On the $L^r$-differentiability of two Lusin classes and a full descriptive characterization of the $HK_r$-integral

Мұқаба

Дәйексөз келтіру

Толық мәтін

Ашық рұқсат Ашық рұқсат
Рұқсат жабық Рұқсат берілді
Рұқсат жабық Тек жазылушылар үшін

Аннотация

It is proved that any function in a Lusin-type class, the class of $ACG_r$-functions, is differentiable almost everywhere in the sense of a derivative defined in the space $L^r$, $1\leqslant r < \infty$. 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.

Авторлар туралы

Paul Musial

Chicago State University, Chicago, IL, USA

Хат алмасуға жауапты Автор.
Email: paul.musial@gmail.com

Valentin Skvortsov

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia; Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia

Email: vaskvor2000@yahoo.com
Doctor of physico-mathematical sciences, Professor

Piotr Sworowski

Institute of Mathematics, Casimirus the Great University, Bydgoszcz, Poland

Email: piotrus@ukw.edu.pl
PhD, no status

Francesco Tulone

University of Palermo, Palermo, Italy

Email: francesco.tulone@unipa.it

Әдебиет тізімі

  1. С. Сакс, Теория интеграла, ИЛ, М., 1949, 496 с.
  2. А. Я. Хинчинъ, “О процессе интегрированiя Denjoy”, Матем. сб., 30:4 (1918), 543–557
  3. V. Ene, Real functions – current topics, Lecture Notes in Math., 1603, Springer-Verlag, Berlin, 1995, xiv+310 pp.
  4. W. F. Pfeffer, “The Lebesgue and Denjoy–Perron integrals from a descriptive point of view”, Ricerche Mat., 48:2 (1999), 211–223
  5. D. Preiss, B. S. Thomson, “The approximate symmetric integral”, Canad. J. Math., 41:3 (1989), 508–555
  6. V. 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
  7. R. A. Gordon, The integrals of Lebesgue, Denjoy, Perron, and Henstock, Grad. Stud. Math., 4, Amer. Math. Soc., Providence, RI, 1994, xii+395 pp.
  8. F. Tulone, V. Skvortsov, “Multidimensional $mathcal P$-adic integrals in some problems of harmonic analysis”, Minimax Theory Appl., 2:1 (2017), 153–174
  9. B. S. Thomson, “On VBG functions and the Denjoy–Khintchine integral”, Real Anal. Exchange, 41:1 (2016), 173–226
  10. P. M. Musial, Y. Sagher, “The $L^{r}$ Henstock–Kurzweil integral”, Studia Math., 160:1 (2004), 53–81
  11. A. P. Calderon, A. Zygmund, “Local properties of solutions of elliptic partial differential equations”, Studia Math., 20 (1961), 171–225
  12. L. Gordon, “Perron's integral for derivatives in $L^{r}$”, Studia Math., 28 (1966/1967), 295–316
  13. P. Musial, V. Skvortsov, F. Tulone, “The $HK_r$-integral is not contained in the $P_r$-integral”, Proc. Amer. Math. Soc., 150:5 (2022), 2107–2114
  14. P. Musial, V. Skvortsov, F. Tulone, “Comparison of the $P_r$-integral with Burkill's integrals and some applications to trigonometric series”, J. Math. Anal. Appl., 523:1 (2023), 127019, 10 pp.
  15. P. Musial, F. Tulone, “Integration by parts for the $L^r$ Henstock–Kurzweil integral”, Electron. J. Differential Equations, 2015 (2015), 44, 7 pp.
  16. P. Musial, F. Tulone, “Dual of the class of $operatorname{HK}_r$-integrable functions”, Minimax Theory Appl., 4:1 (2019), 151–160
  17. F. Tulone, P. Musial, “The $L^r$-variational integral”, Mediterr. J. Math., 19:3 (2022), 96, 10 pp.
  18. П. Мущал, В. А. Скворцов, Ф. Тулоне, “О дескриптивных характеристиках интеграла, восстанавливающего функцию по ее производной в $L^r$”, Матем. заметки, 111:3 (2022), 411–421
  19. V. A. Skvortsov, P. Sworowski, “On the relation between Denjoy–Khintchine and $operatorname{HK}_r$-integrals”, Sib. Math. J., 65:2 (2024), 441–447
  20. Cheng-Ming Lee, “An analogue of the theorem of Hake–Alexandroff–Looman”, Fund. Math., 100:1 (1978), 69–74

Қосымша файлдар

Қосымша файлдар
Әрекет
1. JATS XML
Жүктеу

© Musial P., Skvortsov V.A., Sworowski P.A., Tulone F., 2025

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).