Abstract
We study a generalized contour integral along a nonrectifiable path and its applications.
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Gakhov F. D., Boundary Value Problems, Dover Publications, Inc., New York (1990).
Muskhelishvili N. I., Singular Integral Equations, Wolters–Noordhoff Publishing, Groningen (1967).
Jian–Ke Lu, Boundary Value Problems for Analytic Functions, World Sci., Singapore (1993).
Soldatov A. P., One–Dimensional Singular Operators and Boundary Value Problems of the Theory of Functions [Russian], Vysshaya Shkola, Moscow (1991).
Chibrikova L. I., The Basic Boundary Value Problems for Analytic Functions [Russian], Izdat. Kazansk. Univ., Kazan (1977).
Dynkin E. M., “On the smoothness of Cauchy type integrals,” Zap. Nauchn. Sem. LOMI, vol. 92, 115–133 (1979).
Salimov T., “A direct bound for the singular Cauchy integral along a closed curve,” in: Nauchn. Tr. MV and SSO Azerb. SSR, Izdat. Akad. Nauk AzSSR, Baku, 1979, vol. 5, 59–75.
Kats B. A., “The Riemann boundary value problem on a nonrectifiable Jordan curve,” Dokl. Akad. Nauk SSSR, vol. 267, No. 4, 789–792 (1982).
Kats B. A., “The Riemann problem on a closed Jordan curve,” Izv. Vyssh. Uchebn. Zaved. Mat., no. 4, 68–80 (1983).
Kats B. A., “The Riemann problem on an open Jordan curve,” Soviet Math. (Iz. VUZ), vol. 27, No. 12, 33–44 (1983).
Kats B. A., “The Riemann boundary value problem on non–rectifiable curves and related questions,” Complex Var. Elliptic Equ., vol. 59, No. 8, 1053–1069 (2014).
Kats B. A. and Katz D. B., “Marcinkiewicz exponents and integrals over non–rectifiable paths,” Math. Methods Appl. Sci., vol. 39, No. 12, 3402–3410 (2016).
Katz D. B. and Kats B. A., “Interactions of germs with applications,” Math. Methods Appl. Sci., vol. 40, No. 13, 4974–4981 (2017).
Falkoner F. J., Fractal Geometry, Wiley and Sons, Chichester (2014).
Dolzhenko E. P., “The removability of singularities of analytic functions,” Uspekhi Mat. Nauk, vol. 112, No. 4, 135–142 (1963).
Katz D., “Some new metric characteristics of nonrectifiable curves with applications,” in: Tr. Mat. Tsentra imeni N. I. Lobachevskogo [Russian], Izdat. Kazan Univ., Kazan, 2013, vol. 46, 235–236.
Katz D., “The Marcinkiewicz exponent with applications,” in: Current Trends in Analysis and Its Applications: Proc. 9th ISAAC Congress (Krakow, 2013). Trends in Mathematics, Research Perspectives, Birkhäuser, Basel, 2015, 106–107.
Katz D. B., “Local and weighted Marcinkiewicz exponents with applications,” J. Math. Anal. Appl., vol. 440, No. 1, 74–85 (2016).
Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton (1970).
Seifullaev R. K., “The Riemann boundary value problem on a nonsmooth open curve,” Math. USSR–Sb., vol. 40, No. 2, 135–148 (1981).
Vekua I. N., Generalized Analytic Functions, Pergamon Press, Oxford, London, New York, and Paris (1962).
Katz D. B. and Kats B. A., “Riemann boundary value problem on nonrectifiable curves for certain Beltrami equations,” Math. Methods Appl. Sci., vol. 41, No. 6, 2507–2514 (2018).
Evgrafov M. A., Asymptotic Estimates and Entire Functions [Russian], Gostekhizdat, Moscow (1957).
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The authors were partially supported by the Russian Foundation for Basic Research (Grant 18-31-00060) and is performed according to the Russian Government Program of Competitive Growth of Kazan Federal University.
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Translated from Sibirskii Matematicheskii Zhurnal, vol. 60, no. 1, pp. 95–108, January–February, 2019; DOI: 10.17377/smzh.2019.60.108.
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Kats, B.A., Katz, D.B. Integration over Nonrectifiable Paths with Applications. Sib Math J 60, 71–81 (2019). https://doi.org/10.1134/S0037446619010087
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DOI: https://doi.org/10.1134/S0037446619010087