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Hyperbolic mean growth of bounded holomorphic functions in the ball

Author: E. G. Kwon
Journal: Trans. Amer. Math. Soc. 355 (2003), 1269-1294
MSC (2000): Primary 30D45, 32A35, 47B33
Published electronically: November 5, 2002
MathSciNet review: 1938757
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Abstract: We consider the hyperbolic Hardy class $\varrho H^{p}(B)$, $0<p<\infty $. It consists of $\phi $ holomorphic in the unit complex ball $B$ for which $\vert \phi \vert < 1$ and

\begin{displaymath}\sup _{0<r<1} \, \int _{\partial B} \left \{ \varrho (\phi (r\zeta ), 0)\right \}^{p} \, d\sigma (\zeta ) ~<~ \infty ,\end{displaymath}

where $\varrho $denotes the hyperbolic distance of the unit disc. The hyperbolic version of the Littlewood-Paley type $g$-function and the area function are defined in terms of the invariant gradient of $B$, and membership of $\varrho H^{p}(B)$ is expressed by the $L^{p}$ property of the functions. As an application, we can characterize the boundedness and the compactness of the composition operator $\mathcal{C}_{\phi }$, defined by $\mathcal{C}_{\phi }f = f\circ \phi $, from the Bloch space into the Hardy space $H^{p}(B)$.

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Additional Information

E. G. Kwon
Affiliation: Department of Mathematics Education, Andong National University, Andong 760-749, S. Korea

Keywords: $H^{p}$ space, Bloch space, hyperbolic Hardy class, composition operator, Littlewood-Paley $g$-function, invariant gradient
Received by editor(s): May 15, 2001
Published electronically: November 5, 2002
Additional Notes: This work was supported by grant No. R01-2000-000-00001-0 from the Basic Research Program of the Korea Science & Engineering Foundation.
Article copyright: © Copyright 2002 American Mathematical Society

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