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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!

JT_NL
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    "independent on the usual things" actually is not entirely clear in this context. It should clearly be independent of $z,w$. Can it be dependent on $t,r$? How'bout $\alpha$? – Willie Wong Aug 22 '11 at 14:51
  • @Willie: That is a good remark. I've added it (only independent on $z$, $w$ and $t$). – JT_NL Aug 22 '11 at 14:53
  • 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 C=1. – Did Aug 22 '11 at 15:18
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    @Didier: Whoah. That is horrible. Thanks. – JT_NL Aug 22 '11 at 15:20
  • Oh, $\alpha$ can be assumed to be positive? I had assumed that you meant it is large in modulus. – Willie Wong Aug 22 '11 at 16:09
  • @Willie: No, it is a real number. I'm working on $\mathbf R^d$. The strange thing about this problem is that I was thinking about it for some time and then went to my supervisor and we got a reasonably complex solution but I couldn't believe that it should be that hard. It seems like I was right ;-). – JT_NL Aug 22 '11 at 16:13
  • @Didier: Thanks a lot. Now I'm a bit ashamed. Could you post it as an answer? – JT_NL Aug 22 '11 at 18:57
  • Heh, now I understand why it was so easy... I forgot a $(1 - e^{-2t^2})^{d/2}$ term. – JT_NL Aug 30 '11 at 19:39

1 Answers1

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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.

Did
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