A note on homogeneous Sobolev spaces of fractional order

  • Lorenzo BrascoEmail author
  • Ariel Salort


We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev–Slobodeckiĭ norm. We compare it to the fractional Sobolev space obtained by the K-method in real interpolation theory. We show that the two spaces do not always coincide and give some sufficient conditions on the open sets for this to happen. We also highlight some unnatural behaviors of the interpolation space. The treatment is as self-contained as possible.


Nonlocal operators Fractional Sobolev spaces Real interpolation Poincaré inequality 

Mathematics Subject Classification

46E35 46B70 



The first author would like to thank Yavar Kian and Antoine Lemenant for useful discussions on Stein’s and Jones’ extension theorems. Simon Chandler-Wilde is gratefully acknowledged for some explanations on his paper [11]. This work started during a visit of the second author to the University of Ferrara in October 2017.


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© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di FerraraFerraraItaly
  2. 2.Departamento de MatemáticaFCEN Universidad de Buenos Aires and IMAS, CONICETBuenos AiresArgentina

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