Weighted Composition Operators from Banach Spaces of Holomorphic Functions to Weighted-Type Banach Spaces on the Unit Ball in $$\mathbb {C}^n$$Cn

Let X be a Banach space of holomorphic functions on the unit ball \(\mathbb {B}_n\) in \(\mathbb {C}^n\) whose point-evaluation functionals are bounded. In this work, we characterize the bounded weighted composition operators from X into a weighted-type Banach space \(H^\infty _\mu (\mathbb {B}_n)\), where the weight \(\mu \) is an arbitrary positive continuous function on \(\mathbb {B}_n\). We determine the norm of such operators in terms of the norm of the point-evaluation functionals. Under some restrictions on X, we characterize the compact weighted composition operators mapping X into \(H^\infty _\mu (\mathbb {B}_n)\). Under an alternative set of conditions, we provide essential norm estimates. We apply our results to the cases when X is the Hardy space \(H^p(\mathbb {B}_n)\), the weighted Bergman space \(A_\alpha ^p(\mathbb {B}_n)\) for \(\alpha >-1\) and \(1\le p<\infty \), the Bloch space \(\mathcal {B}\) and the little Bloch space \(\mathcal {B}_0\). In all these cases we obtain precise formulas of the essential norm.