Compact embeddings of Bessel Potential Spaces

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.