Van der Corput lemma (harmonic analysis) explained

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

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

Suppose that a real-valued function

\phi(x) is smooth in an open interval (a, b) ,and that |\phi ^{(k)} (x)|\ge 1 for all x \in (a, b) .Assume that either k \ge 2 , or thatk = 1 and \phi'(x) is monotone for x \in \R .Then there is a constant c _{k} , which does not depend on \phi ,such that| \int ^{b} _{a} e ^{iλ\phi(x)} | \le c ^{-1 /k} _{k λ} for any λ \in \R .

Sublevel set estimates

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

\epsilon .

Suppose that a real-valued function

\phi(x) is smoothon a finite or infinite interval I \subset \R ,and that |\phi ^{(k)} (x)| \ge 1 for all x \in I .There is a constant c _{k} , which does not depend on \phi ,such thatfor any \epsilon \ge 0 the measure of the sublevel set\{x \in I: |\phi(x)| \le \epsilon\} is bounded by c ^{1 /k} _{k\epsilon} .

References

1. Elias Stein, Harmonic Analysis: Real-variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, 1993.
2. M. Christ, Hilbert transforms along curves, Ann. of Math. 122 (1985), 575–596