On the boundedness and compactness of extended Ces\`aro composition operators between weighted Bloch-type spaces

Let $\psi \in H(\BB_n),$ the space of all holomorphic functions on the unit ball $\BB_n$ of $\C^n,$ $\varphi = (\varphi_1, \ldots, \varphi_n) \in S(\BB_n)$ the set of holomorphic self-maps of $\BB_n.$ Let $C_{\psi, \varphi}: \mathcal B_{\nu}$ (and $ \mathcal B_{\nu,0}$) $\to \mathcal B_{\mu} $ (and $ \mathcal B_{\mu,0}$) be weighted extended Ces\`aro operators induced by products of the extended Ces\`aro operator $ C_\varphi $ and integral operator $T_\psi.$ In this paper, we characterize the boundedness and compactness of $ C_{\psi,\varphi} $ via the estimates for either $ |\varphi| $ or $ |\varphi_k| $ \textit{for some $ k\in \{1,\ldots,n\}. $} At the same time, we also give the asymptotic estimates of the norms of these operators.