Weak version of restriction estimates for spheres and paraboloids in finite fields

TL;DR

This paper investigates restriction estimates for spheres and paraboloids over finite fields, proving a specific conjectured L^p-L^2 estimate for certain classes of functions using connections to higher-dimensional homogeneous varieties.

Contribution

It establishes the conjectured restriction estimate in finite fields for functions restricted to coordinate or homogeneous functions, linking it to higher-dimensional homogeneous varieties.

Findings

The L^{(2d+4)/(d+4)}-L^2 restriction estimate holds for specific functions.

The result is achieved by connecting restriction phenomena in d and (d+1) dimensions.

The conjecture is verified in particular cases, advancing understanding of restriction estimates in finite fields.

Abstract

We study L^p-L^r restriction estimates for algebraic varieties in d-dimensional vector spaces over finite fields. Unlike the Euclidean case, if the dimension $d$ is even, then it is conjectured that the L^{(2d+2)/(d+3)}-L^2 Stein-Tomas restriction result can be improved to the L^{(2d+4)/(d+4)}-L^2 estimate for both spheres and paraboloids in finite fields. In this paper we show that the conjectured L^p-L^2 restriction estimate holds in the specific case when test functions under consideration are restricted to d-coordinate functions or homogeneous functions of degree zero. To deduce our result, we use the connection between the restriction phenomena for our varieties in $d$ dimensions and those for homogeneous varieties in (d+1)dimensions.