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Multi-linear forms, graphs, and $L^p$-improving measures in ${Bbb F}_q^d$ | | Pablo Bhowmick
; Alex Iosevich
; Doowon Koh
; Thang Pham
; | | Date: |
1 Jan 2023 | | Abstract: | The purpose of this paper is to introduce and study the following graph
theoretic paradigm. Let $$T_Kf(x)=int K(x,y) f(y) dmu(y),$$ where $f: X o
{Bbb R}$, $X$ 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 $$ Lambda_G(f_1,f_2, dots,
f_n)=int_{x^1, dots, x^n in X} prod_{(i,j) in {mathcal E}(G)}
K(x^i,x^j) prod_{l=1}^n f_l(x^l) dmu(x^l), $$ 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 $f_i$, $1le ile n$, 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. | | Source: | arXiv, 2301.00463 | | Services: | Forum | Review | PDF | Favorites | |
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