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 $\mathrm{\Phi }:A\to \mathrm{\Omega }$, with $\mathrm{\Omega }$ bounded open subset of ${\mathbb{R}}^n$, a Sobolev embedding theorem for the Sobolev space $W_{{\mathrm{\Phi }}_{\ast }\mu }^{k,p}(\mathrm{\Omega })$ of the push-forward measure ${\mathrm{\Phi }}_{\ast }\mu$. We derive an embedding of $W_{{\mathrm{\Phi }}_{\ast }\mu }^{k,p}(\mathrm{\Omega })$ into the space of inner Lipschitz functions and give an application to kernel theory.