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Reinforcing a Philosophy: A counting approach to square functions over local fields

Kirsti D. Biggs, Julia Brandes, Kevin Hughes

TL;DR

This work develops a field-agnostic counting framework to prove square-function estimates for extension operators along planar curves $\gamma(T)=(T,\phi(T))$ with $\phi$ a polynomial of degree $k\ge 2$, uniformly across local fields of arbitrary characteristic not dividing $k$. Central to the method is Wooley's Second Order Differencing (S.O.D.) approach, encapsulated in the polynomial $\psi_\phi$, and the novel notion of special subvarieties that capture off-diagonal near-solutions, enabling bounds without Kakeya-type hypotheses. The main result yields a uniform $L^4$ bound $\| E_{\mathfrak{o}} f \|_{L^4(W_B)} \le 5\sqrt[4]{k}\,\| S_{\mathcal{P}_{R^{-1}}(\mathfrak{o})} f \|_{L^4(W_B)}$ for balls with diameter $\ge C_\phi R^k$, with extensions to Archimedean, non-Archimedean, and complex settings, including monomial curves $\phi(T)=T^k$ ($k\ge3$) and finite-type cases under curvature conditions. The paper further analyzes the $p$-adic geometry of S.O.D.ing polynomials, showing how torsion and curvature interact differently across local fields, and discusses obstructions to smooth $p$-adic analogues via Rolle-type arguments and interpolation. Overall, the work unifies counting arguments with harmonic-analytic square-function estimates across ground fields, providing new bounds and structural insights for both degenerate and non-degenerate finite-type curves. The results have potential implications for Diophantine counting and restriction-type analyses in broader arithmetic-harmonic contexts.

Abstract

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,φ(T))$ where $φ(T)$ is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials $φ(T) = T^k$ for $k \geq 3$ which are uniform over all local fields whose characteristic is coprime to \(k\). Key to our approach is a systematic analysis of the second order differencing polynomial and its geometry in local fields.

Reinforcing a Philosophy: A counting approach to square functions over local fields

TL;DR

This work develops a field-agnostic counting framework to prove square-function estimates for extension operators along planar curves with a polynomial of degree , uniformly across local fields of arbitrary characteristic not dividing . Central to the method is Wooley's Second Order Differencing (S.O.D.) approach, encapsulated in the polynomial , and the novel notion of special subvarieties that capture off-diagonal near-solutions, enabling bounds without Kakeya-type hypotheses. The main result yields a uniform bound for balls with diameter , with extensions to Archimedean, non-Archimedean, and complex settings, including monomial curves () and finite-type cases under curvature conditions. The paper further analyzes the -adic geometry of S.O.D.ing polynomials, showing how torsion and curvature interact differently across local fields, and discusses obstructions to smooth -adic analogues via Rolle-type arguments and interpolation. Overall, the work unifies counting arguments with harmonic-analytic square-function estimates across ground fields, providing new bounds and structural insights for both degenerate and non-degenerate finite-type curves. The results have potential implications for Diophantine counting and restriction-type analyses in broader arithmetic-harmonic contexts.

Abstract

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form where is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials for which are uniform over all local fields whose characteristic is coprime to . Key to our approach is a systematic analysis of the second order differencing polynomial and its geometry in local fields.
Paper Structure (19 sections, 15 theorems, 70 equations)

This paper contains 19 sections, 15 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Background
  3. An indication of our results
  4. Overview of our approach: The Second Order Differencing Method
  5. Outline of the paper
  6. Set-up for local fields
  7. Non-Archimedean local fields
  8. Archimedean local fields
  9. Smooth localization
  10. Statement of the main theorem
  11. The main proposition
  12. Proof of Theorem \ref{['theorem:main:2D']}
  13. The Rolle of curvature I: Archimedean fields
  14. The proof of Theorem \ref{['theorem:real:2D']}
  15. Weakening the regularity assumption
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $k \geq 2$. Suppose that $\gamma(T) := (T,\phi(T))$ with $\phi(T) \in C^{k}([-1/2,1/2])$ is such that $|\partial^{k}{\phi}(t)| \neq 0$ for all $t$ in $[-1,1]$. There exists a positive constant $C_{\phi}$, depending only on $\phi$, such that for all $R \in \mathbb{N}$ and all balls $B$ of diamete

Theorems & Definitions (30)

  • Theorem 1.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • ...and 20 more