and spherical harmonics decomposition that will be used throughout this paper.

Norms and basic estimates {#norms}
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The definition of $Z$ norm
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Recall the notations defined in Section \[notatkg\], and also the vector field set $\Gamma$ as in (\[normkg\]). We will define the $Z$ norm as follows.

\[normdef\]Fix $(j,k)\in\mathcal{J}$, define the norm$$\label{part}\|f\|_{Z_{jk}}=
\begin{dcases}\sup_{|\mu|\leq N/2+2}\langle j\rangle^{N_{0}}2^{\min(k/2,0)}2^{j}\|\Gamma^{\mu}f\|_{L^{2}},&|k|\geq K_{0};\\
\sup_{|\mu|\leq N/2+2}\big(2^{5j/6}\langle j\rangle^{-N_{0}}\|\Gamma^{\mu}f\|_{L^{2}}+\langle j\rangle^{N_{0}}2^{j}\|\widehat{\Gamma^{\mu}f}\|_{L^{1}}\big),&|k|<K_{0}.\end{dcases}$$Also define the full $Z$ norm by$$\label{fullz}\|f\|_{Z}=\sup_{(j,k)\in\mathcal{J}}\|f_{jk}\|_{Z_{jk}},$$ and the $X$ norm by$$\|f\|_{X}=\sup_{|\mu|\leq N}\|\Gamma^{\mu}f\|_{L^{2}}+\|f\|_{Z}.$$

Notice that by (\[part\]) we have $$\label{normcomp}\|f\|_{Z_{jk}}\lesssim \|f\|_{Z_{jk'}},\qquad |k|<K_{0}\leq |k'|\leq O_{K_{0}}(1),$$ if $\widehat{f}$ is supported in $|\xi|\leq O_{K_0}(1)$; this simple observation will be convenient later.

Linear and bilinear estimates
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\[ips0\]Suppose $K,\lambda\geq 1$ and $\epsilon_{j}$ are positive parameters, and $h(\eta)$ is some compact supported function on $\mathbb{R}^{3}$, verifying $$\label{ibp1}\|\partial_{\eta}^{\mu}h(\eta)\|_{L^{1}}\lesssim (CN)!\lambda^{N}$$ for some $C\geq 1$, and $N$ and all $|\mu|\leq N$. Moreover, let $\Phi=\Phi(\eta)\in\mathcal{G}_{C}$ be a function such that $$\label{ibp2}|\partial_{\eta}\Phi(\eta)|\geq\epsilon_{1};\qquad|\partial_{\eta}^{\mu}\Phi(\eta)|\leq \epsilon_{|\mu|}$$ holds for each $2\leq |\mu|\leq n$ and at each point $\eta$ where $h$ or one of its derivatives is nonzero, then we have the estimate $$\label{ipsp}\bigg|\int_{\mathbb{R}^{3}}e^{{i}K\Phi(\eta)}h(\eta)\,\mathrm{d}\eta\bigg|\lesssim e^{-\gamma M^{\gamma}},$$where$$M=\min(K\epsilon_{1}^{2}/\epsilon_{2},K\epsilon_{1}\epsilon_{2}/\epsilon_{3},\cdots,K\epsilon_{1}\epsilon_{n-1}/\epsilon_{n},K\epsilon_{1}\epsilon_{n},K\epsilon_{1}/\lambda),$$and $\gamma$ is small enough depending on $C$. In particular, if we can choose $\epsilon_{j}=\epsilon^{\frac{n-j+1}{l}}$, then we have$$M=\min(K\epsilon^{\frac{n+1}{n}},K\epsilon/\lambda).$$

If we fix a direction $\theta\in\mathbb{S}^{2}$ and replace the $\partial_{\eta}$ in (\[ibp1\]) 