I wondering if you could help me with a problem on integration. I would like to know if the function $\frac{1}{\vert x -y \vert^{n-2}}$ is integrable in $\mathbb{R}^n$. That is, is well defined (in the Lebesgue sense) the integral $\int_{\mathbb{R}^n} \frac{1}{\vert x - y \vert^{n-2}}$ dy where $x\in \mathbb{R}^n$ is given. Is possible to know its result?
Thank in advance!
By translation variance it suffices to look at $x=0$. Then, changing to polar coordinates one is bound to calculate (in the radial direction) the integral (up to a constant) $$ \int_0^{+\infty}\frac{1}{r^{n-2}}r^{n-1}\,dr, $$ which is not convergent (for two reasons: both that the singularity is to strong and the decay is to slow). Thus, your original integral is divergent.
In fact, even if you change the $n-2$ to any other constant, you will not get something that is integrable. Either the function is too singular at $y=x$ or it decays to slowly at infinity.