Connections between $\mathcal{S}$-operators and restriction estimates for spheres over finite fields

TL;DR

This paper introduces a new operator related to the restriction problem for spheres over finite fields and establishes a connection between the operator's boundedness and restriction estimates, proving the conjecture for homogeneous functions.

Contribution

The paper defines the $ ext{S}$ operator and links its boundedness to restriction estimates, proving the $L^2$ restriction conjecture for homogeneous functions over finite fields.

Findings

Established the relationship between $ ext{S}$ operator boundedness and restriction estimates.

Proved the $L^2$ restriction conjecture for spheres in all dimensions for homogeneous functions.

Extended the restriction problem understanding in finite field settings.

Abstract

In this paper, we introduce a new operator, $\mathcal{S}$, which is closely related to the restriction problem for spheres in $\mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field $\mathbb{F}_q$ with $q$ elements. The $\mathcal{S}$ operator is considered as a specific operator that maps functions on $\mathbb{F}_q^d$ to functions on $\mathbb{F}_q^{d+1}$. We explore a relationship between the boundedness of the $\mathcal{S}$ operator and the restriction estimate for spheres in $\mathbb{F}_q^d$. Consequently, using this relationship, we prove that the $L^2$ restriction conjectures for spheres hold in all dimensions when the test functions are restricted to homogeneous functions of degree zero.