• Title/Summary/Keyword: inner-outer factorisation

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ON ABSOLUTE VALUES OF 𝓠K FUNCTIONS

  • Bao, Guanlong;Lou, Zengjian;Qian, Ruishen;Wulan, Hasi
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.2
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    • pp.561-568
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    • 2016
  • 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.