On the limit as $s\to 1^{-}$ of possibly non-separable fractional Orlicz–Sobolev spaces

Abstract

Extended versions of the Bourgain–Brezis–Mironescu theorems on the limit as $s\to 1^{-}$ of the Gagliardo–Slobodeckij fractional seminorm are established in the Orlicz space setting. Our results hold for fractional Orlicz–Sobolev spaces built upon general Young functions, and complement those of [13], where Young functions satisfying the ${\Delta }_2$ and the ${\nabla }_2$ conditions are dealt with. The case of Young functions with an asymptotic linear growth is also considered in connection with the space of functions of bounded variation.