The trilinear embedding theorem

TL;DR

This paper characterizes a trilinear embedding theorem involving measures, dyadic cubes, and a kernel function, using discrete potential theory and checking conditions, extending understanding of multilinear integral inequalities.

Contribution

It provides a new characterization of the trilinear embedding theorem using discrete Wolff's potential and Sawyer's condition, advancing multilinear analysis theory.

Findings

Characterization of the inequality in terms of discrete Wolff's potential.

Use of Sawyer's checking condition for the theorem.

Applicable for exponents with sum of reciprocals at least one.

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 $$ \sum_{Q\in\mathcal{D}} K(Q)\prod_{i=1}^3\left|\int_{Q}f_i\,d\sigma_i\right| \le C \prod_{i=1}^3 \|f_i\|_{L^{p_i}(d\sigma_i)} $$ in terms of discrete Wolff's potential and Sawyer's checking condition, when $1<p_1,p_2,p_3<\infty$ and $\frac1{p_1}+\frac1{p_2}+\frac1{p_3}\ge 1$.