Timeline for Sobolev space is spanned by distributions supported on half-lines?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Aug 28, 2022 at 21:51 | comment | added | Tian Xia | It just occurred to me writing the previous comment that this is related to the question of zero-extension for members of $H^{s}_0$. | |
| Aug 28, 2022 at 21:49 | comment | added | Tian Xia | A related question is whether $H^s_- + H^s_+$ is closed in $H^s$. This is equivalent to these closed subspaces (quotient by intersect) having a positive gap. In the case $s > -1/2$, another equivalent statement is $H^s(\mathbb{R})$ and $H^s([0,\infty))$ induce the same topology on $H^s_+$. If they do, then $H^s_+$ can be taken as an alternative definition of $H^s_0([0,\infty))$. In view of Prof. Remling's answer and $H^{1/2}_0([0,\infty)) = H^{1/2}([0,\infty))$, I suspect that $H^{1/2}_+ \neq H^{1/2}_0([0,\infty))$. | |
| Aug 27, 2022 at 21:13 | answer | added | Christian Remling | timeline score: 3 | |
| Aug 27, 2022 at 19:38 | comment | added | Christian Remling | Since everything is a function, the only way to obtain such a decomposition $u=u_++u_-$ is the obvious one, by restricting to the half lines. But this is the same as applying (something like) $1\pm H$ to $\widehat{u}$, $H$ denoting the Hilbert transform. So your question is closely related to this one: mathoverflow.net/questions/378388/… | |
| Aug 26, 2022 at 20:41 | history | edited | Glorfindel | CC BY-SA 4.0 |
added 14 characters in body
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| S Aug 26, 2022 at 20:01 | review | First questions | |||
| Aug 26, 2022 at 20:41 | |||||
| S Aug 26, 2022 at 20:01 | history | asked | Tian Xia | CC BY-SA 4.0 |