We all know that, given any bounded open set $\Omega\subset \mathbb R$ $$\frac{1}{x}\in L^p(\Omega)\iff p<1 \qquad \frac{1}{x}\in L^p(\Omega^c)\iff p>1$$
what do these conditions become for a dimension greater than $1?$
Is there a formula to say if $$\frac{1}{|x|}\in L^p(\Omega)$$ for $\Omega\in \mathbb R^n$ in the two cases $0\in \Omega$ and $\Omega$ unbounded?
For $\Omega$ a neighborhood of $0$, say $\Omega=B(0;1)$ in $\mathbb{R}^d$ one has
$$ \int_{B(0;1)}\frac{1}{\|x\|^p}\,dx =\sigma_n\int^1_0\frac{r^{n-1}}{r^p}\,dr=\sigma_n \int^1_0\frac{1}{r^{p-n-1}}\,dr $$
Similarly $$ \int_{B(0;1)^c}\frac{1}{\|x\|^p}\,dx =\sigma_n \int^\infty_1\frac{1}{r^{p-n-1}}\,dr $$
where $\sigma_n$ is the surphace area of the $\mathbb{S}^{n-1}$ sphere. From here you can draw conclusions.