Compact embeddings of Bessel Potential Spaces

TL;DR

The paper addresses the problem of compact embeddings for Bessel potential spaces $H^{s,p}({\mathbb R}^n)$ into Lebesgue and Hölder spaces on bounded Lipschitz domains, with a focus on the subcritical, critical, and supercritical regimes determined by $sp$ relative to $n$. It develops and compares three proof strategies: (i) an abstract interpolation-theory approach leveraging complex/real methods and endpoint compactness, (ii) a translation-estimate approach based on the Riesz fractional gradient and the Fréchet–Kolmogorov theorem, and (iii) a contiguity-based approach using the relation between Bessel and Gagliardo spaces. The main result establishes compact embeddings $H^{s,p}({\mathbb R}^n)\hookrightarrow\hookrightarrow L^q(\Omega)$ for $sp<n$ and $q<p_s^*$, $sp=n$ giving $L^q(\Omega)$ for $q<\infty$, and $sp>n$ giving $C^{0,\mu}(\overline{\Omega})$ for $0<\mu<\mu_s^*$; these extend prior continuous-embedding results and provide robust tools for fractional PDE analysis. The work deepens the functional-analytic understanding of Bessel potential spaces, offers multiple proof techniques, and has implications for regularity theory and variational problems in fractional PDEs.

Abstract

Bessel potential spaces have gained renewed interest due to their robust structural properties and applications in fractional partial differential equations (PDEs). These spaces, derived through complex interpolation between Lebesgue and Sobolev spaces, are closely related to the Riesz fractional gradient. Recent studies have demonstrated continuous and compact embeddings of Bessel potential spaces into Lebesgue spaces. This paper extends these findings by addressing the compactness of continuous embeddings from the perspective of abstract interpolation theory. We present three distinct proofs, leveraging compactness results, translation estimates, and the relationship between Gagliardo and Bessel spaces. Our results provide a deeper understanding of the functional analytic properties of Bessel potential spaces and their applications in fractional PDEs.

Theorems & Definitions (28)

• Theorem 1.1: Compact embeddings for Bessel potential spaces
• Theorem 2.1: Krasnoselskii's theorem
• Theorem 2.2: Lions-Peetre
• Theorem 2.3: Functorial first and second Lions-Petree Theorem
• Corollary 2.4: Functorial Lions-Peetre Third Theorem
• Theorem 2.5: Hayakawa
• Theorem 2.6: Cobos-Edmunds-Potter
• Theorem 2.7: Cobos-Edmunds-Fernández-Potter
• Theorem 2.8: Cwikel
• Theorem 2.9: Cobos-Kühn-Schonbek