Predual of \(Q_K\) spaces (Q1951060)

Let \(K: [0,\infty) \to [0,\infty)\) be a right-continuous and nondecreasing function. The space \(Q_K\) consists of all analytic functions \(f \in H(\mathbb D)\) satisfying \[ \|f\|^2_{Q_K} = \sup_{a\in\mathbb D} \int _{\mathbb D} |f'(z)|^2 K(g(a,z)) \;dA(z) < \infty, \] where \(g(z,a)\) is the Green function in \(\mathbb D\) with singularity at \(a \in\mathbb D\), and \(dA(z)\) is the Euclidean area element on \(\mathbb D\) so that \(A(\mathbb D) = 1\). The space \(Q_K\) is a Banach space with the norm \(|f(0)| + \|f\|_{Q_K}\). It is clear that \(Q_K\) is Möbius-invariant. In the case \(K(t) = t^p\), \(0 < p < 1\), the space \(Q_K\) gives the \(Q_p\) space. Especially, \(Q_K\) coincides with BMOA if \(K(t) = t\) and it is known that \(Q_K\) spaces are contained in the Bloch space \(\mathcal B\). In the article under review, the author first obtains a necessary and sufficient condition that a positive measure \(\mu\) on \(\mathbb D\) is a \(K\)-Carleson measure. This result is then used to prove Theorem 10, which is the main result of the article: The predual of the \(Q_K\) space is given by a space of analytic functions in \(H(\mathbb D)\).