Maximum principles, Liouville theorem and symmetry results for the fractional $g-$Laplacian

We study different maximum principles for non-local non-linear operators with non-standard growth that arise naturally in the context of fractional Orlicz-Sobolev spaces and whose most notable representative is the fractional $g-$Laplacian: $$(-{\mathrm{\Delta }}_g{)}^{s}u(x):=\text{p.v.}{\int }_{{\mathbb{R}}^n}g\left(\frac{u(x)-u(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^s}\right)\frac{dy}{\mathrm{\mid }x-y{\mathrm{\mid }}^{n+s}},$$ being $g$ the derivative of a Young function. We further derive qualitative properties of solutions such as a Liouville type theorem and symmetry results and present several possible extensions and some interesting open questions. These are the first results of this type proved in this setting.