Sharp superlevel set estimates for small cap decouplings of the parabola

Abstract

We prove sharp bounds for the size of superlevel sets {x ∈ R : |f(x)| > α} where α > 0 and f : R → C is a Schwartz function with Fourier transform supported in an R-neighborhood of the truncated parabola P. These estimates imply the small cap decoupling theorem for P from [DGW20] and the canonical decoupling theorem for P from [BD15]. New (l, L) small cap decoupling inequalities also follow from our sharp level set estimates. In this paper, we further develop the high/low frequency proof of decoupling for the parabola [GMW20] to prove sharp level set estimates which recover and refine the small cap decoupling results for the parabola in [DGW20]. We begin by describing the problem and our results in terms of exponential sums. The main results in full generality are in §1. For N ≥ 1, R ∈ [N,N2], and 2 ≤ p, let D(N,R, p) denote the smallest constant so that