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Geometric characterizations for conformal mappings in weighted Bergman spaces

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Abstract

We prove that a conformal mapping defined on the unit disk belongs to a weighted Bergman space if and only if certain integrals involving the harmonic measure converge. With the aid of this theorem, we give a geometric characterization of conformal mappings in Hardy or weighted Bergman spaces by studying Euclidean areas. Applying these results, we prove several consequences for such mappings that extend known results for Hardy spaces to weighted Bergman spaces. Moreover, we introduce a number which is the analogue of the Hardy number for weighted Bergman spaces. We derive various expressions for this number and hence we establish new results for the Hardy number and the relation between Hardy and weighted Bergman spaces.

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References

  1. A. Baernstein, The size of the set on which a univalent function is large, J. Anal. Math. 70 (1996), 157–173.

    Article  MathSciNet  MATH  Google Scholar 

  2. A. Baernstein and D. Girela and J. Á. Peláez, Univalent functions, Hardy spaces and spaces of Dirichlet type, Illinois J. Math. 48 (2004), 837–859.

    Article  MathSciNet  MATH  Google Scholar 

  3. D. Betsakos and C. Karafyllia and N. Karamanlis, Hyperbolic metric and membership of conformal maps in the Bergman space, Canad. Math. Bull. 64 (2021), 174–181.

    Article  MathSciNet  MATH  Google Scholar 

  4. P. Duren, Theory of HpSpaces, Academic Press, New York—London, 1970.

    MATH  Google Scholar 

  5. P. Duren and A. Schuster, Bergman Spaces, American Mathematical Society, Providence, RI, 2004.

    Book  MATH  Google Scholar 

  6. M. Essén, On analytic functions which are in Hpfor some positive p, Ark. Mat. 19 (1981), 43–51.

    Article  MathSciNet  MATH  Google Scholar 

  7. M. Essén and D.F. Shea and C.S. Stanton, A value distribution criterion for the class L log L and some related questions, Ann. Inst. Fourier (Grenoble) 35 (1985), 127–150.

    Article  MathSciNet  MATH  Google Scholar 

  8. J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge University Press, Cambridge, 2005.

    Book  MATH  Google Scholar 

  9. L. J. Hansen, Hardy classes and ranges of functions, Michigan Math. J. 17 (1970), 235–248.

    Article  MathSciNet  MATH  Google Scholar 

  10. G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals II, Math. Z. 34 (1932), 403–439.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Hedenmalm and B. Korenblum and K. Zhu, Theory of Bergman Spaces, Springer, New York, 2000.

    Book  MATH  Google Scholar 

  12. C. Karafyllia, Hyperbolic metric and membership of conformal maps in the Hardy space, Proc. Amer. Math. Soc. 147 (2019), 3855–3858.

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Karafyllia, On a relation between harmonic measure and hyperbolic distance on planar domains, Indiana Univ. Math. J. 69 (2020), 1785–1814.

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Karafyllia, On the Hardy number of a domain in terms of harmonic measure and hyperbolic distance, Ark. Mat. 58 (2020), 307–331.

    Article  MathSciNet  MATH  Google Scholar 

  15. Y. C. Kim and T. Sugawa Hardy spaces and unbounded quasidisks, Ann. Acad. Sci. Fenn. Math. 36 (2011), 291–300.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. E. Marshall and W. Smith, The angular distribution of mass by Bergman functions, Rev. Mat. Iberoam. 15 (1999), 93–116.

    Article  MathSciNet  MATH  Google Scholar 

  17. F. Pérez-González and J. Rättyä, Univalent functions in Hardy, Bergman, Bloch and related spaces, J. Anal. Math. 105 (2008), 125–148.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Poggi-Corradini, Geometric Models, Iteration and Composition Operators, Ph.D. Thesis, University of Washington, WA, 1996.

    MATH  Google Scholar 

  19. P. Poggi-Corradini, The Hardy Class of Geometric Models and the Essential Spectral Radius of Composition Operators, J. Funct. Anal. 143 (1997), 129–156.

    Article  MathSciNet  MATH  Google Scholar 

  20. C. Pommerenke, Univalent Functions, Vandenhoeck & Ruprecht, Göttingen, 1975.

    MATH  Google Scholar 

  21. C. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschränkten Oszillation, Comment. Math. Helv. 52 (1977), 591–602.

    Article  MathSciNet  MATH  Google Scholar 

  22. C. Pommerenke, Boundary Behaviour of Conformal Maps, Springer, Berlin, 1992.

    Book  MATH  Google Scholar 

  23. H. Prawitz, Über Mittelwerte analytischer Funktionen, Ark. Mat. Astr. Fys. 20 (1927), 1–12.

    MATH  Google Scholar 

  24. W. Smith, Composition operators between Bergman and Hardy spaces, Trans. Amer. Math. Soc. 348 (1996), 2331–2348.

    Article  MathSciNet  MATH  Google Scholar 

  25. E. M. Stein and R. Shakarchi, title=Real Analysis, Princeton University Press, Princeton, NJ, 2005.

    Book  MATH  Google Scholar 

  26. K. Zhu, Translating inequalities between Hardy and Bergman spaces, Amer. Math. Monthly 111 (2004), 520–525.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY, 11794, USA

    Christina Karafyllia

  2. Department of Mathematics and Applied Mathematics, University of Crete, Heraklion, 70013, Crete, Greece

    Nikolaos Karamanlis

Authors
  1. Christina Karafyllia
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  2. Nikolaos Karamanlis
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Corresponding author

Correspondence to Nikolaos Karamanlis.

Additional information

The second author is supported by the Hellenic Foundation for Research and Innovation, Project HFRI-FM17-1733.

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Karafyllia, C., Karamanlis, N. Geometric characterizations for conformal mappings in weighted Bergman spaces. JAMA 150, 303–324 (2023). https://doi.org/10.1007/s11854-023-0274-3

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  • DOI: https://doi.org/10.1007/s11854-023-0274-3

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