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.
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References
B. S. Thomson, Monogr. Textbooks Pure Appl. Math., Vol. 183: Symmetric Properties of Real Functions, (Marcel Dekker, New York, 1994).
V. A. Skvortsov and F. Tulone, “Kurzweil–Henstock type integral on zero-dimensional group and some of its applications,” Czech. Math. J. 58 (4), 1167–1183 (2008).
V. A. Skvortsov and F. Tulone, “Henstock–Kurzweil type integral in Fourier analysis on zero-dimensional group,” Tatra Mt. Math. Publ. 44, 41–51 (2009).
V. A. Skvortsov and F. Tulone, “Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series,” J. Math. Anal. Appl. 421 (2), 1502–1518 (2015).
V. A. Skvortsov and F. Tulone, “On the coefficients of multiple series with respect to Vilenkin system,” Tatra Mt. Math. Publ. 68, 81–92 (2017).
V. A. Skvortsov and F. Tulone, “Multidimensional \(P\)-adic integrals in some problems of harmonic analysis,” Minimax Theory Appl. 2 (1), 153–174 (2017).
G. Oniani and 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), 12–15 (2018).
G. Oniani and F. Tulone, “On the almost everywhere convergence of multiple Fourier–Haar series,” J. Contemp. Math. Anal. 54 (5), 288–295 (2019).
F. Tulone, “Generality of Henstock–Kurzweil type integral on a compact zero-dimensional metric space,” Tatra Mt. Math. Publ. 49, 81–88 (2011).
R. A. Gordon, Grad. Stud. Math., Vol. 4: The Integrals of Lebesgue, Denjoy, Perron, and Henstock (Amer. Math. Soc., Providence, RI, 1994).
A. Boccuto, V. A. Skvortsov, and F. Tulone, “A Hake-type theorem for integrals with respect to abstract derivation bases in the Riesz space setting,” Math. Slovaca 65 (6), 1319–1336 (2015).
V. A. Skvortsov and F. Tulone, “Generalized Hake property for integrals of Henstock type,” Moscow Univ. Math. Bull. 68 (6), 270–274 (2013).
V. Skvortsov and F. Tulone, “A version of Hake’s theorem for Kurzweil–Henstock integral in terms of variational measure,” Georgian Math. J. 28 (3), 471–476 (2021).
A. Bokkuto, V. A. Skvortsov, and F. Tulone, “Integration of functions ranging in complex Riesz space and some applications in harmonic analysis,” Math. Notes 98 (1), 25–37 (2015).
V. A. Skvortsov and F. Tulone, “Henstock type integral in compact zero-dimensional metric space and quasi-measures representations,” Moscow Univ. Math. Bull. 67 (2), 55–60 (2012).
B. S. Thomson, Mem. Amer. Math. Soc., Vol. 452: Derivates of Interval Functions (Amer. Math. Soc., Providence, RI, 1991).
B. Bongiorno, L. di Piazza, and V. Skvortsov, “A new full descriptive characterization of Denjoy–Perron integral,” Real Anal. Exchange 20 (2), 656–663 (1996).
T. P. Lukashenko, V. A. Skvortsov, and A. P. Solodov, Generalized Integrals (URSS, Moscow, 2011) [in Russian].
V. Ene, Lecture Notes in Math., Vol. 1603: Real Functions – Current Topics (Springer- Verlag, Berlin, 1995).
S. Schwabik, “Variational Measures and the Kurzweil–Henstock integral,” Math. Slovaca 59 (6), 731–752 (2009).
V. A. Skvortsov, “Variations and variational measures in integration theory and some applications,” J. Math. Sci. 91 (5), 3293–3322 (1998).
A. P. Calderon and A. Zygmund, “Local properties of solutions of elliptic partial differential equations,” Studia Math. 20, 171–225 (1961).
L. Gordon, “Perron’s integral for derivatives in \(L^{r}\),” Studia Math. 28, 295–316 (1967).
P. Musial and Y. Sagher, “The \(L^r\) Henstock–Kurzweil integral,” Studia Math. 160 (1), 53–81 (2004).
P. Musial, V. Skvortsov and F. Tulone, “The \(H\!K_r\)-integral is not contained in the \(P_r\)-integral,” Proc. Amer. Math. Soc. (2022) (in press).
P. Musial and F. Tulone, “Integration by parts for the \(L^r\) Henstock–Kurzweil integral,” Electron. J. Differential Equations 2015 (44), 1–7 (2015).
P. Musial and F. Tulone, “Dual of the class of \(H\!K_r\)-integrable functions,” Minimax Theory Appl. 4 (1), 151–160 (2019).
P. Musial and F. Tulone, “The \(L^r\)-variational integral,” Mediterr. J. Math. (2022) (in press).
V. Skvortsov and Yu. Zherebyov, “On Classes of functions generating absolutely continuous variational measures,” Real Anal. Exchange 30 (1), 361–372 (2005).
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This work was supported by the Russian Foundation for Basic Research under grant 20-01-00584.
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Translated from Matematicheskie Zametki, 2022, Vol. 111, pp. 411-421 https://doi.org/10.4213/mzm13284.
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Musial, P., Skvortsov, V.A. & Tulone, F. On Descriptive Characterizations of an Integral Recovering a Function from Its \(L^r\)-Derivative. Math Notes 111, 414–422 (2022). https://doi.org/10.1134/S0001434622030099
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DOI: https://doi.org/10.1134/S0001434622030099