This article is cited in 7 papers

On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative

P. Musial^a, V. A. Skvortsov^b, F. Tulone<sup>c

a</sup> Chicago State University
^b Moscow Center for Fundamental and Applied Mathematics
^c Università degli Studi di Palermo

Abstract: The notion of $L^r$-variational measure generated by a function $F\in L^r[a,b]$ is introduced and, in terms of absolute continuity of this measure, a descriptive characterization of the $H\!K_r$-integral recovering a function from its $L^r$-derivative is given. It is shown that the class of functions generating absolutely continuous $L^r$-variational measure coincides with the class of $ACG_{r}$-functions which was introduced earlier, and that both classes coincide with the class of the indefinite $H\!K_{r}$-integrals under the assumption of $L^r$-differentiability almost everywhere of the functions consisting these classes.

Keywords: $L^r$-derivative, Henstock–Kurzweil-type integral, $L^r$-variational measure, absolutely continuous measure, generalized absolute continuity of a function.

UDC: 517.518.126

Received: 07.09.2021

DOI: 10.4213/mzm13284

 English version: Mathematical Notes, 2022, 111:3, 414–422

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