On the Limit as s -> 0+of Fractional Orlicz-Sobolev Spaces

Abstract

An extended version of the Maz'ya-Shaposhnikova theorem on the limit as s -> 0(+) of the Gagliardo-Slobodeckij fractional seminorm is established in the Orlicz space setting. Our result holds in fractional Orlicz-Sobolev spaces associated with Young functions satisfying the Delta(2)-condition, and, as shown by counterexamples, it may fail if this condition is dropped.

Keywords

• Fractional Orlicz-Sobolev space
• Limits of smoothness parameters