Timeline for When is $W^{1,p}(\Omega)$ a Banach algebra?
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Sep 20, 2023 at 7:13 | vote | accept | Bogdan | ||
| Sep 19, 2023 at 9:35 | comment | added | Romain Gicquaud | This is proven in the book by Adams and Fournier, "Sobolev spaces" | |
| Sep 19, 2023 at 9:32 | answer | added | Michele Caselli | timeline score: 4 | |
| Sep 18, 2023 at 4:29 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
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| Sep 17, 2023 at 1:23 | comment | added | Terry Tao | A general rule of thumb (but not a formal theorem) is that a Sobolev space should be a Banach algebra iff it embeds into $L^\infty$ (because the spectrum should coincide with the essential range). | |
| Sep 16, 2023 at 23:04 | comment | added | Deane Yang | The point is that if $p > d$, $$ W^{1,p}(\Omega) \subset L^\infty(\Omega)$$ and there exists $C > 0$ such that $$ \|f\|_\infty \le C\|f\|_{W^{1,p}}. $$ Therefore, \begin{align*} \|\nabla(fg)\|_{W^{1,p}} &= \|f\nabla g + g\nabla f\|_{W^{1,p}}\\ & \le \|f\|_\infty\|g\|_{W^{1,p}} + \|g\|_{\infty}\|f\|_{W^{1,p}} \\ & \le 2C\|f\|_{W^{1,p}}\|g\|_{W^{1,p}}. \end{align*} I do not believe you can do better than that and can find counterexamples. | |
| Sep 16, 2023 at 20:09 | history | asked | Bogdan | CC BY-SA 4.0 |