Fuglede's theorem in generalized Orlicz--Sobolev spaces

Abstract

In this paper, we show that Orlicz--Sobolev spaces $W^{1,\varphi }(\Omega )$ 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 $W^{1,\varphi }(\Omega )$, and $\varphi (x,\beta )\ge 1$ 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.

12 pages; submitted to Collectanea Mathematica