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Fractional Cauchy transforms, multipliers and Cesàro operators

Author: Evgueni Doubtsov
Journal: Proc. Amer. Math. Soc. 138 (2010), 663-673
MSC (2000): Primary 32A26, 32A37, 47B38; Secondary 46E15, 46J15
Published electronically: October 5, 2009
MathSciNet review: 2557183
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Abstract: Let $ B_n$ denote the unit ball in $ {\mathbb{C}}^n$, $ n\ge 1$. Given an $ \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, \zeta \rangle)^{-\alpha}$ against a complex Borel measure on the sphere $ \{\zeta\in{\mathbb{C}}^n: \vert\zeta\vert=1\}$. We study properties of the holomorphic functions $ g$ such that $ fg\in{\mathcal K}_\alpha(n)$ for all $ f\in{\mathcal K}_\alpha(n)$. Also, we investigate extended Cesàro operators on $ {\mathcal K}_\alpha (n)$.

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Additional Information

Evgueni Doubtsov
Affiliation: St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia

Keywords: Fractional Cauchy transform, pointwise multiplier, extended Ces\`aro operator
Received by editor(s): March 8, 2009
Received by editor(s) in revised form: June 11, 2009
Published electronically: October 5, 2009
Additional Notes: This research was supported by RFBR (grant no. 08-01-00358-a)
Communicated by: Franc Forstneric
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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