Sharp superlevel set estimates for small cap decouplings of the parabola

Abstract

We prove sharp bounds for the size of superlevel sets $\{x\in \mathbb{R}^2:|f(x)|>\alpha\}$ 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.