Bessel Potential Spaces and Complex Interpolation: Continuous embeddings

TL;DR

The paper addresses identifying Bessel potential spaces $H^{s,p}(\,\mathbb{R}^n\,)$ as complex interpolation spaces between $L^p(\,\mathbb{R}^n\,)$ and $W^{1,p}(\,\mathbb{R}^n\,)$, and derives core properties using abstract interpolation theory. It proves a direct norm-equivalence $H^{s,p}(\,\mathbb{R}^n\,) = [L^p(\,\mathbb{R}^n\,), W^{k,p}(\,\mathbb{R}^n\,)]_\theta$ with $s=k\theta$ and shows density, lifting, and reiteration results within this framework. The work further establishes continuous embeddings, including fractional Sobolev embeddings into $L^{p_s^*}$ and critical-case embeddings into $ extbf{BMO}$, and clarifies the relation between Bessel and Gagliardo spaces, notably $H^{s,2}=W^{s,2}$ in the Hilbert case. It also discusses contiguity and nesting between these spaces and poses an open problem on interpolation for the case $p=1$, setting the stage for subsequent investigations linking Bessel spaces to the Riesz fractional gradient and compactness properties.

Abstract

Bessel potential spaces, introduced in the 1960s, are derived through complex interpolation between Lebesgue and Sobolev spaces, making them intermediate spaces of fractional differentiability order. Bessel potential spaces have recently gained attention due to their identification with the Riesz fractional gradient. This paper explores Bessel potential spaces as complex interpolation spaces, providing original proofs of fundamental properties based on abstract interpolation theory. Main results include a direct proof of norm equivalence, continuous embeddings, and the relationship with Gagliardo spaces.

Theorems & Definitions (40)

• Theorem 2.1: Retraction Theorem
• Theorem 2.2: Properties of real interpolation spaces
• Theorem 2.3: Reiteration theorem for the real method
• Theorem 2.5: Properties of Complex interpolation spaces
• Theorem 2.6
• Definition 2.8
• Lemma 2.9
• Theorem 2.10
• Remark 2.11
• Definition 3.1