The van der Corput lemma for oscillatory integrals

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The van der Corput lemma for oscillatory integrals

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[QUICK DESCRIPTION]

If $\phi: [a,b] \to \R$ 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

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

where $C_k$ depends only on $k$.<comment thread="185" />

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 $\phi: [a,b] \to \R$ is as above and $\psi: [a,b] \to \C$ is a function such that $\psi ' \in L^1 ([a,b])$. Then

[math] \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).[/math]

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

[PREREQUISITES]

Harmonic analysis

[EXAMPLE]

[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]].

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The van der Corput lemma for oscillatory integrals

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