Fractional Cauchy transforms, multipliers and Cesàro operators
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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 $\{\zeta \in {\mathbb C}^n:\ |\zeta |=1\}$. 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 ${\mathcal K}_\alpha (n)$.References
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Additional Information
- Evgueni Doubtsov
- Affiliation: St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, St. Petersburg 191023, Russia
- MR Author ID: 361869
- Email: dubtsov@pdmi.ras.ru
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 138 (2010), 663-673
- MSC (2000): Primary 32A26, 32A37, 47B38; Secondary 46E15, 46J15
- DOI: https://doi.org/10.1090/S0002-9939-09-10122-3
- MathSciNet review: 2557183