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Carleson measures and some classes of meromorphic functions

Authors:	Rauno Aulaskari Hasi Wulan Ruhan Zhao

Institution:	Department of Mathematics, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland ; Department of Mathematics, Inner Mongolia Normal University, Hohhot 010022, People's Republic of China Ruhan Zhao ; Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-8502, Japan

Abstract:	For 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 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 , $0<p<\infty$ , 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 , $0<p<\infty$ . This gives a clear difference as compared to the analytic case where the corresponding function spaces ( $M_{p}$ and $Q_{p}$ ) are same.

Keywords:	Carleson measure normal function the $Q_{p}$ space

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