Multi-linear forms, structure of graphs and Lebesgue spaces

Abstract

Consider the operator $$T_{K}f(x)={\int }_{{\mathbb{R}}^d}K(x,y)f(y)dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$\hat{I}{›}_G^K(f_1,\dots ,f_n)=\int \cdots \int \prod _{\{(i,j):1\le i<j\le n;E(i,j)=1\}}K(x^i,x^j)\prod _{i=1}^n{f}_i(x^i)d{x}^i,$$ where $G$ is a connected graph on $n$ vertices, $E$ is the edge map on $G$, i.e $E(i,j)=1$ if and only if the $i$'th and $j$'th vertices are connected by an edge, $K$ is the aforementioned kernel, and $f_i:{\mathbb{R}}^d\to \mathbb{R}$, measurable. This paper establishes multi-linear inequalities of the form $$\hat{I}{›}_G^K(f_1,f_2,\dots ,f_n)\le C{||f_1||}_{L^{p_1}({\mathbb{R}}^d)}{||f_2||}_{L^{p_2}({\mathbb{R}}^d)}\dots {||f_n||}_{L^{p_n}({\mathbb{R}}^d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$.