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Topics in the theory of \(A^p_\alpha\) spaces. (English) Zbl 0667.30032

Teubner-Texte zur Mathematik, 105. Leipzig: BSB B. G. Teubner Verlagsgesellschaft. 199 p. DM 21.00 (1988).
This book is the first attempt in the mathematical literature to treat from a unified position many facts of the theory of the area integrable analytic functions. The \(A^p_{\alpha}\) spaces mentioned in the title are the spaces of functions \(f\) analytic in the unit disc \(D\) for which \[ \int_{D}| f(z)|^ p(1-| z|)^{\alpha} \,dm_{\alpha}(z)<\infty, \] where \(dm_2\) is the plane Lebesgue measure. Such spaces are called “weighted Bergman” spaces in many recent publications.
The book conditionally may be divided into two parts. The first three chapters form the “textbook” part, the remaining four chapters reflect the author’s interests and are formed almost of the results of them.
Table of contents: 1. Spaces of area integrable functions in the unit disk. 2. Bounded projections, conjugate harmonic functions and bounded linear functionals in \(A^p_{\alpha}\). 3. Zeros of \(A^p_{\alpha}\) functions. 4. On zeros of analytic functions with restricted growth. 5. Division in \(X^{\infty}_{\phi}\) and description of closed ideals. 6. \(A^p_{\alpha}\) spaces of holomorphic functions in higher dimensions. 7. \(A^p_{\alpha}\) spaces of harmonic functions in the unit ball of \(R^n\).

MSC:

30D99 Entire and meromorphic functions of one complex variable, and related topics
30H20 Bergman spaces and Fock spaces
30-02 Research exposition (monographs, survey articles) pertaining to functions of a complex variable
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
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