Timeline for On the weak convergence of measures and Sobolev spaces with negative index
Current License: CC BY-SA 4.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Mar 28 at 17:57 | vote | accept | sharpe | ||
| Mar 28 at 16:38 | history | edited | Patrick Li | CC BY-SA 4.0 | Fixed labelling issue with one of the concentric radii. |
| Mar 28 at 16:11 | comment | added | Patrick Li | Hi @Hannes, thanks so much for pointing out my massive mistake. Evans' result was addressing Rellich-Kondrachov on bounded open sets $U$, and I blindly followed his proof without picking up on the issue of a noncompact domain. It seems that for norm-convergence on all of $\mathbb{R}^d$, some tightness of the sequence appears to be required. I have modified my argument to assume this and obtain convergence in that specific setting. For OP's original problem, the sequence is tight, so this should still help with their question. | |
| Mar 28 at 16:07 | history | edited | Patrick Li | CC BY-SA 4.0 | Addressed issue with compactness of the embedding. |
| Mar 28 at 8:50 | comment | added | Hannes | @PatrickLi Yes sure, that is what one wants to do, yet $\mathbb{R}^d$ is not not compact, so I suppose you propose to leverage the vanishing at infinity of $C_0$, yes? Would you mind explaining a bit or do you happen to have a reference? | |
| Mar 27 at 23:20 | comment | added | Patrick Li | @Hannes Yep, by Arzelà-Ascoli. | |
| Mar 27 at 16:44 | comment | added | sharpe | Thanks four your answer! I'll take a look at the Evans' book! | |
| Mar 27 at 9:09 | comment | added | Hannes | Wait a second, $C_0^{1-d/q'}(\mathbb{R}^d) \hookrightarrow C_0(\mathbb{R}^d)$ compactly? | |
| Mar 26 at 19:13 | history | edited | Patrick Li | CC BY-SA 4.0 | Corrected typos with placement of dual indices q and q'. |
| Mar 26 at 19:06 | history | edited | Patrick Li | CC BY-SA 4.0 | Gave a more-accurate reference. |
| Mar 26 at 6:42 | history | edited | Patrick Li | CC BY-SA 4.0 | added 55 characters in body |
| Mar 26 at 5:34 | history | edited | Patrick Li | CC BY-SA 4.0 | Fixed issue with tightness argument |
| Mar 26 at 5:21 | history | edited | Patrick Li | CC BY-SA 4.0 | Corrected typo (missing Fourier transform hat on one term) |
| Mar 26 at 5:12 | history | answered | Patrick Li | CC BY-SA 4.0 |