Van der Corput lemma (harmonic analysis)

In mathematics, in the field of harmonic analysis, the van der Corput lemma is an estimate for oscillatory integrals named after the Dutch mathematician J. G. van der Corput.

The following result is stated by E. Stein:^[1]

Suppose that a real-valued function ${\displaystyle \phi (x)}$ is smooth in an open interval ${\displaystyle (a,b)}$, and that ${\displaystyle |\phi ^{(k)}(x)|\geq 1}$ for all ${\displaystyle x\in (a,b)}$. Assume that either ${\displaystyle k\geq 2}$, or that ${\displaystyle k=1}$ and ${\displaystyle \phi '(x)}$ is monotone for ${\displaystyle x\in \mathbb {R} }$. Then there is a constant ${\displaystyle c_{k}}$, which does not depend on ${\displaystyle \phi }$, such that

${\displaystyle {\bigg |}\int _{a}^{b}e^{i\lambda \phi (x)}{\bigg |}\leq c_{k}\lambda ^{-1/k}}$

for any ${\displaystyle \lambda \in \mathbb {R} }$.