I am working on proving the following: Let $\rho(x)= \frac{2}{2+x^2}$, $\theta >1$ (assumed integer here) and $B \subset H^1\_{ul}$,(uniformly local Sobolev space), be any subset which is bounded in $H\_{ul}^\theta$. Then $B$ is pre-compact in


$$H^1\_\rho = \{ u: \mathbb R \to \mathbb R : u, \partial\_x u \in L^2\_{loc} (\mathbb R),\quad \| u\|\_{H^1\_\rho}=\| u\rho\|\_{H^1} < \infty\}.$$


I need to show that $B$ admits the finite covering in $H^1\_\rho$ by balls of radius less than $\epsilon $. Taking $\chi\_\beta$ as a smooth cut off function and decomposing $u \in B$ into two parts $u= v+u$ where $v= \chi\_\beta u$ and $w= u \chi\_\beta$, and


$$\chi\_\beta(X) = 
\begin{cases}
1,& |x| \ge \beta +1\\
0, & |x| \le \beta
\end{cases}.$$


Taking the following norm


$$\|v\|^2\_{H^1\_\rho}=\|u\chi\_\beta\rho\|^2\_{H^1} =\|u\chi\_\beta\rho(x)\|^2\_{L^2} + \| \partial\_x (u\chi\_\beta\rho)\|^2\_{L^2}.$$


Here I can choose $\beta$ sufficiently large to get the first $L^2$ small as I want but in the second norm after using the product rule I obtain the derivative of the cut-off function which I do not know what it is and how to deal with it? Could enlighten me, please.



First derivative of cut off function
I constructed a function that satisfies what I want. Let $f(x) = e^{-1/x},\,\, x>0$ and vanishes everywhere. Now we define $\varphi\_\beta(x) = f(\beta +1 - |x|)$ and $\psi\_\beta (x)=f(|x|- \beta)$. Therefore the cut-off function is


$$\chi\_\beta(x) = \frac{\psi\_\beta (x)}{\psi\_\beta (x) + \varphi\_\beta (x)}.$$


