Limits of fractional Orlicz-Sobolev spaces

Extended versions of the Bourgain-Brezis-Mironescu theorems on the limit as s->1^- of the Gagliardo-Slobodeckij fractional seminorm are established in the Orlicz space setting. The results hold for fractional Orlicz-Sobolev spaces built upon general Young functions, as well. The case of Young functions with an asymptotic linear growth is also considered in connection with the space of functions of bounded variation. 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. The result holds in fractional Orlicz-Sobolev spaces associated with Young functions fulfilling the \Delta_2-condition, and, as shown by counterexamples, it may fail if this condition is dropped. This is a joint work with Andrea Cianchi, Lubos Pick and Lenka Slavikova.