I am reading a paper '**[Periodic Nonlinear Schrodinger Equation and Invariant Measures](https://math.mit.edu/classes/18.158/gibbs-1D.pdf)**' written by **J.Bourgain**. And I am wondering if I can have some help from this website.


My question is an inequality at **(3.18)** of the paper. The inequality is, with $\lambda>1$
$$\sum\_{M>M\_0} e^{CM-c\sigma\_M^2 M^{1+\frac{2}{p}}\lambda^2}< e^{-cM\_0^{1+\frac{2}{p}}\lambda^2}$$
where the index $M$ is a dyadic [the form of $2^k$] and an arbitrary $M\_0>0$ be given. It is mentioned in the paper that he let $\sigma\_M=M^{-\frac{1}{p}}+\left( \frac{M\_0}{M} \right)^{\frac{1}{2}}$ for $M>M\_0$. Here, I also assume that the positive constants $c$ and $C$ are changing.


I have been trying to figure out this inequality and, using $\sigma\_M$, I was only able to figure out the inequality below.
\begin{align\*}
\sum\_{M>M\_0} e^{CM-c\sigma\_M^2 M^{1+\frac{2}{p}}\lambda^2}
&= \sum\_{M>M\_0} e^{CM-c(M+M\_0M^{2/p}+2M\_0M^{1/2+1/p})\lambda^2} \\
&< \sum\_{M>M\_0} e^{-c(M\_0M^{2/p}+2M\_0^{1+1/p})\lambda^2}\\
&< \sum\_{M>M\_0} e^{-c(M\_0^{1+2/p}+2M\_0^{1+1/p})\lambda^2}
\end{align\*}


where the first inequality is due to the fact that $\lambda>1$ and $c$ and $C$ can be modified. And the second inequality is due to the fact that $M>M\_0$.


Actually, I don't think the second inequality is good to use for this estimate as it would be independent of $M$ so the sum would be infinity.


I hope to figure out it... I thank in advance for the answer or any hints.



An inequality between sum of exponential functions wrt dyadic index
$\newcommand\si\sigma\newcommand\la\lambda$It seems to be (tacitly) assumed in the paper that $p\ge2$ -- see the display between (3.11) and (3.12).
Also, $c$ and $C$ seem to be (tacitly) assumed in the paper to be positive real constants.


Note that
$$M^{-2/p}+M\_0/M\le\si\_M^2\le2M^{-2/p}+2M\_0/M.$$
So,
$$\begin{aligned}
&\sum\_{M>M\_0} \exp\{CM-c\si\_M^2 M^{1+2/p}\la^2\} \\ 
&\le\sum\_{M>M\_0} \exp\{CM-c\la^2(M+M\_0M^{2/p})\} \\ 
&<\exp\{-c\la^2 M\_0^{1+2/p}\}\sum\_{M>M\_0} \exp\{CM-c\la^2 M\} \\ 
&< \exp\{-c\la^2 M\_0^{1+2/p}\},
\end{aligned}$$
as desired, if $\la>0$ is large enough (as assumed at the end of the proof, right after (3.19)).




---


On the other hand, if $C\ge2c\la^2(1+M\_0)$ , then
$$\begin{aligned}
&\sum\_{M>M\_0} \exp\{CM-c\si\_M^2 M^{1+2/p}\la^2\} \\ 
&\ge\sum\_{M>1+M\_0} \exp\{CM-2c\la^2(M+M\_0M^{2/p})\} \\ 
&\ge\sum\_{M>1+M\_0} \exp\{CM-2c\la^2(1+M\_0)M)\}=\infty.
\end{aligned}$$


