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On subspaces and subsets of BMOA and UBC. (English) Zbl 0835.30027
Let \(\Delta\) denote the unit disk in the complex plane, let \(a \in \Delta\) and let \(g(z, a) = \log |(1- \overline {a} z/(z- a)|\) be the Green’s function on \(\Delta\) with singularity at \(a\). The following function sets are introduced for \( 0\leq p< \infty\): \[ \begin{aligned} Q_p &=\{f:\;f\text{ analytic in \(\Delta\) and } \sup_{a\in \Delta} \iint_\Delta |f' (z)|^2 (g(z,a))^p dx dy<\infty\},\\ Q_{p,0} &= \{f:\;f \text{ analytic in \(\Delta\) and } \lim_{|a|\to 1} \iint_\Delta |f' (z) |^2 (g(z, a))^p dx dy =0\},\\ Q_p^\# &= \{f:\;f \text{ meromorphic in \(\Delta\) and } \sup_{a\in \Delta} \iint_\Delta (f^\# (z) )^2 (g (z, a))^p dx dy< \infty \}, \qquad \text{and}\\ Q^\#_{p,0}&=\{f:\;f\text{ meromorphic in \(\Delta\) and }\lim_{|a|\to 1}\iint_\Delta (f^\# (z))^2 (g(z, a))^p dx dy= 0\}, \end{aligned} \] where \(f^\# (z)\) denotes the spherical derivative of the function \(f\). It is well known that \(Q_1= \text{BMOA} (\Delta)\), \(Q_{1,0}= \text{VMOA} (\Delta)\), and \(Q_0=\) the usual Dirichlet space. The first author and the reviewer [Pitman Res. Notes Math. Ser. 305, 136-146 (1994; Zbl 0826.30027)] proved that, for \(p>1\), \(Q_p\) is the Bloch space \(B\), \(Q_{p,0}\) is the “little Bloch space” \(B_0\), \(Q_p^\#\) is the collection of normal functions \(N\), and \(Q^\#_{p,0}\) is the collection of “little” normal functions \(N_0\). In the present paper, the authors show various containment relationships between the various spaces. For example, for \(0< p\leq 1\), \(Q_{p,0} \subset Q_q\) and \(Q^\#_{p,0} \subset Q^\#_p\). Further, for \(0\leq p< q\), \(Q_p \subset Q_q\) and \(Q^\#_p \subset Q^\#_q\), and, for \(0< p< q\), \(Q_0 \subset Q_{p,0} \subset Q_{q,0}\) and \(Q^\#_0 \subset Q^\#_{p,0} \subset Q^\#_{q,0}\). Further, all of the containments mentioned here are proper containments for \(q\leq 1\), as are shown by examples making some clever uses of gap series. In fact, it is shown that a number of containments, of which \(Q_p \subset \bigcap_{p< q\leq 1} Q_q\), \(0\leq p<1\), is typical, are strictly containments. Some other characterizations of these spaces are obtained by replacing \(g(z, a)\) by \(1-|(z- a)/ (1- \overline {a} z)|^2\) in the defining integrals.

MSC:
30D50 Blaschke products, etc. (MSC2000)
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