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 sin (0,1) 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.
Highlights
One of the available notions of Sobolev spaces of fractional order calls into play the Gagliardo–Slobodeckij seminorm
A companion result holds if Rn is replaced by any bounded open set with a sufficiently regular boundary ∂, for any function u ∈ W s,p( ), provided that the seminorm |u|s,p,Rn is replaced by the norm u W s,p( )
We present an application of Theorem 3.1 to a family of Young functions whose behaviour near zero and near infinity is of power-logarithmic type
Summary
One of the available notions of Sobolev spaces of fractional order calls into play the Gagliardo–Slobodeckij seminorm. The fractional Orlicz–Sobolev space, of order s ∈ (0, 1), associated with a Young function A, will be denoted by W s,A( ), and is built upon the Luxemburg type seminorm | · |s,A, given by. A companion result holds if Rn is replaced by any bounded open set with a sufficiently regular boundary ∂ , for any function u ∈ W s,p( ), provided that the seminorm |u|s,p,Rn is replaced by the norm u W s,p( ). Sharp extensions of these Sobolev type inequalities and ensuing embeddings to the spaces W s,A( ) are presented in Sect. A version of Eq (1.6) with Rn replaced by a bounded regular domain can be found in [32]
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