Restriction of averaging operators to algebraic varieties over finite fields

TL;DR

This paper investigates $L^p$ to $L^r$ bounds for averaging operators restricted to algebraic varieties over finite fields, providing optimal estimates for spheres, paraboloids, and cones in various dimensions.

Contribution

It establishes sharp $L^p$ to $L^r$ estimates for restricted averaging operators on algebraic varieties over finite fields, including spheres, paraboloids, and cones, with some endpoint exceptions.

Findings

Optimal $L^p$ to $L^r$ estimates for spheres and paraboloids in dimensions $d\\ge2$

Sharp estimates for cones in odd dimensions $d\\ge 3$

Almost sharp estimates for cones in even dimensions with specific conditions

Abstract

We study $L^p\to L^r$ estimates for restricted averaging operators related to algebraic varieties $V$ of $d$-dimensional vector spaces over finite fields $\mathbb F_q$ with $q$ elements. We observe properties of both the Fourier restriction operator and the averaging operator over $V\subset \mathbb F_q^d.$ As a consequence, we obtain optimal results on the restricted averaging problems for spheres and paraboloids in dimensions $d\ge2,$ and cones in odd dimensions $d\ge 3.$ In addition, when the variety $V$ is a cone lying in an even dimensional vector space over $\mathbb F_q$ and $-1$ is a square number in $\mathbb F_q$, we also obtain sharp estimates except for two endpoints.