On a Characterization of the Rellich-Kondrachov Theorem on Groups

TL;DR

This paper explores the Sobolev spaces on groups, establishing an equivalence with solutions to variational problems and characterizing the Rellich-Kondrachov Theorem to aid in solving eigenvalue problems in this context.

Contribution

It introduces a characterization of the Rellich-Kondrachov Theorem for Sobolev spaces on groups, linking group structure with variational problem solutions.

Findings

Established an equivalence between locally compact Abelian groups and solution spaces.

Provided conditions characterizing the Rellich-Kondrachov Theorem on groups.

Applied results to eigenvalue-eigenfunction problems in product spaces.

Abstract

Motivated by an eigenvalue-eigenfunction problem posed in IR^n x {\Omega}, where {\Omega} is a probability space, we are concerned in this paper with the Sobolev space on groups. Hence it is established an equivalence between locally compact Abelian groups and the space of solutions to the associated variational problem. Then, we study some conditions which characterize in a precisely manner the Rellich-Kondrachov Theorem, the principal ingredient to solve the variational problem.