On fractional Orlicz–Sobolev spaces
Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková
Abstract
Abstract Some recent results on the theory of fractional Orlicz–Sobolev spaces are surveyed. They concern Sobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, and criteria for compact embeddings. The limits of these spaces when the smoothness parameter $$s\in (0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>s</mml:mi><mml:mo>∈</mml:mo><mml:mo>(</mml:mo><mml:mn>0</mml:mn><mml:mo>,</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:math> tends to either of the endpoints of its range are also discussed. This note is based on recent papers of ours, where additional material and proofs can be found.