Suppose I have an integral on the boundary of a Lipschitz domain $\Omega \subset \mathbb{R}^n$ $$\int_{\partial\Omega} f(x-y)dS$$. where $dS$ is the surface element $dS = g(x)dx$.
Can I do a substitution like this: let $z:=x-y$. Then $dz = dx$, so we have $$\int_{\partial\Omega} f(x-y)dS = \int_{\partial\Omega} f(z)dS'$$ where $dS = g(z+y)dz$, because $dS = g(x)dx = g(z+y)dz$. Is this all I need to think about? I guess not, do I need to change the domain of integration too?