Sobolev embedding theorem and subanalytic measures

Abstract

We focus on Borel measures that have a globally subanalytic density function. We prove, given such a measure $\hat{I}¼$ on a set $A$ and a globally subanalytic mapping $\hat{I}¦:A\to \hat{I}\text{c◯}$, with $\hat{I}\text{c◯}$ bounded open subset of ${\mathbb{R}}^n$, a Sobolev embedding theorem for the Sobolev space $W_{\hat{I}{¦}_{\ast }\hat{I}¼}^{k,p}(\hat{I}\text{c◯})$ of the push-forward measure $\hat{I}{¦}_{\ast }\hat{I}¼$. We derive an embedding of $W_{\hat{I}{¦}_{\ast }\hat{I}¼}^{k,p}(\hat{I}\text{c◯})$ into the space of inner Lipschitz functions and give an application to kernel theory.