On mappings generating embedding operators in Sobolev classes on metric measure spaces

TL;DR

This paper investigates how certain homeomorphisms between metric measure spaces induce embedding operators in Sobolev spaces, leading to new Sobolev embedding theorems for various bounded domains.

Contribution

It introduces conditions under which homeomorphisms generate embedding operators in Sobolev classes on metric measure spaces, extending Sobolev embedding results.

Findings

Characterization of homeomorphisms generating embedding operators

Sobolev embedding theorems for bounded domains in metric measure spaces

Conditions for Sobolev space embeddings in complex geometries

Abstract

In this article, we study homeomorphisms $\varphi: \Omega \to \widetilde{\Omega}$ that generate embedding operators in Sobolev classes on metric measure spaces $X$ by the composition rule $\varphi^{\ast}(f)=f\circ\varphi$. In turn, this leads to Sobolev type embedding theorems for a wide class of bounded domains $\widetilde{\Omega}\subset X$.