Bulletin of the Korean Mathematical Society (λŒ€ν•œμˆ˜ν•™νšŒλ³΄)

Korean Mathematical Society (λŒ€ν•œμˆ˜ν•™νšŒ)
ꢌ호 ν‘œμ§€
DOI QRμ½”λ“œ

DOI QR Code

ON ABSOLUTE VALUES OF 𝓠K FUNCTIONS

  • 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
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

Acknowledgement

Supported by : NSF of China, NSF of Guangdong Province

References

  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. https://doi.org/10.1007/s13348-014-0113-z
  2. B. Boe, A norm on the holomorphic Besov space, Proc. Amer. Math. Soc. 131 (2003), no. 1, 235-241. https://doi.org/10.1090/S0002-9939-02-06529-2
  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. https://doi.org/10.1307/mmj/1030132088
  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. https://doi.org/10.1016/j.jfa.2005.07.004
  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. https://doi.org/10.1016/j.jmaa.2007.05.069
  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. https://doi.org/10.1080/17476933.2015.1037747
  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. https://doi.org/10.1515/form.2005.17.4.637
  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, https://doi.org/10.4153/CJM-2017-005-1