Sharp Van Der Corput Estimates and Minimal Divided Differences

1. Sublevel set estimates The importance of sublevel set estimates in van der Corput lemmas was highlighted by A. Carbery, M. Christ, and J. Wright [3], [4]. We find the sharp constant in the following sublevel set estimate. We will use this in Section 4 to prove an asymptotically sharp van der Corput lemma. The constants Cn take different values in each lemma. Lemma 1. Suppose that f : (a, b) → R is n times differentiable with n ≥ 1 and |f (n)(x)| ≥ λ > 0 on (a, b). Then |{x ∈ (a, b) : |f(x)| ≤ α}| ≤ Cn (α/λ) , where Cn = (n!2 ). We note that Cn ≤ 2n for all n ≥ 1, and by Stirling’s formula, lim n→∞ Cn − 4n/e = 0.