Multiplicative structure in weighted BMOA spaces (Q1271294)

Suppose \(\varphi: \mathbb{R}_+ \to\mathbb{R}_+\) is a nondecreasing continuous function, \(\mathbb{R}_+\) is the positive real axis. The space BMO consists of those functions \(f\in L^1 (\partial D)\) for which \[ \int_{\partial D} \bigl| f(e^{i \theta}) -f(z) \bigr| d\mu_z (\theta) \leq c_f\varphi \bigl(1- | z | \bigr), \quad z\in D \] where \(D\) is the unit disk, \(c_f>0\) is a constant, \[ f(z)= \int_{\partial D} f(e^{i\theta}) d\mu_z (\theta), \quad d\mu_z (\theta)= {1-| z |^2 \over | e^{i\theta} -z|^2} {d\theta \over 2\pi}, \] then \(\text{BMOA}_\varphi\) is defined by \(\text{BMOA}_\varphi: =\text{BMO}_\varphi \cap H^1\). In this paper, the inner and outer factors occurring in the canonical factorization of \(\text{BMOA}_\varphi\) functions are completely characterized by explicit quantitative conditions.