Calculus of Variations and Geometric Measure Theory

L. Brasco - A. Salort

A note on homogeneous Sobolev spaces of fractional order

created by brasco on 20 Apr 2018
modified on 31 Dec 2018


Accepted Paper

Inserted: 20 apr 2018
Last Updated: 31 dec 2018

Journal: Ann. Mat. Pura Appl.
Pages: 33
Year: 2018


We consider a homogeneous fractional Sobolev space obtained by completion of the space of smooth test functions, with respect to a Sobolev--Slobodecki\u{\i} 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.

Keywords: Poincare inequality, fractional Sobolev spaces, Nonlocal operators, real interpolation