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Original Articles

Riemann–Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue Spaces

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Pages 1233-1257 | Received 10 Mar 2017, Accepted 07 Jul 2017, Published online: 28 Aug 2017

References

  • Riemann BF . Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse. Leipzig: Werke; 1867.
  • Gakhov FD . Boundary value problems(Essential Russian edition Izdat. Translation edited by I. N. Sneddon. Oxford: Pergamon Press; Reading (MA): Addison-Wesley; 1966. p. 1966. (Essential Russian edition Izdat. Moscow: ‘Nauka’; 1977).
  • Muskhelishvili NI . Singular integral equations. Boundary problems of function theory and their application to mathematical physics. Translation by Radok JRM. Groningen: P. Noordhoff N. V.; 1953.
  • Khvedelidze BV . The method of Cauchy type integrals in discontinuous boundary value problems of the theory of holomorphic functions of a complex variable. (Russian) Current problems in mathematics, Vol. 7 (Russian), pp. 5--162 (errata insert). Moscow: Akad Nauk SSSR Vsesojuz Inst Naučn. i Tehn. Informacii; 1975. translation in J Sovr Math. 1977;7:309–414.
  • Simonenko IB . The Riemann boundary value problem for n pairs of functions with measurable coefficients and its application to the study of singular integrals in Lp spaces with weights. Izv Akad Nauk SSSR Ser Mat. 1964;28:277–306. Russian.
  • Góhberg I , Krupnik N . One-dimensional linear singular integral equations. Vol. II. General theory and applications. Translated from the 1979 German translation by S. Roch and revised by the authors. Basel: Birkhäuser Verlag; 1992. (Vol. 54, Operator theory: advances and applications).
  • Danilyuk II . Irregular boundary value problems in the plane. Moscow: Nauka; 1975.
  • Böttcher A , Karlovich YI . Carleson curves, Muckenhoupt weights, and Toeplitz operators. Vol. 154, Progress in mathematics. Basel: Birkhäuser Verlag; 1997.
  • Câmara MC , Diogo C , Karlovich YI , Spitkovsky IM . Factorizations, Riemann-Hilbert problems and the corona theorem. J Lond Math Soc (2). 2012;86(3):852–878.
  • Duduchava R , Silbermann B . Boundary value problems in domains with peaks. Mem Differ Equ Math Phys. 2000;21:1–122.
  • Maz’ya VG , Soloviev AA . Boundary integral equations on contours with peaks. Vol. 196, Operator theory: advances and applications. Basel: Birkhäuser Verlag; 2010.
  • Kokilashvili V , Paatashvili V . Boundary value problems for analytic and harmonic functions in nonstandard Banach function spaces. New York (NY): Nova Science Publishers; 2012.
  • Iwaniec T , Sbordone C . On the integrability of the Jacobian under minimal hypotheses. Arch Rational Mech Anal. 1992;119(2):129–143.
  • Greco L , Iwaniec T , Sbordone C . Inverting the p-harmonic operator. Manuscripta Math. 1997;92(2):249–258.
  • Ryazanov V . On the Riemann-Hilbert problem without index. Ann Univ Buchar Math Ser. 2014;5(LXIII):169–178.
  • Kokilashvili V , Paatashvili V . The Riemann and Dirichlet problems with data from the grand Lebesgue spaces. Vol. 229, Advances in harmonic analysis and operator theory, Operator theory: advances and applications. Basel: Birkhäuser/Springer Basel AG; 2013. p. 233–251.
  • Kokilashvili V , Meskhi A , Paatashvili V . The Riemann-Hilbert problem in the class of Cauchy type integrals with densities of grand Lebesgue spaces. Trans A Razmadze Math Inst. 2016;170(2):208–211.
  • Khuskivadze G , Kokilashvili V , Paatashvili V . Boundary value problems for analytic and harmonic functions in domains with nonsmooth boundaries. Applications to conformal mappings. Mem Differ Equ Math Phys. 1998;14:195 pp.
  • David G . Opérateurs intégraux singuliers sur certaines courbes du plan complexe. [Singular integral operators over certain curves in the complex plane]. Ann Sci École Norm Sup (4). 1984;17(1):157–189. French.
  • Simonenko IB . Stability of weight properties of functions, with respect to a singular Cauchy integral. Mat Zametki. 1983;33(3):409–416. Russian.
  • Havin VP . Boundary properties of integrals of Cauchy type and of conjugate harmonic functions in regions with rectifiable boundary. Mat Sb (NS). 1965;68(110):499–517. Russian.
  • Stein EM . Interpolation of linear operators. Trans Am Math Soc. 1956;83:482–492.
  • Fiorenza A , Gupta B , Jain P . The maximal theorem for weighted grand Lebesgue spaces. Studia Math. 2008;188(2):123–133.
  • Kokilashvili V . Boundedness criteria for singular integrals in weighted grand Lebesgue spaces, Problems in mathematical analysis. J Math Sci (NY). 2010;170(1):20–33. No. 49.
  • Kokilashvili V . Singular integrals and strong maximal functions in weighted grand Lebesgue spaces. Nonlinear analysis, function spaces and applications. Vol. 9, Prague: Institute of Mathematics, Czech Academy of Sciences; 2011. p. 261–269.
  • Privalov II . Boundary properties of analytic functions. 2nd ed., Moscow-Leningrad: State Publishing House of Technical and Theoretical Literature; 1950. Russian.

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