I want to establish the bounds for the Airy function \begin{equation} K(x) = \frac{1}{2\pi} \int_\mathbb{R} e^{i(x\xi+\xi^3)} \,d\xi. \end{equation} I want to show that $K(x)=O_N(<x>^{-N})$ for any $N\geq0$ and $x>0$; and $K(x)=O(<x>^{-1/4})$ for $x\leq0$, where \begin{equation} <x>=(1+x^2)^{1/2}. \end{equation}
I was told to consider 3 cases:
When $x\geq 1$, use repeated integration by parts.
When $-1<x<1$, use Van der Corput Lemma.
When $x\leq-1$, split the integral into pieces and use Van der Corput Lemma, or integration by parts on each piece.
My attempt:
When $x\geq1$, \begin{equation} K(x) = \frac{1}{2\pi} \int_\mathbb{R} e^{i(x\xi+\xi^3)} \,d\xi = \frac{1}{2\pi} \int_\mathbb{R} \frac{1}{i(x+3\xi^2)} i(x+3\xi^2) e^{i(x\xi+\xi^3)} \,d\xi \end{equation} \begin{equation} = \frac{-1}{2\pi} \int_\mathbb{R} \partial_\xi\left( \frac{1}{i(x+3\xi^2)} \right) e^{i(x\xi+\xi^3)} \,d\xi = \frac{-1}{2\pi} \int_\mathbb{R} \frac{6i\xi}{(x+3\xi^2)^2} e^{i(x\xi+\xi^3)} \,d\xi \end{equation} \begin{equation} = \frac{1}{2\pi} \int_\mathbb{R} \partial_\xi\left( \frac{1}{i(x+3\xi^2)} \frac{6i\xi}{(x+3\xi^2)^2} \right) e^{i(x\xi+\xi^3)} \,d\xi \end{equation} \begin{equation} = \frac{1}{2\pi} \int_\mathbb{R} \partial_\xi\left( \frac{6\xi}{(x+3\xi^2)^3} \right) e^{i(x\xi+\xi^3)} \,d\xi. \end{equation} I don't see how this turns out to the desired bound $O_N(<x>^{-N})$.
When $-1<x<1$, how can I apply the Lemma? I need a compact interval $I$ for $\xi$. Should I use \begin{equation} \int_I e^{i\phi(\xi)} \,d\xi \qquad or \qquad \int_I e^{i\phi(\xi)}\Psi(\xi) \,d\xi\:? \end{equation} What should I choose $\phi$?
When $x\leq-1$, I don't see how to split up the integral. I don't see how the value of $x$ can affect the proof.
Thank you.
