Averaging Operators Over Homogeneous Varieties Over Finite Fields

Koh, Doowon; Shen, Chun-Yen; Shparlinski, Igor E.

doi:10.1007/s12220-015-9595-5

Cited by 2 publications

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References 20 publications

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“…In the finite field setting, Carbery-Stones-Wright [5] initially studied the L p → L r estimates for the averaging operator. Much attention has been given to this problem (for example, see [12,14,16]).…”

Section: It Clearly Follows Thatmentioning

confidence: 99%

Restricted averaging operators to cones over finite fields

2018

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We investigate the sharp L p → L r estimates for the restricted averaging operator A C over the cone C of the d-dimensional vector space F d q over the finite field Fq with q elements. The restricted averaging operator A C for the cone C is defined by the relation that A C f = f * σ| C , where σ denotes the normalized surface measure on the cone C, and f is a complex valued function on the space F d q with the normalized counting measure dx. In the previous work [15], the sharp boundedness of A C was obtained in odd dimensions d ≥ 3 but partial results were only given in even dimensions d ≥ 4. In this paper we prove the optimal estimates in even dimensions d ≥ 6 in the case when the cone C ⊂ F d q contains a d/2 dimensional subspace.

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Section: It Clearly Follows Thatmentioning

confidence: 99%

Restricted averaging operators to cones over finite fields

2018

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Multi-linear forms, graphs, and -improving measures in

Bhowmik,

Iosevich,

Koh

et al. 2023

Can. J. Math.-J. Can. Math.

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The purpose of this paper is to introduce and study the following graph-theoretic paradigm. Let $$ \begin{align*}T_Kf(x)=\int K(x,y) f(y) d\mu(y),\end{align*} $$ 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 $$ \begin{align*}\Lambda_G(f_1,f_2, \dots, f_n)=\int_{x^1, \dots, x^n \in X} \ \prod_{(i,j) \in {\mathcal E}(G)} K(x^i,x^j) \prod_{l=1}^n f_l(x^l) d\mu(x^l),\end{align*} $$ 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 (0.1) $$ \begin{align} \Lambda_G(f_1, \dots, f_n) \leq C \prod_{i=1}^n {||f_i||}_{L^{p_i}(X, \mu)} \end{align} $$ holds for all nonnegative 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 (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.

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Averaging Operators Over Homogeneous Varieties Over Finite Fields

Cited by 2 publications

References 20 publications

Restricted averaging operators to cones over finite fields

Restricted averaging operators to cones over finite fields

Multi-linear forms, graphs, and -improving measures in

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