Mathematische Annalen

, Volume 351, Issue 2, pp 305–326

Nevanlinna counting function and Carleson function of analytic maps

  • Pascal Lefèvre
  • Daniel Li
  • Hervé Queffélec
  • Luis Rodríguez-Piazza
Article

Abstract

We show that the maximal Nevanlinna counting function and the Carleson function of analytic self-maps of the unit disk are equivalent, up to constants.

Mathematics Subject Classification (2000)

Primary: 47B33 Secondary: 30C80 47B10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bourdon P.S.: Rudin’s orthogonality problem and the Nevanlinna counting function. Proc. Am. Math. Soc. 125(4), 1187–1192 (1997)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Carleson L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76(2), 547–559 (1962)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Choa J.S., Kim H.O.: On function-theoretic conditions characterizing compact composition operators on H 2. Proc. Jpn. Acad. Ser. A 75, 109–112 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Choe B.R.: The essential norms of composition operators. Glasg. Math. J. 34(2), 143–155 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Essén M., Shea D.F., Stanton C.S.: A value-distribution criterion for the class L log L and some related questions. Ann. Inst. Fourier (Grenoble) 35, 125–150 (1985)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kim H.O.: Averages of Nevanlinna counting functions of holomorphic self-maps of the unit disk. Hokkaido Math. J. 33(3), 697–706 (2004)MathSciNetMATHGoogle Scholar
  7. 7.
    Kim H.O.: Another look at average formulas of Nevanlinna counting functions of holomorphic self-maps of the unit disk. Kyungpook Math. J. 48(1), 155–163 (2008)MathSciNetMATHGoogle Scholar
  8. 8.
    Lefèvre P., Li D., Queffélec H., Rodríguez-Piazza L.: Opérateurs de composition sur les espaces de Hardy-Orlicz. C. R. Math. Acad. Sci. Paris 344(1), 5–10 (2007)MathSciNetMATHGoogle Scholar
  9. 9.
    Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Composition operators on Hardy-Orlicz spaces. Mem. Am. Math. Soc. 207(974) (2010)Google Scholar
  10. 10.
    Lefèvre P., Li D., Queffélec H., Rodríguez-Piazza L.: Some examples of compact composition operators on H 2. J. Funct. Anal. 255(11), 3098–3124 (2008)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Lefèvre P., Li D., Queffélec H., Rodríguez-Piazza L.: Compact composition operators on H 2 and Hardy-Orlicz spaces. J. Math. Anal. Appl. 354, 360–371 (2009)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Some revisited results about composition operators on Hardy spaces. Preprint. arXiv:1001.3328Google Scholar
  13. 13.
    Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Compact composition operators on Bergman-Orlicz spaces. Preprint. arXiv: 0910.5368Google Scholar
  14. 14.
    Littlewood J.E.: On inequalities in the theory of functions. Proc. Lond. Math. Soc. 23(2), 481–519 (1925)CrossRefGoogle Scholar
  15. 15.
    Liu P., Saksman E., Tylli H.-O.: Small composition operators on analytic vector-valued function spaces. Pac. J. Math. 184(2), 295–309 (1998)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Luecking D.H.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73, 345–368 (1987)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Luecking D.H., Zhu K.: Composition operators belonging to the Schatten ideals. Am. J. Math. 114, 878–906 (1992)MathSciNetCrossRefGoogle Scholar
  18. 18.
    McCluer B.: Compact composition operators on H p (B N). Mich. Math. J. 32, 237–248 (1985)CrossRefGoogle Scholar
  19. 19.
    McCluer B., Shapiro J.: Angular derivatives and compact composition operators on the Hardy and Bergman spaces. Can. J. Math. 38(4), 878–906 (1986)CrossRefGoogle Scholar
  20. 20.
    Nakazi T.: The Nevanlinna counting functions for Rudin’s orthogonal functions. Proc. Am. Math. Soc. 131(4), 1267–1271 (2003)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Nevanlinna R.: Le Théorème de Picard-Borel et la Théorie des Fonctions Méromorphes, second edition. Chelsea Publishing Company, New York (1974)Google Scholar
  22. 22.
    Nieminen P.J., Saksman E.: Boundary correspondence of Nevanlinna counting functions for self-maps of the unit disc. Trans. Am. Math. Soc. 356(8), 3167–3187 (2004)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Power S.C.: Vanishing Carleson measures. Bull. Lond. Math. Soc. 12(3), 207–210 (1980)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Rubel L.A.: Entire and Meromorphic Functions, Universitext, Tracts in Mathematics. Springer-Verlag, New York (1996)CrossRefGoogle Scholar
  25. 25.
    Shapiro J.H.: Composition Operators and Classical Function Theory, Universitext, Tracts in Mathematics. Springer-Verlag, New York (1993)Google Scholar
  26. 26.
    Shapiro J.H.: The essential norm of a composition operator. Ann. Math. 125, 375–404 (1987)CrossRefMATHGoogle Scholar
  27. 27.
    Shapiro J.H., Sundberg C.: Compact composition operators on L 1. Proc. Am. Math. Soc. 108(2), 443–449 (1990)MathSciNetMATHGoogle Scholar
  28. 28.
    Stanton C.S.: Counting functions and majorization for Jensen measures. Pac. J. Math. 125(2), 459–468 (1986)MATHGoogle Scholar
  29. 29.
    Zhu K.: Operator Theory in Function Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139. Marcel Dekker, New York (1990)Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Pascal Lefèvre
    • 1
    • 2
  • Daniel Li
    • 1
    • 2
    • 3
  • Hervé Queffélec
    • 1
    • 4
  • Luis Rodríguez-Piazza
    • 5
  1. 1.Univ Lille Nord de FranceLilleFrance
  2. 2.U-Artois, Laboratoire de Mathématiques de Lens EA 2462, Fédération CNRS Nord-Pas-de-Calais FR 2956LensFrance
  3. 3.Faculté des Sciences Jean PerrinLensFrance
  4. 4.USTL, Laboratoire Paul Painlevé U.M.R. CNRS 8524Villeneuve d’Ascq CedexFrance
  5. 5.Departamento de Análisis Matemático, Facultad de MatemáticasUniversidad de SevillaSevillaSpain

Personalised recommendations