Let \(\mathbb D\) be the unit disc in \(\mathbb C\), \(H^\infty(\mathbb D)\) the class of bounded analytic functions on \(\mathbb D\). For \(w\in\mathbb D\), define \(g(z,w)=\log\frac{1}{|\sigma_w(z)|}\), where \(\sigma_w(z)=\frac{w-z}{1-\overline wz}\). Let \(k:[0,\infty)\to [0,\infty)\), and define \(Q_K\) to be the set of all \(f\) analytic in \(\mathbb D\) for which \(\| f\|^2_{Q_K}=\sup_{w\in\mathbb D}\int_{\mathbb D}\| f'(z)\|^2 K(g(z,w))\,dA(z)<\infty\), where \(dA(z)\) is the Euclidean area element on \(\mathbb D\). Different choices of \(k\) give familiar spaces such as the Bloch space and BMOA. The authors establish necessary and sufficient conditions for analytic functions to belong to \(Q_K\), where \(K\) satisfies seven conditions, which will not be given here.
One characterization of \(Q_K\) involves the concept of a \(K\)-Carleson measure: a positive measure \(d\mu\) is said to be a \(K\)-Carleson measure on \(\mathbb D\) if \(\sup_{I\subset\partial\mathbb D}\int_{S(I)}K\left(\frac{1-|z|}{|I|}\right)\,d\mu(z)<\infty\), where \(S(I)=\{rw\in\mathbb D: 1-|I|<r<1\), \(w\in I\}\). Then the authors prove: \(f\in Q_K\) if \(|f'(z)|^2 dA\) is a \(K\)-Carleson measure on \(\mathbb D\). This is used to give necessary and sufficient conditions for membership in \(Q_K\) based on behavior of boundary values: \(f\in H^2\) has \(f\in Q_K\) if and only if \(\sup_{I\subset\partial\mathbb D}\int_I\int_I\frac{|f(w)-f(z)|^2}{|w-z|^2}K\left(\frac{|w-z|}{I}\right)|dw||dz|<\infty.\) In addition, necessary and sufficient conditions are given for inner functions to belong to \(Q_K\). One final characterization is given in terms of \(|f(z)|\), \(z\in\mathbb D\cup \partial \mathbb D\).