On interval based generalizations of absolute continuity for functions on $\mathbb{R}^n$

TL;DR

This paper explores generalized notions of absolute continuity for functions on 11n, confirming a conjecture that 1-absolutely continuous functions need not be differentiable a.e., and establishing relations among various classes of such functions.

Contribution

It introduces and analyzes interval-based generalizations of absolute continuity, confirming a key conjecture and clarifying the relationships among different classes of these functions.

Findings

1-absolutely continuous functions need not be differentiable a.e.

Established inclusion relations among classes of absolutely continuous functions.

Provided examples of pathological functions within these classes.

Abstract

We study notions of absolute continuity for functions defined on $\mathbb{R}^n$similar to the notion of $\alpha$-absolute continuity in the sense of Bongiorno. We confirm a conjecture of Mal\'y that 1-absolutely continuous functions do not need to be differentiable a.e., and we show several other pathological examples of functions in this class. We establish containment relations of the class $1-AC_{\rm WDN}$ which consits of all functions in $1-AC$ which are in the Sobolev space $W^{1,2}_{loc}$, are differentiable a.e. and satisfy the Luzin (N) property, with previously studied classes of absolutely continuous functions.