I have a question about an example of functions in Sobolev space. But I think you can give a hint without knowing the Sobolev space because I just want to know how to integrate a function with n-dimensional domain.
Question : If $u(x)=|x|^{-\alpha}$, $x \in U=$ the open punctured unit ball in $\bf{R}^n$, then for which values of $\alpha >0, n, p$ does $u$ belong to $W^{1,p}(U)$?
Clearly, $u$ is locally summable. So I want to show that $||u||_p+\sum \int_U|u_{x_i}|^pdx$ is finite.
But I don't know how to integrate a function over $U$. I tried the iterated integral but it is so cumbersome. Please give me any hints. Thanks in advance.