proposition is the energy inequality , which depends on the $Z'$ norm defined in . This norm has to be chosen strong enough to allow for the energy inequality to hold; in particular $$\|\partial_iR_j f\|_{L^\infty}+\|R_j f\|_{Z'}\lesssim \|f\|_{Z'},\qquad j,k\in\{1,2\}.$$ On the other hand, the $Z'$ norm has to be chosen small enough, in such a way that the function $t\to\|U(t)\|_{Z'}$ is almost integrable in time; the main inequality we need is $$\label{basic}
 \|e^{-it\Lambda}f\|_{Z'}\lesssim (1+t)^{-1}\|f\|_{Z},\qquad \text{ for any }t\in[0,\infty),$$ which is an easy consequence of .

To prove the global result in Theorem \[MainThm2\] we also need the following bootstrap estimate:

\[Norm\] Assume that $U\in X^N_T$ is a solution of the equation $$(\partial_t+i\Lambda)U=\frac{i}{4}\sum_{j=1}^2\Lambda R_j\big[|\nabla|\Lambda^{-1}(U+\overline{U})\cdot R_j(U-\overline{U})\big]+\frac{i}{8}\sum_{j=1}^2|\nabla|\big[R_j(U-\overline{U})\cdot R_j(U-\overline{U})\big]$$ with the property that $$\label{sec2}
\|U\|_{X^N_T}\leq\varepsilon_0\leq 1.$$ Then $$\label{sec3}
\sup_{t\in[0,T]}\|e^{it\Lambda}U(t)-U(0)\|_{Y^{N_0}}\lesssim \varepsilon^2_0.$$

Proof of Theorem \[MainThm2\] {#Uproof}
-----------------------------

Theorem \[MainThm2\] is a consequence of Proposition \[LocExLemma\] and Proposition \[Norm\]. Indeed, assume that we start with data $U_0$ satisfying $\|U_0\|_{H^N}+\|U_0\|_{Y^{N_0}}=\epsilon\leq\overline{\varepsilon}$, for some $\overline{\varepsilon}$ sufficiently small relative to the value of $\delta_0$ in Proposition \[LocExLemma\] (i). Assume that we have constructed a solution $U\in C([0,T]:H^N)$ (using Proposition \[LocExLemma\] (ii)), for some $T\geq 1$, such that $$\label{cons1}
 \sup_{t\in[0,T]}\|e^{it\Lambda}U(t)\|_{Y^{N_0}}\leq \epsilon^{3/4}.$$ Then, using , $$\|U(t)\|_{Z'}\lesssim (1+t)^{-1}\epsilon^{3/4},\qquad\text{ for any }t\in[0,T].$$ Then, using , 