Weak Differentiability for the Absolute Value Function on the Unitball

Question

Could use some help here, as I am not familiar with the concept of weak differentiability enough.

Let $A = B_1(0) \subset \mathbb{R}^n$ be the unit ball, $\alpha \in \mathbb{R}, \alpha \neq 0$, $f(x) = \vert x \vert^\alpha, \, f(0) = 0$, then $f$ is weakly differentiable on $A$ iff $\alpha > -\left( n - 1 \right)$ for $n \geq 2$.

Answer 1

$f$ is weakly differentiable if it defines a regular distribution. So it should be in $L_{loc}^1(B(0,1))$. We passe to the polar coordinates we can see then that $f$ is weakly differentiable if and only if $\int\limits_0^1 {{r^\alpha }{r^{n - 1}}dr} $ converges.