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Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces

Ryan Alvarado, Przemysław Górka, Artur Słabuszewski

Abstract

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.

Compact embeddings of Sobolev, Besov, and Triebel-Lizorkin spaces

Abstract

We establish necessary and sufficient conditions guaranteeing compactness of embeddings of fractional Sobolev spaces, Besov spaces, and Triebel-Lizorkin spaces, in the general context of quasi-metric-measure spaces. Although stated in the setting of quasi-metric spaces, the main results in this article are new, even in the metric setting. Moreover, by considering the more general category of quasi-metric spaces we are able to obtain these characterizations for optimal ranges of exponents that depend (quantitatively) on the geometric makeup of the underlying space.

Paper Structure

This paper contains 20 sections, 50 theorems, 246 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The class of quasi-metric spaces
  4. Fractional Sobolev, Triebel--Lizorkin, and Besov spaces
  5. Topological and measure-theoretic considerations
  6. Separability and total boundedness in quasi-metric-measure spaces
  7. Doubling-type conditions for measures
  8. Integrable measures
  9. Borel regularity and Lusin's theorem
  10. Total boundedness in $L^p$ spaces with $p\geq0$
  11. Compact embeddings of $M^{\alpha}_{p,q}$ and $N^{\alpha}_{p,q}$ into $L^{\tilde{p}}$ spaces
  12. Embeddings into $L^0$
  13. Embeddings into $L^p$
  14. Sufficient Conditions
  15. Necessary Conditions
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(X,d,\mu)$ be a quasi-metric-measure space and suppose that $\mu$ is integrable, in the sense that for each fixed $r\in(0,\infty)$, the mapping $x\mapsto\mu(B_d(x,r))$ is measurableIf $(X,d)$ is a metric space then the mapping $x\mapsto\mu(B_d(x,r))$ is always measurable. and where $B_d(x,r)\subseteq X$ denotes the (quasi-)metric ball with center $x\in X$ and radius $r\in(0,\infty)$. Then, f

Figures (1)

  • Figure 1: The distance between two points in the infinite comb

Theorems & Definitions (118)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 108 more