I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem 3.2. The goal of this theorem is to estimate $\frac{d}{dt}E(Iu(t))$, where $I=I_N$ is the smooth Fourier multiplier given by its symbol $$m(\xi)=\begin{cases}1,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\qquad |x|\le N \\ \left(\frac{N}{|\xi|}\right)^{1-s}, \qquad |x|\ge 2N, \end{cases}$$ $u$ solves $$i\partial_t u + \Delta u= u|u|^4$$ in $d=1$ and $$E(Iu(t))=\int \frac{1}{2}|\nabla Iu|^2+ \frac{1}{6}\int |Iu|^6\,dx.$$ It turns out that we can write $$\frac{d}{dt}E(Iu(t))=-\operatorname{Re}\int_{\xi_1+\cdots\xi_6=0 }i|\xi_1|^2\widehat{\overline{Iu}}(t,\xi_1) \left[1-\frac{m(\xi_2+\cdots \xi_6)}{m(\xi_2) \cdots m(\xi_6)}\right] \\ \times \widehat{Iu}(t,\xi_2)\widehat{\overline{Iu}}(t,\xi_3)\widehat{Iu}(t,\xi_4)\widehat{\overline{Iu}}(t,\xi_5)\widehat{Iu}(t,\xi_6)d\xi_1\cdots d\xi_6 +\operatorname{similar \,term.}$$ where $\widehat{f}$ denotes the Fourier transform of $f$ (note the alternating complex conjugates). As is standard, he makes a Littlewood-Paley decomposition, restricting each factor $Iu(\xi_j, t)$ to be frequency-localized in the annulus $|\xi_j|\sim N_j$: $$-\operatorname{Re}\int_{\xi_1+\cdots\xi_6=0 }i|\xi_1|^2\widehat{Iu_1}(t,\xi_1) \left[1-\frac{m(\xi_2+\cdots \xi_6)}{m(\xi_2) \cdots m(\xi_6)}\right] \\ \times \widehat{Iu_2}(t,\xi_2)\widehat{Iu_3}(t,\xi_3)\widehat{Iu_4}(t,\xi_4)\widehat{Iu_5}(t,\xi_5)\widehat{Iu_6}(t,\xi_6)d\xi_1\cdots d\xi_6 $$ before taking absolute values, integrating in time and summing at the end over $N_j$. He also uses a multilinear estimate to bound the integral over frequency space of the product of these various functions $I(\xi_j, t)$ by the $L^1_x$ norm of the product of these functions in physical space.
Where I first get confused is when he estimate the variation of the energy in the case that $N_2 \ge N_3 \gtrsim N \gg N_4$ and $N_1\sim N_2 \sim N_3$ (we may assume that $N_2 \ge \cdots \ge N_6)$.
He must estimate the following norm: $$\| P_{\le 10 N_4} ((P_{N_1} \nabla Iu_1)(P_{N_2} \nabla Iu_2)(P_{N_3} \nabla Iu_3)(P_{N_4} \nabla Iu_4)) \|_{L^1_{t,x}},$$ where $P_{N_j}$ is a Littlewood-Paley projection to frequencies $|\xi_j|\sim N_j$.
However, he says that he must consider various cases depending on if $u_i=u$ or $u_i=\overline{u}$, with one of the cases being $$\| P_{\le 10 N_4} ((P_{N_1} \overline{\nabla Iu})(P_{N_2} \overline{\nabla Iu})(P_{N_3} \overline{\nabla Iu})(P_{N_4} \nabla Iu_4)) \|_{L^1_{t,x}}.$$ But why is this term relevant? In the original expression for $d/dt E(Iu(t))$, the complex conjugates are alternating, so I would expect that we only need to consider the case $$\| P_{\le 10 N_4} ((P_{N_1} \overline{\nabla Iu})(P_{N_2} \nabla Iu)(P_{N_3} \overline{\nabla Iu})(P_{N_4} \nabla Iu_4)) \|_{L^1_{t,x}},$$ or its variant where the complex conjugate signs are all reversed. Any insight is appreciated.
