(\[decaykg\]).

To overcome this obstacle, we will introduce the rotation vector fields, as included the vector field set $\Gamma$ in (\[normkg\]). The idea of using such vector fields goes back to Klainerman [@Kl85], [@Kl86] (see also [@Kl87]); in the current situation, we have $$\label{impr}\|e^{-it\Lambda}f_{jk}\|_{L^{\infty}}\lesssim (1+t)^{-1-\delta}2^{(1/2+\delta)j}(\|f_{jk}\|_{L^{2}}+\|\Omega f_{jk}\|_{L^{2}}),$$ at least when $j\leq (1-\delta^2)m$ and $|k|\lesssim 1$, where $\Omega$ is the rotational part of $\Gamma$. Note that we gain a power $2^{(1/2-\delta)j}$ at the price of using one $\Omega$, which in particular allows us to close the argument with the weak $2^{5j/6}$ power in the $Z$ norm. For the proof of the improved linear dispersion inequality (\[impr\]), see Proposition \[disper0\] below.

### A bilinear lemma {#bililemma}

The above arguments are only for the second iteration. In general, the input functions $f_{\mu}$ and $f_{\nu}$ are not really Schwartz, which causes trouble when trying to localize $\eta$ to the vicinity of the “space resonant set” where $\nabla_{\eta}\Phi=0$, in the study of the integral (\[duhamel\]). In [@IP12], this was solved by using the (strong) $Z$ norm and an orthogonality trick; these arguments will not be sufficient here, due to our $Z$ norm being strictly weaker.

This issue is solved in the current paper by exploiting further 