down (thus uniformly bounded) for long time, and also enjoy localization in space. This localization is captured in a carefully designed “$Z$-norm” which was used in [@IP12]; see also similar localization bounds in [@G11] and [@GM11]. One then has the linear dispersion estimate $$\label{decaykg}\|v_{\sigma}(t)\|_{L^{\infty}}=\|e^{-it\Lambda_{\sigma}}f_{\sigma}(t)\|_{L^{\infty}}\lesssim (1+|t|)^{-1-\delta}\|f_{\sigma}(t)\|_{Z},$$ provided that the $Z$ norm is chosen correctly.

Proving that $\|f_{\sigma}(t)\|_{Z}$ is uniformly bounded is achieved by using the Duhamel formula$$\label{duhamel}\widehat{f_{\sigma}}(t,\xi)=\widehat{f_{\sigma}}(0,\xi)+\int_{0}^{t}\int_{\mathbb{R}^{3}}e^{{{i}}s\Phi_{\sigma\mu\nu}(\xi,\eta)}m(\xi,\eta)\widehat{f_{\mu}}(s,\xi-\eta)\widehat{f_{\nu}}(s,\eta)\,\mathrm{d}\eta\mathrm{d}s.$$ Here one can notice that, at least in the approximation when the inputs $f_{\mu}$ and $f_{\nu}$ are Schwartz, the main contribution of the integral (\[duhamel\]) comes from the vicinity of the stationary set$$\mathcal{R}:=\{(\xi,\eta):\Phi_{\sigma\mu\nu}(\xi,\eta)=\nabla_{\eta}\Phi_{\sigma\mu\nu}(\xi,\eta)=0\}.$$ This set $\mathcal{R}$, which is called the *spacetime resonance* set, was first introduced in [@GMS09] and plays a crucial role in the analysis.

Using this strategy, in [@IP12], the authors were able to perform a robust stationary phase analysis near $\mathcal{R}$, and proved small data scattering of (\[nlkg\]) under the additional assumptions that $$\label{assm1}(c_{\alpha}-c_{\beta})(c_{\alpha}^{2}b_{\alpha}-c_{\beta}^{2}b_{\beta})\geq 0,\qquad1\leq\alpha,\beta\leq d;$$ $$\label{assm2}b_{\alpha}+b_{\beta}-b_{\gamma}\neq 0,\qquad1\leq\alpha,\beta,\gamma\leq d.$$ Below we will briefly discuss why these two assumptions are needed in [@IP12], and how we are able to remove them in the current paper.

### Spherical symmetry and rotation vector fields

It is clear that the choice of the $Z$ norm 