Sharp van der Corput estimates and minimal divided differences

TL;DR

This work establishes sharp constants in sublevel set estimates and van der Corput-type lemmas for oscillatory integrals. It introduces a sharp $C_n=(n!2^{2n-1})^{1/n}$ bound for sublevel sets under $|f^{(n)}|\ge\lambda$, derived via Chebyshev polynomials and a complex mean-value framework, and proves the Chebyshev extrema uniquely minimize nth divided differences. A complex second mean-value theorem for integrals is developed, enabling precise oscillatory integral bounds such as $|\int_a^b e^{if(x)}dx|\le 2/\lambda$ when $f'\ge\lambda>0$, and related Fourier- transform estimates for increasing $f$. Finally, the van der Corput lemma is sharpened with asymptotically optimal constants $C_n\to 4/e$ as $n\to\infty$, with improved $n=2$ bounds and discussion of polynomial-phase scenarios, providing tighter, near-optimal oscillatory integral estimates useful in harmonic analysis.

Abstract

We find the nodes that minimise divided differences and use them to find the sharp constant in a sublevel set estimate. We also find the sharp constant in the first instance of the van der Corput Lemma using a complex mean value theorem for integrals. With these sharp bounds we improve the constant in the general van der Corput Lemma, so that it is asymptotically sharp.

Theorems & Definitions (17)

• Lemma 1
• Lemma 2
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Corollary 5
• proof
• Lemma 6