Reinforcing a Philosophy: Littlewood--Paley theory for the moment curve over generic local fields

TL;DR

This paper presents new proofs for square function estimates related to the moment curve over generic local fields, employing algebraic and geometric methods to improve understanding of harmonic analysis in these settings.

Contribution

It introduces simplified and sharper proofs for isotropic square function estimates using algebraic geometry and polynomial curve analysis.

Findings

Established isotropic square function estimates over generic local fields

Provided sharper proofs for real and complex polynomial curves

Enhanced understanding of harmonic analysis techniques in algebraic settings

Abstract

Using the Girard--Newton formulae, I give a simple proof of isotropic square function estimates for extension operators along the moment curve in generic local fields. Using Bezout's Theorem and the Implicit Function Theorem, I give an alternate, sharper proof for real or complex non-degenerate, polynomial curves.