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

Michał Dymek

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. Górka in [P. Górka, 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.

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

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. Górka in [P. Górka, 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.
Paper Structure (19 sections, 38 theorems, 206 equations)

This paper contains 19 sections, 38 theorems, 206 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Metric measure spaces
  4. Measure-preserving isometries
  5. The space of measurable functions
  6. Variable exponent Lebesgue spaces
  7. Mixed Lebesgue sequence spaces
  8. Variable exponent Sobolev, Triebel-Lizorkin and Besov spaces on metric spaces
  9. Hölder space with variable exponent
  10. Main Results
  11. Compact embeddings on metric spaces with finite or integrable measure
  12. Embeddings into the space of measurable functions
  13. Embeddings into variable exponent Lebesgue spaces
  14. Rellich-Kondrachov type compactness theorem
  15. Compact embeddings of group invariant Besov and Triebel-Lizorkin spaces
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $(X,d)$ be a metric space and $r\in (0,\infty)$. The following statements are equivalent:

Theorems & Definitions (66)

  • Lemma 2.1
  • Lemma 2.2
  • Definition 2.1
  • Example 1
  • Example 2
  • Theorem 2.3
  • Lemma 2.4
  • Proposition 2.5
  • Lemma 2.6
  • Lemma 2.7
  • ...and 56 more