Timeline for Limit case of Sobolev space in $1$-D
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Apr 24, 2020 at 13:04 | comment | added | Jean Van Schaftingen | This implies that (f) is constant (see the paper by Haïm Brezis, “How to recognize constant functions. Connections with Sobolev spaces”). | |
| Apr 8, 2020 at 20:19 | comment | added | Denis Serre | @MichaelRenardy. Indeed, you are right. I had an exchange, after posting the Q, with Petru Mironescu, who confirmed the fact. | |
| Apr 8, 2020 at 19:40 | comment | added | Michael Renardy | If f is smooth with nonzero derivative, the integrand is of order $1/|x-y|$, which is not integrable. I suspect the condition implies that f is constant. | |
| Apr 8, 2020 at 17:24 | answer | added | LL 3.14 | timeline score: 3 | |
| Apr 8, 2020 at 9:40 | comment | added | Hannes | If I see it correctly, then by the "modulus of continuity-characterization" of Besov spaces, $f$ will lie in $B^1_{3,3}(\mathbb{R})$. Unfortunately, this a slightly larger space than $W^{1,3}(\mathbb{R})$. | |
| Apr 7, 2020 at 10:08 | history | edited | YCor |
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| Apr 7, 2020 at 8:41 | history | asked | Denis Serre | CC BY-SA 4.0 |