Sobolev embedding theorem and subanalytic measures

TL;DR

This paper establishes a Sobolev embedding theorem for measures with subanalytic densities and applies it to the push-forward measure under a subanalytic map, with implications for kernel theory.

Contribution

It proves a new Sobolev embedding theorem for subanalytic measures and explores its applications to kernel theory and Lipschitz functions.

Findings

Established Sobolev embedding for subanalytic measures.

Derived embedding into inner Lipschitz functions.

Applied results to kernel theory.

Abstract

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.