Analytic \(Q_{\log ,p}\) Spaces

  • S. Luo
  • J. XiaoEmail author


As a novel bridge between the Dirichlet space \({\mathcal {D}}\), the John–Nirenberg space \(\mathcal {BMOA}\), and the Bloch space \({\mathcal {B}}\) on the unit disk, the Moebius invariant analytic function space \(Q_{\log ,p}\) founded directly on a Moebius invariant isoperimetry is discovered in accordance with the Moebius invariant inclusion chain \({\mathcal {H}}^\infty \subsetneq \mathcal {BMOA}\subsetneq {\mathcal {B}}\), where \({\mathcal {H}}^\infty \) is the Hardy algebra of all bounded analytic functions on the unit disk.


\(Q_{\log , p}\) Bloch space BMOA Hardy algebra 

Mathematics Subject Classification

30H05 30H25 30H80 



  1. 1.
    Earl, J.P.: On the interpolation of bounded sequences by bounded functions. J. Lond. Math. Soc. 2, 544–548 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Essén, M.: A survey of \(Q\)-spaces and \(Q^\#\)-classes. In: Analysis and Applications-ISAAC 2001 (Berlin), Int. Soc. Anal. Appl. Comput. vol. 10, pp. 73-87. Kluwer Acad. Publ., Dordrecht (2003)Google Scholar
  3. 3.
    Essén, M., Wulan, H.: On analytic and meromorphic functions and spaces of \(Q_K\)-type. Illinois J. Math. 46, 1233–1258 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Essén, M., Xiao, J.: Some results on \(Q_p\) spaces, \(0<p<1\). J. Reine Angew. Math. 485, 173–195 (1997)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Essén, M., Wulan, H., Xiao, J.: Several function-theoretic characterizations of Moebius invariant \(Q_K\) spaces. J. Funct. Anal. 230, 78–115 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Garnett, J.: Bounded Analytic Functions. Academic Press, New York (1981)zbMATHGoogle Scholar
  7. 7.
    Garsia, A.M.: On the smoothness of functions satisfying certain integral inequalities. In: 1969 Functional Analysis (Proc. Sympos. Monterey, Calif., 1969), pp. 127–C162. Academic Press, New York (1970)Google Scholar
  8. 8.
    Janson, S.: On the space \(Q_p\) and its dyadic counterpart. In: Proc. Symposium on Complex Analysis and Differential Equations, June 15–18, 1997. Acta Univ. Upsaliensis Skr. Uppsala Univ. C. Organ. Hist., vol. 64, pp. 194–205Google Scholar
  9. 9.
    John, F., Nirenberg, L.: On functions of bounded mean oscillation. Commun. Pure Appl. Math. 14, 415–426 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jones, P.: \(L^\infty \) estimates for the \({\overline{\partial }}\) problem in a half-plane. Acta Math. 150, 137–152 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kufner, A., Persson, L.-E.: Weighted Inequalities of Hardy Type. World Scientific, River Edge, NJ (2003)CrossRefzbMATHGoogle Scholar
  12. 12.
    Nicolau, A., Xiao, J.: Bounded functions in Möbius invariant Dirichlet spaces. J. Funct. Anal. 150, 383–425 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)zbMATHGoogle Scholar
  14. 14.
    Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton, NJ (1993)zbMATHGoogle Scholar
  15. 15.
    Wulan, H., Zhu, K.: Möbius Invariant \({\cal{Q}}_K\) Spaces. Springer, New York (2017)CrossRefGoogle Scholar
  16. 16.
    Xiao, J.: The \({\overline{\partial }}\)-problem for multipliers of the Sobolev space. Manuscr. Math. 97, 217–232 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Xiao, J.: Some essential properties of \(Q_p(\partial \Delta )\)-spaces. J. Fourier Anal. Appl. 6, 311–323 (2000)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Xiao, J.: Holomorphic \({\cal{Q}}\) Classes. Lecture Notes in Math, vol. 1767. Springer, Berlin (2001)CrossRefGoogle Scholar
  19. 19.
    Xiao, J.: Geometric \(Q_p\) Functions. Birkhauser Verlag, Basel (2006)Google Scholar
  20. 20.
    Xiao, J., Zhang, J.: N-S systems via \({\cal{Q}}\)-\({\cal{Q}}^{-1}\) spaces. J. Geom. Anal.
  21. 21.
    Zhang, X., Guo, Y., Shang, Q., Li, S.: The Gleason’s Problem on \(F(p, q, s)\) Type Spaces in the Unit Ball of \({{\mathbb{C}}}^n\). Complex Anal. Oper. Theory 12, 1251–1265 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhou, J.: Interpolating sequences in \({\cal{Q}}_K\) spaces. J. Math. (PRC) 36, 511–518 (2016)Google Scholar
  23. 23.
    Zhou, J., Bao, G.: Analytic version of \({\cal{Q}}_1(\partial {\mathbb{D}})\) space. J. Math. Anal. Appl. 422, 1091–1102 (2015)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Zhu, K.: Operator Theory in Function Spaces. Mathematical Surveys and Monographs, 2nd edn. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaPeople’s Republic of China
  2. 2.Department of Mathematics and StatisticsMemorial University of NewfoundlandSt. John’sCanada

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