On the $L^r$-differentiability of two Lusin-type classes and a full descriptive characterization of the $\mathrm{HK}_r$-integral

• Published in: Sbornik. Mathematics Vol. 216; no. 6; pp. 780 - 790
• Main Authors: Musial, Paul, Skvortsov, Valentin Anatol'evich, Sworowski, Piotr, Tulone, Francesco
• Format: Journal Article
• Language: English
• Published: 2025
• ISSN: 1064-5616; 1468-4802
• DOI: 10.4213/sm10129e

It is proved that any function of a Lusin-type class, the class of $\mathrm{ACG}_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space $L^r$, $1\le r<\infty$. This leads to a full descriptive characterization of a Henstock-Kurzweil-type integral, the $\mathrm{HK}_r$-integral, which serves to recover functions from their $L^r$-derivatives. The class $\mathrm{ACG}_r$ is compared with the classical Lusin class $\mathrm{ACG}$, and it is shown that continuous $\mathrm{ACG}$-functions can fail to be $L^r$-differentiable almost everywhere. Bibliography: 20 titles.