On an integral-type operator between weighted-type spaces and Bloch-type spaces on the unit ball (Q613237)

Let \(H(B_n)\) be the space of all holomorphic functions on \(B_n\) and let \(S(\mathbb{B})\) be the collection of all holomorphic self mappings of \(B_n\), where \(B_n\) denotes the open unit ball of the complex \(n\)-dimensional Euclidean space \(\mathbb C^n\). Assuming that \(g\in H(B_n)\) with \(g(0)=0\) and \(\varphi\in S(B_n),\) the integral operator on the unit ball is defined by \[ P_{\varphi}^{g}(f)(z)=\int _{0}^{1}f(\varphi(tz))g(tz)\frac{dt}{t} , \quad f\in H(B_n),\;z \in B_n. \] In this paper, the authors investigate the boundedness and compactness of this integral-type operator between weighted-type spaces and Bloch-type spaces on the unit ball \(B_n\).