The $n$ linear embedding theorem

Abstract

Let ${\sigma }_i$, $i=1,\dots ,n$, 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 $$\sum _{Q\in \mathcal{D}}K(Q)\prod _{i=1}^n\left|{\int }_Q{f}_{i}d{\sigma }_i\right|\le C\prod _{i=1}^n\Vert f_i{\Vert }_{L^{p_i}(d{\sigma }_i)}$$ in terms of multilinear Sawyer's checking condition and discrete multinonlinear Wolff's potential, when $1<p_i<\infty$.

References in corpus (2)

• The A_2 theorem: Remarks and complements
• The trilinear embedding theorem