Fractional order Orlicz-Sobolev spaces

In this paper we define the fractional order Orlicz-Sobolev spaces, and prove its convergence to the classical Orlicz-Sobolev spaces when the fractional parameter $s\uparrow 1$ in the spirit of the celebrated result of Bourgain-Brezis-Mironescu. We then deduce some consequences such as $\mathrm{\Gamma }-$convergence of the modulars and convergence of solutions for some fractional versions of the ${\mathrm{\Delta }}_g$ operator as the fractional parameter $s\uparrow 1$.