A regularity condition in Sobolev spaces $W^{1,p}_{\mathrm loc}({\mathbb R}^n)$ with $1 ≤ p \lt n$

Abstract

Extending Malý's geometric definition of absolutely continuous functions of $n$ variables (in a sense equivalent to that of Rado-Reichelderfer), we define classes of $p$-absolutely continuous functions $(1\leq p \lt n)$ and show that this weaker notion of absolute continuity still implies differentiability almost everywhere, although it does not imply continuity or Lusin's condition (N).