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

  • Roman PukhtaievychEmail author
  • Sergiy Plaksa


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.


Commutative Banach algebra Monogenic function Cauchy type integral Sokhotskii–Plemelj formulas 

Mathematics Subject Classification

30G35 32A26 32A55 



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).


  1. 1.
    Babaev, A.A., Salaev, V.V.: Boundary value problems and singular equations on a rectifiable contour. Mat. Zametki 31(4), 571–580 (1982)MathSciNetzbMATHGoogle Scholar
  2. 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)MathSciNetzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 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)MathSciNetGoogle Scholar
  6. 6.
    Douglis, A.: A function-theoretic approach to elliptic systems of equations in two variables. Commun. Pure Appl. Math. 6, 259–289 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gakhov, F.D.: Boundary Value Problems. Dover Publications Inc., New York (1990)zbMATHGoogle Scholar
  8. 8.
    Gerus, O.F.: Finite-dimensional smoothness of Cauchy-type integrals. Ukr. Math. J. 29(5), 490–493 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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. 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)MathSciNetzbMATHGoogle Scholar
  13. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kamke, E.: Das Lebesgue-Stieltjes-Integral. 2te, verbesserte Aufl. B. G. Teubner Verlagsgesellschaft, Leipzig (1960)Google Scholar
  15. 15.
    Ketchum, P.W.: Analytic functions of hypercomplex variables. Trans. Am. Math. Soc. 30(4), 641–667 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lehto, O., Virtanen, K.I.: Quasiconformal Mappings in the Plane, 2nd edn. Springer, New York (1973)CrossRefzbMATHGoogle Scholar
  18. 18.
    Magnaradze, L.: On a generalization of the theorem of Plemelj–Privalov. Soobščeniya Akad. Nauk Gruzin. SSR. 8, 509–516 (1947)MathSciNetzbMATHGoogle Scholar
  19. 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)Google Scholar
  20. 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. 21.
    Muskhelishvili, N.I.: Singular Integral Equations. Dover Publications Inc., New York (1992)Google Scholar
  22. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 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)MathSciNetzbMATHGoogle Scholar
  24. 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)MathSciNetzbMATHGoogle Scholar
  25. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Plemelj, J.: Ein Ergänzungssatz zur Cauchyschen Integraldarstellung analytischer funktionen, randwerte betreffend. Monatsh. Math. Phys. 19(1), 205–210 (1908)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Privaloff, I.I.: Sur l’intégrale du type de Cauchy–Stieltjes. C. R. (Doklady) Acad. Sci. URSS (N.S.) 27, 195–197 (1940)MathSciNetzbMATHGoogle Scholar
  28. 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. 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)Google Scholar
  30. 30.
    Salaev, V.V.: Direct and inverse estimates for a singular Cauchy integral along a closed curve. Mat. Zametki 19(3), 365–380 (1976)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Shpakivskyi, V.S.: Curvilinear integral theorems for monogenic functions in commutative associative algebras. Adv. Appl. Clifford Algebras 26(1), 417–434 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ward, J.A.: A theory of analytic functions in linear associative algebras. Duke Math. J. 7, 233–248 (1940)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zygmund, A.: Trigonometric Series. Cambridge Mathematical Library, Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematics “Tullio Levi-Civita”University of PadovaPaduaItaly
  2. 2.Department of Complex Analysis and Potential TheoryInstitute of Mathematics of the National Academy of Sciences of UkraineKievUkraine

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