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

TL;DR

The paper studies the finite-field restriction problem for spheres and introduces the $\mathcal{S}$-operator, which links functions on $\mathbb{F}_q^d$ to $\mathbb{F}_q^{d+1}$ and connects to sphere restriction estimates. It proves that the $L^2$ restriction conjectures for spheres hold in all dimensions when test functions are homogeneous of degree zero, by reducing the problem to $j$-homogeneous varieties $H_j^d$ in $\mathbb{F}_q^{d+1}$ and deriving sharp $L^2$ bounds via explicit Fourier analysis and Gauss-sum techniques. The work combines a reduction lemma, Gauss-sum calculations, and a three-case analysis to obtain the $L^2$ bounds for $H_j^d$, which in turn yield the desired sphere restriction results. This approach provides structural insights into finite-field restriction phenomena and has potential implications for related problems such as distance sets in finite fields.

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.

Theorems & Definitions (27)

• Definition 1.1
• Definition 1.2: Restriction problem for spheres
• Conjecture 1.3
• Conjecture 1.4
• Theorem 1.5
• Lemma 2.1
• Conjecture 2.2
• Lemma 2.3
• Definition 3.1: The $\mathcal{S}$-operator
• Lemma 3.2