Compactness of Sobolev embeddings and decay of norms

TL;DR

This paper explores the conditions under which Sobolev space embeddings into rearrangement-invariant spaces are compact, revealing surprising cases where enlarging the target space does not lead to compactness.

Contribution

It introduces new methods to analyze the noncompactness of Sobolev embeddings by enlarging the target space or the associated Marcinkiewicz space.

Findings

Compactness depends on the decay rate of measures on balls.

Enlarging the target space does not guarantee compactness.

Two approaches to demonstrate noncompactness are developed.

Abstract

We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed of its decay on balls. We show that the gateway to compactness of such embeddings, while formally describable by means of optimal embeddings and almost-compact embeddings, is quite elusive. It is known that such a Sobolev embedding is not compact when its target space has the optimal fundamental function. We show that, quite surprisingly, such a target space can actually be "fundamentally enlarged", and yet the resulting embedding remains noncompact. In order to do that, we develop two different approaches. One is based on enlarging the optimal target space itself, and the other is based on enlarging the Marcinkiewicz space corresponding to the optimal fundamental function.