Timeline for Is Schwartz space $\mathbb R^n$ contained in every fractional Sobolev space on $\mathbb R^n$? [closed]
Current License: CC BY-SA 4.0
13 events
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| May 23, 2022 at 13:12 | history | closed |
Gro-Tsen Michael Renardy Mark Wildon Alexandre Eremenko Amir Sagiv |
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| May 17, 2022 at 8:38 | comment | added | dohmatob | Well, $\int_{\mathbb R^n}\|\nabla f(x)\|^2 dx = \int_{\mathbb R^n}\|z\|^2 |\hat{f}(z)|^2dx \asymp \|f\|_{H^1(\mathbb R^n)}^2$. On the other hand, $f \in \mathcal S(\mathbb R^n)$ implies that $\|\nabla f(x)\|^2 \lesssim (1+\|x\|)^{-m}$ for any $m$. No ? | |
| May 17, 2022 at 8:28 | comment | added | Dirk | Your definition of the $H^1$-norm does not make sense. The first inequality is wrong and the other terms do eben contain $f$. | |
| May 17, 2022 at 6:15 | vote | accept | dohmatob | ||
| May 16, 2022 at 20:45 | history | edited | dohmatob | CC BY-SA 4.0 |
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| May 16, 2022 at 20:32 | comment | added | Michael Renardy | The answer is yes. And it is the Schwartz space, not the Schwarz space. | |
| May 16, 2022 at 19:56 | review | Close votes | |||
| May 23, 2022 at 13:12 | |||||
| May 16, 2022 at 19:20 | answer | added | terceira | timeline score: 4 | |
| May 16, 2022 at 18:55 | history | edited | dohmatob | CC BY-SA 4.0 |
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| May 16, 2022 at 18:52 | comment | added | Willie Wong | If $H^k$ is defined by $\|f\|_{H^k}^2 = \int (1 + |\xi|^2)^k |\hat{f}(\xi)|^2 ~d\xi$, then yes. | |
| May 16, 2022 at 18:36 | comment | added | dohmatob | Laplacian: $Lap(f)(x) := \|\nabla f(x)\|^2$. | |
| May 16, 2022 at 18:21 | comment | added | Daniele Tampieri | Could you clarify what is the meaning of the $Lap$ notation? | |
| May 16, 2022 at 18:09 | history | asked | dohmatob | CC BY-SA 4.0 |