Di Nezza, Eleonora; Palatucci, Giampiero; Valdinoci, Enrico Hitchhiker’s guide to the fractional Sobolev spaces. (English) Zbl 1252.46023 Bull. Sci. Math. 136, No. 5, 521-573 (2012). 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. Cited in 1 ReviewCited in 2685 Documents MathOverflow Questions: What is the relationship between Hölder spaces and differentiability? 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 Keywords:fractional Sobolev spaces; Gagliardo norm; fractional Laplacian; nonlocal energy; Sobolev embeddings × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0314.46030 [2] Alberti, G.; Bouchitté, G.; Seppecher, P., Phase transition with the line-tension effect, Arch. Ration. Mech. Anal., 144, 1, 1-46 (1998) · Zbl 0915.76093 [3] Aronszajn, N., Boundary values of functions with finite Dirichlet integral, Tech. Report of Univ. of Kansas, 14, 77-94 (1955) · Zbl 0068.08201 [4] Bates, P. W., On some nonlocal evolution equations arising in materials science, (Nonlinear Dynamics and Evolution Equations. 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