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On families of Cauchy transformations. (English) Zbl 0917.30025

Let \(\Delta\) denote the unit disk, and let \(B\) be the set of holomorphic functions \(\varphi\) on \(\Delta\) with \(|\varphi(z)|<1\) and \(\varphi(0)=0\). Let \(B\) be provided with the topology of uniform convergence on compact subsets of \(\Delta\), and let \({\mathcal N}\) denote the set of complex-valued Borel measures on \(B\). For \(\alpha>0\), \(A_\alpha\) denotes the set of functions \(f\) for which there exists a measure \(\mu\) in \({\mathcal N}\) such that (*) \(f(z)= \int(1- \varphi(z))^{-\alpha} d\mu(\varphi)\), the integral being taken over \(B\). For \(\alpha=0\), (*) is replaced by \(f(z)=\int \log(1/(1-\varphi(z))) d\mu (\varphi)\). The more familiar classes \(F_\alpha\) of Cauchy transforms have \(\overline xz\) in place of \(\varphi(z)\) in these formulas, and the measures are supported on the set \(\{xz:| x|=1\}\). \(A_\alpha\) is a Banach space under the norm \(\| f\|_{A_\alpha}= \inf\| \mu\|\) for \(\alpha>0\) and \(\| f\|_{A_\alpha}=\inf\|\mu\|+| f(0)|\) for \(\alpha=0\), the results are: (i) If \(0\leq\alpha<\beta\), then \(A_\alpha\subset A_\beta\), and \(\| f\|_{A_\beta} \leq\| f\|_{A_\alpha}\), (ii) \(A_0=\text{BMOA}\); the norm on \(A_0\) is equivalent to the standard norm on BMOA, (iii) \(\| z^n \|_{A_\alpha}\) is bounded for \(n\geq 1\) by a constant \(k\leq 4\pi/3\), (iv) if \(f(z)= \sum a_nz^n\) is holomorphic on \(\Delta\) and \(\sum| a_n|\) converges, then \(f\) is an \(A_\alpha\) for all \(\alpha\geq 0\).

MSC:

30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
30D99 Entire and meromorphic functions of one complex variable, and related topics

Keywords:

BMOA
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