Sobolev embedding theorem and subanalytic measures

We focus on Borel measures that have a globally subanalytic density function. We prove, given such a measure $\mu$ on a set $A$ and a globally subanalytic mapping $\Phi:A\to \Omega$, with $\Omega$ bounded open subset of $\mathbb{R}^n$, a Sobolev embedding theorem for the Sobolev space $W^{k,p}_{\Phi_*\mu}(\Omega)$ of the push-forward measure $\Phi_*\mu$. We derive an embedding of $W^{k,p}_{\Phi_*\mu}(\Omega)$ into the space of inner Lipschitz functions and give an application to kernel theory.