Consider the oscillatory integral
$$I(\lambda):=\int_{0}^{\infty}\int_{0}^{\infty} \psi(x,y) \, \mathrm{e}^{\dot{\imath}\phi(x,y)}dxdy$$ where
1) the phase $\phi$ and amplitude $\psi$ are smooth
2) $\lim_{y\rightarrow 0}\psi(x,y)=\lim_{y\rightarrow \infty}\psi(x,y)=0$
3) $\psi$ and all its derivatives are uniformly bounded
4) there exists a unique point $(x_0,0)$ such that $\phi(x_0,0)=0$, $(\nabla \phi)(x_0,0)=0$ but $|(D^{2}\phi) (x_0,0)|=c\neq 0$.
5)$(\nabla \phi)(x,y)\neq 0$ in the domain $x>0$, $y>0$ BUT the nonstationary phase asymptotic is not useful because the gradient is not uniformly bounded.
How get an asymptotic of this oscillatory integral?
Thanks a lot