Optimal embedding results for fractional Sobolev spaces

TL;DR

This paper establishes optimal conditions for continuous and compact embeddings of fractional Sobolev spaces, improving classical results without relying on Besov spaces, and applies to both unbounded and bounded Lipschitz domains.

Contribution

It provides new optimal embedding conditions for fractional Sobolev spaces using interpolation, enhancing classical results without involving Besov spaces.

Findings

Enhanced continuous embedding conditions for $W^{s,p}(bla)$.

Improved classical compact embeddings for bounded Lipschitz domains.

Results are proved to be optimal.

Abstract

This paper deals with the fractional Sobolev spaces $W^{s, p}(\Omega)$, with $s\in (0, 1]$ and $p\in[1,+\infty]$. Here, we use the interpolation results in [4] to provide suitable conditions on the exponents $s$ and $p$ so that the spaces $W^{s, p}(\Omega)$ realize a continuous embedding when either $\Omega=\mathbb R^N$ or $\Omega$ is any open and bounded domain with Lipschitz boundary. Our results enhance the classical continuous embedding and, when $\Omega$ is any open bounded domain with Lipschitz boundary, we also improve the classical compact embeddings. All the results stated here are proved to be optimal. Also, our strategy does not require the use of Besov or other interpolation spaces.