Stationary-phase expansion at one simple interior critical point

Use [F5] for a compact neighborhood of $0$ inside $(-R,R)$ and symmetrize the resulting bump to obtain the even $\eta$ in the Statement. By [F1], $\chi$ is $1$ near $x_0$, vanishes near $a,b$, and $A$ is compactly supported and $C^4$; moreover $A(0)=f(x_0)\mathrm{/}\sqrt{\mathrm{\mid }{g}^{\mathrm{\prime }\mathrm{\prime }}(x_0)\mathrm{\mid }}$. Since $g^{\mathrm{\prime }}$ has no other zero, its modulus has a positive minimum on the two closed intervals outside a smaller neighborhood where $\chi =1$.

Choose $r<R$ with $\mathrm{supp}\eta \subset (-r,r)$ and put $K={\mathrm{\Phi }}^{-1}([-r,r])$, a compact interval inside $J$. Since $\chi f$ vanishes off $K$, interval additivity [F6] identifies its integral on $[a,b]$ with its integral on $K$. Apply [F4] on $K$ separately to the real and imaginary parts. The transformed amplitude is $A(u)$, supported in $(-r,r)$, so [F6] also identifies its integral on $[-r,r]$ with that on $[-R,R]$. The phase exponential factors as $e^{i\omega g(x_0)}{e}^{i\mathrm{sgn}(\omega g^{\mathrm{\prime }\mathrm{\prime }}(x_0))\mathrm{\mid }\omega \mathrm{\mid }{u}^2\mathrm{/}2}$. Then [F2], with $\lambda =\mathrm{\mid }\omega \mathrm{\mid }$ and sign $\sigma$, gives the stationary term in the Statement and a local remainder at most $C_{R,\eta }\mathrm{\parallel }A{\mathrm{\parallel }}_{C^4}\mathrm{\mid }\omega {\mathrm{\mid }}^{-3\mathrm{/}2}$.

The complementary amplitude $h$ vanishes on a neighborhood of $x_0$. Apply [F3] with $r=2$ on each side of that neighborhood; interval additivity [F6] combines the two formulas, and the internal endpoint terms vanish because $h$ and its derivatives are zero there. Since $\chi$ vanishes near $a,b$, the first endpoint term is exactly $\frac{1}{i\omega }[f{e}^{i\omega g}\mathrm{/}{g}^{\mathrm{\prime }}{]}_a^b$. The complex triangle inequality [F7] combines the remaining side estimates into a total modulus at most $(\mathrm{\mid }{T}_{g}h(a)\mathrm{/}{g}^{\mathrm{\prime }}(a)\mathrm{\mid }+\mathrm{\mid }{T}_{g}h(b)\mathrm{/}{g}^{\mathrm{\prime }}(b)\mathrm{\mid }+(b-a)\mathrm{\parallel }{T}_g^{2}h{\mathrm{\parallel }}_{\mathrm{\infty }})\mathrm{\mid }\omega {\mathrm{\mid }}^{-2}\mathrm{.}$

By componentwise linearity [F6], the localized and complementary integrals sum to $I_{\omega }(f,g;[a,b])$. The complex triangle inequality [F7] and $\mathrm{\mid }\omega {\mathrm{\mid }}^{-2}\le \mathrm{\mid }\omega {\mathrm{\mid }}^{-3\mathrm{/}2}$ for $\mathrm{\mid }\omega \mathrm{\mid }\ge 1$ combine the two remainder bounds and give the stated ${\mathcal{R}}_{\omega }$ bound and endpoint term.