Timeline for Extension theory with bump function
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 15, 2011 at 16:21 | comment | added | Nilima Nigam | Yakov, the answer is 'yes'. One defines $H^s(\Omega)$ for any open set as the distributions which are the restrictions to $\Omega$ of some $U\in H^s(\mathbb{R}^n)$,with the norm being the induced norm. This provides the continuous inclusion of $H^s(\Omega) \subset W^{s,2}(\Omega)$ for $s\geq 0$. [To obtain set equality, one needs $\Omega$ to permit an extension operator. Set equality holds for any non-empty open $\Omega$, and $s$ the negative integers.] One then uses Peetre's inequality on the object $U$. MacLean's book has good discussion. | |
| Jun 15, 2011 at 13:50 | comment | added | Yakov Shlapentokh-Rothman | Is approach (b) supposed to work for $\Omega \neq \mathbb{R}^n$? It is not clear to me how the Fourier transform will be useful unless $\Omega = \mathbb{R}^n$, since that is the only case in which the Sobolev spaces are defined directly in terms of the Fourier transform. | |
| Jun 15, 2011 at 7:35 | vote | accept | alext87 | ||
| Jun 15, 2011 at 7:35 | history | bounty ended | alext87 | ||
| Jun 15, 2011 at 4:04 | history | answered | Nilima Nigam | CC BY-SA 3.0 |