In this article we define the first Dirichlet eigenvalue for the fractional $g-$Laplacian and we prove diverse properties on it, including isolation, positivity of its eigenfunctions and its behaviour as $s\to 1^+$. In the second part of this manuscript we prove some modular and norm Hardy inequalities in fractional Orlicz-Sobolev spaces, which provide for lower bounds of eigenvalues in certain configurations. 查看全文>>