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 (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.