Skip to main content
SpringerLink
Log in

Operators and multipliers on weighted Bergman spaces

  • Published:
Afrika Matematika Aims and scope Submit manuscript

Abstract

In 1992, Blasco considered a weighted Bergman space using Dini-weight function. He proved that a linear operator T on weighted Bergman space to any Banach space X is bounded if and only if certain one parameter fractional derivative of a single X-valued analytic function satisfy some growth condition. In this paper, we use a three parameters fractional derivative of a single X-valued analytic function and provide an equivalent condition. Our technique uses the Gaussian hypergeometric functions. Furthermore we supply some conditions on the parameters a, b and c under which the Gaussian hypergeometric functions F(a, b; c; z) are Dini-weight.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  2. Axler, S.: Bergman spaces and their operators, surveys on some recent results on operator theory. In: Conway, J.B., Morrel, B.B. (eds.) Pitman Research Notes in Math., chapter I, vol. 171, pp. 1–50 (1988)

  3. Blasco, O.: Operators on weighted Bergman spaces (\(0<p\le 1\)) and application. Duke Math. J. 66(3), 443–467 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blasco, O.: Spaces of vector valued analytic functions and applications, geometry of Banach spaces. In: Proceedings of conference held in Strobl, Austria 1989, London Math. Soc. Lecture Note Ser. 158, pp. 33–48. Cambridge University Press (1990)

  5. Blasco, O., de Souza, G.S.: Spaces of analytic functions on the disc where the growth of \(M_{p}(F, r)\) depends on a weight. J. Math. Anal. Appl. 147, 580–598 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Sh, Ponnusamy, S., Wang, X.: The isoperimetric type and Fejer–Riesz type inequalities for pluriharmonic mappings. Scientia Sinica Mathematica (Chinese version) 44(2), 127–138 (2014)

    Article  Google Scholar 

  7. Chen, Sh, Ponnusamy, S., Wang, X.: Area integral means, Hardy and weighted Bergman spaces of planar harmonic mappings. Kodai Math. J. 36, 313–324 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Sh, Ponnusamy, S., Wang, X.: Harmonic mappings in Bergman spaces. Monatsh. Math. 170(3–4), 325–342 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duren, P.L.: Theory of \(H^p\)-Spaces. Academic Press, New York (1970)

    MATH  Google Scholar 

  10. Duren, P.L., Shields, A.L.: Coefficient multipliers of \(H^p\) and \(B^p\) spaces. Pac. J. Math. 32, 69–78 (1970)

    Article  MATH  Google Scholar 

  11. Duren, P.L., Romberg, B.W., Shields, A.L.: Linear functionals on \(H^p\)-spaces with \(0<p<1\). J. Reigne Angew. Math. 238, 32–60 (1969)

    MATH  Google Scholar 

  12. Duren, P.L., Schuster, A.P.: Bergman spaces, Math. Surveys Monogr., vol. 100. Amer. Math. Soc., Providence, Rhode Island (2004)

  13. Flett, T.M.: The dual of an inequality of Hardy and Littlewood and some related inequalities. J. Math. Anal. Appl. 38, 746–765 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  14. Flett, T.M.: Lipschitz spaces of functions on the circle and the disc. J. Math. Anal. Appl. 39, 125–158 (1972)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Kalton, N.: Analytic functions in non locally convex spaces and applications. Stud. Math. 83, 275–303 (1986)

    Article  MATH  Google Scholar 

  17. Kalton, N.: Some applications of vector-valued analytic and harmonic functions, probability and banach spaces proceedings, Zaragoza 1985. Lecture Notes in Math. vol. 1221, pp. 114–140 (1986)

  18. Shapiro, J.H.: Mackey topologies, reproducing kernels and diagonal maps on the Hardy and Bergman spaces. Duke Math. J. 43, 187–202 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Shields, A.L., Williams, D.L.: Bounded projections, duality and multipliers in spaces of analytic functions. Trans. Am. Math. Soc. 162, 287–302 (1971)

    MATH  Google Scholar 

  20. Temme, N.M.: Special Functions: An Introduction to the Classical Functions of Mathematical Physics. Wiley, New York (1996)

    Book  MATH  Google Scholar 

  21. Wojtaszcyk, P.: Multipliers into Bergman spaces and Nevalinna classes. Can. Math. Bull. 33, 151–161 (1990)

    Article  Google Scholar 

  22. Zhu, K.: Operator theory in function spaces. Marcel Dekker Inc, New York (1990)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Sciences, Gauhati University, Guwahati, 781014, India

    S. Naik & K. Rajbangshi

Authors
  1. S. Naik
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. Rajbangshi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Naik.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Naik, S., Rajbangshi, K. Operators and multipliers on weighted Bergman spaces. Afr. Mat. 30, 269–277 (2019). https://doi.org/10.1007/s13370-018-0645-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13370-018-0645-6

Keywords

Mathematics Subject Classification

Search

Navigation