Sharp superlevel set estimates for small cap decouplings of the parabola

Abstract

We prove sharp bounds for the size of superlevel sets ${x \in \mathbb{R}^2\colon |f(x)| >\alpha}$, where $\alpha>0$ and $f\colon\mathbb{R}^2\to\mathbb{C}$ 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 (2020) and the canonical decoupling theorem for $\mathbb{P}^1$ of Bourgain and Demeter (2015). New $(\ell^q,L^p)$ small cap decoupling inequalities also follow from our sharp level set estimates.