were to choose $\|f\|_{Z_{jk}}=2^{j}2^{\lambda k}\|f\|_{L^{2}}$ as in [@IP12], then we *must* have $\lambda\geq 1/2$. On the other hand, it can be proved that when $\lambda\geq 1/2$, the best decay rate for $\|e^{-it\Lambda_{\nu}}f\|_{L^{\infty}}$ is precisely $t^{-1}$, which is not integrable. Note that this cannot be helped by the use of rotation vector fields $\Omega$, since it does nothing to rescaled Schwartz functions.

In this paper, we circumvent this difficulty by adding a log factor and choosing $$\|f\|_{Z_{jk}}=2^{j}2^{k/2}\langle j\rangle^{N_{0}}\|f\|_{L^{2}}.$$ Note that this makes the decay rate integrable, and does *not* violate the above intuitions. The drawback is that, when integrating by parts with low frequencies, we have to be precise “up to log factors”. In order to achieve this, we use a sharp integration by parts lemma, Proposition \[ips0\], which requires that we integrate by parts $A$ times with $A$ *depending on* the functions themselves. This lemma is proved in Section \[norms\], and the analysis of low frequencies is carried out in Section \[lowlowtolow\].

### Remarks on the two dimensional case

The study of (\[nlkg\]) in two dimensions will be much harder, and one should not expect a general result as the one in this paper. In fact, with carefully chosen speeds 