How to show this integral on boundary of Lipschitz domain is finite?

Question

Sorry for asking a basic question but this did not get answered on M.SE.

Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that $$\int_{\partial\Omega} \frac{1}{|y|^{n-2}} dS(y) < \infty$$ where $dS$ is the surface measure.

My problem is that there is no easy way (for me) to convert this integral via chart maps etc because it's a Lipschitz domain (and not a graph which would be easy). If the integral were over an open subset of $\mathbb{R}^d$ (i.e. a normal integral) then it's easy to show using polar coordinates.

Answer 1

The integral over the complement of any nbd of the origin is finite. You can bound the contribution to the integral in a nbd of the origin, changing variable with a bi-lipschitz local chart: This reduces to the case of a flat boundary, that gives a finite value. Note that the area formula is not even needed, but only the elementary inequality.

Answer 2

You can use the dyadic decomposition. The integral is controlled by $$\sum_{k=0}^{+\infty}2^{k(n-2)}\mbox{Area}(\partial\Omega\cap (B_{2^{-k+1}}\setminus B_{2^{-k}})).$$ The next problem is to give an upper bound of $\mbox{Area}(\partial\Omega\cap B_{2^{-k+1}})$.