Let $U^n$ be the unit polydisc of ${\mathbb C}^n$ and $\varphi(z)=(\varphi_1(z),\ldots,\varphi_n(z))$ a holomorphic self-map of $U^n.$ Let $H(U^n)$ denote the space of all holomorphic functions on $U^n,$ $H^\infty(U^n)$ the space of all bounded holomorphic functions on $U^n,$ and ${\cal B}^a(U^n),$ $a>0,$ the $a$\-Bloch space, i.e.,
$$
{\cal B}^a(U^n)=\bigg\{ f\in H(U^n)\, |\, \|f\|_{{\cal B}^a}=|f(0)|+\sup\limits_{z\in U^n}\sum\limits^n_{k=1} \left|\frac{\partial f} {\partial z_k}(z)\right|\left(1- |z_k|^2\right)^a<+\infty\bigg\}. \hspace{-0.3cm}
$$
 We give a necessary and sufficient condition for the composition operator $C_{\\varphi}$ induced by $\varphi$ to be bounded and compact between $H^\infty(U^n)$ and $a$\-Bloch space ${\cal B}^a(U^n),$ when $a\geq 1.$

Composition Operators between $H^{\infty}$ and $α$ -Bloch Spaces on the Polydisc

Stevo Stevic

Abstract

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