Hitchhiker's guide to the fractional Sobolev spaces

TL;DR

This paper explores the properties and relationships of fractional Sobolev spaces, providing original proofs of embeddings, extension domains, and regularity results without using interpolation or Besov space theory.

Contribution

It offers new proofs of fundamental properties of fractional Sobolev spaces, avoiding traditional interpolation methods and including counterexamples in non-Lipschitz domains.

Findings

Continuous and compact embeddings established

Extension domain regularity analyzed

Counterexamples in non-Lipschitz domains provided

Abstract

This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.