I have problems understanding two (for my research important) details in the Proof of Theorem 4 (page 14) in this paper: <http://arxiv.org/abs/1209.1518>. Notation is very straightforward and is found on page 4.
I expect that both issues are in reality the same problem:


1. In the first line of page 14 Schottdorf claims:
$$S\_1=\sum\_NN^{2s}||P\_N \sum\_H \sum\_{L<<H}I^+(u\_L,v\_H)||\_{U^2\_+}^2\color{red}{\lesssim}\sum\_HH^{2s}||P\_H \sum\_{L<<H}I^+(u\_L,v\_H)||\_{U^2\_+}^2\lesssim||u||\_{Y^s}^2||v||\_{Y^s}^2$$
2. The line below he claims for $S\_2$ (I just plugged in the Definition of $X^s$)
$$S\_2\leq \sum\_H\sum\_{H\sim H'}(\sum\_LL^{2s}||P\_LI^+(u\_{H'},v\_H)||\_{U^2\_+}^2)^{1/2}\color{red}{\lesssim} \sum\_H\sum\_{H\sim H'}(\sum\_{L\lesssim H}L^{2s}||P\_LI^+(u\_{H'},v\_H)||\_{U^2\_+}^2)^{1/2}$$


In both problems some Frequencies are just discarded, that should mean that the corresponding terms are 0 and that is probably because the Littlewood Paley Projectors $P\_L$ and $P\_H$ are on different Frequencies. However if that would always work then the considererd integral term $I^+(u\_{L},v\_H)$ would always be 0 which wouldn't make sense. Can somebody help me understand the highlighted inequalities?



Estimating sums with restrictions to different Frequencies
To answer your question 1 (and I expect the answer to question 2 is similar):


Note that $I^+(f,g)$ first take the product $fg$ and act on it as a Fourier multiplier. So the frequency support of $I^+(f,g)$ is the same as that of $fg$. 


Now, when $L \ll H$ you have that the frequency support of $u\_L v\_H$ is $\approx H$, using that $2^H \pm 2^L \approx 2^H$ in this case. (You actually have the support being in $[H-3,H+3]$ or something similar depending on how you defined the cut-off functions.) So this tells you that 


$$ P\_N I^+(u\_L, v\_H) \neq 0 \iff N \approx H $$


which means in the sum over $N$ and $H$, you only need to sum "diagonally" over the $N\approx H$ terms since all the rest are zero. 


