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 \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\}. 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.