Restriction of averaging operators to algebraic varieties over finite fields

Doowon Koh, Seongjun Yeom

Published 2016-01-28Version 1

We study the mapping properties of restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. Given an algebraic variety $V \subset \mathbb F_q^d,$ let us denote by $dx$ and $\sigma$ the normalized counting measure on $\mathbb F_q^d$ and the normalized surface measure on $V$, respectively. For functions $f: \mathbb F_q^d \to \mathbb C$, a restricted averaging operator $A_V$ is defined by restricting $f\ast \sigma$ to the variety $V$, that is $A_Vf=f\ast \sigma|_V.$ The main purpose of this paper is to investigate $L^p\to L^r$ estimates of the restricted averaging operator $A_V.$ We relate this problem to both the Fourier restriction problem and the averaging problem over $V\subset \mathbb F_q^d.$ As a consequence, we obtain the optimal results on the restricted averaging problems for regular varieties such as the spheres, the paraboloids for dimensions $d\ge2,$ and the cones for odd dimensions $d\ge 3.$ In addition, when the variety $V$ is the cone lying in an even dimensional vector space over $\mathbb F_q$, we obtain the sharp weak-type estimates for restricted averaging operators $A_V.$