An inequality between sum of exponential functions wrt dyadic index

Question

I am reading a paper 'Periodic Nonlinear Schrodinger Equation and Invariant Measures' 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.

Answer 1

$\newcommand\si\sigma\newcommand\la\lambda$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)).

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On the other hand, if $C\ge2c\la^2(1+M_0)$ , then $$\begin{aligned} &\sum_{M>M_0} \exp\{CM-c\si_M^2 M^{1+2/p}\la^2\} \\ &\ge\sum_{M>1+M_0} \exp\{CM-2c\la^2(M+M_0M^{2/p})\} \\ &\ge\sum_{M>1+M_0} \exp\{CM-2c\la^2(1+M_0)M)\}=\infty. \end{aligned}$$