Timeline for The translation is continuous in $L^1(\mathbb{R}^n,d\mu)$, $d\mu=\frac{1}{1+|y|^{n+a}}dy$,$ a>0$
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| Nov 13, 2020 at 5:47 | comment | added | Ayman Moussa | Well it's continuous and tends to 1 when $|y|\rightarrow +\infty$. I think your question has nothing to do in MO and should have probably be posted on math.stackexchange.com | |
| Nov 12, 2020 at 14:59 | comment | added | inoc | @AymanMoussa How i can prove that, for any $h\in\mathbb{R}^n$, the function $y\mapsto\frac{1+|y|^{n+a}}{1+|y-h|^{n+a}}$ is bounded on the whole $\mathbb{R}^n$? | |
| Nov 12, 2020 at 13:20 | comment | added | Ayman Moussa | 1. you can check that for any $h\in\mathbf{R}^n$, $y\mapsto \frac{1+|y|^{n+a}}{1+|y-h|^{n+a}}$ is bounded and conclude. 2. the second point is done as usual by the density argument (done in any textbooks) | |
| Nov 12, 2020 at 11:25 | history | asked | inoc | CC BY-SA 4.0 |