Inner-outer factorization on \(\mathcal Q_p\) spaces (Q2346156)

Given \(0<p\leq 1\), let \(\mathcal Q_p(\mathbb T)\) be the set of all \(f\in L^2(\mathbb T)\) such that \[ \sup_{I\subseteq\mathbb T}|I|^{-p}\int_I\int_I \frac{|f(\xi)-f(\eta)|^2}{|\xi-\eta|^{2-p}} |d\xi|\;|d\eta|<\infty, \] and let \(Q_p\) be the set of all holomorphic functions \(f\) in \(\mathbb D\) for which \[ \sup_{a\in\mathbb D}\int_{\mathbb D} |f '(z)|^2 \log^p\frac{|1-\overline a z|} {|a-z|}\;dA(z)<\infty. \] The authors show that for an inner function \(\theta\in Q_p\), and \(f\in Q_p\), one has that \(f\theta\in Q_p\) if and only if \(f\overline \theta\in \mathcal Q_p(\mathbb T)\) and that this happens if and only if \(\sup\{|f(z)|: |\theta(z)|<\epsilon\}<\infty\) for some/all \(\epsilon>0\), which is equivalent to \(\sup_{z\in\mathbb D} |f(z)|^2(1-|\theta(z)|^2)<\infty\). Similar results are derived for the associated multiplier algebra \(M(Q_p)\) and the space \(Q_p\cap AB^s_q\), where \(AB^s_q\) is the space of all \(f\in H(\mathbb D)\) such that \[ \int_{\mathbb D} |f '(z)|^q (1-|z|)^{(1-s)q-1}\;dA(z)<\infty, \] with \(0<p\leq 1\), \(1\leq q<\infty\) and \(0<s<1/q\).