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Some properties of a Cauchy type integral in a three-dimensional commutative algebra with one-dimensional radical

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Abstract

In the paper we consider a certain analog of the Cauchy type integral taking values in a three-dimensional commutative algebra over the field of complex numbers with one-dimensional radical. We have established sufficient conditions for the existence of limiting values for such an integral. It is also shown that analogues of Sokhotskii–Plemelj formulas hold.

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References

  1. Babaev, A.A., Salaev, V.V.: Boundary value problems and singular equations on a rectifiable contour. Mat. Zametki 31(4), 571–580 (1982)

    MathSciNet  MATH  Google Scholar 

  2. Blaya, R.A., Peña, D.P., Reyes, J.B.: Conjugate hyperharmonic functions and Cauchy type integrals in Douglis analysis. Complex Var. Theory Appl. 48(12), 1023–1039 (2003)

    MathSciNet  MATH  Google Scholar 

  3. Blaya, R.A., Reyes, J.B., Kats, B.: Cauchy integral and singular integral operator over closed Jordan curves. Monatsh. Math. 176(1), 1–15 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blaya, R.A., Reyes, J.B., Peña, D.P.: Riemann boundary value problem for hyperanalytic functions. Int. J. Math. Math. Sci. 17, 2821–2840 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Davydov, N.A.: The continuity of an integral of Cauchy type in a closed region. Doklady Akad. Nauk SSSR (N.S.) 64, 759–762 (1949)

    MathSciNet  Google Scholar 

  6. Douglis, A.: A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6, 259–289 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  7. Gakhov, F.D.: Boundary Value Problems. Dover Publications Inc., New York (1990)

    MATH  Google Scholar 

  8. Gerus, O.F.: Finite-dimensional smoothness of Cauchy-type integrals. Ukr. Math. J. 29(5), 490–493 (1977)

    Article  MATH  Google Scholar 

  9. Gerus, O.F.: An estimate for the modulus of continuity of a Cauchy-type integral in a domain and on its boundary. Ukr. Math. J. 48(10), 1321–1328 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gerus, O.F.: On the modulus of continuity of solid derivatives of a Cauchy-type integral. Ukr. Math. J. 50(4), 476–484 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gilbert, R.P., Buchanan, J.L.: First Order Elliptic Systems, Mathematics in Science and Engineering, vol. 163. Academic Press Inc, Orlando, FL (1983)

    Google Scholar 

  12. Gilbert, R.P., Zeng, Y.S.: Hyperanalytic Riemann boundary value problems on rectifiable closed curves. Complex Var. Theory Appl. 20(1–4), 277–288 (1992)

    MathSciNet  MATH  Google Scholar 

  13. Gryshchuk, S.V., Plaksa, S.A.: Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations. Open Math. 15, 374–381 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kamke, E.: Das Lebesgue-Stieltjes-Integral. 2te, verbesserte Aufl. B. G. Teubner Verlagsgesellschaft, Leipzig (1960)

  15. Ketchum, P.W.: Analytic functions of hypercomplex variables. Trans. Am. Math. Soc. 30(4), 641–667 (1928)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kunz, K.S.: Application of an algebraic technique to the solution of Laplaces equation in three dimensions. SIAM J. Appl. Math. 21, 425–441 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lehto, O., Virtanen, K.I.: Quasiconformal Map**s in the Plane, 2nd edn. Springer, New York (1973)

    Book  MATH  Google Scholar 

  18. Magnaradze, L.: On a generalization of the theorem of Plemelj–Privalov. Soobščeniya Akad. Nauk Gruzin. SSR. 8, 509–516 (1947)

    MathSciNet  MATH  Google Scholar 

  19. Mel’nichenko, I.P.: A method of description of potential fields with axial symmetry. In: Current Problems in Real and Complex Analysis. Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, pp. 98–102 (1984)

  20. Mel’nichenko, I.P., Plaksa, S.A.: Commutative Algebras and Spatial Potential Fields, p. 231. Inst. Math. NAS Ukraine, Kyiv (2008). (in Russian)

    Google Scholar 

  21. Muskhelishvili, N.I.: Singular Integral Equations. Dover Publications Inc., New York (1992)

    Google Scholar 

  22. Plaksa, S.A.: Riemann boundary problem with infinite index of logarithmic order on a spiral-form contour. I. Ukr. Math. J. 42(11), 1351–1358 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Plaksa, S.A., Shpakivskyi, V.S.: Limiting values of the Cauchy type integral in a three-dimensional harmonic algebra. Eur. Math. J. 3(2), 120–128 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Plaksa, S.A., Shpakivskyi, V.S.: On limiting values of Cauchy type integral in a harmonic algebra with two-dimensional radical. Ann. Univ. Mariae Curie-Skłodowska Sect. A 67(1), 57–64 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Plaksa, S.A., Shpakovskii, V.S.: On the logarithmic residues of monogenic functions in a three-dimensional harmonic algebra with two-dimensional radical. Ukr. Math. J. 65(7), 1079–1086 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  26. Plemelj, J.: Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer funktionen, randwerte betreffend. Monatsh. Math. Phys. 19(1), 205–210 (1908)

    Article  MathSciNet  MATH  Google Scholar 

  27. Privaloff, I.I.: Sur l’intégrale du type de Cauchy–Stieltjes. C. R. (Doklady) Acad. Sci. URSS (N.S.) 27, 195–197 (1940)

    MathSciNet  MATH  Google Scholar 

  28. Privalov, I.I.: Graničnye svoĭstva analitičeskih funkciĭ(Boundary Properties of Analytic Functions), 2nd edn. Gosudarstv. Izdat. Tehn.-Teor. Lit, Moscow (1950)

    Google Scholar 

  29. Pukhtaievych, R.P., Plaksa, S.A.: Cauchy type integral on a straight line in a three-dimensional harmonic algebra with one-dimensional radical. In: Zb. Pr. Inst. Mat. NAN Ukr., vol. 12, pp. 212–219. Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine (2015)

  30. Salaev, V.V.: Direct and inverse estimates for a singular Cauchy integral along a closed curve. Mat. Zametki 19(3), 365–380 (1976)

    MathSciNet  MATH  Google Scholar 

  31. Shpakivskyi, V.S.: Curvilinear integral theorems for monogenic functions in commutative associative algebras. Adv. Appl. Clifford Algebras 26(1), 417–434 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  32. Ward, J.A.: A theory of analytic functions in linear associative algebras. Duke Math. J. 7, 233–248 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  33. Zygmund, A.: Trigonometric Series. Cambridge Mathematical Library, Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

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Acknowledgements

Pukhtaievych R. is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). This research was supported by Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO) and partially by Ministry of Education and Science of Ukraine (Project No. 0116U001528).

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Authors and Affiliations

  1. Department of Mathematics “Tullio Levi-Civita”, University of Padova, Via Trieste 63, 35121, Padua, Italy

    Roman Pukhtaievych

  2. Department of Complex Analysis and Potential Theory, Institute of Mathematics of the National Academy of Sciences of Ukraine, 3, Tereshchenkivska st., Kiev, 01004, Ukraine

    Roman Pukhtaievych & Sergiy Plaksa

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  1. Roman Pukhtaievych
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  2. Sergiy Plaksa
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Correspondence to Roman Pukhtaievych.

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Communicated by G. Teschl.

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This research was partially supported by Ministry of Education and Science of Ukraine (Project No. 0116U001528).

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Pukhtaievych, R., Plaksa, S. Some properties of a Cauchy type integral in a three-dimensional commutative algebra with one-dimensional radical. Monatsh Math 189, 523–548 (2019). https://doi.org/10.1007/s00605-019-01288-4

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