigma}_{i})} $$

in terms of the multilinear Sawyer testing conditions and the n weight discrete Wolff potential conditions, when 1 < p _i < ∞.