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.