Sharp superlevel set estimates for small cap decouplings of the parabola (Q6172747)

Summary: 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 \textit{C. Demeter} et al. [Geom. Funct. Anal. 30, No. 4, 989--1062 (2020; Zbl 1454.42008)] and the canonical decoupling theorem for \(\mathbb{P}^1\) of \textit{J. Bourgain} and \textit{C. Demeter} [Ann. Math. (2) 182, No. 1, 351--389 (2015; Zbl 1322.42014)]. New \((\ell^q , L^p )\) small cap decoupling inequalities also follow from our sharp level set estimates.