Timeline for Explicit expression for the fractional Laplacian of $1/(1+|x|^2)^s$
Current License: CC BY-SA 4.0
15 events
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| Aug 7, 2021 at 11:45 | comment | added | Mateusz Kwaśnicki | Not sure if I understand the question: if $f$ is continuous on $[0,\tfrac12]$, then it is bounded on this interval, and hence $|f(|x|^{-2})|$ is bounded on $\mathbb R^n \setminus B(0,2)$. Combined with the hypergeometric identity from the answer, this gives $|{_2F_1}(a,b,c,-|x|^2)|\leqslant C (|x|^{-2a} + |x|^{-2b})$ when $|x|\geqslant 2$. And, of course, ${_2F_1}(a,b,c,-|x|^2)$ is bounded on $B(0, 2)$. Does this answer your question? | |
| Aug 7, 2021 at 11:31 | comment | added | Student | Sure will do, I had a quick question though. I don't the inequality is deduced from the continuity of the function. Could you please elaborate on that? | |
| Aug 7, 2021 at 10:32 | comment | added | Mateusz Kwaśnicki | Great, thanks! I just corrected the one-half typo. If you spot anything else feel free to edit the post. | |
| Aug 7, 2021 at 10:31 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Aug 7, 2021 at 9:30 | comment | added | Student | I did the computations and indeed the constant $C_{n,s}>0$ and the number of zeros is equal to $0$, one just needs to fix the expression for the number of zeros by adding 1/2 to the identity. | |
| Jul 30, 2021 at 9:40 | vote | accept | Student | ||
| Jul 28, 2021 at 18:53 | comment | added | Student | Thank you very much for your detailed answer. This is really helpful! | |
| Jul 28, 2021 at 18:21 | comment | added | Mateusz Kwaśnicki | A added a comment that addresses your other questions. | |
| Jul 28, 2021 at 18:20 | history | edited | Mateusz Kwaśnicki | CC BY-SA 4.0 |
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| Jul 28, 2021 at 11:05 | comment | added | Mateusz Kwaśnicki | This DLMF entry seems to be an option. | |
| Jul 28, 2021 at 10:41 | comment | added | Student | Thanks for your comment, in my case $n>2s$. Could you suggest me a reference to obtain the estimate $c(n,s)(1+|x|^2)^{-2s}$? | |
| Jul 26, 2021 at 19:13 | comment | added | Mateusz Kwaśnicki | 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 $-2s$). | |
| Jul 26, 2021 at 17:39 | comment | added | Student | Thank you for your answer. Is there a way to estimate these functions from above? | |
| Jul 26, 2021 at 17:03 | comment | added | Mateusz Kwaśnicki | 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. | |
| Jul 26, 2021 at 17:02 | history | answered | Mateusz Kwaśnicki | CC BY-SA 4.0 |