Consider the oscillatory integral
$$I(\lambda):=\int_{\mathbb{R}^{n}} \mathrm{e}^{\lambda\dot{\imath}\phi{(x)}} \psi{(x)}dx $$ where the phase $\phi$ and the amplitude $\psi$ are smooth real-valued functions and $\psi$ has compact support.
If the phase $\phi$ is not stationary in the support of $\psi$ then
$$I(\lambda)=O(\lambda^{-N})\label{1}\tag{1}$$
for all $N\geq 0$.
The phase is stationary at the point $x_{0}$ if $$ (\nabla \phi)(x_{0})=0. $$ In practice, one may need to know the exact contribution of the derivatives of $\phi$ and $\psi$ to the asymptotic \eqref{1}. The coefficients in the one dimensional case can be easily calculated. It can be immediately seen that the factor ${1}/{\phi^{\prime}}$ plays an important role.
The question is:
Does any one know how precisely the implicit constant in the asymptotic \eqref{1} depends on the derivatives of $\phi$ and $\psi$ and the support of $\psi$ ?
Any reference ?