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Harmonic Conjugates in Weighted Bergman Spaces of Quaternion-Valued Functions

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Abstract

In this paper we study the harmonic conjugation problem in weighted Bergman spaces of quaternion-valued functions on the unit ball. For a scalar-valued harmonic function belonging to a Bergman space, harmonic conjugates in the same Bergman space are found.

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References

  1. K. L. Avetisyan, Fractional integro-differentiation in harmonic mixed norm spaces on a half-space, Comment. Math. Univ. Carolinae 42 (2001), 691–709.

    MathSciNet  MATH  Google Scholar 

  2. K. L. Avetisyan, A maximal theorem for a weighted space on the unit ball of ℝn, Int. Conf. Harmonic Analysis and Approx. III, 2005, Tsaghkadzor, Armenia, Abstracts, Yerevan (2005) 10–11; http://math.sci.am/conference/sept2005/Abstracts.pdf.

  3. S. Axler, P. Bourdon and W. Ramey, Harmonic Function Theory, 2nd Edition, Springer-Verlag, New York, 2001.

    MATH  Google Scholar 

  4. S. Bernstein, K. Gürlebeck, L. F. Reséndis and L. M. Tovar, Dirichlet and Hardy spaces of harmonic and monogenic functions, Z. Anal. Anwendungen 24 (2005), 763–789.

    Article  MathSciNet  Google Scholar 

  5. F. Brackx and N. Van Acker, A conjugate Poisson kernel in Euclidean space — Maple procedures for explicit calculation, Simon Stevin 67 no. 1–2 (1993), 3–14.

    MATH  Google Scholar 

  6. F. Brackx and R. Delanghe, On harmonic potential fields and the structure of monogenic functions, Z. Anal. Anwendunge 22 (2003), 261–273.

    Article  MathSciNet  MATH  Google Scholar 

  7. F. Brackx, R. Delanghe and F. Sommen, Clifford Analysis. Research Notes in Mathematics, 76. Boston — London — Melbourne: Pitman Advanced Publishing Program, 1982.

    MATH  Google Scholar 

  8. F. Brackx, R. Delanghe and F. Sommen, On conjugate harmonic functions in Euclidean space, Math. Methods Appl. Sci. 25 (2002), 1553–1562.

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Brackx, B. De Knock, H. De Schepper and D. Eelbode, On the interplay between the Hilbert transform and conjugate harmonic functions, Math. Methods Appl. Sci. 29 no.12 (2006), 1435–1450.

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Brackx and H. De Schepper, Conjugate harmonic functions in Euclidean space: a spherical approach, Comput. Methods Funct. Theory 6 no.1 (2006), 165–182.

    MathSciNet  MATH  Google Scholar 

  11. D. Constales, A conjugate harmonic to the Poisson kernel in the unit ball of ℝn, Simon Stevin 62 no.3–4 (1988), 289–291.

    MathSciNet  MATH  Google Scholar 

  12. O. Djordjević and M. Pavlović, Equivalent norms on Dirichlet spaces of polyharmonic functions on the ball in ℝN, Bol. Soc. Mat. Mexicana 13 no.2 (2007), 307–319.

    MathSciNet  MATH  Google Scholar 

  13. O. Djordjević and M. Pavlović, On a Littlewood-Paley type inequality, Proc. Amer. Math. Soc. 135 (2007), 3607–3611.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Fefferman and E. M. Stein, Hp spaces of several variables, Acta Math. 129 (1972), 137–193.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Gürlebeck, K. Habetha and W. Sprössig, Holomorphic Functions in the Plane and n-Dimensional Space, Birkhäuser Verlag, Basel, 2007.

    Google Scholar 

  16. K. Gürlebeck and W. Sprössig, Quaternionic and Clifford Calculus for Engineers and Physicists, John Wiley & Sons, Chichester, 1997.

    MATH  Google Scholar 

  17. G. H. Hardy and J. E. Littlewood, Some properties of conjugate functions, J. Reine Angew. Math. 167 (1931), 405–423.

    Google Scholar 

  18. M. Jevtić and M. Pavlović, Harmonic Bergman functions on the unit ball in ℝn, Acta Math. Hungar. 85 (1999), 81–96.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Miao, Reproducing kernels for harmonic Bergman spaces of the unit ball, Monatsh. Math. 125 (1998), 25–35.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Pavlović, Decompositions of Lp and Hardy spaces of polyharmonic functions, J. Math. Anal. Appl. 216 (1997), 499–509.

    Article  MathSciNet  MATH  Google Scholar 

  21. G. Ren, Harmonic Bergman spaces with small exponents in the unit ball, Collect. Math. 53 (2002), 83–98.

    MathSciNet  MATH  Google Scholar 

  22. G. Ren and U. Kähler, Weighted harmonic Bloch spaces and Gleason’s problem, Complex Variables 48 (2003), 235–245.

    Article  MATH  Google Scholar 

  23. G. Ren and U. Kähler, Hardy-Littlewood inequalities and Qp-spaces, Z. Anal Anwendungen 24 (2005), 375–388.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Rudin, Function Theory in the Unit Ball of Cn, Springer-Verlag, Berlin, Heidelberg, New York, 1980.

    Book  MATH  Google Scholar 

  25. M. Shapiro, On the conjugate harmonic functions of M. Riesz — E. Stein — G. Weiss, in: Dimiev, Stancho et al. (eds.), Topics in complex analysis, differential geometry and mathematical physics, Third international workshop on complex structures and vector fields, St. Konstantin, Bulgaria, August 23–29, 1996; Singapore: World Scientific. (1997), 8–32.

    Google Scholar 

  26. J. H. Shi, Inequalities for the integral means of holomorphic functions and their derivatives in the unit ball of Cn, Trans. Amer. Math. Soc. 328 (1991), 619–637.

    Article  MathSciNet  MATH  Google Scholar 

  27. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, New Jersey, 1970.

    MATH  Google Scholar 

  28. S. Stević, A note on polyharmonic functions, J. Math. Anal. Appl. 278 (2003), 243–249.

    Article  MathSciNet  MATH  Google Scholar 

  29. S. Stević, A Littlewood-Paley type inequality, Bol. Soc. Brasil Math. 34 no.2 (2003), 211–217.

    Article  MATH  Google Scholar 

  30. A. Sudbery, Quaternionic analysis, Math. Proc. Camb. Phil. Soc. 85 (1979), 199–225.

    Article  MathSciNet  MATH  Google Scholar 

  31. Z. Xu, J. Chen and W. Zhang, A harmonic conjugate of the Poisson kernel and a boundary value problem for monogenic functions in the unit ball of ℝn (n ≥ 2), Simon Stevin 64 no.2 (1990), 187–201.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Physics, Yerevan State University, Alex Manoogian St. 1, Yerevan, 375025, Armenia

    Karen Avetisyan

  2. Institut für Mathematik/Physik, Bauhaus-Universität Weimar, Coudray-Str. 13 B, 99423, Weimar, Germany

    Klaus Gürlebeck

  3. Institut für Angewandte Analysis, Technische Universität Bergakademie Freiberg, 09596, Freiberg, Germany

    Wolfgang Sprössig

Authors
  1. Karen Avetisyan
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  2. Klaus Gürlebeck
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  3. Wolfgang Sprössig
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Correspondence to Karen Avetisyan.

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Avetisyan, K., Gürlebeck, K. & Sprössig, W. Harmonic Conjugates in Weighted Bergman Spaces of Quaternion-Valued Functions. Comput. Methods Funct. Theory 9, 593–608 (2009). https://doi.org/10.1007/BF03321747

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  • DOI: https://doi.org/10.1007/BF03321747

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