The $n$ linear embedding theorem

TL;DR

This paper characterizes the conditions under which a multilinear embedding inequality involving measures, dyadic cubes, and a kernel function holds, using multilinear Sawyer's condition and Wolff's potential.

Contribution

It provides a new characterization of the $n$ linear embedding theorem through multilinear Sawyer's condition and Wolff's potential, extending previous results.

Findings

Characterization of the inequality in terms of multilinear Sawyer's condition.

Use of discrete multinonlinear Wolff's potential for the characterization.

Applicable for measures and functions with $1<p_i< finite$.

Abstract

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 $1<p_i<\infty$.