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:
Suppose that a real-valued function formula_1 is smooth in an open interval formula_2,
and that formula_3 for all formula_4.
Assume that either formula_5, or that
formula_6 and formula_7 is monotone for formula_8.
There is a constant formula_9, which does not depend on formula_10,
such that
for any formula_12.
Sublevel set estimates.
The van der Corput lemma is closely related to the sublevel set estimates
(see for example
which give the upper bound on the measure of the set
where a function takes values not larger than formula_13.
Suppose that a real-valued function formula_1 is smooth
on a finite or infinite interval formula_15,
and that formula_3 for all formula_17.
There is a constant formula_9, which does not depend on formula_10,
such that
for any formula_20
the measure of the sublevel set
formula_21
is bounded by formula_22.