as local coordinates, so that the integral (\[polar\]) reduces to$$\label{fin}I=\int_{\mathbb{R}\times\mathbb{R}^{2}}e^{{i}\big(t\Lambda(\rho)+\alpha\rho\sqrt{1-(\theta^{2})^{2}-(\theta^{3})^{2}}\big)}\psi_{1}(\rho)\psi_{2}(\theta^{2},\theta^{3})h(\rho,\theta^{2},\theta^{3})\,\mathrm{d}\rho\mathrm{d}\theta_{2}\mathrm{d}\theta_{3},$$ where $\psi_{1}$ and $\psi_{2}$ are cutoff functions, and $h$ is obtained from $g_{1}$ after change of variables.

Now, recall $\alpha\sim 2^{r}$, let $$\label{seteps}\epsilon'=2^{-r/2}\langle m\rangle^{2A};\qquad\epsilon''=2^{\max(j-m,-m/2)}\langle m\rangle^{2A},$$ we proceed to estimate (\[fin\]) in the region where $|\theta^{2}|+|\theta^{3}|\gtrsim \epsilon'$. After inserting a cutoff $(1-\varphi_{0})((\epsilon')^{-1}\theta^{2},(\epsilon')^{-1}\theta^{3})$, we will use Proposition \[ips0\] to integrate by parts in $(\theta^{2},\theta^{3})$, for any fixed $\rho$, choosing $$K=\alpha, \quad n=1, \quad \epsilon\sim\epsilon', \quad\lambda\sim(\epsilon')^{-1},$$ so that $I$ is bounded by $\exp(-\gamma\langle m\rangle^{A\gamma/2})$, which is $\lesssim 2^{-10m}$ if $A$ is large enough. Here we have used that $2^{l}\leq 2^{7\delta m}\leq(\epsilon')^{-1}$ and $\|\partial_{\theta}^{\mu}h(\rho,\theta)\|_{L_{\theta}^{2}}\lesssim 2^{|\mu|l}$, which is because $g_{1}(\rho,\cdot)$ is still a linear combination of spherical harmonics of degree $\lesssim 2^{l}$.

Now, in (\[fin\]), we will restrict to the region where $|\theta^{2}|+|\theta^{3}|\ll \epsilon'$, and then fix $\theta^{2}$ and $\theta^{3}$. Let$$\Xi(\rho):=\partial_{\rho}\big(\Lambda(\rho)+t^{-1}\alpha\rho\theta^{1}\big)=\Lambda'(\rho)+t^{-1}\alpha\theta^{1},$$ we will then consider the part where $|\Xi(\rho)|\gtrsim \epsilon''$. In this case, we will use Proposition \[ips0\], and set$$K=t,\quad n=1,\quad\epsilon\sim\epsilon'',\quad\lambda\sim\max(M_{0},(\epsilon'')^{-1})$$ to bound $I\lesssim 2^{-10m}$. Here we have used $\|\partial_{\rho}^{\mu}h(\rho,\theta)\|_{L^{2}}\lesssim M_{0}^{|\mu|}$, because $\mathcal{F}_{\rho}g_{1}(\rho,\theta)(\tau)$ is supported in $|\tau|\lesssim M_{0}$.

Therefore, we can restrict to the region $|\Xi(\rho)|\lesssim \epsilon''$. Using Hölder and (\[sobolev2\]), we have$$|I|\lesssim (\epsilon')^{2}\min\big(\epsilon''\|g_{1}\|_{L_{\theta}^{\infty}L_{\rho}^{\infty}},(\epsilon'')^{1/2}\|g_{1}\|_{L_{\theta}^{\infty}L_{\rho}^{2}}\big)\lesssim(\epsilon')^{2}2^{l}\min\big(\epsilon''M_{0}^{1/2},(\epsilon'')^{1/2}\big)\|g_{1}\|_{L_{\rho}^{2}L_{\theta}^{2}},$$ which implies$$\label{extraomega}|I|\lesssim \langle m\rangle^{A}2^{(j-3m)/2}2^{l}\|F\|_{L^{2}}\lesssim 2^{l}\langle m\rangle^{N_{0}+A}2^{-3m/2-j/3}$$ by (\[part\]). Since we trivially 