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 is smooth in an open interval,
and that for all.
Assume that either, or that
and is monotone for.
There is a constant, which does not depend on,
such that
for any.

Sublevel set estimates

The van der Corput lemma is closely related to the sublevel set estimates
,
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,
and that for all.
There is a constant, which does not depend on,
such that
for any
the measure of the sublevel set
is bounded by.