On absolute values of QK functions

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 $f\in {\mathcal{Q}}_K(\partial \mathbb{D})\Rightarrow |f|\in {\mathcal{Q}}_K(\partial \mathbb{D})$, 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 this condition together with $|f|\in {\mathcal{Q}}_K(\partial \mathbb{D})$ 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 ${\mathcal{Q}}_p$ spaces.