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

Abstract

Let $U^n$ be the unit polydisc of ${\mathbb{C}}^n$ and $\phi (z)=({\phi }_1(z),\dots ,{\phi }_n(z))$ 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 ${\mathcal{B}}^a(U^n),$ $a>0,$ the $a$-Bloch space, i.e.,

We give a necessary and sufficient condition for the composition operator $$\begin{gathered} C_{ \\ varphi} \end{gathered}$$ induced by $\phi$ to be bounded and compact between $H^{\infty }(U^n)$ and $a$-Bloch space ${\mathcal{B}}^a(U^n),$ when $a\ge 1.$