Geodesic
On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative
Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 411-421.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{MZM_2022_111_3_a7,
     author = {P. Musial and V. A. Skvortsov and F. Tulone},
     title = {On {Descriptive} {Characterizations} of an {Integral} {Recovering} a {Function} from {Its} $L^r${-Derivative}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {411--421},
     publisher = {mathdoc},
     volume = {111},
     number = {3},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a7/}
}
TY  - JOUR
AU  - P. Musial
AU  - V. A. Skvortsov
AU  - F. Tulone
TI  - On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative
JO  - Matematičeskie zametki
PY  - 2022
SP  - 411
EP  - 421
VL  - 111
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a7/
LA  - ru
ID  - MZM_2022_111_3_a7
ER  - 
%0 Journal Article
%A P. Musial
%A V. A. Skvortsov
%A F. Tulone
%T On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative
%J Matematičeskie zametki
%D 2022
%P 411-421
%V 111
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a7/
%G ru
%F MZM_2022_111_3_a7
P. Musial; V. A. Skvortsov; F. Tulone. On Descriptive Characterizations of an Integral Recovering a Function from Its $L^r$-Derivative. Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 411-421. http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a7/

[1] B. S. Thomson, Symmetric Properties of Real Functions, Monogr. Textbooks Pure Appl. Math., 183, Marcel Dekker, New York, 1994 | MR | Zbl

[2] V. A. Skvortsov, F. Tulone, “Kurzweil–Henstock type integral on zero-dimensional group and some of its applications”, Czech. Math. J., 58:4 (2008), 1167–1183 | DOI | MR | Zbl

[3] V. A. Skvortsov, F. Tulone, “Henstock–Kurzweil type integral in Fourier analysis on zero-dimensional group”, Tatra Mt. Math. Publ., 44 (2009), 41–51 | MR | Zbl

[4] V. A. Skvortsov, F. Tulone, “Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series”, J. Math. Anal. Appl., 421:2 (2015), 1502–1518 | DOI | MR | Zbl

[5] V. A. Skvortsov, F. Tulone, “On the coefficients of multiple series with respect to Vilenkin system”, Tatra Mt. Math. Publ., 68 (2017), 81–92 | MR | Zbl

[6] V. A. Skvortsov, F. Tulone, “Multidimensional $P$-adic integrals in some problems of harmonic analysis”, Minimax Theory Appl., 2:1 (2017), 153–174 | MR | Zbl

[7] G. Oniani, F. Tulone, “On the possible values of upper and lower derivatives with respect to differentiation bases of product structure”, Bull. Georgian Natl. Acad. Sci. (N.S.), 12:1 (2018), 12–15 | MR | Zbl

[8] G. Oniani, F. Tulone, “On the Almost Everywhere Convergence of Multiple Fourier–Haar Series”, J. Contemp. Math. Anal., 54:5 (2019), 288–295 | DOI | MR | Zbl

[9] F. Tulone, “Generality of Henstock–Kurzweil type integral on a compact zero-dimensional metric space”, Tatra Mt. Math. Publ., 49 (2011), 81–88 | MR | Zbl

[10] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Grad. Stud. Math., 4, Amer. Math. Soc., Providence, RI, 1994 | DOI | MR | Zbl

[11] A. Boccuto, V. A. Skvortsov, F. Tulone, “A Hake-type theorem for integrals with respect to abstract derivation bases in the Riesz space setting”, Math. Slovaca, 65:6 (2015), 1319–1336 | DOI | MR | Zbl

[12] V. A. Skvortsov, F. Tulone, “Obobschennoe svoistvo Khake dlya integralov khenstokovskogo tipa”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2013, no. 6, 9–13 | MR | Zbl

[13] V. Skvortsov, F. Tulone, “A version of Hake's theorem for Kurzweil–Henstock integral in terms of variational measure”, Georgian Math. J., 28:3 (2021), 471–476 | DOI | MR | Zbl

[14] A. Bokkuto, V. A. Skvortsov, F. Tulone, “Integrirovanie funktsii so znacheniyami v kompleksnom prostranstve Rissa i nekotorye prilozheniya v garmonicheskom analize”, Matem. zametki, 98:1 (2015), 12–26 | DOI | MR

[15] V. A. Skvortsov, F. Tulone, “Integral khenstokovskogo tipa na kompaktnom nul-mernom metricheskom prostranstve i predstavlenie kvazi-mery”, Vestn. Mosk. un-ta. Ser. 1. Matem., mekh., 2012, no. 2, 11–17 | MR

[16] B. S. Thomson, Derivates of Interval Functions, Mem. Amer. Math. Soc., 452, Amer. Math. Soc., Providence, RI, 1991 | MR | Zbl

[17] B. Bongiorno, L. di Piazza, V. Skvortsov, “A new full descriptive characterization of Denjoy–Perron integral”, Real Anal. Exchange, 20:2 (1996), 656–663 | DOI | MR

[18] T. P. Lukashenko, V. A. Skvortsov, A. P. Solodov, Obobschennye integraly, URSS, M., 2011

[19] V. Ene, Real Functions – Current Topics, Lecture Notes in Math., 1603, Springer-Verlag, Berlin, 1995 | DOI | MR | Zbl

[20] S. Schwabik, “Variational Measures and the Kurzweil–Henstock integral”, Math. Slovaca, 59:6 (2009), 731–752 | DOI | MR | Zbl

[21] V. A. Skvortsov, “Variations and variational measures in integration theory and some applications”, J. Math. Sci., 91:5 (1998), 3293–3322 | DOI | MR | Zbl

[22] A. P. Calderon, A. Zygmund, “Local properties of solutions of elliptic partial differential equations”, Studia Math., 20 (1961), 171–225 | DOI | MR | Zbl

[23] L. Gordon, “Perron's integral for derivatives in $L^{r}$”, Studia Math., 28 (1967), 295–316 | DOI | MR | Zbl

[24] P. Musial, Y. Sagher, “The $L^r$ Henstock–Kurzweil integral”, Studia Math., 160:1 (2004), 53–81 | DOI | MR | Zbl

[25] P. Musial, V. Skvortsov, F. Tulone, “The $HK_r$-integral is not contained in the $P_r$-integral”, Proc. Amer. Math. Soc., 2022 (to appear)

[26] P. Musial, F. Tulone, “Integration by parts for the $L^r$ Henstock–Kurzweil integral”, Electron. J. Differential Equations, 2015:44 (2015), 1–7 | MR

[27] P. Musial, F. Tulone, “Dual of the Class of $HK_r$-Integrable Functions”, Minimax Theory Appl., 4:1 (2019), 151–160 | MR | Zbl

[28] P. Musial, F. Tulone, “The $L^r$-Variational Integral”, Mediterr. J. Math., 2022 (to appear)

[29] V. Skvortsov, Yu. Zherebyov, “On Classes of Functions Generating Absolutely Continuous Variational Measures”, Real Anal. Exchange, 30:1 (2005), 361–372 | DOI | MR | Zbl