Skip to main content
SpringerLink
Log in

Coefficients Multipliers of Weighted Spaces of Harmonic Functions

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

Let \({h_g^\infty}\) be the space of harmonic functions in the unit ball that are bounded by some increasing radial function that tends to infinity as r goes to one; these spaces are called growth spaces. We describe functions in growth spaces by the Cesàro means of their expansions in harmonic polynomials and apply this characterization to study coefficient multipliers between growth spaces. Further, we introduce spaces of harmonic functions of regular growth and show that integral operators considered recently in connection to boundary oscillation of harmonic functions in weighted spaces, can be realized as multipliers that map growth spaces to corresponding spaces of regular growth.

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. Anderson J.M.: Coefficient multipliers and solid spaces. J. Anal. 1, 13–19 (1993)

    MathSciNet  MATH  Google Scholar 

  2. Anderson J.M., Shields A.L.: Coefficient multipliers of Bloch functions. Trans. Am. Math. Soc. 224(2), 255–265 (1976)

    Article  MathSciNet  Google Scholar 

  3. Bennett G., Stegenga D.A., Timoney R.M.: Coefficients of Bloch and Lipschitz functions. Ill. J. Math. 25(3), 520–531 (1981)

    MathSciNet  MATH  Google Scholar 

  4. Blasco O.: Multipliers on spaces of analytic functions. Can. J. Math. 47(1), 44–64 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bonami A., Clerc J.-L.: Sommes de Cesàro et multiplicateurs des dé-ve-lop-pe-ments en harmoniques sphériques. Trans. Am. Math. Soc. 183, 223–262 (1973)

    MathSciNet  MATH  Google Scholar 

  6. Buckley S.M.: Mixed norms and analytic function spaces. Math. Proc. R. Ir. Acad. 100A(1), 1–9 (2000)

    MathSciNet  Google Scholar 

  7. Buckley S.M.: Relative solidity for spaces of holomorphic functions. Math. Proc. R. Ir. Acad. 104A(1), 83–97 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Buckley S.M., Koskela P., Vukotić D.: Fractional integration, differentiation, and weighted Bergman spaces. Math. Proc. Cambr. Philos. Soc. 126(2), 369–385 (1999)

    Article  MATH  Google Scholar 

  9. Buckley S.M., Ramanujan M.S., Vukotić D.: Bounded and compact multipliers between Bergman and Hardy spaces. Integral Equ. Oper. Theory 35(1), 1–19 (1999)

    Article  MATH  Google Scholar 

  10. Dai F., Xu Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York (2013)

    Book  MATH  Google Scholar 

  11. de Boor C.: Divided differences. Surv. Approx. Theory 1, 46–69 (2005)

    MathSciNet  MATH  Google Scholar 

  12. de Leeuw K., Katznelson Y., Kahane J.-P.: Sur les coefficientes de Fourier des functions continues. C.R. Acad. Sci. Paris Sér. A-B 285(16), A1001–A1003 (1977)

    Google Scholar 

  13. Doubtsov, E.: Characterisations of Hardy growth spaces with doubling weights. Bull. Aust. Math. Soc. 90(2), 275–282 (2014)

  14. Eikrem, K.S.: Characterization and boundary behavior of harmonic functions in growth spaces. Ph.D. thesis, Norwegian University of Science and Technology (NTNU) (2013)

  15. Eikrem K.S.: Hadamard gap series in growth spaces. Collect. Math. 64(1), 1–15 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Eikrem K.S., Malinnikova E., Mozolyako P.: Wavelet decomposition of harmonic functions in growth spaces. J. Anal. Math. 122, 87–111 (2014)

    Article  MathSciNet  Google Scholar 

  17. Girela D., Pavlović M., Peláez J.Á.: Spaces of analytic functions of Hardy–Bloch type. J. Anal. Math. 100, 53–81 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kellogg C.N.: An extension of the Hausdorff–Young theorem. Mich. Math. J. 18, 121–127 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  19. Kogbetliantz E.: Recherches sur la sommabilité de séries ultraphériques par le méthode des moyennes arithmétiques. J. Math. Pure Appl. Ser. 9 3, 107–187 (1924)

    MATH  Google Scholar 

  20. Lusky W.: On weighted spaces of harmonic and holomorphic functions. J. Lond. Math. Soc. 51, 309–320 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lusky W.: On the isomorphism classes of weighted spaces of harmonic and holomorphic functions. Studia Math. 175(1), 19–45 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lyubarskii Yu., Malinnikova E.: Radial oscillation of harmonic functions in the Korenblum space. Bull. Lond. Math. Soc. 44(1), 68–84 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Nowak M.: Coefficient multipliers of spaces of analytic functions. Ann. Univ. Mariae Curie-Skł odowska Sect. A 52(1), 107–119 (1998)

    MATH  Google Scholar 

  24. Pavlović M.: Mixed norm spaces of analytic and harmonic functions, I. Publ. Inst. Math. 40(54), 117–141 (1986)

    Google Scholar 

  25. Pavlović M.: Mixed norm spaces of analytic and harmonic functions, II. Publ. Inst. Math. 41(55), 97–110 (1987)

    Google Scholar 

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

    MathSciNet  Google Scholar 

  27. Shields A.L., Williams D.L.: Bounded projections, duality, and multipliers in spaces of harmonic functions. J. Reine Angew. Math. 299/300, 256–279 (1978)

    MathSciNet  Google Scholar 

  28. Szegö, G.: Orthogonal Polynomilas, 4th edn. American Mathematical Society, Providence (1975)

  29. Vukotić. D.: On the coefficient multipliers of Bergman spaces. J. Lond. Math. Soc. II. Ser. 50(2), 341–348 (1994)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Kjersti Solberg Eikrem & Eugenia Malinnikova

Authors
  1. Kjersti Solberg Eikrem
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eugenia Malinnikova
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Eugenia Malinnikova.

Additional information

The second author is supported by Project 213638 of the Research Council of Norway.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Eikrem, K.S., Malinnikova, E. Coefficients Multipliers of Weighted Spaces of Harmonic Functions. Integr. Equ. Oper. Theory 82, 555–573 (2015). https://doi.org/10.1007/s00020-015-2221-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-015-2221-x

Mathematics Subject Classification

Keywords

Search

Navigation