Trying to determine when $f(x)=|x|^{-\lambda}\in W^{1,p}(B)$ where $B\subset\mathbb{R}^n$ is the unit ball and $\lambda >0$. I've computed the distributional derivatives as $\partial_i f(x)=-\lambda x_i |x|^{-\lambda-2}$. It remains to check when $\partial_i f(x)\in L^p$. I note that $\partial_i f(x)<-\lambda |x|^{-\lambda-2}$ so I can use the fact that
$f\in L^p \Leftrightarrow \sum_{k\in\mathbb{Z}}2^{kp}m(|f|>2^k)$
Of course, this gives a weaker result which may not account for all cases. I'm wondering if there's a better way to go about this?
If a function $u$ is actually a radially symmetric function, then we have the integration formula $$ \int_{B(0,R)} u(x)\, dx = |\mathbb{S}^{n-1}| \int_0^R u(r)r^{n-1}\, dr, $$ where $n$ is the dimension of the space where $u$ is defined. In your case, $u(x)=|x|^{-\lambda}$, and it is straightforward to apply the formula and check the integrability of $u$.