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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

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

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

Acknowledgement

Supported by : NSF of China, NSF of Guangdong Province

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