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For $s\in (0,1)$, is there are an explicit expression for $$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$

Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\right)^s}$ defined on the ball $B(0,R)$ where $R>1$ we have $(-\Delta)^s u(x)\geq c(n,s) u^2(x)$ where $c(n,s)>0$ is constant dependent on $n$ and $s.$ Any references/hints on how to do this will be much appreciated.

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    $\begingroup$ In a comment to my answer you asked for an upper bound for $(-\Delta)^s u$, while now you seem to be concerned with the lower bound. Which is correct? (The lower bound should work unless the dimension is too low; intuition suggests that one needs $n > 4s$, but I did not even try to check it rigorously.) $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 28, 2021 at 15:29
  • $\begingroup$ Thanks for your reply, I apologize for not adding this before. I want a lower bound for the problem I am trying to solve. I work with the assumption that $n\geq 6s.$ I have been trying to use the integral representation for the fractional laplacian and the trouble comes when $|y|<2|x|$ because in the other case the integrand is positive. $\endgroup$
    – Student
    Commented Jul 28, 2021 at 16:18

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Unless I am making a typo, the result is: $$ 2^{2s} \frac{\Gamma(\tfrac n2+s) \Gamma(2s)}{\Gamma(\tfrac n2)^2} {_2F_1}(\tfrac n2+s, 2s, \tfrac n2, -|x|^2), $$ where $n$ is the dimension and ${_2F_1}$ is the Gauss's hypergeometric function. See Table 1 on page 168 in my survey [1], or Corollary 2 in the original paper [2].

References:

[1] Mateusz Kwaśnicki, Fractional Laplace Operator and its Properties, in: A. Kochubei, Y. Luchko, Handbook of Fractional Calculus with Applications. Volume 1: Basic Theory, De Gruyter Reference, De Gruyter, Berlin, 2019, https://doi.org/10.1515/9783110571622-007

[2] Bartłomiej Dyda, Alexey Kuznetsov, Mateusz Kwaśnicki, Fractional Laplace operator and Meijer G-function, Constructive Approx. 45(3) (2017): 427–448, https://doi.org/10.1007/s00365-016-9336-4

Edit: The hypergeometric function is continuous on $(-\infty, 1)$, and — as described, for example, in this DLMF entry — we have $$ {_2F_1}(a, b, c, -|x|^2) = \frac{p}{|x|^{2a}} \, {_2F_1}(a', b', c', |x|^{-2}) + \frac{q}{|x|^{2b}} \, {_2F_1}(a'', b'', c'', |x|^{-2}) $$ for some (explicit) constants $p, q, a', b', c', a'', b'', c''$. In particular, $$ {_2F_1}(a, b, c, -|x|^2) \sim \frac{p}{|x|^{2a}} + \frac{q}{|x|^{2b}} $$ as $|x| \to \infty$. Therefore, as long as $n > 4s$, we have $$ (-\Delta)^s [(1 + |x|^2)^{-2s}] \sim C_{n,s} |x|^{-4s} $$ as $|x| \to \infty$, for some (explicit) constant $C_{n,s}$. Obviously, $(-\Delta)^s [(1 + |x|^2)^{-2s}]$ is continuous, so we immediatiely get $$ |(-\Delta)^s [(1 + |x|^2)^{-2s}]| \leqslant C_{n,s}' (1 + |x|^2)^{-2s} . $$ In order to get a similar lower bound, one needs to check that $C_{n,s} > 0$ (which seems to be the case when $n > 4s$, as long as I did not make any mistake), and that $(-\Delta)^s [(1 + |x|^2)^{-2s}]$ is never zero. I am no expert here, but another DLMF entry seems to suggest that the total number of zeroes of the hypergeometric function is $$\lfloor s \rfloor + \tfrac12 + \tfrac12 \operatorname{sign}(\Gamma(-s) \Gamma(2s) \Gamma(\tfrac n2+s) \Gamma(\tfrac n2-2s)) = 0 ,$$ as long as, again, $n > 4s$.

(Double check for typos.)

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    $\begingroup$ I remember this has been asked here before, but I cannot find the previous question. If you know where it is, feel free to add a link and mark this as a duplicate. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 26, 2021 at 17:03
  • $\begingroup$ Thank you for your answer. Is there a way to estimate these functions from above? $\endgroup$
    – Student
    Commented Jul 26, 2021 at 17:39
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    $\begingroup$ Sure, estimates of $_2F_1$ are well-known. This will be something like $c(n,s) (1+|x|^2)^{-2s}$ (unless $2s > n$, in which case the exponent is $-\tfrac n2-s$ rather than $-2s$). $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 26, 2021 at 19:13
  • $\begingroup$ Thanks for your comment, in my case $n>2s$. Could you suggest me a reference to obtain the estimate $c(n,s)(1+|x|^2)^{-2s}$? $\endgroup$
    – Student
    Commented Jul 28, 2021 at 10:41
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    $\begingroup$ This DLMF entry seems to be an option. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 28, 2021 at 11:05

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