I am having a little confusion in verifying the two dimensional oscillatory integral in Lemma 2.1 in [This paper](https://epubs.siam.org/doi/epdf/10.1137/080739173), namely


$$I\_t (x,y) = \int\_{\mathbb{R}^2} |\xi|^{\epsilon + i \beta} e^{i t(\xi^3 + \xi \eta^2 + x \xi + y \eta)} d\xi\, d\eta.$$


I verified the integration with respect to $\eta$ using the inverse of Fourier transform. I reached the same result on the paper, namely


$$\int\_{\mathbb{R}^2} |\xi|^{\epsilon} e^{i t(\xi^3 + \xi \eta^2 + x \xi + y \eta)} d\xi \,d\eta = \sqrt{\pi} \lim\_{a \to \infty} \int\_\mathbb{R}\frac{|\xi|^\epsilon}{ \sqrt{| t\xi|}} e^{i t(\xi^3+ \xi \eta^2) + x \xi ) -\frac{y^2}{4 t \xi} + \frac{\pi}{4} \operatorname{sgn}{(t \xi)}} \chi\_a(\xi)\, d \xi,$$


where $\chi\_a(\xi) = \chi\_{\{ \xi: |\xi| \leq a\}}(\xi)$.


The author stated that the right hand side of the above equation is bounded uniformly by $|t|^{- \frac{2 + \epsilon}{3}}$.


I think there is a change of variable has involved which made the Ven der Corput's lemma applicable. Can the lemma be used directly? what about the term on the phase function which has $\xi$ on denominator? Any explanation is appreciated. Thanks in advance.



Two dimensional oscillatory integral
Just perform the change of variable $t^{\frac{1}{3}} \xi\mapsto \xi$, then change the variable $a$ such that the integration becomes as in the paper.


