the rotation vector fields $\Omega$; here note that at least for medium frequencies (i.e. frequencies $\sim 1$, which is where spacetime resonance takes place), we have$$\sup_{\theta\in\mathbb{S}^{2}}\|f(\rho\theta)\|_{L_{\rho}^{2}}\lesssim\sup_{|\alpha|\leq 2}\|\Omega_{0}^{\alpha}f\|_{L^{2}},$$ thus we can gain a factor of $\delta^{2}$, provided that we can somehow restrict the direction vector of $\eta$ in (\[duhamel\]) to a region of size $\delta$ in $\mathbb{S}^{2}$. This motivates the second main technical tool in this paper, namely the bilinear integration lemma, Lemma \[angular\] below, which holds under very mild conditions, and effectively allows one to restrict to $|\sin\angle (\xi,\eta)|\lesssim 2^{-(1/2-\epsilon)m}$ for medium frequencies. With this additional gain, we can then close the $Z$ norm estimate using Schur’s bound; for details see Section \[med\].

We remark that the use of rotation vector fields in this paper, and in particular the gain in both linear (Proposition \[disper0\]) and bilinear (Lemma \[angular\]) estimates, is a quite general feature and has since been applied to different dispersive models, see [@DIP14], [@DIPP16]; see also [@IP-ww] and [@IP-cp] for slightly different uses of vector fields.

### Low frequencies, and sharp integration by parts

The other assumption (\[assm2\]) in [@IP12] was used to ensure that $(0,0)\not\in\mathcal{R}$. This greatly simplifies the analysis by allowing to integrate by parts 