A sharp square function estimate for the moment curve in $\mathbb{R}^n$

TL;DR

This paper establishes sharp square function estimates for the moment curve in high-dimensional space using advanced decoupling techniques, improving understanding of oscillatory integral behavior.

Contribution

It introduces a novel inductive approach that combines decoupling with auxiliary conical set estimates to achieve sharp bounds for the moment curve in $R^n$.

Findings

Proves sharp square function estimates for the moment curve.

Develops an inductive scheme leveraging lower-dimensional information.

Utilizes high-low frequency methods from decoupling theory.

Abstract

We use high-low frequency methods developed in the context of decoupling to prove sharp (up to $C_\epsilon R^\epsilon$) square function estimates for the moment curve $(t,t^2,\ldots,t^n)$ in $\mathbb{R}^n$. Our inductive scheme incorporates sharp square function estimates for auxiliary conical sets, which allows us to fully exploit lower dimensional information.