of (\[ibp2\]) holds, we can bound$$\begin{aligned}
\big\|\partial_{\eta}^{\mu}(\varphi_{0}(2\epsilon_{1}^{-1}\partial_{2}\Phi))\big\|_{L^{\infty}}&\lesssim& ((3C+8)N)!\cdot\epsilon_{1}^{-r}\prod_{j=0}^{n-2}\epsilon_{j+2}^{r_{j}}\lesssim ((3C+8)N)!\epsilon_{1}^{-r}\prod_{j=0}^{n-2}(\epsilon_{1}(\lambda')^{j+1})^{r_{j}}\nonumber\\
&\lesssim & ((3C+8)N)!\epsilon_{1}^{-\rho}(\lambda')^{N-n\rho}\nonumber\\
&\lesssim& ((3C+8)N)!(\lambda')^{N}\nonumber,\end{aligned}$$ where $\lambda'$ is the maximum of $\epsilon_{j+1}/\epsilon_{j}$, since we have $\epsilon_{1}^{-1}\lesssim (\lambda')^{n}$. This completes the proof.

\[remark0\] When $n=1$, we will also use a slightly more general version that allow$$|\partial_{\eta}^{\mu}\Phi(\eta)|\lesssim (CN)!(\lambda')^{N}$$ for some $\lambda'\geq 1$. In this case we simply rescale to reduce to the case proved above, and obtain that (\[ipsp\]) holds with $M=\min(K(\lambda')^{-2}\epsilon^{2},K\epsilon/\lambda)$.

\[disper0\] Suppose $\|f\|_{X}\lesssim 1$, and that $\langle t\rangle\sim 2^{m}$. Also let $\Lambda=\Lambda_{\nu}$ with $\nu\in\mathcal{P}$ and fix $(j,k)\in\mathcal{J}$.

1.  If $k\leq -K_{0}$, we have$$\label{disper1}\sup_{|\mu|\leq N/2+2}\big\|e^{{i}t\Lambda}\Gamma^{\mu}S_{l}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim\langle j\rangle^{-N_{0}}\min(2^{-j+k},2^{-(3m-j+k)/2}).$$

2.  If $|k|<K_{0}$ and $ |j-m|\leq A\log m$, we have $$\label{disper4}\sup_{|\mu|\leq N/2+2}\big\|e^{{i}t\Lambda}\Gamma^{\mu}S_{l}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim 2^{-m}\langle m\rangle^{-N_{0}+A}.$$

3.  If $|k|<K_{0}$, and $|j-m|\geq A\log m$, we have $$\label{disper3}\sup_{|\mu|\leq N/2+2}\big\|e^{{i}t\Lambda}\Gamma^{\mu}S_{l}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim 2^{-3m/2-j/3}\min(2^{l}\langle m\rangle^{N_0+A},2^{4\delta(m+j)}).$$

4.  If $k\geq K_{0}$ and $|j-m|\leq A\log m$, we have $$\label{disper2}\sup_{|\mu|\leq N/2+2}\big\|e^{{i}t\Lambda}\Gamma^{\mu}S_{l}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim\langle m\rangle^{-N_{0}+A}2^{-m+3k/2}.$$

5.  If $k\geq K_{0}$ and $|j-m|\geq A\log m$, we have $$\label{disper5}\sup_{|\mu|\leq N/2+2}\big\|e^{{i}t\Lambda}\Gamma^{\mu}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim 2^{-(3m+j)/2}\min(2^{4k+l}\langle m\rangle^{-N_{0}+A},2^{2\delta(3m+j)}).$$

First we recall the standard dispersion estimate for the Klein-Gordon flow$$\label{stkg}\|P_{k}e^{it\Lambda}f\|_{L^{\infty}}\lesssim 2^{-3m/2}(1+2^{3k})\|f\|_{L^{1}},$$ see [@IP12], Lemma 5.2.

Now (\[disper4\]) is a direct consequence of Hausdorff-Young and the definition of the $Z$ norm; the bounds (\[disper1\]) and (\[disper2\]) are also easily deduced. In fact, when $(j,k)\in\mathcal{J}$ and $k\leq -K_{0}$, from Hausdorff-Young we have $$\big\|e^{{i}t\Lambda}\Gamma^{\mu}f_{jk}^{*}\big\|_{L^{\infty}}\lesssim\big\|\widehat{\Gamma^{\mu}f_{jk}^{*}}\big\|_{L^{1}}\lesssim\|\varphi_{[k-2,k+2]}(\xi)\|_{L^{2}}\|\widehat{\Gamma^{\mu}f_{jk}}\|_{L^{2}}\lesssim 