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 annual $A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y \in \mathbb{R}$, $t >0$ and $\eta \in \mathbb{N}$.

The estimate I found using Van der Corput 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

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

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 annual $A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y, t \in \mathbb{R}$, and $\eta \in \mathbb{Z}$.

The estimate I found using Van der Corput 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

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 annual $A:= \{ 2^{k-1} \leq | \xi| \leq 2^{k+1} \}$, $y \in \mathbb{R}$, $t >0$ and $\eta \in \mathbb{N}$.

The estimate I found using Van der Corput 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