space,$$\mathcal{G}_{\sigma}(\mathbb{R}^{d})=\bigg\{f\in C^{\infty}(\mathbb{R}^{d}):\sup_{x\in K}|\partial^{\alpha}f(x)|\leq C_{K}^{|\alpha|}(\sigma|\alpha|)!\bigg\},\qquad\forall\textrm{ compact }K\subset\mathbb{R}^{d}.$$ Abusing notation, we will regard $\varphi$ as a radial function on $\mathbb{R}^{3}$ by $\varphi(x)=\varphi(|x|)$; We then have $\varphi\in \mathcal{G}_{6}(\mathbb{R}^{3})$. Next, define $$\varphi_k(x):=\varphi(x/2^k)-\varphi(x/2^{k-1})\quad\text{ for any }k\in\mathbb{Z},\qquad \varphi_I:=\sum_{m\in I\cap\mathbb{Z}}\varphi_m\quad\text{ for any }I\subseteq\mathbb{R},$$ and define functions like $\varphi_{\leq B}$ similarly. For any $x\in\mathbb{Z}$ define $$x_+:=\max(x,0),\qquad x_-:=-\min(x,0).$$ Let the set $$\mathcal{J}:=\{(k,j)\in\mathbb{Z}\times\mathbb{Z}_+:\,k+j\geq 0\},$$ and for any $(k,j)\in\mathcal{J}$ let $$\varphi_{j}^{(k)}(x)=\left\{\begin{aligned}
&\varphi_{j}(x),&\quad\text{if }j&>\max(0,-k);\\
&\varphi_{\leq j}(x),&\quad\text{if }j&=\max(0,-k).
\end{aligned}\right.$$ Let $P_k$ denote the operator on $\mathbb{R}^3$ defined by the Fourier multiplier $\xi\to \varphi_k(\xi)$; let $P_{\leq B}$ (respectively $P_{I}$) denote the operators on $\mathbb{R}^3$ defined by the Fourier multipliers $\xi\to \varphi_{\leq B}(\xi)$ (respectively $\xi\to \varphi_{I}(\xi)$). For $(k,j)\in\mathcal{J}$ let $$\label{qjk}
(Q_{jk}f)(x):=\varphi^{(k)}_j(x)\cdot P_kf(x);\qquad f_{jk}:=Q_{jk}f, \qquad f_{jk}^{*}:=P_{[k-2,k+2]}Q_{jk}f.$$

Let $\Lambda_{\alpha}=\sqrt{b_{\alpha}^{2}-c_{\alpha}^{2}\Delta}$ be the linear phase, and define $$\Lambda_{\alpha}(\xi):=\sqrt{c_{\alpha}^{2}|\xi|^{2}+b_{\alpha}^{2}},\quad1\leq\alpha\leq d;\qquad b_{-\alpha}=-b_{\alpha},\quad c_{-\alpha}=c_{\alpha},\quad\Lambda_{-\alpha}=-\Lambda_{\alpha}.$$ Let $\mathcal{P}=\mathbb{Z}\cap([1,d]\cup[-d,-1])$; for $\sigma,\mu,\nu\in \mathcal{P}$, we define the associated nonlinear phase $$\label{phasedef}
\Phi_{\sigma\mu\nu}(\xi,\eta):=\Lambda_{\sigma}(\xi)-\Lambda_{\mu}(\xi-\eta)-\Lambda_{\nu}(\eta),$$ and the corresponding parallel phase $$\Phi^{+}_{\sigma\mu\nu}(\alpha,\beta):=\Phi_{\sigma\mu\nu}(\alpha e,\beta e)=\Lambda_{\sigma}(\alpha)-\Lambda_{\mu}(\alpha-\beta)-\Lambda_{\nu}(\beta),$$ where $e\in\mathbb{S}^{2}$ and $\alpha,\beta\in\mathbb{R}$.

For later purposes, we also need the spherical harmonics decomposition, which is described below. Let $(r,\theta,\varphi)$ be polar coordinates for $\xi$ and $Y_{q}^{m}$ be spherical harmonics, and note $\Delta_{\mathbb{S}^{2}}Y_{q}^{m}=-q(q+1)Y_{q}^{m}$. For any function $f$, we have the expansion$$\label{decomp}f(\xi)=\sum_{q=0}^{\infty}\sum_{m=-q}^{q}f_{q}^{m}(r)Y_{q}^{m}(\theta,\varphi),$$ and may define accordingly$$\label{spherical}S_{l}f:=\sum_{q=0}^{\infty}\sum_{m=-q}^{q}\varphi_{l}(q)f_{q}^{m}(r)Y_{q}^{m}(\theta,\varphi)$$ for $l\geq 1$, and with $\varphi_{0}$ replaced by $\varphi_{\leq0}$ for $l=0$. Note 