Fuglede's theorem in generalized Orlicz--Sobolev spaces

TL;DR

This paper characterizes Orlicz--Sobolev spaces using ACL and ACC properties, extending classical results to more general spaces under specific density and growth conditions.

Contribution

It introduces ACL and ACC characterizations for Orlicz--Sobolev spaces, including new results for special cases like Orlicz and double phase growth.

Findings

Characterization of $W^{1,}$ via ACL and ACC

Results hold under density and growth assumptions

New insights even for classical Orlicz and double phase spaces

Abstract

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.