My question is about the $L^1_x$ norm of an oscillatory integral like $$ \int_{\mathbb{R}^n} e^{i(y\cdot x+\lambda \phi(y))}f(y)dy,$$ where $\lambda \in \mathbb{R}$, $f\in C^{\infty}_c(\mathbb{R}^n)$ is compactly supported and $\phi$ is real-valued and smooth (I have in mind $\phi(y)=\sqrt{1+|y|^2}$ but feel free to assume what you need). Of course, I am interested in the decay rate for $\lambda$ in this norm. Is there any known result or some hint?
The classical $L^{\infty}$ van der Corput bound is too "strong" to be integrated and I already used Bernstein-type lemmas, but I would like to have a refined estimate if any. I tried to look for results in this area following the traces of the classical papers by Hormander and Phong-Stein on $L^p \to L^{p'}$ boundedness, but nothing useful came out.