Consider some bounded domain $\Omega\subset \mathbb{R}^n$, s.t. the boundary $\partial\Omega$ is a smooth submanifold of dimension $(n-1)$ and fix some point $x_0\in \partial\Omega$. Now I want to prove, that
$$\int\limits_{\partial\Omega}\frac{1}{\Vert x-x_0 \Vert^{n-\frac{3}{2}}}d\sigma(x)<\infty,$$ whereas $d\sigma(x)$ denotes the standard induced measure on $\partial\Omega$.
I'm wondering if one can rewrite this integral as a volume integral over some bounded domain. Because the function under the integral sign is locally integrable with respect to the Lebesgue measure. But I'm worried about the singularity at the boundary point $x_0$...
Would you be so kind and help me?
After a translation and a rotation, we can assume without lost of generality that $x_0=0$ and there is a function $g:(-\alpha,\alpha)^{N-1}\to \mathbb{R}$ such that $\partial\Omega \cap B(0,r)$ is the graph of $g$ for some $r>0$ and $g(0)=0$, $g'(0)=0$.
Let $x'=(x_1,x_2,...,x_{N-1})$, $x=(x',x_N)$ and $Q(x)=(x',g(x')+x_n)$ for $x'\in (-\alpha,\alpha)^{N-1}$ and $x_n\in (-\beta,\beta)$. We have that $$\int_{\partial\Omega \cap B(0,r)}\frac{1}{|y|^{N-3/2}}d\sigma=\int_{(-\alpha,\alpha)^{N-1}}\frac{1}{|Q(x',0)|^{N-3/2}}\left(1+\sum_{i=1}^{N-1}\left(\frac{\partial g(x')}{\partial x'_i}\right)^2\right)^{1/2}dx'\tag{1}$$
By one hand, there is a constanct $C>0$ such that $$\left(1+\sum_{i=1}^{N-1}\left(\frac{\partial g(x')}{\partial x'_i}\right)^2\right)^{1/2}\le C,\ \forall\ x'\in (-\alpha,\alpha)^{N-1}.\tag{2}$$
On the other hand $$|(x',0)|\le |Q(x',0)|,\ x'\in (-\alpha,\alpha)^{N-1}.\tag{3}$$
Therefore, combining $(1)$-$(3)$, we conlcude that $$\int_{\partial\Omega \cap B(0,r)}\frac{1}{|y|^{N-3/2}}d\sigma \le C\int_{(-\alpha,\alpha)^{N-1}}\frac{1}{|x'|^{N-3/2}}dx'.$$
From here, as you have said in you post, the conclusion is straigthforward.