A Survey of Q—Spaces and Q#-Classes

  • Matts Essén
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 10)


We study Q-spaces and Q #-classes which are Möbius-invariant, weighted Dirichlet spaces or classes of analytic or meromorphic functions in the unit disc in the plane. Generalizations to half-spaces in higher dimensions are also possible. In the analytic case, they are subspaces of BMOA or of the Bloch space B. In the meromorphic case, they are subclasses of the class of normal functions N or of the class of spherical Bloch functions B #. In the first part of the survey, we discuss concrete examples where different kinds of p-Carleson measures (0 < p < 1) are important. In the last section, we discuss a more general theory which gives both new results and new proofs of several results from the first part.


Meromorphic Function Besov Space Blaschke Product Carleson Measure Dirichlet Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Arazy, J., Fisher, S., Peetre, J.: Möbius invariant function spaces. J. Reine Angew. Math. 363 (1985), 110–145.MathSciNetMATHGoogle Scholar
  2. [2]
    Aulaskari, R., Lappan, P.: Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal. Complex analysis and its applications Pitman Research Notes in Mathematics 305. Longman Scientific Technical Harlow 1994, 136–146.Google Scholar
  3. [3]
    Aulaskari, R., Stegenga, D., Xiao, J.: Some subclasses of BMOA and their characterization in terms of Carleson measures. Rocky Mountain J. Math. 26 (1996), 485–506.MathSciNetMATHGoogle Scholar
  4. [4]
    Aulaskari, R., Wulan, H., Zhao, R.: Carleson measure and some classes of meromorphic functions. Proc. Amer. Math. Soc. 128 (2000), 2329–2335.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Aulaskari, R., Xiao, J., Zhao,R.: On subspaces and subsets of BMOA and UBC. Analysis 15 (1995), 101–121.MathSciNetMATHGoogle Scholar
  6. [6]
    Baernstein, A. II: Analytic functions of bounded mean oscillation. Aspects of contemporary complex analysis. Academic Press, London, 1980, 3–36.Google Scholar
  7. [7]
    Carleson, L.: Interpolations by bounded analytic functions and the corona problem. Ann. Math. 76 (1962), 547–559.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Duren, P.: Theory of H“ spaces. Academic Press, New York and London, 1970.MATHGoogle Scholar
  9. [9]
    Essén, M.: Qp-spaces. “Complex Function Spaces”, Mekrijärvi 1999 (ed.Aulaskari), Department of Mathematics, University of Joensuu, Finland, Report no 4, 2001, 9–40.Google Scholar
  10. [10]
    Essén, M., Janson, S., Peng, L., Xiao, J.: Q-spaces of several real variables. Ind. Univ. Math. J. 49 (2000), 575–615.MATHGoogle Scholar
  11. [11]
    Essén, M., Wulan, H.: Carleson type measures and their applications. Complex Variables, Theory Appl. 42 (2000), 67–88.Google Scholar
  12. [12]
    Essén, M., Wulan, H.: On analytic and meromorphic functions and spaces of Qx-type. Department of Mathematics, Uppsala University, Report 2000: 32.Google Scholar
  13. [13]
    Essén, M., Xiao, J.: Some results on Qp-spaces, 0 p 1. J. Reine Angew. Math. 485 (1997), 173–195.MathSciNetMATHGoogle Scholar
  14. [14]
    Essén, M., Xiao, J.: QP-spaces–a survey. Complex Function Spaces, Mekrijärvi 1999, (ed. Aulaskari), Department of Mathematics, University of Joensuu, Finland, Report no 4, 2001, 41–60.Google Scholar
  15. [15]
    Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Amer. Math. Soc. 77 (1971), 587–588.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Garnett, J.: Bounded analytic functions. Academic Press, New York 1981.MATHGoogle Scholar
  17. [17]
    Lappan, P.: A non-normal locally uniformly univalent function. Bull. London Math. Soc. 5 (1973), 291–294.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Nicolau, A.: The corona property for bounded analytic functions in some Besov spaces. Proc. Amer. Math. Soc. 110 (1990), 135–140.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    Ortega, J.M., Fàbrega, J.: The corona type decomposition in some Besov spaces. Math. Scand. 78 (1996), 93–111.MathSciNetGoogle Scholar
  20. [20]
    Pommerenke, Ch.: Boundary behaviour of conformal maps. Springer, Berlin 1992.MATHCrossRefGoogle Scholar
  21. [21]
    Tolokonnikov, V.A.: The corona theorem in algebras of bounded analytic functions. Amer. Math. Soc. Trans. 149 (1991), 61–93.Google Scholar
  22. [22]
    Wulan, H.: On some classes of meromorphic functions. Ann. Acad. Sci. Fenn. Math. Diss. 116 (1998), 1–57.Google Scholar
  23. [23]
    Wu, P., Wulan, H.: Characterizations of QT spaces. J. Math. Anal. Appl. 254 (2001), 484–497.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Xiao, J.: The Qp corona theorem. Pac. J. of Math. 194 (2000), 491–509.MATHCrossRefGoogle Scholar
  25. [25]
    Xiao, J.: Some essential properties of Qp(ÔO)-spaces. J. of Fourier Analysis and Applications 6 (2000), 311–323.MATHCrossRefGoogle Scholar
  26. [26]
    Xiao, J.: Outer functions in Q p and Qp,o. Preprint 1999.Google Scholar
  27. [27]
    Xiao, J.: Holomorphic Q classes. Lecture Notes in Mathematics 1767. Springer, Berlin 2001.Google Scholar
  28. [28]
    Yamashita, S.: Functions of uniformly bounded characteristic. Ann. Acad. Sci. Fenn. Ser. A I Math. 7 (1982), 349–367.MathSciNetMATHGoogle Scholar

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© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • Matts Essén
    • 1
  1. 1.Department of MathematicsUppsala UniversityUppsalaSweden

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