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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Carleson measures and some classes
of meromorphic functions

Authors: Rauno Aulaskari, Hasi Wulan and Ruhan Zhao
Journal: Proc. Amer. Math. Soc. 128 (2000), 2329-2335
MSC (1991): Primary 30D50
Published electronically: December 7, 1999
MathSciNet review: 1657750
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Abstract: For $|a|<1$ let $\varphi _{a}$ be the Möbius transformation defined by $\varphi _{a}(z)=\frac{a-z}{1-\bar az}$, and let $g(z,a)=\log |\frac{1-\bar az}{z-a}|$ be the Green's function of the unit disk $\mathcal{D}$. We construct an analytic function $f$ belonging to $M_{p}^{\#}=\{f:\text{$f$ meromorphic in $\mathcal{D}$ and\,} \sup _{a\in \mathcal{D}} \iint _{\mathcal{D}}(f^{\#}(z))^{2}(1-|\varphi _{a}(z)|^{2})^{p}\,dA(z)<\infty \}$ for all $p$, $0<p<\infty $, but not belonging to $Q_{p}^{\#}=\{f:f$ meromorphic in $\mathcal{D}$ and $\sup _{a\in \mathcal{D}}\iint _{\mathcal{D}}(f^{\#}(z))^{2}(g(z,a))^{p}\,dA(z)<\infty \}$ for any $p$, $0<p<\infty $. This gives a clear difference as compared to the analytic case where the corresponding function spaces ($M_{p}$ and $Q_{p}$) are same.

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Additional Information

Rauno Aulaskari
Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland

Hasi Wulan
Affiliation: Department of Mathematics, Inner Mongolia Normal University, Hohhot 010022, People’s Republic of China

Ruhan Zhao
Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Keywords: Carleson measure, normal function, the $Q_{p}$ space
Received by editor(s): April 20, 1998
Received by editor(s) in revised form: September 15, 1998
Published electronically: December 7, 1999
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society