On absolute values of QK functions

TL;DR

This paper explores how absolute values influence functions in $\\mathcal{Q}_K$ spaces, providing new criteria for function classification and inner-outer factorization based on modulus properties.

Contribution

It introduces conditions involving only the modulus of functions that characterize membership in $\\mathcal{Q}_K$ spaces and offers a novel criterion for inner-outer factorization.

Findings

Established that $f\in \mathcal{Q}_K$ implies $|f|\in \mathcal{Q}_K$, but not vice versa.

Provided a Hardy space $H^2$-based condition linking $|f|$ and $f$ in $\\mathcal{Q}_K$.

Presented new criteria for inner-outer factorization in $\\mathcal{Q}_K$ and $\\mathcal{Q}_p$ spaces.

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.