Explanation of a step in a work by C. E. Kenig and A.D. Ionescu I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. Lectures of the CMI/IAS workshop on mathematical aspects of nonlinear PDEs, Princeton, NJ, USA, 2004. Princeton, NJ: Princeton University Press (ISBN 978-0-691-12955-6/pbk; 978-0-691-12860-3/hbk). Annals of Mathematics Studies 163, 181-211 (2007). ZBL1387.35528.
I have one particular question for which I wasn't able to find an answer.

In equation (9.3.14), why did they assume that $l \in [j, 7j/4]$?

Thanks in advance.
 A: $\newcommand{\ep}{\epsilon}$Formula (9.3.14) in the linked paper states that
\begin{equation}
    L:=\Big|\sum_{m\ge0}(m/2^j)^{1/2}\psi_1^2(m/2^j)e^{i(mx'+m^3t)}\Big|
    \le C_\ep R, \tag{1}\label{1}
\end{equation}
where
\begin{equation}
R:=2^{(3/4+2\ep)j}2^{2(l-j)/5},     
\end{equation}
$\ep>0$, $x'$ and $t$ are real numbers, $j$ and $l$ are integers such that $l\ge j$, and (by a description at the top of p. 187 in the paper) the function $\psi_1$ is such that $0\le\psi_1\le1$ and $\psi_1(x)=0$ if $|x|\ge4$.
So,
\begin{equation}
    L\le\sum_{0\le m\le2^{j+2}}(m/2^j)^{1/2}. \tag{2}\label{2}
\end{equation}
It follows that, if $j\le-3$, then $L=0$ and hence \eqref{1} trivially holds. If $-3<j\le0$, then $L\ll1\ll e^{4\ep}R$ and hence \eqref{1} still holds. (We write $A\ll B$ if $A\le CB$ for some universal real constant $C>0$.)
Finally, consider the case $j>0$. Then, by \eqref{2}, $L\ll2^j$. On the other hand, if $l>7j/4$, then $R>2^{(3/4)j}2^{2\times3j/(4\times5)}=2^{21j/20}>2^j$. So, \eqref{1} holds in this case as well.
Thus, we may assume that $l\le7j/4$ and hence $l\in[j,7j/4]$.
