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
(see for example
[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 ,
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 .
References[]
^Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993. ISBN0-691-03216-5
^M. Christ, Hilbert transforms along curves, Ann. of Math. 122 (1985), 575--596