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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



Families of fractional Cauchy transforms in the ball

Author: E. S. Dubtsov
Translated by: The author
Original publication: Algebra i Analiz, tom 21 (2009), nomer 6.
Journal: St. Petersburg Math. J. 21 (2010), 957-978
MSC (2010): Primary 32A26, 32A37
Published electronically: September 22, 2010
MathSciNet review: 2604545
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Abstract: Let $ B_n$ denote the unit ball in $ {\mathbb{C}}^n$, $ n\ge 1$. Given $ \alpha > 0$, let $ {\mathcal K}_\alpha (n)$ denote the class of functions defined for $ z\in B_n$ by integrating the kernel $ (1- \langle z, \za \rangle)^{-\alpha}$ against a complex-valued Borel measure on the sphere $ \{\zeta\in{\mathbb{C}}^n : \vert\zeta\vert=1\}$. The families $ {\mathcal K}_\alpha (1)$ of fractional Cauchy transforms have been investigated intensively by several authors. In the paper, various properties of $ \caa(n)$, $ n\ge 2$, are studied. In particular, relations between $ {\mathcal K}_\alpha (n)$ and other spaces of holomorphic functions in the ball are obtained. Also, pointwise multipliers for the spaces $ {\mathcal K}_\alpha (n)$ are investigated.

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Additional Information

E. S. Dubtsov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Fractional Cauchy transform, Bergman–Sobolev space, pointwise multiplier
Received by editor(s): November 23, 2008
Published electronically: September 22, 2010
Additional Notes: Supported by RFBR (grant no. 08-01-00358-a) and by the Russian Science Support Foundation.
Dedicated: Dedicated to Victor Petrovich Havin on the occasion of his 75th birthday
Article copyright: © Copyright 2010 American Mathematical Society

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