On the boundedness and compactness of extended Cesàro composition operators between weighted Bloch-type spaces

Abstract

Let $\psi \in H(\mathbb{B}_n),$ the space of all holomorphic functions on the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n,$ $\varphi = (\varphi_1, \ldots, \varphi_n) \in S(\mathbb{B}_n)$ the set of holomorphic self-maps of $\mathbb{B}_n.$ Let $C_{\psi, \varphi}: \mathcal B_{\nu}$ (and $ \mathcal B_{\nu,0}$) $\to \mathcal B_{\mu} $ (and $ \mathcal B_{\mu,0}$) be weighted extended Cesàro operators induced by products of the extended Cesàro 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| $ for some $ k\in \{1,\ldots,n\}. $ At the same time, we also give the asymptotic estimates of the norms of these operators.