Timeline for If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Current License: CC BY-SA 4.0
5 events
| when toggle format | what | by | license | comment | |
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| Dec 10 at 13:15 | answer | added | Willie Wong | timeline score: 4 | |
| Dec 10 at 4:44 | comment | added | Willie Wong | If $u$ is in $H^2$, we have that $u$ is continuous by Sobolev embedding, and so $|x|^{-1} u$ is in $L^p$ for any $p \in [2,3)$. The 3 here is sharp, as even taking $u$ in $C^\infty_c$ we see that $L^3$ cannot be achieved. Using the Sobolev embedding we see that you can certainly not hope for $\alpha \geq \frac12$, as for such values $H^\alpha$ embeds into $L^3$. | |
| Dec 10 at 1:44 | comment | added | Christian Remling | The FT of $|x|^{-1}$ is $|\xi|^{-2}$ in $d=3$. | |
| Dec 10 at 0:03 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| Dec 9 at 23:57 | history | asked | JZS | CC BY-SA 4.0 |