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On a general family of function spaces. (English) Zbl 0851.30017

Annales Academiæ Scientiarum Fennicæ. Mathematica. Dissertationes. 105. Helsinki: Suomalainen Tiedeakatemia. 56 p. (1996).
Let \(A (\Delta)\) denote the collection of all functions analytic in the unit disk \( \Delta\), the following general family of function spaces was introduced and studied in great detail by the author in this paper. For \(0 < p < \infty\), \(- 2 < q < \infty\) and \(0 < s < \infty\), set \[ F(p, q,s) : = \Bigl \{f \in A (\Delta) : \sup_{a \in \Delta} I_a^{p,q,s} (f) < \infty \Bigr\} \] where \[ I_a^{p, q,s} (f) = \int_\Delta \bigl |f'(z) \bigr |^p \bigl( 1 - |z |^2 \bigr)^q \bigl( g(z,a) \bigr )^s dm (z) \] and \(g (z,a)\) is the Green’s function of \(\Delta\) with logarithmic singularity at \(a \in \Delta\). The author proves some basic properties of this space and shows some relations between \(F(p,q,s)\) and several classical spaces, such as \(\alpha\)-Bloch space, BMOA, etc. By introducing the notion of the \(\alpha\)-Möbius invariance, it follows that \(F(p,q,s)\) is \({q + 2 \over p}\)-Möbius invariant and \(\alpha\)-Bloch spaces are maximal along certain \(\alpha\)-Möbius invariant spaces. This result generalizes a theorem for the Bloch space by L. A. Rubel and R. M. Timoney [Proc. Am. Math. Soc. 75, 45-49 (1979; Zbl 0405.46020)], and gives an answer to a question by K. Zhu [Rocky Mt. J. Math. 23, No. 3, 1143-1177 (1993; Zbl 0787.30019)]. The estimate of the radial growth of the derivatives for the functions in \(F(p,q,s)\) is given. The author studies inclusion relations among the spaces \(F(p,q,s)\) and proves that all spaces \(F(p,q,s)\) are different for different \(s \in [0,1]\).
Reviewer: He Yuzan (Beijing)

MSC:

30D45 Normal functions of one complex variable, normal families
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