Compact composition operators on the Bloch space in polydiscs (Q1609637)

Using some results in \textit{J.-H. Shi} and \textit{L. Lou} [Acta Math. Sin., Engl. Ser. 16, 85-98 (2000; Zbl 0967.32007)], the authors prove that for a holomorphic self-map \(\phi=(\phi_1, \cdots, \phi_n)\) of the polydisc \(U^n\), the composition operator \(C_\phi\) is compact on the Bloch space \(\beta(U^n)\) if and only if for every \(\varepsilon >0\), there exists a \(\delta>0\), such that \[ \sum_{k,l=1}^n \Bigl|\frac{\partial \phi_l(z)}{\partial z_k} \Bigr|\frac{1-|z_k|^2}{1-|\phi_l(z)|^2} < \varepsilon, \] whenever \(\text{dist}(\phi(z), \partial U^n) <\delta\). This is an extension of result by \textit{K. Madigan} and \textit{A. Matheson} [Trans. Am. Math. Soc. 347, 2679-2687 (1995; Zbl 0826.47023))], to \(n \geq 1\).