DOI QRμ½”λ“œ



  • Bao, Guanlong (Department of Mathematics, Shantou University) ;
  • Lou, Zengjian (Department of Mathematics, Shantou University) ;
  • Qian, Ruishen (School of Mathematics and Computation Science, Lingnan Normal University) ;
  • Wulan, Hasi (Department of Mathematics, Shantou University)
  • Received : 2015.03.17
  • Published : 2016.03.31


In this paper, the effect of absolute values on the behavior of functions f in the spaces $\mathcal{Q}_K$ is investigated. It is clear that $g{\in}\mathcal{Q}_K({\partial}{\mathbb{D}}){\Rightarrow}{\mid}g{\mid}{\in}\mathcal{Q}_K({\partial}{\mathbb{D}})$, but the converse is not always true. For f in the Hardy space $H^2$, we give a condition involving the modulus of the function only, such that the condition together with ${\mid}f{\mid}{\in}\mathcal{Q}_K({\partial}{\mathbb{D}})$ is equivalent to $f{\in}\mathcal{Q}_K$. As an application, a new criterion for inner-outer factorisation of $\mathcal{Q}_K$ spaces is given. These results are also new for $Q_p$ spaces.


$\mathcal{Q}_K$ spaces;absolute values;inner-outer factorisation


Supported by : NSF of China, NSF of Guangdong Province


  1. G. Bao, Z. Lou, R. Qian, and H. Wulan, Improving multipliers and zero sets in QK spaces, Collect. Math. 66 (2015), no. 3, 453-468.
  2. B. Boe, A norm on the holomorphic Besov space, Proc. Amer. Math. Soc. 131 (2003), no. 1, 235-241.
  3. P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
  4. K. Dyakonov, Besov spaces and outer functions, Michigan Math. J. 45 (1998), no. 1, 143-157.
  5. M. Essen and H. Wulan, On analytic and meromorphic function and spaces of ${\mathcal{Q}}_K$-type, Illionis J. Math. 46 (2002), no. 4, 1233-1258.
  6. M. Essen, H. Wulan, and J. Xiao, Several function-theoretic characterizations of Mobius invariant ${\mathcal{Q}}_K$ spaces, J. Funct. Anal. 230 (2006), no. 1, 78-115.
  7. J. Garnett, Bounded Analytic Functions, Springer, New York, 2007.
  8. D. Girela, Analytic functions of bounded mean oscillation, In: Complex Function Spaces, Mekrijarvi 1999, 61-170, Editor: R. Aulaskari. Univ. Joensuu Dept. Math. Rep. Ser. 4, Univ. Joensuu, Joensuu, 2001.
  9. J. Pau, Bounded Mobius invariant ${\mathcal{Q}}_K$ spaces, J. Math. Anal. Appl. 338 (2008), no. 2, 1029-1042.
  10. H.Wulan and F. Ye, Some results in Mobius invariant ${\mathcal{Q}}_K$ spaces, Complex Var. Elliptic Equ. 60 (2015), no. 11, 1602-1611.
  11. J. Xiao, Holomorphic $\mathcal{Q}$ Classes, Springer, LNM 1767, Berlin, 2001.
  12. J. Xiao, Some results on ${\mathcal{Q}}_p$ spaces, 0 < p < 1, continued, Forum Math. 17 (2005), no. 4, 637-668.
  13. J. Xiao, Geometric ${\mathcal{Q}}_p$ Functions, Birkhauser Verlag, Basel-Boston-Berlin, 2006.
  14. K. Zhu, Operator Theory in Function Spaces, American Mathematical Society, Providence, RI, 2007.

Cited by

  1. On Dirichlet Spaces With a Class of Superharmonic Weights vol.70, pp.04, 2018,