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.

24 pages

References in corpus (1)

• Improved decoupling for the parabola

Cited by in corpus (1)

• Incidence estimates for $\alpha$-dimensional tubes and $\beta$-dimensional balls in ${\mathbb{R}}^2$