at scale $R_j^{-1/2 + \delta}$, and so if we abbreviate $\eta(y_1) := \eta_T(y_1, h(y_1)) J(y_1)$, then $\eta$ obeys

$$| \eta^{(k)} | \lesssim (R_j^{1/2 - \delta})^k ,$$

and $\eta$ is supported on $I(T)$. We let $\phi(y_1) = 2 \pi \omega_2 h(y_1)$. We now have to bound the following integral:

$$\int_{I(T)} \eta(y_1) e^{ i \phi(y_1)} dy_1.$$

This integral can be bounded using stationary phase. The method is essentially the same as in [@St], Chapter 8, Proposition 1. Here is a sketch. We note that on $I(T)$,

$$| \phi'(y_1)| = |\omega_2| |h'(y_1)| \gtrsim R_j^{1/2 + \delta}, \textrm{ and }$$

$$| \phi^{(k)} (y_1)| = | \omega_2| |h^{(k)}(y_1)| \lesssim R_j.$$

Next we note that

$$\frac{1}{i \phi'} \frac{d}{dy_1} e^{i \phi(y_1)} = e^{ i \phi(y_1)}.$$

We define $D =  \frac{1}{i \phi'} \frac{d}{dy_1} $, so our integral becomes $\int_{I(T)} \eta D^N e^{i \phi} dy_1$, where $N$ is an arbitrary integer. Next we expand out $D^N e^{i \phi}$ and we integrate by parts many times so that none of the derivatives actually lands on $e^{i \phi}$. Using our lower bound on $|\phi'|$ and our upper bounds on the higher derivatives of $\phi$ and the derivatives of $\eta$, it follows that our integral is bounded by $C_N R_j^{- 2 \delta 