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

TL;DR

This paper develops new estimates for square functions associated with polynomial curves over local fields, using a systematic analysis of second order differencing polynomials and their geometric properties.

Contribution

It introduces a novel counting approach for square functions over local fields, providing uniform estimates for polynomial curves of degree at least 2.

Findings

New bounds for square functions on polynomial curves

Uniform estimates over local fields with certain characteristics

Analysis of second order differencing polynomials in local fields

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 $(T,\phi(T))$ where $\phi(T)$ is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials $\phi(T) = T^k$ 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.