Fuglede's theorem in generalized Orlicz--Sobolev spaces

In this paper, we show that Orlicz–Sobolev spaces $W^{1,\varphi }(\mathrm{\Omega })$ can be characterized with the ACL- and ACC-characterizations. ACL stands for absolutely continuous on lines and ACC for absolutely continuous on curves. Our results hold under the assumptions that $C^1(\mathrm{\Omega })$ functions are dense in $W^{1,\varphi }(\mathrm{\Omega })$, and $\varphi (x,\beta )\ge 1$ for some $\beta >0$ and almost every $x\in \mathrm{\Omega }$. The results are new even in the special cases of Orlicz and double phase growth.