Carleson measures and some classes of meromorphic functions
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- by Rauno Aulaskari, Hasi Wulan and Ruhan Zhao PDF
- Proc. Amer. Math. Soc. 128 (2000), 2329-2335 Request permission
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 $M_{p}^{\#} = \{ f : {}$ $f$ 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.References
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Additional Information
- Rauno Aulaskari
- Affiliation: Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland
- Email: Rauno.Aulaskari@joensuu.fi, wulan@cc.joensuu.fi
- Hasi Wulan
- Affiliation: Department of Mathematics, Inner Mongolia Normal University, Hohhot 010022, People’s Republic of China
- ORCID: 0000-0001-6771-7311
- Ruhan Zhao
- Affiliation: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan
- Email: zhao@kusm.kyoto-u.ac.jp
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2329-2335
- MSC (1991): Primary 30D50
- DOI: https://doi.org/10.1090/S0002-9939-99-05273-9
- MathSciNet review: 1657750