Fuglede's theorem in generalized Orlicz--Sobolev spaces

In this paper, we show that Orlicz--Sobolev spaces $W^{1,\phi}(\Omega)$ 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 $C^1(\Omega)$ functions are dense in $W^{1,\phi}(\Omega)$, and $\phi(x,\beta) \geq 1$ for some $\beta > 0$ and almost every $x \in \Omega$. The results are new even in the special cases of Orlicz and double phase growth.