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The van der Corput lemma for oscillatory integrals

Quick description

If  [a,b] \to \R is a phase obeying the bound |\phi^{(k)}(x)| \geq \lambda for all x \in [a,b] and some k \geq 2 and \lambda > 0, then

 \bigg|\int_a^b e^{i \phi(x)}\ dx\bigg| \leq C_k / \lambda^{1/k},

where C_k depends only on k.

The claim also holds for k=1 provided that \phi'(x) is monotone.

Multidimensional analogues are known.

An easy consequence of van der Corput's lemma (proven by an application of "use integration by parts to exploit cancellation") is the following statement. Suppose that  [a,b] \to \R is as above and  [a,b] \to \C is a function such that \psi ' \in L^1 ([a,b]). Then

 \bigg|\int_a^b e^{i \phi(x)}\psi(x) \ dx\bigg| \leq C_k / \lambda^{1/k} \big(\ |\psi(b)|+\int_a ^b |\psi'(x)|dx \ \big).

Typically in applications the function \psi is a smooth bump function supported in  (a,b) .

Prerequisites

Harmonic analysis

Example 1

General discussion

This bound is cruder than the asymptotics provided by the method of stationary phase.

Not to be confused with the van der Corput lemma for equidistribution.

Comments

Inline comments

The following comments were made inline in the article. You can click on 'view commented text' to see precisely where they were made.

The constant in van der Corput's lemma

For the constant in van der Corput's lemma we have  C_k \simeq k. The fact that this is best possible up to numerical constants can be seen by testing against the function \phi(x) =\frac{1}{k!}x^k.

sub-level set estimates

The general principle that sub-level set estimates imply van der Corput type estimates should be somewhere here. Also the nice trick that van der Corput type estimates imply sub-level set estimates would be very useful. However, all these things cannot be under 'Quick description'. Any ideas about how to structure this?

yannis

Use general discussion

I guess one could put all this in the general discussion section (and perhaps divide into subsections). All of these observations are indeed worth putting in the main page. (There will also be some connections with the "control level sets" page and the "linearize the phase" page: making the phase linearisation transformation t = \phi(x) one sees that the van der Corput integrals are essentially Fourier transforms of the level sets.)

Feel free to make a start on these things; I will try to come back to this page later and add more to it (I'm working my way through various other integration techniques at the moment).