Norm of the composition operator mapping Bloch space into Hardy or Bergman space

Commun. Korean Math. Soc. 2003 Vol. 18, No. 4, 653-659 Printed December 1, 2003

Ern Gun Kwon, Jinkee Lee Andong National University, Andong National University

Abstract : Let $1 \leq p < \infty$ and $\alpha > -1$. If $f$ is a holomorphic self-map of the open unit disc $U$ of $\Bbb C$ with $f(0)=0$, then the quantity $$ \int_U \left\{\frac {\vert f'(z)\vert}{1 - \vert f (z)\vert^2 } \right\}^p (1-|z|)^{\alpha+p} dx dy $$ is equivalent to the operator norm of the composition operator $C_f : {\Cal B} \rightarrow A^{p,\alpha} $ defined by $C_f h = h \circ f -h(0) $, where ${\Cal B}$ and $A^{p,\alpha}$ are the Bloch space and the weighted Bergman space on $U$ respectively.

Keywords : Bloch space, composition operator, $A^{p,\alpha}$, space$H^p$ space