  The purpose of this paper is to introduce and study the following graph
theoretic paradigm. Let $$T_Kf(x)=\int K(x,y) f(y) d\mu(y),$$ where $f: X \to
{\Bbb R}$, $X$ a set, finite or infinite, and $K$ and $\mu$ denote a suitable
kernel and a measure, respectively. Given a connected ordered graph $G$ on $n$
vertices, consider the multi-linear form $$ \Lambda_G(f_1,f_2, \dots,
f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)}
K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l), $$ where ${\mathcal E}(G)$ is the
edge set of $G$. Define $\Lambda_G(p_1, \ldots, p_n)$ as the smallest constant
$C>0$ such that the inequality $$ \Lambda_G(f_1, \dots, f_n) \leq C
\prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)}$$ holds for all non-negative
real-valued functions $f_i$, $1\le i\le n$, on $X$. The basic question is, how
does the structure of $G$ and the mapping properties of the operator $T_K$
influence the sharp exponents. In this paper, this question is investigated
mainly in the case $X={\Bbb F}_q^d$, the $d$-dimensional vector space over the
field with $q$ elements, and $K(x^i,x^j)$ is the indicator function of the
sphere evaluated at $x^i-x^j$. This provides a connection with the study of
$L^p$-improving measures and distance set problems.

Classical Analysis and ODEs