Inclusion relations among fractional Orlicz-Sobolev spaces and a Littlewood-Paley characterization

Embeddings among fractional Orlicz-Sobolev spaces with different smoothness are characterized. The equivalence of their Gagliardo-Slobodeckij norms to norms defined via Littlewood-Paley decompostions, via oscillations, or via Besov type difference quotients is also established. These equivalences, of independent interest, are a key tool in the proof of the relevant embeddings. They also rest upon a new optimal inequality for convolutions in Orlicz spaces.