I am interested in learning about quantitative refinements of the method of stationary phase which allow to treat small perturbations in the phase of an oscillatory integral (of the first kind, in Stein's terminology).
I have the following concrete problem, which originated from studying the restriction problem for the Fourier transform on the 2-sphere. Let $f\in C_0^\infty (\mathbb{R}^2)$ be supported in $B(0,R_1)$, and consider the phase $\phi_\epsilon (y)=\frac{|y|^2}{2}+\epsilon\frac{1}{8}|y|^4.$ Look at the integrals
$$I_\epsilon(x,t)=\int_{\mathbb{R}^2}e^{-ix\cdot y}e^{-it \phi_\epsilon(y)}f(y)dy$$ and
$$J_\epsilon(x,t)=\int_{\mathbb{R}^2}e^{-ix\cdot y}e^{-it (\phi_\epsilon(y)+\epsilon^2 |y|^6)}f(y)dy.$$
Let $R_2>0$ be sufficiently large. Is it true that, for $|(x,t)|\geq R_2$,
$$|I_\epsilon(x,t)|^4-|J_\epsilon(x,t)|^4=C\epsilon^8 $|I_\epsilon(x,t)|^4-|J_\epsilon(x,t)|^4=C\epsilon^2 |(x,t)|^{-4} \;\;\textrm{as }\;\; \epsilon\rightarrow 0^+,$$ where the constant $C$ is allowed to depend on $R_1$ and $R_2$ (and possibly on some appropriate norm of $f$ and its derivatives), but on nothing else?
As a consequence one would have that
$\|J_\epsilon\|_{L^4_{x,t}(\mathbb{R}^3)}^4=\|I_\epsilon\|_{L^4_{x,t}(\mathbb{R}^3)}^4+O(\epsilon^2)$ (which is what I am really interested in).
Thank you!
