Essential norm of weighted composition operator between \(\alpha\)-Bloch space and \(\beta\)-Bloch space in polydiscs (Q1777913)

Let \(\varphi(z)=(\varphi_{1}(z),\dots,\varphi_n(z))\) be a holomorphic self-map of \(D^n\) and \(\psi(z)\) a holomorphic function on \(D^n\), where \(D^n\) is the unit polydiscs of \(\mathbb{C}^n\). For \(0<\alpha,\beta<1\), using the same methods as [\textit{X.--J.\ Zhang}, Chin.\ Ann.\ Math., Ser.\ A 24, No. 6, 711--720 (2003; Zbl 1076.47019)], the present paper gives lower and upper estimates of the essential norm of a bounded weighted composition operator \(\psi C_{\varphi}\) between the \(\alpha\)-Bloch-space \(\mathcal{B}^\alpha(D^n)\) and the \(\beta\)-Bloch-space \(\mathcal{B}^\beta (D^{n})\).