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Mapping properties of the $S$-operator

Hunseok Kang, Doowon Koh, Changhun Yang

Abstract

In this paper, we study the $\ell^p\to \ell^r$ estimates for the $S$-operator arising in restriction problems for spheres over finite fields. We establish a necessary and sufficient condition for the boundedness of the $S$-operator. Furthermore, we investigate this problem under certain restrictions on test functions. In particular, we address the sharp results when test functions are restricted to radial functions.

Mapping properties of the $S$-operator

Abstract

In this paper, we study the estimates for the -operator arising in restriction problems for spheres over finite fields. We establish a necessary and sufficient condition for the boundedness of the -operator. Furthermore, we investigate this problem under certain restrictions on test functions. In particular, we address the sharp results when test functions are restricted to radial functions.
Paper Structure (17 sections, 13 theorems, 96 equations, 1 figure)

This paper contains 17 sections, 13 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Historical background of finite field restriction theory
  3. Development of spherical restriction theory over finite fields
  4. Application of finite spherical restriction problems to the Erdős-Falconer distance problem
  5. Main problem and its motivation
  6. Statement of main results
  7. Results on the $\mathcal{S}$-operator problem
  8. Results on the restricted $\mathcal{S}$-operator problem
  9. Structure of the paper
  10. Preliminaries
  11. Sharp result for $\mathcal{S}(p\to s)\lesssim 1$ (proof of Theorem \ref{['MMain']})
  12. Proof of Theorem \ref{['T1']}
  13. Proof of Theorem \ref{['T2']}
  14. Proof of Corollary \ref{['SadCor']}
  15. The $\mathcal{S}$-operator problem for the radial test functions (proof of Theorem \ref{['mainRR']})
  16. ...and 2 more sections

Key Result

Proposition 1.2

Suppose that there exists a constant $C$ independent of $q$ such that for all $j\in \mathbb F_q^*,$ it satisfies that $R_{S_j^{d-1}} \left(p\to 2\right)\leq C$ for some $1\le p\le \infty.$ Then for any $E\subset \mathbb F_q^d, d\ge2,$ we have In particular, if $|E|\ge q^{\frac{dp}{3p-2}},$ then $|\Delta(E)|\gtrsim q.$

Figures (1)

  • Figure 1: Sharp boundedness regions for the $\mathcal{S}$-operator (red shaded) and $\mathcal{S}_{\mathcal{R}_q}$-operator on radial functions (red + blue shaded). Boundaries are optimal: boundedness fails outside these regions. The radial restriction allows extension from $(1, 1/2)$ to $(1, d/(d+1))$.

Theorems & Definitions (19)

  • Conjecture 1.1
  • Proposition 1.2: Lemma 4.1, CKP19
  • Corollary 1.3
  • Definition 1.4: The $\mathcal{S}$-operator
  • Lemma 1.6: Lemma 3.4, KK26
  • Theorem 1.7
  • Remark 1.8
  • Corollary 1.9
  • Remark 1.10
  • Theorem 1.12
  • ...and 9 more