Fractional Orlicz-Sobolev embeddings

On the Maz'ya-Shaposhnikova formula in Orlicz-Sobolev spaces on Carnot groups

“…The proof of Theorem 1.1 follows the path tracked in [1], which in turn is based on the original argument used in [23]. We notice that as a byproduct of the lower bound of the liminf (see Lemma 3.1), we are able to recover a Hardy-type inequality for small enough s (see (3.6)), akin to the one proved in [2]. We also stress that, coherently with our previous results in [10], the proof of Theorem 1.1 actually works if the homogeneous norm there considered satisfies the classical triangular inequality…”

Maximum principles, Liouville theorem and symmetry results for the fractional $g-$Laplacian

“…This operator was introduced in [14] and has received an increasing attention in the last years since it allows to model non-local problems obeying a non-power behavior. See for instance [1,2,3,4,15,12,21,20] and the references therein.…”

“…Throughout the paper, solutions of equations involving the fractional g−Laplacian will be assumed to be of class C 1,1 loc (R n ) ∩ L g , being L g the tail space defined as…”

Boundedness of functions in fractional Orlicz-Sobolev spaces

“…A sound theory of fractional Orlicz-Sobolev spaces -a natural functional framework for these problems -is of course crucial in connection with their study. Properties of fractional Orlicz-Sobolev spaces are the subject of the contributions [2,3,4,5,10,16,31,39]. They provide extensions of some aspects of the theory of the classical fractional Sobolev spaces, which has been developed over the years -see e.g.…”