Boundedness of functions in fractional Orlicz-Sobolev spaces

A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^{\mathrm{\infty }}({\mathbb{R}}^n)$ fails. They also augment a classical embedding theorem for standard fractional Sobolev spaces.