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

is smooth in an open interval

Assume that either

is monotone for

Then there is a constant

, which does not depend on

λ ∈

The van der Corput lemma is closely related to the sublevel set estimates,[2] which give the upper bound on the measure of the set where a function takes values not larger than

Suppose that a real-valued function

is smooth on a finite or infinite interval

There is a constant

, which does not depend on

ϵ ≥ 0

the measure of the sublevel set

≤ ϵ }