This article is cited in 1 paper

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

P. Musial^a, V. A. Skvortsov^bc, P. Sworowski^d, F. Tulone<sup>e

a</sup> Chicago State University, Chicago, IL, USA
^b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia
^c Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
^d Institute of Mathematics, Casimirus the Great University, Bydgoszcz, Poland
^e University of Palermo, Palermo, Italy

Abstract: 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.

Keywords: $L^r$-derivative, $L^r$-Henstock–Kurzweil integral, Denjoy integral, Lusin's class $\mathrm{ACG}$, class $\mathrm{ACG}_r$.

MSC: 26A24, 26A39

Received: 28.05.2024 and 11.09.2024

DOI: 10.4213/sm10129

 English version: Sbornik: Mathematics, 2025, 216:6, 780–790

Bibliographic databases:

• 
• 
• 
•