Compact embeddings of variable exponent Sobolev, Besov, and Triebel-Lizorkin spaces on metric measure spaces

TL;DR

This paper investigates the conditions under which variable exponent Sobolev, Besov, and Triebel-Lizorkin spaces embed compactly into other spaces on metric measure spaces, addressing an open question and considering symmetry effects.

Contribution

It provides sufficient and necessary conditions for compact embeddings of variable exponent function spaces on metric measure spaces, and explores the impact of isometry group actions.

Findings

Established criteria for compactness of embeddings.

Proved a Berestycki-Lions type theorem for noncompact metric spaces.

Answered an open question by Gorka regarding compactness conditions.

Abstract

We study compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces with variable exponents on both bounded and unbounded metric measure spaces. We establish sufficient conditions for compactness, and under additional assumptions, we show that they are also necessary. Moreover, we investigate the influence of isometry group actions on the compactness of embeddings. In particular, we answer the open question posed by P. G\'orka in [P. G\'orka, Looking For Compactness In Sobolev Spaces On Noncompact etric Spaces, Ann. Acad. Sci. Fenn., Vol 43, 2018, 531-540], proving a Berestycki-Lions type theorem.