  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.
