The trilinear embedding theorem

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. In this paper we give a characterization of the trilinear embedding theorem. That is, we give a characterization of the inequality $$\sum _{Q\in \mathcal{D}}K(Q)\prod _{i=1}^3\left|{\int }_Q{f}_{i}d{\sigma }_i\right|\le C\prod _{i=1}^3\Vert f_i{\Vert }_{L^{p_i}(d{\sigma }_i)}$$ in terms of discrete Wolff's potential and Sawyer's checking condition, when $1<p_1,p_2,p_3<\infty$ and $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}\ge 1$.

References in corpus (2)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I
• The A_2 theorem: Remarks and complements

Cited by in corpus (1)

• The $n$ linear embedding theorem