I'm trying to get an upper bound in the following way: $$ I_{\alpha} = \int_{\mathbb{R}} \frac{dx}{(|x^2 - \alpha| + 1)^{1+ \epsilon}} \leq C \frac{1}{\alpha^{\gamma}}, $$ where $\alpha >0$ and $0 < \varepsilon <<1$, for some $\gamma >0$. My attempt was to write

$$ I_{\alpha} = \int_{|x| > \sqrt{\alpha}} \frac{dx}{(|x^2 - \alpha| + 1)^{1+ \epsilon}} + \int_{|x| \leq \sqrt{\alpha}} \frac{dx}{(|x^2 - \alpha| + 1)^{1+ \epsilon}}. $$ However, it is not clear how to obtain an upper bound of form $C \frac{1}{\alpha^{\gamma}}$ for the second integral above. Any suggestions or ideas are welcome. Thanks!


So you can get $$ I_α < 6 \,\min(1,\alpha^{-\gamma}) \leqslant \frac{6}{\alpha^\gamma} $$ with $γ = \frac{\varepsilon}{2(2+\varepsilon)}$.

  • First notice that if $α$ is small, say $α \leqslant 1$, then $$ \int_{|x|>1} \frac{\mathrm d x}{(\,|x^2-\alpha|+1)^{1+\varepsilon}} \leqslant \int_{|x|>1} \frac{\mathrm d x}{|x|^{2(1+\varepsilon)}} = \frac{2}{1+2\,\varepsilon} < 2 \\ \int_{|x|<1} \frac{\mathrm d x}{(\,|x^2-\alpha|+1)^{1+\varepsilon}} \leqslant \int_{|x|<1} \mathrm d x = 2 $$ so in this case $$ I_α \leqslant 4. $$

  • Hence we can focus on the case $\alpha>1$. Let me start by doing the change of variable $x = \sqrt \alpha\, y$ to get $$ I_α = \int_{\mathbb R} \frac{\sqrt α\,\mathrm d y}{(\alpha\,|y^2-1|+1)^{1+\varepsilon}} = \int_0^\infty \frac{2\,\sqrt α\,\mathrm d y}{(\alpha\,|y^2-1|+1)^{1+\varepsilon}}. $$ Here we see clearly that the difficulty lies in the region where $y^2 = 1$. Let put this zone around $0$ by taking $z = y^2-1$. Then $$ I_α = \int_{-1}^\infty \frac{\sqrt α\,\mathrm d z}{\sqrt{1+z}\,(\alpha\,|z|+1)^{1+\varepsilon}}. $$ As said before, we need to isolate the region near $z=0$ and say that it is small. Hence we cut the integral in three parts: for $r,R\geqslant 0$ we have $$ \begin{align*} I_{α,1}=\int_{-1}^{-r} \frac{\sqrt α\,\mathrm d z}{\sqrt{1+z}\,(\alpha\,|z|+1)^{1+\varepsilon}} &\leqslant \frac{\sqrt α}{(\alpha\,r+1)^{1+\varepsilon}} \int_{0}^{1-r} \frac{\mathrm d w}{\sqrt{w}} = \frac{2\,\sqrt{(1-r)\,α}}{(\alpha\,r+1)^{1+\varepsilon}} \\ I_{α,2}=\int_{-r}^{R} \frac{\sqrt α\,\mathrm d z}{\sqrt{1+z}\,(\alpha\,|z|+1)^{1+\varepsilon}} &\leqslant \sqrt α \int_{1-r}^{1+R} \frac{\mathrm d w}{\sqrt{w}} = 2\,\sqrt{\alpha}\,(\sqrt{1+R}-\sqrt{1-r}) \\ I_{α,3} = \int_{R}^{\infty} \frac{\sqrt α\,\mathrm d z}{\sqrt{1+z}\,(\alpha\,|z|+1)^{1+\varepsilon}} &<\alpha^{-1/2-\varepsilon} \int_{R}^{\infty} \frac{\mathrm d z}{z^{3/2+\varepsilon}} < \frac{2}{(\alpha\,R)^{1/2+\varepsilon}} \end{align*} $$ Now we just have to optimize on $r$ and $R$ to get $I_{\alpha,2}$ small without making the other integrals too large. To get the best rate of decay for each part, we can take $R=\alpha^{-\frac{3}{3+2\varepsilon}}$ and $r=\alpha^{-\frac{1+\varepsilon}{2+\varepsilon}}$. This gives $$ \begin{align*} I_{α,1} &\leqslant \frac{\sqrt{2\alpha}}{(\alpha r)^{1+\varepsilon}} = \frac{\sqrt 2}{\alpha^\gamma} \\ I_{α,2} &\leqslant 2\,\sqrt{\alpha}\,(\sqrt{1-R}-\sqrt{1-r}) \leqslant \sqrt{\alpha}\,(r+R) = \alpha^{-\gamma} + \alpha^{-\beta} \\ I_{α,3} &< \frac{2}{\alpha^{\beta}}, \end{align*} $$ where $β = \frac{1}{2\,(3+2\varepsilon)} > \gamma$ if $\varepsilon$ is small.

  • $\begingroup$ LL3.14, thank you very much! Your solution was very enlightening to me. $\endgroup$
    – Marcelo Ng
    Sep 15 at 12:13
  • $\begingroup$ Oh actually I did a small mistake, we cannot take $R=0$ ... I corrected it. Also, where is this integral coming from? $\endgroup$
    – LL 3.14
    Sep 15 at 16:28
  • $\begingroup$ Dear @LL 3.14, thanks for the correction. This integral comes from a theory involving a perturbation of the nonlinear Schrödinger equation. $\endgroup$
    – Marcelo Ng
    Sep 17 at 18:28
  • 1
    $\begingroup$ Oh, interesting, thank you :) If your question is solved, you should indicate it as so. $\endgroup$
    – LL 3.14
    Sep 18 at 7:15
  • $\begingroup$ Dear @LL3.14. The perturbation of the Schrodinger equation mentioned above is known as the Schrödinger-Debye system :). $\endgroup$
    – Marcelo Ng
    2 days ago

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