On the $L^r$-differentiability of two Lusin-type classes and a full descriptive characterization of the $HK_r$-integral - Musial, Paul; Skvortsov, Valentin Anatol'evich; Sworowski, Piotr; Tulone, Francesco (Sbornik. Mathematics, 2025)
It is proved that any function of a Lusin-type class, the class of $ACG_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space $L^r$, $1\le r<∞$. This leads to a full descriptive characterization of a Henstock-Kurzweil-type integral, the $HK_r$-integral, which serves to recover functions from their $L^r$-derivatives. The class $ACG_r$ is compared with the classical Lusin class $ACG$, and it is shown that continuous $ACG$-functions can fail to be $L^r$-differentiable almost everywhere. Bibliography: 20 titles.