Four proofs of cocompacness for Sobolev embeddings

TL;DR

This paper presents four different proofs demonstrating the cocompactness property of Sobolev embeddings, using methods from PDE, potential theory, and harmonic analysis, highlighting the structural behavior of bounded sequences.

Contribution

It provides a comprehensive exposition of multiple proofs of cocompactness for Sobolev embeddings, connecting various mathematical techniques.

Findings

Cocompactness implies structured bubble decompositions for bounded sequences.

Four distinct proofs of cocompactness are detailed.

Methods include classical PDE, potential theory, and harmonic analysis.

Abstract

Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the domain space) have subsequences that can be represented as a sum of a well-structured "bubble decomposition" (or defect of compactness) plus a remainder vanishing in the target space. This note is an exposition of different proofs of cocompactness for Sobolev-type embeddings, which employ methods of classical PDE, potential theory, and harmonic analysis.