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On the Solvability of Singular Integral Equations with Reflection on the Unit Circle

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Abstract

The solvability of a class of singular integral equations with reflection in weighted Lebesgue spaces is analyzed, and the corresponding solutions are obtained. The main techniques are based on the consideration of certain complementary projections and operator identities. Therefore, the equations under study are associated with systems of pure singular integral equations. These systems will be then analyzed by means of a corresponding Riemann boundary value problem. As a consequence of such a procedure, the solutions of the initial equations are constructed from the solutions of Riemann boundary value problems. In the final part of the paper, the method is also applied to singular integral equations with the so-called commutative and anti-commutative weighted Carleman shifts.

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References

  1. Bastos M.A., Fernandes C.A., Karlovich Y.I.: Spectral measures in C*-algebras of singular integral operators with shifts. J. Funct. Anal. 242, 86–126 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baturev A.A., Kravchenko V.G., Litvinchuk G.: Approximate methods for singular integral equations with a non-Carleman shift. J. Integral Equations Appl. 8, 1–17 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Boiarskii B.V.: On generalized Hilbert boundary value problems, Soobsh. Akad. Nauk Gruz. SSR 25(4), 385–390 (1960)

    Google Scholar 

  4. Carleman T.: Application de la théorie des équations intégrales linéaires aux systémes d’ équations différentielles non linéaires. Acta Math. 59, 63–87 (1932)

    Article  MathSciNet  Google Scholar 

  5. Castro L.P., Rojas E.M.: Reduction of singular integral operators with flip and their Fredholm property. Lobachevskii J. Math. 29, 119–129 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Castro L.P., Rojas E.M.: Explicit solutions of Cauchy singular integral equations with weighted Carleman shift. J. Math. Anal. Appl. 371, 128–133 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  7. Castro, L.P., Rojas, E.M.: Invertibility of singular integral operators with flip through explicit operator relations. In: Integral Methods in Science and Engineering, vol. 1, pp. 105–114. Birkhäuser, Boston, (2010)

  8. Chaun L.H., Mau N.V., Tuan N.M.: On a class of singular integral equations with the linear fractional Carleman shift and the degenerate kernel. Complex Var. Elliptic Equ. 53, 117–137 (2008)

    Article  MathSciNet  Google Scholar 

  9. Chaun L.H., Tuan N.M.: On singular integral equations with the Carleman shifts in the case of the vanishing coefficient. Acta Math. Vietnam. 28, 319–333 (2003)

    MathSciNet  Google Scholar 

  10. Clancey K.F., Gohberg I.: Factorization Matrix Functions and Singular Integral Operators. Operator Theory: Advances and Applications, vol. 3. Birkhäuser, Basel-Boston-Stuttgart (1981)

  11. Ferreira J., Litvinchuk G.S., Reis M.D.L.: Calculation of the defect numbers of the generalized Hilbert and Carleman boundary value problems with linear fractional Carleman shift. Integr. Equ. Oper. Theory 57, 185–207 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gakhov F.D.: Boundary Values Problems. Dover, New York (1990)

    Google Scholar 

  13. Haseman, C.: Anwendung der Theorie der Integralgleichungen auf einige Randwertaufgaben. Gottingen (1911)

  14. Hilbert, D.: Über eine Anwendung der Integralgleichungen auf ein Problem der Funktionentheorie. Verh. d. 3. intern. Math. Kongr. Heidelb. pp 233–240 (1905, in German)

  15. Karapetiants N., Samko S.: Singular integral equations on the real line with a fractional-linear Carleman shift. Proc. A. Razmadze Math. Inst. 124, 73–106 (2000)

    MATH  MathSciNet  Google Scholar 

  16. Karapetiants N., Samko S.: Equations with Involutive Operators. Birkhäuser, Boston (2001)

    MATH  Google Scholar 

  17. Karelin A.A.: Applications of operator equalities to singular integral operators and to Riemann boundary value problems. Math. Nachr. 280, 1108–1117 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kravchenko V.G., Lebre A.B., Litvinchuk G.S., Teixeira F.S.: A normalization problem for a class of singular integral operators with Carleman shift and unbounded coefficients. Integr. Equ. Oper. Theory 21, 342–354 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kravchenko V.G., Lebre A.B., Rodríguez J.S.: Factorization of singular integral operators with a Carleman shift and spectral problems. J. Integr. Equ. Appl. 13, 339–383 (2001)

    Article  MATH  Google Scholar 

  20. Kravchenko V.G., Lebre A.B., Rodríguez J.S.: Factorization of singular integral operators with a Carleman shift via factorization of matrix functions: the anticommutative case. Math. Nachr. 280, 1157–1175 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kravchenko V.G., Litvinchuk G.S.: Introduction to the Theory of Singular Integral Operators with Shift. Kluwer, Amsterdam (1994)

    MATH  Google Scholar 

  22. Litvinchuk, G.S.: Solvability Theory of Boundary Value Problems and Singular Integral Equations with Shift. Mathematics and its Applications, vol. 523. Kluwer, Dordrecht (2000)

  23. Litvinchuk, G.S., Spitkovsky, I.M.: Factorization of Measurable Matrix Functions. Operator Theory: Advances and Applications, vol. 25. Birkhäuser, Basel-Boston (1987)

  24. Markouchevitch, A.: Quelques remarques sur les intégrales du type de Cauchy. Učenye Zapiski Moskov. Gos. Univ. 100. Matematika 1, 31–33 (1946, in Russian, French summary)

  25. Mikhlin S.G., Prössdorf S.: Singular Integral Operators. Springer, Berlin (1980)

    Google Scholar 

  26. Riemann, B.: Lehrsätze aus der analysis situs für die Theorie der Integrale von zweigliedrigen vollständigen Differentialen. J. Reine Angew. Math. 54, 105–110 (1857, in German)

  27. Riemann, B.: Gesammelte mathematische Werke, wissenschaftlicher Nachlass und Nachträge [Collected mathematical works, scientific Nachlass and addenda] Based on the edition by Heinrich Weber and Richard Dedekind. Teubner-Archiv zur Mathematik [Teubner Archive on Mathematics], Suppl. 1. BSB B. G. Teubner Verlagsgesellschaft, Leipzig; Springer, Berlin (1990, in German)

  28. Tuan N.M.: On a class of singular integral equations with rotation. Acta Math. Vietnam. 21, 201–211 (1996)

    MATH  MathSciNet  Google Scholar 

  29. Vekua, N.P.: On a generalized system of singular integral equations. Soobščeniya Akad. Nauk Gruzin. SSR. 9, 153–160 (1948, in Russian)

    Google Scholar 

  30. Vekua, N. P.: The Carleman boundary problem for several unknown functions. Soobščeniya Akad. Nauk Gruzin. SSR. 13, 9–14 (1952, in Russian)

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Authors and Affiliations

  1. Department of Mathematics, University of Aveiro, 3810-193, Aveiro, Portugal

    L. P. Castro

  2. Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia

    E. M. Rojas

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  1. L. P. Castro
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  2. E. M. Rojas
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Correspondence to L. P. Castro.

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Dedicated to the memory of Professor Diómedes Bárcenas

This work was supported in part by Center for R&D in Mathematics and Applications, University of Aveiro, Portugal, through FCT—Portuguese Foundation for Science and Technology.

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Castro, L.P., Rojas, E.M. On the Solvability of Singular Integral Equations with Reflection on the Unit Circle. Integr. Equ. Oper. Theory 70, 63–99 (2011). https://doi.org/10.1007/s00020-011-1871-6

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  • DOI: https://doi.org/10.1007/s00020-011-1871-6

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