Sharp superlevel set estimates for small cap decouplings of the parabola

TL;DR

This paper establishes sharp bounds for superlevel sets of functions with Fourier support near the parabola, leading to improved decoupling theorems and inequalities in harmonic analysis.

Contribution

It provides the first sharp superlevel set estimates for small cap decouplings of the parabola, enhancing existing decoupling results and inequalities.

Findings

Proved sharp bounds for superlevel sets of functions near the parabola.

Derived small cap decoupling theorems for the parabola.

Established new $(\,ell^q,L^p)$ decoupling inequalities.

Abstract

We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>\alpha\}$ where $\alpha>0$ and $f:\mathbb{R}^2\to\mathbb{C}$ is a Schwartz function with Fourier transform supported in an $R^{-1}$-neighborhood of the truncated parabola $\mathbb{P}^1$. These estimates imply the small cap decoupling theorem for $\mathbb{P}^1$ of Demeter, Guth, and Wang, and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter. New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.