I want to show
$$\Delta \frac{1}{n(n-1)\alpha(n)}\int_{\mathbb{R}^n}\frac{f(y)}{|x-y|^{n-2}}dy=f(x)$$
Where the $\alpha(n)$ is the volume of $n-$dimension unit ball.
I tried using the sphere coordination and failed. I think maybe it's relevant to Fourier transform somehow.
Thanks in advance.