4
$\begingroup$

First let $z, w \in \mathbf R^d$ with $|z - w| < \min\{1, |z|^{-1}\}$. Further let $0 < t < r < \infty$. I wish to obtain an inequality of the form

$\exp \left (- \frac{|e^{-t^2} z - w|^2}{1 - e^{-2t^2}} \right )\exp \left (-\alpha\frac{|e^{-t^2} w - z|^2}{1 - e^{-2t^2}} \right ) \lesssim \frac1{|z - w|^d}$ where our friend $\alpha > 0$ is very very large. $a \lesssim b$ means that there is a constant $C > 0$ independent on the usual things (in this case only $z$, $w$ and $t$) such that $a \leq C b$.

So my question is if such an inequality works and if so how do we derive it? I've looked at this for a while, but I can't seem to figure this out. Any suggestions would be great!

  • 0
    Heh, now I understand why it was so easy... I forgot a $(1 - e^{-2t^2})^{d/2}$ term.2011-08-30

1 Answers 1

4

The LHS is at most 1 (as the exponential of a negative real number) and the RHS is at least 1 (because |z-w|<1). Hence the inequality holds with C=1.