Bellwethers of composition operators for typical-growth spaces (Q2925286)

Let \(\mathbb D^k\) be the polydisk in \(\mathbb C^k\) for \(k \geq 1\), and \(H^\infty_v(\mathbb D^k)\) the weighted space of bounded analytic functions \(f: \mathbb D^k \to \mathbb C\), where \(v: \mathbb D^k \to (0,\infty)\) is a bounded continuous weight function. For two classes of weight functions \(v\), the author finds a specific analytic self-map \(\alpha: \mathbb D^k \to \mathbb D^k\) (a bellwether in the terminology of \textit{P. S. Bourdon} [J. Aust. Math. Soc. 86, No.~3, 305--314 (2009; Zbl 1173.47015)]) such that the boundedness of the composition operator \(f \mapsto C_\alpha(f) = f \circ \alpha: H^\infty_v(\mathbb D^k) \to H^\infty_v(\mathbb D^k)\) implies that the composition maps \(C_\phi\) are bounded \(H^\infty_v(\mathbb D^k) \to H^\infty_v(\mathbb D^k)\) for any \(\phi: \mathbb D^k \to \mathbb D^k)\) of the form \(\phi(z_1,\dots,z_k) = (\phi_1(z_{\sigma(1)}),\dots, \phi_k(z_{\sigma(k)}))\), where \((z_1,\dots,z_k) \in \mathbb D^k\). Here \(\phi_j: \mathbb D \to \mathbb D\) are analytic maps for \(j = 1, \dots, k\) and \(\sigma\) is any permutation of \(\{1,\dots,k\}\). These results partially extend to the polydisk setting earlier ones for the case \(k = 1\) due to Bourdon [loc.\,cit.]\ and \textit{J. Bonet} et al. [J. Aust. Math. Soc., Ser. A 64, No. 1, 101--118 (1998; Zbl 0912.47014)].