Timeline for Continuous embedding between parabolic Sobolev spaces
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 27, 2019 at 13:01 | comment | added | Hannes | Yep, that's correct. (Watch out that $\theta \in (0,1)$ only though in the interpolation context! You'd need to replace $H^2$ by $H^k$ for $k$ large enough to get the embedding for all, also noninteger, smoothness parameters.) | |
| Jun 27, 2019 at 12:18 | comment | added | John | The above embedding is mentioned in Adams book for $\theta$ being an integer, i.e., $B^m_{p,1}(\Omega)\hookrightarrow W^m_{p}(\Omega)$, and the relation between real and complex interpolation spaces suggests it is true for arbitrary real $\theta\in (0,\infty)$. Is my understanding correct? | |
| Jun 27, 2019 at 12:10 | comment | added | John | I found in Adams' book that $B^{2\theta}_{2,q}(\Omega)=(L^2(\Omega),W^{2}_{2}(\Omega))_{\theta,q}$ for $q\ge 1$. So essentially, the relation between real and complex interpolation spaces suggests that $B^{2\theta}_{2,1}(\Omega)\hookrightarrow H^{2\theta}(\Omega)$. Is it correct? | |
| Jun 27, 2019 at 11:28 | vote | accept | John | ||
| Jun 27, 2019 at 8:17 | comment | added | Hannes | @John I added some explanations, let me know if you need more. | |
| Jun 27, 2019 at 8:07 | history | edited | Hannes | CC BY-SA 4.0 |
Explained the interpolation step more
|
| Jun 27, 2019 at 0:53 | comment | added | John | Thank you for pointing out the reference! I am not quite familiar with the interpolation space. May you kindly suggest a reference about the connection between $(L^2(\Omega),H^2(\Omega))_{\theta,1}$ and the classical Sobolev space? Wiki only states that for $s\in(k,k+1)$, $W^{s,p}(\Omega)=(W^{k,p}(\Omega),W^{k+1,p}(\Omega))_{\theta,p}$ for certain $\theta$. | |
| Jun 26, 2019 at 11:31 | history | answered | Hannes | CC BY-SA 4.0 |