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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Multipliers of families of Cauchy-Stieltjes transforms

Authors: R. A. Hibschweiler and T. H. MacGregor
Journal: Trans. Amer. Math. Soc. 331 (1992), 377-394
MSC: Primary 30E20
MathSciNet review: 1120775
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Abstract: For $ \alpha > 0$ let $ {\mathcal{F}_\alpha }$ denote the class of functions defined for $ \vert z\vert < 1$ by integrating $ 1/{(1 - xz)^\alpha }$ against a complex measure on $ \vert x\vert= 1$. A function $ g$ holomorphic in $ \vert z\vert < 1$ is a multiplier of $ {\mathcal{F}_\alpha }$ if $ f \in {\mathcal{F}_\alpha }$ implies $ gf \in {\mathcal{F}_\alpha }$. The class of all such multipliers is denoted by $ {\mathcal{M}_\alpha }$. Various properties of $ {\mathcal{M}_\alpha }$ are studied in this paper. For example, it is proven that $ \alpha < \beta $ implies $ {\mathcal{M}_\alpha } \subset {\mathcal{M}_\beta }$, and also that $ {\mathcal{M}_\alpha } \subset {H^\infty }$. Examples are given of bounded functions which are not multipliers. A new proof is given of a theorem of Vinogradov which asserts that if $ f^{\prime}$ is in the Hardy class $ {H^1}$, then $ f \in {\mathcal{M}_1}$. Also the theorem is improved to $ f^{\prime} \in {H^1}$ implies $ f \in {\mathcal{M}_\alpha }$, for all $ \alpha > 0$. Finally, let $ \alpha > 0$ and let $ f$ be holomorphic in $ \vert z\vert < 1$. It is known that $ f$ is bounded if and only if its Cesàro sums are uniformly bounded in $ \vert z\vert \leq 1$. This result is generalized using suitable polynomials defined for $ \alpha > 0$.

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Keywords: Cauchy-Stieltjes transforms, complex measures, multipliers
Article copyright: © Copyright 1992 American Mathematical Society