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On the Theory of the Boundary Behavior of Conjugate Harmonic Functions

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Abstract

It is known that if a harmonic function u on the unit disk \({\mathbb {D}}\) in \({\mathbb {C}}\) has angular limits on a measurable set E of the unit circle \(\partial {\mathbb {D}}\), then its conjugate harmonic function v in \({\mathbb {D}}\) also has angular limits a.e. on E and both boundary functions are finite a.e. and measurable on E. This result is extended to arbitrary Jordan domains with rectifiable boundaries in terms of the natural parameter. On this basis, we study various Stieltjes integrals as Poisson-Stieltjes, conjugate Poisson-Stieltjes, Schwartz-Stieltjes and Cauchy-Stieltjes and prove theorems on the existence of their finite angular limits a.e. in terms of the Hilbert-Stieltjes integral. These results hold for arbitrary bounded integrands that are differentiable a.e. and, in particular, for integrands of the class \(\mathcal{{CBV}}\) (countably bounded variation).

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References

  1. Bari, N.K.: Trigonometric series, Gos. Izd. Fiz.–Mat. Lit., Moscow, 1961 (in Russian); transl. as A treatise on trigonometric series, vol. I–II. Macmillan Co., New York (1964)

  2. Borovkov, A.A.: Probability theory; transl. from the 1986 Russian original by A, p. 1998. Gordon and Breach Science Publishers, Amsterdam, Borovkova and revised by the author (1986). Probability theory, Transl. from the 1986

  3. Bourbaki, N.: General Topology. The Main Structures. Nauka, Moscow (1968). [in Russian]

    Google Scholar 

  4. Collingwood, E.F., Lohwater, A.J.: The Theory of Cluster Sets, Cambridge Tracts in Mathematics and Mathematical Physics, 56. Cambridge University Press, Cambridge (1966)

    Google Scholar 

  5. Dovgoshey, O., Martio, O., Ryazanov, V., Vuorinen, M.: The Cantor function. Expo. Math. 24, 1–37 (2006)

    Article  MathSciNet  Google Scholar 

  6. Duren, P.L.: Theory of Hp Spaces, Pure and Applied Mathematics, vol. 38. Academic Press, New York-London (1970)

    Google Scholar 

  7. Dyn’kin, E.M.: Methods of the Theory of Singular Integrals (the Hilbert Transform and Calderton–Zygmund Theory) (Russian),Current Problems in Mathematics. Fundamental Directions, vol. 15 (Russian), 197–292, 303, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow (1987)

  8. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)

    MATH  Google Scholar 

  9. Garnett, J.B., Marshall, D.E.: Harmonic Measure. Cambridge Univ. Press, Cambridge (2005)

    Book  Google Scholar 

  10. Gehring,F.W. : On the Dirichlet problem, Michigan Math. J., 3 201 (1955-1956)

    Article  Google Scholar 

  11. Goluzin, G.M.: Geometric theory of functions of a complex variable, Transl. of Math. Monographs, 26, American Mathematical Society, Providence, R.I. (1969)

  12. Havin, V.P.: Boundary properties of integrals of Cauchy type and of conjugate harmonic functions in regions with rectifable boundary. Mat. Sb. (N.S.) 68(110), 499–517 (1965). (in Russian)

    MathSciNet  Google Scholar 

  13. Henneken, P.L., Tortra, A.: Teoriya veroyatnostei i nekotorye ee prilozheniya (Russian) [Probability theory and some of its applications], Nauka, Moscow, 1974; transl. from Hennequin P.L., Tortrat A., Theorie des probabilites et quelques applications, (French) Masson et Cie, Editeurs, Paris, (1965)

  14. Kislyakov, S.V.: Classical problems of Fourier analysis (Russian), Current problems in mathematics. Fundamental directions, 15 (Russian), 135–195, 303, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, (1987)

  15. Koosis, P.: Introduction to \(H_p\) Spaces. Cambridge Tracts in Mathematics, vol. 115, 2nd edn. Cambridge Univ. Press, Cambridge (1998)

    MATH  Google Scholar 

  16. Luzin, N.N.: K osnovnoi theoreme integral’nogo ischisleniya [On the main theorem of integral calculus]. Mat. Sb. 28, 266–294 (1912). (in Russian)

    Google Scholar 

  17. Luzin, N.N.: Integral i trigonometriceskii ryady [Integral and trigonometric series], Dissertation, Moskwa, (in Russian)(1915)

  18. Luzin, N.N.: Integral i trigonometriceskii ryady [Integral and trigonometric series], In: Editing and commentary by N.K. Bari and D.E. Men’shov. Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, (in Russian) (1951)

  19. Luzin, N.: Sur la notion de l’integrale. Ann. Mat. Pura e Appl. 26(3), 77–129 (1917)

    Article  Google Scholar 

  20. Luzin, N., Priwalow, I.: Sur l’unicite et la multiplicite des fonctions analytiques. C. R. Acad. Sci. Paris 178, 456–459 (1924)

    Google Scholar 

  21. Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer, New York (2009)

    MATH  Google Scholar 

  22. Nevanlinna, R.: Eindeutige Analytische Funktionen. Michigan, Ann Arbor (1944)

    MATH  Google Scholar 

  23. Pfeffer, W.F.: Integration by parts for the generalized Riemann–Stieltjes integral. J. Aust. Math. Soc. Ser. A 34(2), 229–233 (1983)

    Article  MathSciNet  Google Scholar 

  24. Pommerenke, Ch.: Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299. Springer, Berlin (1992)

    Google Scholar 

  25. Privaloff, I.: Sur l’integrale du type de Cauchy-Stieltjes (Russian), Bull. Acad. Sci. URSS. Ser. Math. [Izvestia Akad. Nauk SSSR] 4, 261–276 (1940)

  26. Priwalow, I.I.: Vvedenie v teoriyu funktsii kompleksnogo peremennogo (Russian) [Introduction in the theory of functions of one complex variable], ed. 12, Nauka, Moscow, (1977)

  27. Priwalow, I.I.: Randeigenschaften analytischer Funktionen, Hochschulbücher für Mathematik, 25. Deutscher Verlag der Wissenschaften, Berlin (1956)

    Google Scholar 

  28. Riesz, M.: Sur les fonctions conjuguees. Math. Z. 27, 218–244 (1927)

    Article  MathSciNet  Google Scholar 

  29. Ryazanov, V.: On the boundary behavior of conjugate harmonic functions, Proceedings of Inst. Appl. Math. Mech. of Nat. Acad. Sci. of Ukraine, 31 (2017), 117–123; see also arXiv:1710.00323v3 [math.CV] 3 Mar 2018, 8 pp

  30. Ryazanov, V.: On the Riemann–Hilbert Problem without Index, Ann. Univ. Bucharest, Ser. Math. 5 (LXIII), no. 1 (2014), 169–178

  31. Ryazanov, V.: Infinite dimension of solutions of the Dirichlet problem, Open Math. (the former Central European J. Math.) 13, no. 1 348–350 (2015)

  32. Ryazanov, V.: On Neumann and Poincare problems for Laplace equation. Anal. Math. Phys. 7(3), 285–289 (2017)

    Article  MathSciNet  Google Scholar 

  33. Ryazanov, V., Yefimushkin, A.: On the Riemann–Hilbert problem for the Beltrami equations. Contemp. Math. 667, 299–316 (2016)

    Article  MathSciNet  Google Scholar 

  34. Saks, S.: Theory of the Integral, Warsaw, 1937. Dover Publications Inc., New York (1964)

    Google Scholar 

  35. Smirnoff, V.: Sur les valeurs limites des fonctions, regulieres a l’interieur d’un cercle. J. Soc. Phys. Math. Leningrade 2(2), 22–37 (1929)

    MathSciNet  MATH  Google Scholar 

  36. Yefimushkin, A.: On Neumann and Poincare problems in \(A\)-harmonic analysis, Advances in Analysis, 1, no. 2 (2016), 114-120; see also arXiv:1608.08457v1 [math.CV] 30 Aug 2016, 1–14

  37. Zygmund, A.: Trigonometric Series, Wilno, (1935)

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  1. Institute of Applied Mathematics and Mechanics of National Academy of Sciences of Ukraine, room 417,19 General Batyuk Str, Slavyansk, 84116, Ukraine

    Vladimir Ryazanov

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  1. Vladimir Ryazanov
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Correspondence to Vladimir Ryazanov.

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Communicated by Daniel Aron Alpay.

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Ryazanov, V. On the Theory of the Boundary Behavior of Conjugate Harmonic Functions. Complex Anal. Oper. Theory 13, 2899–2915 (2019). https://doi.org/10.1007/s11785-018-0861-y

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