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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)
##### Keywords:
spherical derivative
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