Bound for a certain integration Let $\psi$ be a smooth function with compact support, and $\phi$ is smooth and $\phi'(x) \neq 0$ for any $x$ in support of $\psi$. Define $$I(\lambda) = \int_\mathbb{R} e^{i\lambda \phi(x)}\psi(x) dx$$ for $\lambda > 0.$ Then there exists a constant $c$ such that $$|I(\lambda)| \leq c\lambda^{-a}$$ for any $a > 0$ as $\lambda \rightarrow \infty.$
I guess I should apply integration by parts to have some term related to $\lambda$, but it does not work. Any suggestion or guidance what to try ?
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First set $$F(\psi)(\lambda) = I(\lambda),$$ then $$F(\psi')(\lambda) = \int_\mathbb{R} e^{i\lambda \phi(x)}\psi'(x) dx = (-i\lambda)F(\phi'\psi).$$
Generally, $$F(\psi^p)(\lambda) = (-i\lambda)^pF((\phi')^p\psi)$$ for $p \in \mathbb{N}.$ Then, by compactness of $\psi$, $$|\int_\mathbb{R} e^{i\lambda \phi(x)}\psi(x)(\phi')^p dx| \leq C.$$
I want to show that, somehow,
$|I(\lambda)| \leq k|\int_\mathbb{R} e^{i\lambda \phi(x)}\psi(x)(\phi')^p dx| $ for some constant $k$, but I get stuck.
 A: The problem is that the quantity you're calculating is not closely connected to the integral you're trying to bound.  The integrand in $I(\lambda)$ is highly oscillatory with a lot of cancellation so it is unlikely you will be able to obtain any correlation between the two integrals where one integrand has been multiplied by $(\phi')^p$.
But note that your identity $F(\psi') = (-i\lambda) F(\phi' \psi)$ is equivalent (by an appropriate substitution) to:
$$F((\psi/\phi')') = (-i \lambda) F(\psi).$$
Can you see how this gives you $|F(\psi)(\lambda)| < C/\lambda$?  Then you just need to iterate this idea to get higher powers of $\lambda$ on the denominator.
A: Suppose that $\phi : \mathbb R \to \mathbb R$ is a diffeomorphism. Then
$$ \int_{\mathbb R}e^{i\lambda \phi (x)}\psi (x)\, dx = \int_{\mathbb R}e^{i\lambda y}\psi (\phi^{-1}(y))\frac{1}{\phi'(\phi^{-1}(y))}\, dy.$$
On the right is the Fourier transform of a function in the Schwartz space. Thus it is in the Schwartz space, and the estimates follow.
Perhaps your problem can be put in this framework.
