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Multi-linear forms, graphs, and 
$L^p$-improving measures in 
${\Bbb F}_q^d$
Published online by Cambridge University Press: 27 December 2023
Abstract
The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let where 
$$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$
$f: X \to {\Bbb R}$, 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 where 
$$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$
${\mathcal E}(G)$ is the edge set of G. Define 
$\Lambda _G(p_1, \ldots , p_n)$ as the smallest constant 
$C>0$ such that the inequality (0.1)holds for all nonnegative real-valued functions 
$$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$
$f_i$, 
$1\le i\le n$, 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 (0.1). 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, 
$K(x^i,x^j)$ is the indicator function of the sphere evaluated at 
$x^i-x^j$, and connected graphs G with at most four vertices.
MSC classification
- Type
- Article
- Information
- Copyright
- © The Author(s), 2023. Published by Cambridge University Press on behalf of Canadian Mathematical Society
Footnotes
A. Iosevich and P. Bhowmik were supported in part by the National Science Foundation (Grant No. HDR TRIPODS-1934962) and the National Science Foundation (Grant No. DMS-2154232). D. Koh was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (Grant No. RS-2023-00249597). T. Pham was supported by the research project QG.23.03 of Vietnam National University, Hanoi.
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