Essential norms of composition operators between Bloch type spaces in the polydisk (Q1934255)

Let \(H(\mathbb{D}^n)\) denote the space of holomorphic functions on the polydisk \(\mathbb{D}^n\) of \(\mathbb{C}^n\), \(n\geq 1\). For \(p>0\), the Bloch type space \(\mathcal{B}^p(\mathbb{D}^n)\) consists of those \(f\in H(\mathbb{D}^n)\) for which \[ \|f\|_p = |f(0)| + \sup_{z\in \mathbb{D}^n} \sum_{k=1}^n (1-|z_k|^2)^p \left|\frac{\partial f}{\partial z_k}(z) \right| <\infty. \] Given a holomorphic map \(\varphi: \mathbb{D}\to \mathbb{D}\), \textit{R.-H. Zhao} [Proc. Am. Math. Soc. 138, No. 7, 2537--2546 (2010; Zbl 1190.47028)] computed the essential norm \(\|C_\varphi\|_e\) of the composition operator \(C_\varphi: \mathcal{B}^p(\mathbb{D})\to \mathcal{B}^q(\mathbb{D})\) in terms of \(\|\varphi^m\|_q\), \(m\geq 1\). The authors show that Zhao's approach is applicable to the composition operators \(C_\varphi: \mathcal{B}^p(\mathbb{D}^n)\to \mathcal{B}^q(\mathbb{D}^n)\), \(n\geq 2\). However, if \(n\geq 2\), then the estimates of \(\|C_\varphi\|_e\) are different for \(p\geq 1\) and \(0<p<1\).