On the modulus of continuity of fractional Orlicz-Sobolev functions

Abstract

Necessary and sufficient conditions are presented for a fractional Orlicz-Sobolev space on \({\mathbb {R}}^n\) to be continuously embedded into a space of uniformly continuous functions. The optimal modulus of continuity is exhibited whenever these conditions are fulfilled. These results pertain to the supercritical Sobolev regime and complement earlier sharp embeddings into rearrangement-invariant spaces concerning the subcritical setting. Classical embeddings for fractional Sobolev spaces into Hölder spaces are recovered as special instances. Proofs require novel strategies, since customary methods fail to produce optimal conclusions.