I am confused in finding the right bound for the following oscillatory integral


$$I = \int\_\mathbb{R} (\psi(2^{-k} \xi))^2 e^{i (y \xi - 3 \eta \xi^2 t)} d\xi.$$


Where $\psi(2^{-k} \xi)$ is a smooth cutt-off function supported on the annulus
$A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y \in \mathbb{R}$, $t >0$ and $\eta \in \mathbb{N}$.


The estimate I found using Van der Corput's lemma is as follows:


$$|I| \leq \frac{ 2^{1-2k}}{|\eta \,t|^{\frac{1}{2}}}.$$


Could you please check if my result is correct? I am going to use this result to build many things on it so the last thing I want is building my solution on a not right estimate. That's why I posted it here. Thanks in Advance



Estimate for an oscillatory integral of the first kind
Write $s=\eta t$ and note that $I(s,y)$ solves the 1D Schrödinger equation $iI\_s-3I\_{yy}=0$. Thus it satisfies the sharp estimate $|I(s,y)|\le c\_0s^{-1/2}$ where $c\_0$ is a multiple of $\int|I(0,y)|dy$. Now, $I(0,y)$ is the Fourier transform of $\chi(2^{-k}\xi)$ where $\chi=\psi^2$, that is to say $I(0,y)=2^k\widehat\chi(2^ky)$, so its $L^1(R)$ norm should be independent of $k$. In other words it seems that the right estimate is $|I(s,y)|\le c\_0|t\eta|^{-1/2}$ with $c\_0$ a constant.


