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Hitchhiker’s guide to the fractional Sobolev spaces. (English) Zbl 1252.46023
Summary: 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.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35S30 Fourier integral operators applied to PDEs
35S05 Pseudodifferential operators as generalizations of partial differential operators
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