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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.

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$\newcommand{\ep}{\epsilon}$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 $l\in[j,7j/4]$.

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  • $\begingroup$ I can not thank you enough. $\endgroup$
    – Mr. Proof
    Commented May 13, 2022 at 0:40
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    $\begingroup$ @SelfLearner : I am glad this was of help. $\endgroup$
    – Iosif Pinelis
    Commented May 13, 2022 at 18:43
  • $\begingroup$ I still have a problem understanding a similar restriction, but with $k$, and $k$, integers in the paragraph after inequality (2.5) on page 8 of the paper:arxiv.org/pdf/2107.01482.pdf. Does it contain a mistake? $\endgroup$
    – Mr. Proof
    Commented May 15, 2022 at 6:51
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    $\begingroup$ @SelfLearner : I think such a question (about a completely different paper) should be posted in a separate post, rather than in a comment. $\endgroup$
    – Iosif Pinelis
    Commented May 15, 2022 at 17:33
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    $\begingroup$ @SelfLearner : What is confusing here? For each $j\in\{-2,-1,0\}$, compare the values of the sum in (2) and $R$, taking into account that $l\ge j$. $\endgroup$
    – Iosif Pinelis
    Commented May 17, 2022 at 4:01

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