Timeline for About radial Sobolev inequality (Strauss Lemma)
Current License: CC BY-SA 4.0
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| Apr 6, 2022 at 5:49 | comment | added | Tao | Thank you so much. The original question is about whether I can get less power of gradient term to control the behavior of radial function at infinity (at cost of less decay: to be honest, in my setting, I do not care much about how much decay it will give us even if $|x|^{-\varepsilon}$ is enough), and the "$\beta< \frac{1}{2}$" part in your answer seems completely solved my question. I am also looking forward to your answer about whether we can get stronger decay condition at infinity if you have figured it out. | |
| Apr 6, 2022 at 2:40 | vote | accept | Tao | ||
| Apr 5, 2022 at 23:56 | history | edited | Willie Wong | CC BY-SA 4.0 |
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| Apr 5, 2022 at 23:11 | comment | added | Willie Wong | Incidentally, in your question you restricted to $|x| > 1$... in this case smaller $\alpha$ is strictly dominated by larger $\alpha$. If your goal is to get stronger decay condition at infinity, you want larger $\alpha$ and not smaller. I am pretty sure there are counterexamples for the estimate when $\beta < \frac12$; will post a second answer if I figure it out. | |
| Apr 5, 2022 at 22:55 | history | answered | Willie Wong | CC BY-SA 4.0 |